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Wolffia is a genus of aquatic plants with a cosmopolitan distribution. They include the smallest flowering plants on Earth. Commonly called watermeal or duckweed, these aquatic plants resemble specks of cornmeal floating on the water. Individuals often float together in pairs or form floating mats with related plants, such as Lemna and Spirodela species.
Description
Wolffia are free-floating aquatic plants with fronds that are nearly spherical to cylindrical in shape and lack airspaces or veins. They do not have roots. Their rarely seen flowers originate from a cavity on the upper surface of the frond, and each flower has one stamen and one pistil.
Although Wolffia can reproduce by seed, they usually use vegetative reproduction. A mother frond has a terminal conical cavity from which it produces daughter fronds.
Physiology
The growth rate of Wolffia varies within and among species. The rates of photosynthesis and respiration also vary proportionately to growth rate. The fastest growth rate (in fact, the fastest growth rate of any flowering plant) is shown by a clone of Wolffia microscopica, with a doubling time of 29.3 hours.
As food
Wolffia are a potential high-protein human food source. One species, W. microscopica, is over 20% protein by dry weight and has high content of essential amino acids. They have historically been collected from the water and eaten as a vegetable in Asia.
Species
, eleven species are accepted on Kew's Plants of the World Online:
Wolffia angusta
Wolffia arrhiza
Wolffia australiana
Wolffia borealis
Wolffia brasiliensis
Wolffia columbiana
Wolffia cylindracea
Wolffia elongata
Wolffia globosa
Wolffia microscopica
Wolffia neglecta
References
External links
The Duckweed Genome Project from Rutgers University
Landolt, E. (1986) Biosystematic investigations in the family of duckweeds (Lemnaceae). Vol. 2. The family of Lemnaceae - A monographic study. Part 1 of the monograph: Morphology; karyology; ecology; geographic distribution; systematic position; nomenclature; descriptions. Veröff. Geobot. Inst., Stiftung Rübel, ETH, Zurich.
Lemnoideae
Araceae genera
Freshwater plants | {
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Thailand participated in the 1962 Asian Games in Jakarta on 24 August to 4 September 1962. Thailand ended the games at 12 overall medals including 2 gold medals.
Nations at the 1962 Asian Games
1962
Asian Games | {
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{"url":"http:\/\/math.stackexchange.com\/questions\/137167\/showing-int-0-infty-frac1x212x24-frac-pi18-via-contou","text":"# Showing $\\int_{0}^{\\infty} \\frac{1}{(x^2+1)^2(x^2+4)}=\\frac{\\pi}{18}$ via contour integration\n\nI want to show that: $$\\int_{0}^{\\infty} \\frac{1}{(x^2+1)^2(x^2+4)}=\\frac{\\pi}{18}$$ so considering: $$\\int_{\\gamma} \\frac{1}{(z^2+1)^2(z^2+4)}$$ where gamma is the curve going from $0$ to $-R$ along the real axis, from $-R$ to R via a semi-circle in the upper plane and then from $R$ to 0 along the real axis.\n\nUsing the residue theorem we have that: $$\\int_{\\gamma} \\frac{1}{(z^2+1)^2(z^2+4)}=2\\pi i \\sum Res$$ so re-writing the integrand as $\\displaystyle\\frac{1}{(z-2i)(z+2i)(z+i)^2(z-i)^2}$\n\nwe can see that there is two simple poles at $2i$,$-2i$ and two poles of order 2 at $i$,$-i$. Calculating the residues: $$Res_{z=2i}=\\lim_{z\\rightarrow 2i} \\displaystyle\\frac{1}{(z+2i)(z+i)^2(z-i)^2}=\\frac{1}{36i}$$\n\n$$Res_{z=-2i}=\\lim_{z\\rightarrow 2i} \\displaystyle\\frac{1}{(z-2i)(z+i)^2(z-i)^2}=\\frac{-1}{36i}$$\n\n$$Res_{z=i}\\lim_{z\\rightarrow i} \\frac{d}{dz} \\frac{1}{(z-2i)(z+2i)(z+i)^2}=\\frac{2i}{36}+\\frac{2}{24i}$$\n\n$$Res_{z=-i}\\lim_{z\\rightarrow -i} \\frac{d}{dz} \\frac{1}{(z-2i)(z+2i)(z-i)^2}=\\frac{-2i}{36}+\\frac{-2}{24i}$$\n\nBut now the sum of the residues is 0 and so when I integrate over my curve letting R go to $\\infty$ (and the integral over top semi-circle goes to 0) I will just get 0?\n\nNot sure what I've done wrong? Thanks very much for any help\n\n-\nNote that only $2i$ and $i$ lie in the region bounded by $\\gamma$. Therefore, the sum of the residue on the right hand side is the sum of the residue at $2i$ and $i$ only. \u2013\u00a0 Paul Apr 26 '12 at 9:00\nOf course, thanks very much! I was staring at my calculations of those residues for ages! That will give me $\\frac{-i}{18}\\times 2\\pi i$ then the 2's cancel when I change the direction of one of the integrals and I get my result. Thanks very much again! \u2013\u00a0 hmmmm Apr 26 '12 at 9:09\n@Paul Maybe you can expand your comment as an answer. \u2013\u00a0 Davide Giraudo Apr 26 '12 at 11:48\nSince your coutour confusingly has winding number -1 around the two residues, you have to put a minus sign in the residue theorem. \u2013\u00a0 GEdgar Jun 8 '12 at 19:11\nAlso note that you don't need contour integration to solve this integral. Partial fractions-like techniques would work. \u2013\u00a0 bartgol Jun 19 '12 at 22:09\n\nConsider the contour $C$ that spans along $-R$ to $R$ and around the arc $Re^{i\\theta}$ for $0\\le\\theta\\le \\pi$.\n\nLetting\n\n$$f(z):=\\frac{1}{(z^2+1)^2(z^2+4)}=\\frac{1}{(z+i)^2(z-i)^2(z+2i)(z-2i)}$$\n\nand we see the poles are located at $\\pm i$ and $\\pm 2i$. Letting $R \\to \\infty$, it is very clear that the denominator explodes, causing the integral around the arc to disappear. Then\n\n$$\\oint_C f(z)\\, dz = 2\\pi i(\\operatorname*{Res}_{z = i}f(z) + \\operatorname*{Res}_{z = 2i}f(z))$$\n\nbecause $2i$ and $i$ are the only poles in $C$.\nThe pole of $i$ is of order 2:\n\n$$\\operatorname*{Res}_{z = i}f(z) = \\lim_{z \\to i} \\frac{1}{1!}\\frac{d}{dz} (z-i)^2 f(z)= \\lim_{z \\to i} \\frac{d}{dz}\\frac{1}{(z+i)^2(z^2+4)}= \\lim_{z \\to i} \\frac{2(2z^2 +iz+4)}{(i+z)^3(4+z^2)^2}=-\\frac{i}{36}$$\n\nThe pole of $2i$ is simple:\n\n$$\\operatorname*{Res}_{z = 2i}f(z) = \\lim_{z \\to 2i} (z-2i)f(z) = \\frac{1}{(-4+1)^2(2i+2i)}=-\\frac{i}{36}$$\n\nSo finally\n\n$$\\int_0^\\infty f(x)\\, dx = \\frac{1}{2}\\int_{-\\infty}^\\infty f(x)\\, dx = \\pi i\\left(-\\frac{i}{36}-\\frac{i}{36}\\right) = \\frac{\\pi}{18}$$\n\n-\n\nWhen using the residue theorem, you only consider residues enclosed by the path you are integrating over. In your case, you only consider residues in the upper half plane, as any point in the upper half plane will be enclosed by the semicircle as $R$ goes to infinity. Only $2i$ and $i$ lie in the upper half plane, out of the four poles of the function, so you only consider residues at those points. Your error comes from summing all of the residues, even ones that don't lie in the region bounded by the contour of integration.\n\n-","date":"2015-04-18 09:22:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9707838296890259, \"perplexity\": 226.7466963478828}, \"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-2015-18\/segments\/1429246634257.45\/warc\/CC-MAIN-20150417045714-00275-ip-10-235-10-82.ec2.internal.warc.gz\"}"} | null | null |
Flock-rustling
Trendy new churches poach worshippers from stuffy ones
Bishops hope "planting" new churches will lure in young agnostics
BritainOct 27th 2018 edition
HELD WITHIN the blackened walls of a disused department store, the evening service at Portsmouth's Harbour Church resembles a gig in a trendy nightclub. Guitars take the place of a church organ; hymn books have been swapped for plasma screens displaying song lyrics. Alex Wood, the vicar, favours skinny black jeans rather than a clerical robe. Swathed in blue light, his congregation of teenagers and 20-somethings sing their way through a playlist of uplifting Christian rock.
Churchgoing has plummeted in Britain. Only 740,000 worshippers regularly make it to Anglican churches on Sundays, half as many as in 1970. To halt this decline, the Church of England has launched an evangelism drive. Part of its strategy is to attract young agnostics by "planting" churches, an American model where members of a healthy church set up a new one elsewhere. Holy Trinity Brompton (HTB), a thriving evangelical church in west London, has been planting churches since the 1980s. Now the clergy's top brass want to emulate its success. According to Ric Thorpe, the Bishop of Islington and a former HTB man, 2,400 church plants are planned by 2030.
New evangelical churches have proved capable of astonishing growth. Harbour Church was planted by 15 missionaries from St Peter's in Brighton in 2016. Now its four services pull in around 600 worshippers each Sunday. St Peter's itself was set up by 30 HTB parishioners in 2009 and now boasts a flock of 1,000.
Their success lies in repackaging Christianity to appeal to the young, says Bishop Thorpe. Harbour Church's main morning service (where worshippers have an average age of 27) starts with pastries and micro-brewed Brazilian coffee; the evening service (average age: 19) is followed by hot dogs and craft beer. Ryan Forey, a trainee priest at the church, is frustrated by perceptions of Christianity as stuffy. "Jesus's first miracle was turning water into wine; he kept the party going," he says.
Harbour Church says it is not a church for people who are already Christian. Yet some worry that these young, vibrant churches are not winning new converts as intended, but rather cannibalising existing congregations. In 2016 a study of five HTB churches in London found that 38% of parishioners had transferred from another church. A show of hands at Harbour Church suggests that about a third left another congregation to join. Another large group are students looking for a term-time church. One worshipper explains that he defected from his old church because it was rigid and traditional. Mr Wood says he does his best to persuade such parishioners to return, but if they claim God called them to his church there is little he can do.
Planting is changing the face of the Church of England. Not so long ago, evangelicals armed with guitars were politely dismissed as an oddball fringe. But Justin Welby (who attended HTB before he was ordained) has staked his career as Archbishop of Canterbury on getting more bums on pews. The missionary zeal of evangelicals and their eagerness to plant churches means they are at the forefront of his push.
This rankles with some vicars. Talk of a takeover is rife. Alarm bells ring when HTB comes to town with a new plant, as priests fear losing the youngsters in their flocks. Rural parishes complain that funding goes to flashy urban ventures. And despite the guitars and coffee, evangelicals tend to be "anti-gay and a bit funny on women", says Linda Woodhead, a sociologist at Lancaster University. She warns that, as the church grows more evangelical, it risks morphing into a sect that appeals to a dwindling pool of conservative enthusiasts.
At Harbour Church, Mr Wood acknowledges Christianity can seem odd to young people. As an ex-BBC producer and former atheist, even he thinks "it's bizarre that I'm a vicar." Bishops are hoping church plants can stem Anglicanism's decline by inspiring similar leaps of faith.
This article appeared in the Britain section of the print edition under the headline "Trendy new churches poach worshippers from stuffy ones"
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What's going wrong here?
Britain's 20,000 new cops won't get the justice ministry celebrating | {
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Q: In the situation that i have a countably generated \sigma -algebra, can i assume the generator to have the shape of an algebra? Or, differently formulated: can a countable generator be completed to an algebra by countable set operations, while still generating the same sigma-alegbra?
since the generator not necessarily has a defined structure it's hard to imagine this should be possible. On the other hand i can't seem to find a counterexample. One idea was to find out whether there is a minimal countable generator and then fill it up with countable missing sets. The other is the converse, find a maximal countable generator and proof the algebra properties. In the first case, there are restrictions to when a minimal generator exists. Sadly i cannot afford those boundaries. In the second case, i have absolutely no idea how to find out whether finite intersections/unions or complements are included in the set, whereas if i have a maximal generator (in case of existence, I haven't found anything confirming or denying this) i have automatically the base set in the generator. Are there any ideas out there, that could help?
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{"url":"https:\/\/codingbobby.xyz\/projects\/chaotic-shapes\/langford\/","text":"# Langford Attractor\n\nThis attractor1 was found by William Finlay Langford in 1984.2 As the title of his paper suggests, the attractor is a bifurcation of a torus. What does that mean? Well, what he has done is taking a system of differential equations that resulted in a torus shape and simply added a bifurcation term to one of them. Jokes aside, he has probably done a lot more work to get to that point but looking at the equations and tinkering with the parameters let you imagine just that. Specifically, this term is $$\\varepsilon\\, z\\, x^3$$ and it is added to $$\\dot{z}$$. If you leave it out by setting $$\\varepsilon = 0$$, you get a spiraly donut shape3:\n\nThe inner tube is very thin here but when decreasing $$\\alpha$$ to something like $$0.65$$, the shape becomes obvious:\n\nNow when increasing the bifurcation parameter (and keeping the original value for $$\\alpha$$), the chaos begins. Here is it with $$\\varepsilon = 0.001$$ and as you can see, this very slight change pushes the system out of symmetry:\n\n## Renders\n\nDifferential system:\n\n$\\dot{x} = x\\, (z - \\beta) - \\omega\\, y$ $\\dot{y} = y\\, (z - \\beta) + \\omega\\, x$ $\\dot{z} = \\lambda + \\alpha\\, z - \\frac{z^3}{3} - (1 + \\varrho\\, z)\\, (x^2 + y^2) + \\varepsilon\\, z\\, x^3$\n\nConstants:\n\n$\\alpha = 0.95$ $\\beta = 0.7$ $\\lambda = 0.6$ $\\omega = 3.5$ $\\varrho = 0.25$ $\\varepsilon = 0.1$\n\n1. Most online resources refer to this attractor as the \u201cAizawa Attractor\u201d despite that Yoji Aizawa (it apparently got named after) seems to have nothing to do with it.4 How this confusion arose is unclear.\n\n2. W.F. Langford, 1984. \"Numerical Studies of Torus Bifurcations\". In: T. K\u00fcpper, H.D. Mittelmann, H. Weber (eds), \"Numerical Methods for Bifurcation Problems\". International Series of Numerical Mathematics, Vol 70. doi:10.1007\/978-3-0348-6256-1_19\n\n3. This and the other simple images were generated using Processing, similar to this one I\u2019ve done for the Lorenz Attractor\n\n4. This was pointed out by: E. Fleurantin, J.D. Mireles James, 2019. \"Resonant tori, transport barriers, and chaos in a vector field with a Neimark-Sacker bifurcation\". Department of Mathematics, Florida Atlantic University.\n\n<- \ud83d\udd78 \ud83d\udc8d ->","date":"2022-05-20 17:31:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7783007025718689, \"perplexity\": 1573.5942069860514}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662533972.17\/warc\/CC-MAIN-20220520160139-20220520190139-00062.warc.gz\"}"} | null | null |
Join our live interactive lessons to be part of the best valued class with an expert IELTS teacher.
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{"url":"https:\/\/projecteuclid.org\/euclid.aop\/1041903215","text":"## The Annals of Probability\n\n### Weak limits of perturbed random walks and the equation $Y_t = B_t + \\alpha\\sup\\{Y_s\\colon s \\leq t\\}+\\beta\\inf\\{Y\\sb s\\colon s \\leq t\\}$\n\nBurgess Davis\n\n#### Abstract\n\nLet $\\alpha$ and $\\beta$ be real numbers and $f \\in C_0 [0, \\infty)$. We study the existence and uniqueness of solutions g of the equation $g(t) = f(t) + \\alpha \\sup_{0 \\leq s \\leq t} g(s) + \\beta \\inf_{0 \\leq s \\leq t} g(s)$. Carmona, Petit, Le Gall, and Yor have shown existence or nonexistence and uniqueness for some $\\alpha, \\beta$. We settle the remaining cases. We study the nearest neighbor walk on the integers, which behaves just like fair random walk unless one neighbor has been visited and the other has not, when it jumps to the unvisited neighbor with probability p. If $p < 2\/3$, we show these processes, scaled, converge to the solution of the equation above for Brownian paths, with $\\alpha = \\beta = (2p - 1)\/p$.\n\n#### Article information\n\nSource\nAnn. Probab., Volume 24, Number 4 (1996), 2007-2023.\n\nDates\nFirst available in Project Euclid: 6 January 2003\n\nhttps:\/\/projecteuclid.org\/euclid.aop\/1041903215\n\nDigital Object Identifier\ndoi:10.1214\/aop\/1041903215\n\nMathematical Reviews number (MathSciNet)\nMR1415238\n\nZentralblatt MATH identifier\n0870.60076\n\n#### Citation\n\nDavis, Burgess. Weak limits of perturbed random walks and the equation $Y_t = B_t + \\alpha\\sup\\{Y_s\\colon s \\leq t\\}+\\beta\\inf\\{Y\\sb s\\colon s \\leq t\\}$. Ann. Probab. 24 (1996), no. 4, 2007--2023. doi:10.1214\/aop\/1041903215. https:\/\/projecteuclid.org\/euclid.aop\/1041903215\n\n#### References\n\n\u2022 BOLTHAUSEN, E. and SCHMOCK, U. 1994. On self-attracting one-dimensional random walks. Preprint. Z.\n\u2022 CARMONA, P., PETIT, F. and YOR, M. 1993. Beta variables as times spent in 0, by certain perturbed reflecting Brownian motions. Bull. London Math Soc. To appear. Z.\n\u2022 CARMONA, P., PETIT, F. and YOR, M. 1994. Probab. Theory Related Fields 100 1 29. Z.\n\u2022 DAVIS, B. 1989. Loss of recurrence in reinforced random walk. In Almost Every where ConverZ. gence: Proceedings of a Conference G. Edgar and L. Sucheston, eds. 179 185. Academic Press, New York. Z.\n\u2022 DAVIS, B. 1990. Reinforced random walk. Probab. Theory Related Fields 84 203 229. Z.\n\u2022 DIACONIS, P. 1988. Recent progress on de Finetti's notions of exchangeability. In Bayesian Z. Statistics J. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds. 111 125. Oxford Univ. Press. Z.\n\u2022 DOOB, J. L. 1951. Stochastic Processes. Wiley, New York. Z.\n\u2022 HARRISON, J. M. and SHEPP, L. A. 1981. On skew-Brownian motion. Ann. Probab. 9 309 313. Z. Y Y\n\u2022 LE GALL, J. F. 1986. L'equation stochastique Y B M I comme limite des equa\u00b4 \u00b4 t t t t tions de Norris Rogers Williams. Unpublished notes. Z.\n\u2022 LE GALL, J. F. AND YOR, M. 1992. Enlacement du mouvement brownien autour des courbes de l'espace. Trans. Amer. Math. Soc. 317 687 722. Z.\n\u2022 NESTER, D. 1994. A random walk with partial reflection or attraction at its extrema. Preprint. Z.\n\u2022 OTHMER, H. and STEVENS, A. 1995. Aggregation, blow-up, and collapse: the ABC's of taxis in reinforced random walks. Preprint. Z.\n\u2022 PEMANTLE, R. 1988. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 1229 1241. Z.\n\u2022 PEMANTLE, R. 1992. Vertex-reinforced random walk. Probab. Theory Related Fields 92 117 136. Z.\n\u2022 PERMAN, M. 1995. An excusion approach to Ray Knight theorems for perturbed reflecting Brownian motion. Stochastic Process. Appl. To appear. Z.\n\u2022 POLLARD, D. 1982. Convergence of Stochastic Processes. Springer, New York. Z.\n\u2022 SELLKE, T. 1994a. Recurrence of reinforced random walk on a ladder. Probab. Theory Related Fields. To appear.\n\u2022 SELLKE, T. 1994b. Reinforced random walk on the d-dimensional lattice. Probab. Theory Related Fields. To appear. Z.\n\u2022 TOTH, B. 1994. True'' self-avoiding walk with generalized bond repulsion on. J. Statist. \u00b4 Phy s. 77 17 33. Z.\n\u2022 TOTH, B. 1995. The true'' self avoiding walk with bond repulsion on : limit theorems. Ann. \u00b4 Probab. 23 1523 1556. Z.\n\u2022 TOTH, B. 1996. Generalized Ray Knight theory and limit theorems for self-interacting random \u00b4 walks on. Ann. Probab. 24 1324 1367. Z.\n\u2022 WERNER, W. 1995. Some remarks on perturbed reflecting Brownian motion. Seminaire de Probabilites XXIX. Lecture Notes in Math. 1613 37 43. Springer, Berlin. \u00b4\n\u2022 WEST LAFAy ETTE, INDIANA 47907 E-MAIL: bdavis@snap.stat.purdue.edu","date":"2020-01-29 02:11:21","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\": 0.8168824911117554, \"perplexity\": 5936.079817761904}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251783621.89\/warc\/CC-MAIN-20200129010251-20200129040251-00161.warc.gz\"}"} | null | null |
Sphaerodactylus randi este o specie de șopârle din genul Sphaerodactylus, familia Gekkonidae, descrisă de Shreve 1968.
Subspecii
Această specie cuprinde următoarele subspecii:
S. r. methorius
S. r. strahmi
S. r. randi
Referințe
Sphaerodactylus | {
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\section{Introduction}
Many real-world data collections are low in quality because of errors (e.g., typographical errors or phonetic misspellings), incomplete or missing data, incompatible formats for recording database fields (e.g., dates, addresses), and temporal inconsistencies. Integrating or querying multiple data sources to identify records that belong to the same real-world entity is a challenging task in the presence of such data. This task is often referred to as the \textit{entity resolution (ER)} problem and appears in many database applications when identifying duplicates, cleansing data, or improving data quality.
ER has been studied for years. However, the issues of efficiently handling large-scale data remain an open research problem~\cite{Liang}. Typical ER algorithms have a quadratic running time, which is computationally prohibitive for large-scale data collections. This performance bottleneck occurs due to the detail-level pairwise comparison step of the ER process.
Consider the motivating example of linking medical records where one entity can have many entries over several years. A public health case study could involve hundreds of thousands of entities with millions of records. Quadratic ER among the records will require trillions of pairwise comparisons in this application, which is computationally infeasible.
Indexing techniques address this problem by grouping similar records into blocks. Usually, groups of similar records within or between datasets are smaller than the total number of records. Hence, many comparisons will be among the non-matching records when applying pairwise comparisons using a brute-force ER algorithm. Indexing techniques aim to reduce the potential number of comparisons by reducing the comparisons between those non-matches~\cite{sur1}. Therefore, indexing-based ER solutions run much faster than brute-force ER solutions~\cite{HARRA}.
However, most existing indexing methods still have quadratic time complexity and are too slow to deal with very large-scale data collections. Also, some non-scaling ER algorithms tend to generate a large number of blocks and a large number of candidate records in each block for large-scale scenarios~\cite{sur1}.
Traditional ER solutions often process databases offline in batch mode, and no further action is required once a pair of matches are determined. However, many organisations are moving online where they have to provide their services through prompt responses. Hence, many newer, real-world scenarios require real-time query processing against large-scale databases. Some of the applications demand entity query matching against large-scale reference databases within a short time. We refer to this as the \textit{query matching} problem in ER. In such configurations, traditional ER solutions become less efficient or unusable.
In order to motivate the problem context and illustrate the usefulness of the approach presented in this paper, we provide the following real-world example. The application is in criminal investigations, where law enforcement officers need to query a database to identify potential suspects. Suspects usually lie to police investigators about their true identities, e.g., names, birth dates, or addresses~\cite{Chen_Chung}. Hence, finding an exact match for falsified data against the real identity recorded in a law enforcement computer system is problematic. Detecting deceptive identities is a time-consuming activity that involves large amounts of manual information processing in real-life scenarios~\cite{cr_1}. Therefore efficient ER solutions that support real-time and approximate query matching against existing datasets can be invaluable.
Toward the challenge of real-time approximate query matching, we present an indexing technique that reduces the number of pairwise comparisons needed in the ER. The proposed indexing method transforms a set of records into a set of vectors in a metric-space, specifically a lower-dimensional Euclidean space using multidimensional scaling. These vectors have two main attractive attributes in the context. First, comparisons between vectors in a metric-space are much cheaper than string comparisons. Second, these vectors support efficient indexing data structures. We utilise these properties to propose an indexing approach that classifies similar vectors in a low-dimensional Euclidean space. We call our method \textit{Em-K indexing}, as our method operates by embedding data in a K-dimensional space.
Exact query matching is easy against a reference database as we can search based on lexicographical order using an efficient data structure, e.g., binary trees. However, exact query matching is impossible with many real-world databases due to various data quality issues, and approximate query matching is the common approach~\cite{Christen2012}. The proposed indexing method address the problem of real-time and approximate query matching. We will be mainly using the term query matching instead of approximate query matching in this paper.
Our main objective is to develop a fast indexing method for query matching. We want our method to be efficient for large-scale data and robust to errors in the data. The proposed method searches for a block of records in the reference database as a set of potential matches to the querying record. Hence we avoid comparing the query against all the records in the referencing database. However, the search is done in the Euclidean space using a Kd-tree data structure where each record in the reference database and a query record become a vector in this space.
Given the block size is~$B$, we search for this number of nearest neighbours for the query point by traversing the Kd-tree nodes that store the reference database. The search has a $O(B \log N)$ complexity, given $N$ is the number of records in the reference database. The method embeds query records in a pre-mapped Euclidean space to search against the reference data points. We propose an out-of-sample embedding that uses a fraction of the original data (defined as landmarks in Section~5) for query record embedding. For $L$ landmarks, the embedding requires $O(L^2)$ operations, and we can choose $L$ such that $L \ll N$.
In this paper, we explore the use of metric-space indexing for efficient and approximate query matching. In particular, our contributions are,
\begin{itemize}
\item We formulate the query matching and deduplication problems in ER to provide metric-space base solutions. First, we propose an indexing approach based on landmarks as a motivating example to explore the basic building blocks needed for query matching.
\item We propose a landmarks based indexing technique for query matching to provide a quick and accurate block of potential matches. Our method can process a stream of queries against a large-scale data set within a short time. By doing so, we obtain as many of the matching records as possible where the processing time of a single query takes a sub-second time. The technique is robust to noisy data that contains errors and allow efficient approximate query matching.
\end{itemize}
\section{Preliminaries}
To describe the problem succinctly, we first describe relevant definitions and some key concepts. The definitions of relevant concepts here follow those in~\cite{schema_agnostic,sur2}.
\subsection{Entity Resolution }
An {\it entity} is a real-world object, e.g., person, place or product, that has a unique identifier to distinguish it from other entities of the same type. An {\it entity profile} describes an entity using a collection of name-value pairs. A set of entity profiles is called {\it entity collection}, denoted by $E$. A pair of similar entity profiles are called {\it duplicates}. A duplicate of an entity can be either an exact copy of the original entity profile or an entity profile that contains an error (e.g., typographical error). A database representation of an entity profile is usually referred to as a \emph{record}.
\newtheorem{mydef1}{Definition}
\begin{mydef1}(Entity Resolution): Given two records $r_i, r_j$ is a {\it match},
if they refer to the same unique real-world object. We denote this as
$r_i \equiv r_j$. The goal of ER is to link different records that describe the same entity within an entity collection or across two or more entity collections.
\end{mydef1}
\subsection{ER Tasks}
Following the above definitions, we distinguish between the following ER tasks.
\begin{enumerate}
\item \textit{Dirty ER:} Given an entity collection $E$ that contains entities $e_1,e_2,..., e_n$, find all duplicates and produces a set of equivalence clusters of distinct entities. It is also known as deduplication in many database applications.
\item \textit{Clean–Clean ER:} Given two duplicate free entity collection $E_1$ and $E_2$ that contains entities $e_1,e_2,..., e_n$, find all records that belong to a single entity. Our indexing method is aimed at the \textit{Clean–Clean ER} problem where one dataset is a stream of queries, and the other is a reference database. However, we use deduplication as a motivating example that explains the relevant building blocks of the proposed method.
\end{enumerate}
\subsection{Record Comparisons}
Similar records pairs of an entity are determined by applying a similarity function over the corresponding attributes of two records. Assume a pair of records $(r_i, r_j)$ and a set of attributes $a_1, a_2, ... , a_x$ that describe them. The similarities $s_1, ..., s_x$ between attribute values are determined by applying a set of similarity functions $sim(r_i.a_k , r_j.a_k )$, with $1 \leq k \leq x$ for each pair of attribute values. Then a total similarity score of $S=\sum_{i=1}^{x}s_i$ is calculated to classify the record pairs as a match or a non-match based on a matching threshold.
Several comparison methods such as edit distance, a.k.a Levenshtein distance, Jaro distance, and q-gram distance are found in the domain of strings~\cite{Loo2014}. In this work, we mainly used Levenshtein distance to measure the similarity at the attribute level. It calculates the minimum number of character insertions, deletions, and replacements necessary to transform a string $s_1$ into a string $s_2$. Minkowski metrics based on $L^p$ norms, ${\parallel x \parallel}_p = (\sum_i |x_i|^p )^{1/p}$, with $p \geqslant 1$ are a common used class of vector spaces. For our vector space, we used the most common Minkowski metric; Euclidean distances $d_E(p=2)$.
\subsection{Indexing or Blocking Techniques:}
Traditional ER requires pairwise comparisons between all the records. For instance, gievn two entity collections $E_1$ and $E_2$, with sizes $|E_1|$ and $|E_2|$, it requires $|E_1| \times |E_2|$ comparisons between entity records. In practice, this is infeasible when, $|E_1|$ and $|E_2|$ are large due to the inherent quadratic complexity of the comparison process.
Indexing or blocking reduces the number of detail-level pairwise comparisons between records by removing pairs that are unlikely to be real matches. The traditional blocking techniques partition the databases into non-overlapping blocks, only comparing the records within blocks. Hence, reducing the number of pairwise comparisons.
Our method is an example of join-based blocking techniques that convert blocking into the nearest neighbour search. The blocks are created by searching the vector space for similar records using the Em-K indexing rather than partitioning the dataset. As a result, similar records are grouped into overlapping blocks by combining spatial joins with block building~\cite{sur2}.
\section{Problem Formulation}
In this section, we formulate our problem using the concepts and the definitions presented in the previous section. The proposed Em-K indexing method functions as a preprocessing approach for a more detailed query-matching ER. We use deduplication as a motivating example that explains the basic building blocks to the query matching problem. Thus we defined the following two problems:
\medskip
\noindent \textbf{Query Matching}: is the problem of finding similar records given two entity collections, $E_r$ a reference database, and $Q$ a stream of queries. The size of each dataset is denoted by $|E_r|$ and $|Q|$ respectively and we assume $|E_r|$ is fixed and $|Q|\to \infty$. Let $Q={q_1,q_2,..,q_{|Q|}}$, where each query $q_i$ in $Q$ represents a record of an entity $e_j$. A query record $q_i$ has the same attribute schema as the records in $E_r$. Hence for each query $q_i$ in $Q$, the records in $E_r$ that belong to the same entity ($e_j$) need to be found. In real-life problems, the query rate might be very fast. There may or may not be a matching record for every $q_i \in Q$ that belongs to the entity $e_j$.
\newtheorem{mydef}{Problem Statement}
\begin{mydef}
(Indexing for Query matching): For two datasets (one is a reference database and the other is a stream of queries) with overlapping records, run the best algorithm for a given amount of time to find the block of records containing as many matches as possible for the querying record. This is a pre-processing step to increase the efficiency of the subsequent detailed query match.
\end{mydef}
We also consider the deduplication problem here, primarily as an explanatory model. We defined deduplication in Section 2.2 under \textit{dirty ER}. In the following, we define the indexing for deduplication.
\begin{mydef}
(Indexing for deduplication): For a given entity collection containing duplicates, group similar records into blocks to reduce the number of comparisons needed in subsequent detailed deduplication while missing as few matches as possible.
\end{mydef}
Traditional indexing techniques split the database into non-overlapping blocks, only comparing the records within any block~\cite{MacCallum, sur1, sur2, Matthew, Naumann}. Blocks of similar pairs are determined by building an indexing structure that takes a set of records as input and classifies them according to some criteria. Usually, this criterion is based on matching a \textit{blocking key} consisting of a single or several attribute values of records~\cite{sur1}. Our method uses blocking values to create blocks of records that transform blocking into a k-nearest neighbour (k-NN) search. We combine the spatial joins with block building to convert blocking values to a similarity preserving Euclidean space. The result is overlapping blocks of records.
The proposed method requires mapping blocking values into multidimensional vectors. Since many comparisons in ER are between strings of characters, we focus on entity attributes that contain string values here. Unlike the similarity between string values, the similarity between numerical values is easy to compute using $L^p$ norms in a metric-space~\cite{HARRA}. Hence, we used this property of the metric-space to propose a scalable indexing technique for query matching.
If we assume strings as elements of a complicated high-dimensional space, the distance between two different strings is typically large. However, misspelled strings tend to be located near correctly spelled strings~\cite{Mazeika}. The embedding of a string database into a metric-space needs to preserve these two properties. Thus, coordinates for blocking values are determined in a Euclidean space such that the associated Euclidean distances approximate the dissimilarities between the original blocking values.
The general problem of assigning coordinates in this manner is one of embedding a metric or non-metric-space into a Euclidean space~\cite{Virtual_L}. Suppose $R$ is a collection of objects, $\delta$ measures the distances between $R$ objects, $X$ represents the coordinates matrix for the $R$ objects in the Euclidean space, and $d$ measures the distances between coordinates.
Embedding of a metric or non-metric-space $(R,\delta)$ into a Euclidean space $(X,d)$ is a mapping $\phi : R \to X$. In this paper $(R,\delta)$ will always be a finite space (\textit{i.e.,} $R$ is a finite set) and $(X,d)$ will always be a Euclidean space.
\begin{mydef}
(The embedding problem): For a given metric or non-metric space, find a $\phi$ that minimizes distortion, stress, or a similar error metric between $(R,\delta)$ and $(X,d)$.
\end{mydef}
A commonly used technique for embedding a set of distances (or dissimilarities) into a Euclidean space is multidimensional scaling (MDS)~\cite{Kruskal1964}. We applied MDS because we can adapt it to achieve good time efficiency and distance preserve capability for large-scale data with a small amount of extra effort.
Among the variants of MDS, we use least-squares multidimensional scaling (LSMDS) for the embedding since it gave the best results compared to other variations such as classical scaling~\cite{herath2020simulating}. We can map a set of blocking values to a lower-dimensional Euclidean space by applying LSMDS such that the distances between vectors preserve the dissimilarities between them. This embedding leaves similar blocking values closer in the Euclidean space allowing efficient, geometric-based indexing.
LSMDS initially maps each item in the non-metric or metric-space to a $K$-dimensional point. Then minimises the discrepancy between the actual dissimilarities and the estimated distances in the $K$-dimensional space by optimisation~\cite{MDS}. This discrepancy is measured using \textit{raw stress}~($\sigma_{raw}$) given by the relative error where ${\delta_{ij}}$ is the dissimilarity between the two objects and ${d_{ij}}$ is the Euclidean distance between their estimated points.
\begin{equation} \label{eq:1}
\sigma_{raw}(\mathbf{X}) ={\sum_{i,j=1}^{n}w_{ij}\Big(d_{ij}(\mathbf{X})-\delta_{ij}\Big)^{2}}.
\end{equation}
Possible weights for each pair of points are denoted by $w_{ij}$. Weights are useful in handling missing values and the default values are \mbox{$w_{ij} = 0$}, if $\delta_{ij}$ is missing and \mbox{$w_{ij} = 1$}, otherwise~\cite{MDS}. We do not apply weights in this work, hence, \mbox{$w_{ij} = 1$} always. We prefer the normalized stress ($\sigma$) in our experiments since it is popular and theoretically justified. The normalized stress ($\sigma$) is obtained by $\sigma= \sqrt{\sigma_{raw}(\mathbf{X})/\delta_{ij}^2 }.$
However, traditional MDS algorithms such as LSMDS require extensive preprocessing and usually are computationally expensive, thus not appropriate for large scale applications. The two main drawbacks are,
\begin{itemize}
\item MDS requires $O(N^2 )$ time, where $N$ is the number of items. Thus, it is impractical for large~$N$.
\item In an out-of-sample setting or a query-by-example setting, a query item has to be mapped to a point in the pre-mapped Euclidean space. Given LSMDS algorithm is $O(N^2)$, an incremental algorithm to search/add a new item in the database would be $O(N)$. Hence a query search would be similar to sequential scanning of a database~\cite{FastMap}.
\end{itemize}
Among the proposed methods of scalable MDS, we are interested in using an out-of-sample embedding approach as a scaling method for LSMDS. We have two main purposes:
\begin{enumerate}
\item To embed large-scale reference databases.
\item To embed previously unseen data to a pre-mapped Euclidean space.
\end{enumerate}
Suppose we have a configuration of $N$ points in a \mbox{$K$-dimensional} Euclidean space obtained by applying LSMDS to a set of $N$ objects. Let $Q$ be out-of-sample objects, with measured pairwise dissimilarities from each of the original $N$ objects. The out-of-sample embedding problem is to embed the new $Q$ objects into the pre-mapped \mbox{$K$-dimensional} Euclidean space.
Our out-of-sample embedding approach uses the stochastic gradient descent algorithm to minimise the following objective function for numerical optimisation. The out-of-sample embedding of a new object $y$ is obtained by minimising the following objective function,
\begin{equation} \label{eq:2}
\hat{\sigma}(y) = {\sum_{i=1}^n}\big( \left\| x_i-\hat{y}\right\|_2-\delta_{iy}\big)^2,
\end{equation}
\noindent where $y$ is the new object and $\hat{y}$ is its position in the Euclidean space. The $\delta_{iy}$ represent the dissimilarities between point $i$ and the new object $y$. The Euclidean distance between the $i^{th}$ point and $y$ in the Euclidean space is given by $\left\| x_i-\hat{y}\right\|_2$. We seek to find a position of $y$ that minimises $\hat{\sigma}(y)$. Here we keep $w_{ij}=1$, similar to LSMDS.
The out-of-sample embedding approach becomes inefficient for large-scale data by comparing each new point with all the existing points. We scale our out-of-sample embedding solution to accommodate large-scale data by only considering a fraction of the pre-mapped Euclidean space. The initially selected subset of pre-mapped data is usually known as \emph{landmarks}. Landmarks or \emph{anchors} have been used with out-of-sample extensions to scale MDS and other embedding techniques~\cite{Silva_2002,Virtual_L}. We discuss the characteristics of a good set of landmarks and landmarks selection methods in Section~5.
Once the Euclidean space consists of all the data required, we then formulate our indexing method to generate blocks that group close-by vectors in the Euclidean space.
MDS maps high dimensional data (in the original space) into a Euclidean space (vector space). The rationale for performing such a mapping is to approximate the distances between objects in an original space in Euclidean space. Searching for similar points in Euclidean space is less expensive and quick since we can use efficient data structures such as Kd-trees. We use Kd-trees and k-NN search to find similar vectors in the Euclidean space.
In general, k-NN search refers to finding the closest elements for a query $q$ within a given set of points $N$, as measured by some distance function ${d}(N, q)$~\cite{Kumar}. The distance function $d$ is a metric, e.g., $L^p$ norm, which satisfies the non-negativity, identity, symmetry, and triangle inequality properties~\cite{JChatfied1980, Loo2014}. We use Euclidean distances $d_E$( where $p=2$) in our calculations. Here we are interested in finding the k-NNs ($k$ nearest neighbours) where k may be moderately large.
\begin{mydef}{k-nearest neighbour (k-NN) search}: The query retrieves the $k$ closest elements to $q$ in $N$. If the collection to be searched consists of fewer than $k$ objects, the query returns the whole database. If the set of the $k$ nearest neighbours of $q$ are $n_c$, then formally, $n_c$ can be defined as follows:
\\
$kNN(q) = {n_c \subseteq N, |n_c| = k \land \forall x \in n_c, y \in N - n_c : d(q, x) \leq d(q, y)}$.
\end{mydef}
An index is a data structure that reduces the number of distance evaluations needed at query time. An efficient and scalable indexing method can facilitate accurate and efficient k-NN search that supports large-scale datasets. We can apply several k-NN search methods for indexing arbitrary metric spaces; for more details, refer to the surveys~\cite{Hjaltason,Edgar}. Distance-based indexing methods use distance computations to build the index. Once the index is created, these can often perform similarity queries with a significantly lower number of distance computations than a sequential scan of the entire dataset~\cite{Hjaltason}.
The decision of which indexing structure to apply depends on several factors, including query type, data type, complexity and the application. Among many data structures, we choose Kd-trees for the k-NN search. It is considered one of the best data structures for indexing multidimensional spaces and is designed for efficient k-NN search~\cite{DES}.
Kd-trees organise $K$-dimensional vectors of numeric data. Each internal node of the tree represents a branching decision in terms of a single attribute's value, called a \textit{split value}~\cite{Talbert}. These internal nodes generate a splitting hyperplane that divides the space into subspaces using this split value, usually the median value along the splitting dimension. We will use the median when constructing the Kd-tree for the data in our experiments. Building a Kd-tree (with the number of dimensions $K$ fixed, and dataset size $N$) has $O(N\log N)$ complexity~\cite{Arya}. For more details on Kd-trees and k-NN search implementations, refer to~\cite{Arya}.
The k-NN search algorithm aims to find a node in the tree closest to a given input vector. Searching a Kd-tree for $k$ nearest neighbours is $O(k\log N)$, which is the key to fast indexing. It uses tree properties to quickly eliminate large portions of the search space~\cite{DES}.
\section{Methods of Em-K Indexing }
\subsection{Indexing for Deduplication}
Deduplication refers to identifying matching records within a single database and has many applications in database and business contexts. For instance, many businesses maintain databases of customer information that are utilised for advertising purposes, e.g., emailing flyers. Duplicate entries might arise because of errors in data entry or address changes. Deduplication techniques are useful to remove duplicate entries and to improve the quality of the collected information. Duplicate-free customer information databases prevent emailing several copies of flyers to the same customer, which reduces the cost of advertising, but there are many other benefits.
Indexing is a preprocessing step to avoid the need to perform detail-level comparisons between $O(N^2)$ pairs of records. The proposed indexing method utilises the properties of vectors and Kd-trees in a Euclidean space. The method has two main steps,
\begin{enumerate}
\item Embedding the blocking values: We select the blocking criteria based on the attributes of the records in a dataset. For instance, given a set of records with entity identifying attributes such as first name, last name, date of birth, or postcode, we may choose one or several values of them in our indexing method. We embed the blocking values of a dataset/database in the Euclidean space by applying LSMDS. The embedding depends on the size of the dataset. We propose two techniques:
\begin{enumerate}
\item\textit{Complete LSMDS}: For a given dataset of size N, apply LSMDS for the blocking values of all the records.
\item \textit{Landmark LSMDS}: Apply complete LSMDS only to a fraction of the dataset (the landmark records). Then the remaining records are embedded using the out-of-sample embedding against the landmark points. We explained this approach in Section 2.
\end{enumerate}
\item Nearest neighbour search: Searching for similar points is a two-step procedure.
\begin{enumerate}
\item The first step is to build the Kd-tree in the Euclidean space using all the points that represent the blocking values in this space.
\item The second step is to create blocks of similar points by searching the nearest-neighbours of the Kd-tree nodes.
\end{enumerate}
\smallskip
Since the Kd-tree construction uses all the available points in the Euclidean space, each record becomes a node in the Kd-tree. Likewise, each node becomes a query against the rest of the nodes in the k-NN search. Each node has a fixed number of nearest neighbours (NNs) allocated for them as we keep the k-NN search fixed for every querying node. Hence, each node in the tree becomes a small block of records that contains its NNs as the members. The block sizes are uniform and depend on the number of NNs allocated for a node.
\end{enumerate}
Once all the blocks are determined by the indexing method, we return the pairs of similar points in each block as potential matching records. Then, we retrieve the original blocking values that correspond to these points and compare them to classify the pairs as candidate matches or non-matches based on a pre-selected threshold.
A detailed level comparison among the other attribute values will be only required between the pairs that indexing identifies as candidate matches. Thus our indexing method act as a filtering step that reduces the total number of detailed comparisons one has to perform when identifying similar records in a dataset.
\subsubsection{Complexity}
We can quantify the complexity of the proposed indexing method. The method has two components: a relatively slow step where the records (blocking values) are mapped to a Euclidean space, followed by a relatively fast step that creates blocks in the Euclidean space.
Assume that we have $N$ records in the underlying database. Complete LSMDS requires calculating the distance between all pairs of blocking values, hence, $O(N^2)$ operations for the embedding. This complexity dominates the LSMDS calculations. However, for large $N$, we can choose a set of landmarks $L$ and apply LSMDS with a complexity of $O(L^2)$. The embedding of the rest of the points has a linear complexity of $O(ML)$ operations, where $M=N-L$. Hence the overall complexity of landmark LSMDS is $O(L^2 + ML)$.
The second phase builds and searches the Kd-tree to create blocks of records. Building the Kd-tree requires $O(N \log N)$ operations, and searching for $k$ nearest neighbours for $N$ points requires $O(Nk \log N)$ operations, where the size of a block ($B$) is equal to $k$. Hence the overall complexity of the indexing step is $O((1+k)N \log N)$.
Since a complete LSMDS requires $O(N^2)$ operations, we recommend using the landmark LSMDS and as small as possible $k$ to reduce the complexity of the proposed method.
\subsection{Indexing for Query Matching}
Query matching in this paper refers to querying a stream of records against a reference database to find records that refer to the same entity as the query (see Section 3). Query processing should be quick and accurate for many ER solutions to increase their usability in real-time applications. In some applications, a stream of queries might need to be processed within a given time, collecting as many matches as possible. In such settings, we have to trade accuracy against speed when detecting matching records. This section presents a scalable indexing method for real-time, approximate query matching against a large-scale database. The ideas presented in Section~4.1 serve as the basic building blocks for the proposed method.
Following the \textit{clean-clean ER} scenario, we consider a large-scale reference database $E_r$ and a batch of streaming queries $Q$. Similar to the previous indexing method, we first embed all the blocking values of the reference database in a Euclidean space. We use a set of landmarks and the out-of-sample embedding of LSMDS to reduce the overhead of embedding the database $E_r$. The embedding is a two-step procedure: 1) apply LSMDS over the landmarks 2) then map the rest of the blocking values using out-of-sample embedding of LSMDS based on the distances to these landmarks. We then construct a Kd-tree using all the points in the Euclidean space, where each data point becomes a node in the tree.
For streaming queries, we process a single query record at a time. First, we embed the blocking value of the query record in the pre-mapped Euclidean space. In general, out-of-sample embedding would require calculating all the distances from the new query to the pre-mapped blocking values in the original string space, which is not desirable with large-scale databases. Hence, we only calculate the distances to the landmarks when mapping a new record. The mapping position is determined by applying the out-of-sample embedding of LSMDS to the new query record. The process is similar to mapping the blocking values that are not landmarks in the reference database.
The only inputs we need for the mapping are the distances from a query to the pre-selected landmark blocking values. Then we search for the k-NNs of the new point using the existing Kd-tree structure of the reference database. A new block of points that contains similar points for the query point will be determined. The block size $B$ depends on the $k$ of the k-NN search. We retrieve the original blocking values for the block of similar points and use a pre-selected threshold to filter out the potential matching pairs. The records that contain these as blocking values are the matching records to the querying record. A detailed level comparison between the candidate matching pairs may be required after the initial step of indexing.
The algorithm needs to process a query within a fixed time, potentially sub-second. Since we are querying a stream of query records against a large-scale database, the processing time of a single query is limited. The overall system performance can be improved by trading accuracy against efficiency and scalability. We will fine-tune our method to trade-off many comparisons against accuracy to offer a greater number of detections within a fixed time. Thus we select a set of optimal parameters that scale our data by considering the trade-off between accuracy and scalability.
\subsubsection{Complexity}
We can quantify the complexity of different stages of the Em-K indexing for query matching. Similar to the previous method, it contains two components: a relatively slow step where a new query is mapped to a pre-mapped Euclidean space, followed by a relatively fast step that creates a block of similar points which contains potential matches for \mbox{the query}. We assume below that the reference database has already been embedded since this cost is amortised across many queries.
Suppose that we have $N$ data points in our reference database. Typically it would require~$O(N)$ operations to compare the existing records with a query point, which is not feasible for larger~$N$. We avoid this complexity by applying out-of-sample embedding of LSMDS using a set of~$L$ landmarks, which requires only~$O(L)$ operations to embed a new data point. The embedding is efficient if $L$ is chosen such that $L \ll N$.
In the second phase, we search for $k$ nearest neighbours for the query. Therefore, the block size $B$ is equal to $k$. It will cost $O(k \log N )$ operations to search the tree and $O(k)$ operations to compare a new query point within a block. The total cost of indexing is $O(L+ k \log N)$. However, \mbox{$k \ll L \ll N$} in large-scale applications. The cost of the k-NN search is insignificant due to the efficient indexing structure of the Kd-tree. Thus out-of-sample embedding step dominates the complexity of the proposed method.
The number of landmarks $L$ will affect both the accuracy and scalability of this method. We will discuss the selection of landmarks and results in the next section.
\section{Experimental Validation}
We evaluated the two proposed Em-K indexing methods under various settings (e.g., different dimensions, varying block sizes and datasets, and error rates). Two main questions to study in the experiments are: (1) Are these proposed methods robust over various settings? (2) Does Em-K indexing achieve high accuracy and good scalability?
All algorithms are implemented in R and executed on a desktop with Intel Core 5 Quad 2.3GHz, 16GB RAM, and MacOS Big Sur.
\subsection{Set Up}
\subsubsection{Data Sets}
We examined the performance of our methods over two synthetic datasets. They can be manipulated to have significant variations in their size and characteristics (e.g., error rates).
\begin{itemize}
\item[-] \textit{\textbf{Dataset-1:}} The first data set contains records with synthetically generated biographic information. Each record has a given name and a surname. They are generated using the tool Geco~\cite{geco}. We introduced duplicate records with errors by slightly modifying the values of randomly selected entries. In record generation, we assumed a record has only one duplicate with a maximum of two typographical errors (substitutions, deletions, insertions, and transpositions) in both attribute values.
For the deduplication datasets, there is one duplicate for a particular record within the dataset. Similarly, in query matching, each query has one duplicate within the reference database, and the reference database is duplicate free.
\item[-] \textit{\textbf{Dataset-2:}} The second dataset is the benchmark dataset presented in~\cite{Saeedi}. It is based on personal records from the North Carolina voter registry and synthetically generated duplicates using Geco~\cite{geco}. Each record has several attribute fields. We cannot control the errors of the duplicate records in this dataset since it has been formulated as a benchmark~dataset. However, after careful analysis of the dataset, we estimated that a duplicate record has a maximum of three edit distance errors for this work. We have only considered the first name and the last name fields in our experiments.
Similarly, we selected the deduplication data such that there is only one duplicate for a particular record within the dataset. In the query matching, we choose the queries to have only one duplicate within the reference database and the reference database to be duplicate free.
\end{itemize}
\subsubsection{Matching Rates}
We control the number of duplicates in our experiments in order to understand how well the method works in different circumstances. We used two matching rates: Consider the datasets $E$, $E_r$ and $Q$ with sizes $|E|$, $|E_r|$ and~$|Q|$,
\begin{enumerate}
\item \textit{Deduplication matching rate ($DMR$)}: the matching rate for deduplication is defined to be the $DMR(E)= \frac{E_d}{|E|}$ where $E_d$ is the number of duplicate records in $E$.
\item \textit{Query matching rate ($QMR$)}: the matching rate for query matching of $Q$ against a reference database $E_r$ is defined to be the $QMR = \frac{M_{RQ}}{|E_r|}$ where $M_{RQ}$ is the number of records in $E_r$ that has a matching record (duplicate) in $Q$.
\end{enumerate}
We control the number of duplicates within and between datasets by changing the $DMR$ and $QMR$. We represent $DMR$ and $QMR$ as percentages in our experiments.
\subsubsection{Performance Evaluation Metrics}
We use two measures to quantify the efficiency and the quality of our indexing method proposed for deduplication~\cite{Christen2007}.
\begin{itemize}
\item Reduction Ratio (RR): Measures the relative reduction of the comparison space, given by $RR=1- \frac{N_b}{|E| (|E|-1)}$ where $N_b$ is the number of potential matching pairs produced by an indexing algorithm. This quantifies how useful the indexing is at reducing the search space for detailed comparisons.
\item Pair Completeness (PC): This is given by $PC=\frac{N_m}{M}$, where $N_m$ denotes the detected number of real matches by the indexing algorithm and $M$ represents the number of all real matches in $E$. This is a measure of how accurate the indexing is.
\end{itemize}
Both PC and RR are defined in the interval $[0, 1]$, with higher values indicating higher recall and efficiency, respectively. However, PC and RR have a trade-off: more comparisons (higher $N_b$) allow high PC but reduce the RR. Therefore, indexing techniques are successful when they achieve a fair balance between PC and RR.
We used two measures to evaluate the performance of our indexing method of query matching. These are different from the standard measures of indexing defined above as we combine indexing with query matching here. Hence, in this context, we are interested in measuring the efficiency of the method in terms of time and speed of processing a query.
The Em-K indexing method returns a block of records that contains both true positive (TP) and false positive (FP) matches per query as a final result. Hence the following measures are used.
\begin{itemize}
\item Number of true positives per computational effort: Measures the number of true matching records determined by the indexing method when processing queries within a set period.
\item Precision: Measures the accuracy of the query matching in terms of precision. The precision $P$ is denoted by $P=\frac{|TP|}{|TP|+|FP|}$, where $|TP|$ is the number of true positives and $|FP|$ is the number of false positives.
\end{itemize}
All the CPU running times are measured in \textit{seconds} and denoted by RT.
\subsection{Choice of Parameters}
Several factors may impact the performance of the proposed
methods; some of them (e.g., dimension ($K$), block size ($B$), landmarks ($L$) ) are control parameters that we rely on to fine-tune the performance, while others (e.g., dataset size or error rate) are parameters determined by data sets.
In this section, We initially discuss the choices of the $K$, $B$, $L$ parameters in our indexing implementations (and their rationale). The robustness of the methods is measured with respect to the varying sizes and matching rates of the data sets in Sections~5.5–5.7.
\textbf{$\mathbf K$}: The dimension of the Euclidean space ($K$) plays an important role in the performance of our indexing methods. We applied LSMDS to a sample of 5000 records selected from Dataset-1. In \autoref{fig:1}, the first y-axis shows the stress ($\sigma$ defined in \autoref{eq:1}) decreases as $K$ increases.
A good value of $K$ should differentiate similar objects from dissimilar ones by approximating the original distances. If $K$ is too small, we will have high-stress values where dissimilar pairs will not fall far enough from each other. It could also place dissimilar pairs closer and similar pairs further apart. Conversely, high $K$ values will have low-stress values. However, in terms of RT, higher dimensions increase the embedding time. In \autoref{fig:1}, the second y-axis represents embedding RT. It takes more than 30 minutes to embed the dataset in 19 or 20 dimensions.
Considering the trade-off between the \textit{stress vs dimension} and the \textit{embedding time vs dimension}, a reasonable value of $\sigma$ is found around 6-8 dimensions. This is consistent with the embedding name strings in the Euclidean space, as discussed in detail in~\cite{herath2020simulating}. We will use $K=7$ here.
\begin{figure}[ht!]
\centering
\includegraphics [height=1.7in, width=3.2in]{02_time_vs_embedding.png}
\caption{The trade-off between the dimension ($K$) vs stress ($\sigma$) and embedding time. The first y-axis represents $\sigma$, while the second y-axis represents the embedding time. The $\sigma$ tends towards a small but non-zero asymptote when $K$ increases. The running time increases linearly when K increases. Higher dimensions allow lower $\sigma$ values for the embedding but increase the embedding time, for marginal benefit.}
\label{fig:1}
\Description{ Line plot with two y-axises. The right y-axis shows the stress values ranging from 0.1 to 0.7, and the left y-axis shows embedding running time in minutes. It ranges from 0-50 minutes. The x-axis represents the dimension which varies from 1 to 20.}
\end{figure}
{$\mathbf{B}$}: Block size is a dominating factor that directly affects the effectiveness and efficiency of many indexing techniques. Large block sizes increase RT in the indexing step and have a low RR and high PC values. In contrast, small block sizes lead to high RR values with fewer comparisons within each block. However, this may result in low PC values due to missing some matches. Blocks of similar points are determined by k-NN search in Em-K indexing methods. Hence, $B$ is equal to $k$ (number of nearest neighbours). We will consider the choice of $B$ in detail in Section~5.2.1.
\textbf{$\mathbf L$}: The number of landmarks ($L$) is another important factor in our indexing methods. Landmarks are utilised for two different purposes in this work. First, we use landmarks for embedding large-scale data into a Euclidean space when the standard LSMDS method becomes inefficient. Second, we use landmarks to embed the out-of-sample queries in a pre-mapped Euclidean space. We discuss the role of landmarks in deduplication and their impact on the proposed indexing method in Section~5.2.2.
In the following experiments we investigated how $B$ and $L$ impact our indexing algorithms.
\subsubsection{Varying Block Sizes ($B$)}
To test the choice of $B$, we used sample datasets containing 5000 records from the two data sources. We set the $DMR=10$\% for the data selected from Dataset-1, which means there are 500 duplicates within the selected 5000 records in the sample dataset. Similarly, we used $DMR=7.5$\% for the second data sample selected from Dataset-2. Hence within the 5000 records, 375 of them are duplicates.
We performed deduplication indexing on the two datasets. \autoref{fig:2} illustrates the trade-off between PC and RR of our indexing method for different block sizes using the Dataset-1. In each instance, we changed $B$ by varying $k$ in the k-NN search. PC increases with the increase of $B$. In contrast, RR decreases due to the increment in the number of comparisons within each block of records.
\begin{figure}[ht!]
\centering
\includegraphics [height=1.8in, width=3.3in]{01_deduplication_dim.png}
\caption{The trade-off between the reduction ratio (RR) and pair completeness (PC) in different dimensions. Ideally, RR=PC=1, hence we prefer methods whose results lie as close to the top-right corner of the graph. The block sizes are 100, 90, 80, 70, 60, 50, 40, 30, 20. Large blocks achieve higher PC but lower RR. The three curves illustrate that the higher the dimension, the better the results, up to the point of diminishing returns.}
\label{fig:2}
\Description{ A line plot with three lines that represent results of different dimensions. The y-axis contains pair completeness values ranging from 0.9 to 1. The x-axis represents the reduction ratio values with a range of 0.7 to 1.}
\end{figure}
The results also indicate that dimension around 7 are good at shifting the PC-RR curve to the top-right corner delivering a good ratio between RR and PC. However, higher dimensions also mean higher computational costs (e.g., high RT) for the embedding. Based on \autoref{fig:1} and \autoref{fig:2}, therefore, we conclude that using $K=7$ and $B=\{50, 60\}$ gives the best compromise PC-RR ratio and RT overall for the given data set. In subsequent experiments, we used $K=7$.
\begin{figure}[ht!]
\centering
\includegraphics [height=1.4in, width=2.6in]{01_deduplication_data.png}
\caption{The trade-off between the reduction ratio (RR) and pair completeness (PC) for two different datasets for the proposed indexing. The block sizes are 100, 90, 80, 70, 60, 50, 40, 30, 20. PC is very low for the second dataset while RR values are closer.}
\label{fig:3}
\Description{ A line plot with two lines that represent results of different datasets. The y-axis contains pair completeness values ranging from 0.7 to 1. The x-axis represents the reduction ratio values with a range of 0.7 to 1.}
\end{figure}
In most comparisons, we observed similar results for both datasets and discussed only the results of Dataset-1. However, we present one comparison that has comparably different results here. \autoref{fig:3} compares the two datasets in a fixed dimension ($K=7$), varying $B$ in each instance. Both data sets achieve similar RR values with different PC values. PC values are comparatively low for Dataset-2 and are around 70\% for most occurrences.
Dataset 1 shows better results compared to Dataset-2 in \autoref{fig:3}. This behaviour is expected since the two datasets have different characteristics, e.g., different matching rates and a different number of errors in each field. Furthermore, we used different pre-selected thresholds ($\theta_m$) when validating candidate matching pairs in the two datasets. These thresholds are selected based on the errors in the two datasets. In Dataset-1, duplicates have a maximum of two typographical errors and therefore $\theta_m=2$. For Dataset-2, we assumed that each duplicate record has a maximum of three typographical errors, and we set $\theta_m=3$ therein.
\subsubsection{The Effect of Landmarks}
The following experiment investigated the effect of the two different embedding techniques on the proposed indexing method. Performance is measured using the PC and RR curves. First, we applied complete LSMDS similar to \autoref{fig:2}, keeping the $K=7$ fixed. Second, we applied LSMDS only to a set of landmarks in the same dimension. The remaining points are embedded using the out-of-sample embedding of LSMDS and the distances to landmarks. Blocksize $B$ is varied as before.
\begin{figure}[ht!]
\centering
\includegraphics [height=2.1in, width=4.1in]{01_compare_fullKDtree}
\caption{The trade-off between the reduction ratio (RR) and pair completeness (PC) for the proposed indexing based on different embeddings.The results are based on a complete LSMDS and landmark based LSMDS for different number of landmarks. The block sizes are 100, 90, 80, 70, 60, 55, 50, 45, 40, 35, 30.}
\label{fig:4}
\Description{ A line plot with five lines that represent results of different LSMDS embeddings. The y-axis contains pair completeness values ranging from 0.95 to 1. The x-axis represents the reduction ratio values with a range of 0.8 to 1.}
\end{figure}
\autoref{fig:4} compares the trade-off between PC and RR for complete LSMDS and landmark LSMDS (for different $L$). Each instance represents a different block size. PC and RR change similarly to the previous experiment for different block sizes. However, \autoref{fig:4} suggests that we can get similar results by choosing an approximate embedding that uses landmarks instead of complete LSMDS. The use of landmarks decreases the distance calculations and we can avoid the inherent complexity and inefficiency of LSMDS when processing large-scale data.
Using our indexing solution, we can solve deduplication applications in ER. The method requires embedding records into a Euclidean space in order to apply the indexing technique. Since complete LSMDS is not suitable for large-scale data, we recommend using the landmark LSMDS. We applied the farthest first sampling~\cite{KAMOUSI20161} for reproducible results in landmarks selection; however, random selection works well in practice.
The optimal parameter setting for our data is $K=7$, $B= 50$ and $L=1500$. We used the proposed indexing method of deduplication as a motivating example for the next set of experiments.
\subsection{Indexing for Query Matching}
In our experiments, we used a reference database with $E_r=5000$ records. The streaming query dataset $Q$ is flexible in size because we only consider streaming queries within a period. Each query in $Q$ has a duplicate record in the reference database, i.e., $QMR=1$. We made this assumption to keep the experiment more efficient instead of mimicking a real-world ER problem. In a real-world scenario, each query may not have a matching record within the reference database, or the same query may appear in the stream to search against the reference database. However, the time required for searching is the same when no match is present.
Similar to the previous method, several factors affect the performance of the proposed indexing method for querying, e.g., $K, L, B$. Since the embedding of the reference database is the same as before, we keep $K=7$. The block size is $B$ is equal to $k$, the number of nearest neighbours in k-NN search.
We used landmarks to support the out-of-sample embedding. The number of landmarks, $L$, directly impacts the running time (RT) of the proposed method since each query needs to be embedded in the Euclidean space. The other costs that contribute to RT are the distance calculations and k-NN search.
First, we embedded the reference database generated from Dataset-1 applying landmark LSMDS. We chose the landmarks based on the farthest-first sampling. Then, we built a Kd-tree using all the reference data points in the Euclidean space. Once the Kd-tree is built, we passed queries to search the tree for k-NNs. Hence, each query in $Q$ needs to be mapped in the Euclidean space. We used the same set of landmarks among the reference data points to map the query points. Distance calculations are required among the new query point and the landmarks when applying the out-of-sample embedding of LSMDS. A block of similar points is determined for a new query by searching k-NNs in the existing Kd-tree.
Our method process a single query at a time. In the following experiment, we processed 500 queries against the reference database. For each query, we measured the embedding RT and the distance calculation RT separately. Then we calculated the mean values for both categories, processing all the 500 queries. \autoref{fig:5} shows the comparison of the mean RT of distance calculations and out-of-sample embedding for varying numbers of landmarks. Increasing $L$ linearly increases the embedding RT of a single query. The distance calculation RT also increases linearly with $L$ but is negligible compared to out-of-sample embedding RT.
\begin{figure}[ht!]
\centering
\includegraphics [height=2.1in, width=4.1in]{01_emd_time_feb}
\caption{The calculation times for distance calculations and out-of-sample embedding for query matching. Both depend on the number of landmarks, but distance calculations are much faster.}
\label{fig:5}
\Description{ A line plot with two lines that represent running times. The y-axis contains running times ranging from 0 to 2 measured in seconds. The x-axis represents the number of landmarks with a range of 300 to 3000.}
\end{figure}
We also measured the cost of the k-NN search when creating a block of records for a new query. This search can be done efficiently in the Euclidean space using the Kd-tree and priority queues. It takes less than a millisecond, which is insignificant compared to the total RT of the embedding process. Moreover, increasing $k$ has a smaller impact due to its efficient implementation with priority queues~\cite{Arya}.
A scalable query matching method should be able to process as many queries as possible within a period. In our indexing method, increasing $L$ limits the number of queries processed within a set period. On the other hand, small $L$ tends to decrease the accuracy of the embedding. As a result, a new query may be not mapped closer to its duplicate, reducing the probability of grouping them as similar points. Hence an optimal $L$ is required to maintain the scalability without degrading the quality of the results.
We measured the scalability and the efficiency of our method using two quantitative measures with respect to time: the number of true positive ($|TP|$) matches detected per computational effort and the precision (P) per computational effort. Hence we processed a stream of queries $Q$ against a reference database $E_r$ within a given period, varying the control parameters such as $L$ and $B$ in different instances. Then we calculated $|TP|$ and $|FP|$ found by our method in each instance. The accuracy of the results is measured in terms of precision. It measures the rate of TP against all the positive results (sum of $|TP|$ and $|FP|$) returned by the method within the given period. Subsequently, we determined an optimal set of parameter values for our data that returns the highest TP matches and precision within a fixed period.
In the following experiments, we used a reference database of $E_r=5000$ records and the stream of query records $Q=500$. We applied landmark LSMDS to embed the records in $E_r$ to the Euclidean space in each experiment. Then the queries are processed using the same set of landmarks. Hence, every instance of a different $L$ has a disparate embedding of the reference database, then used for query embedding and searching.
\begin{figure}[ht!]
\centering
\includegraphics [height=2.1in, width=4.1in]{01_new_query_NN.png}
\caption{The trade-off between the number of true positives and the number of landmarks for three different block sizes. Here the block size is equal to the number of k-NNs. Many landmarks allow fewer queries to be processed, while many k-NNs allow more true match detections.}
\label{fig:6}
\Description{ A line plot with three lines that represent results of a different nearest neighbour search. The y-axis contains the number of true positive values ranging from 0 to 500. The x-axis represents the number of landmarks with a range of 300 to 3000.}
\end{figure}
\autoref{fig:6} shows the comparison of the $|TP|$ against varying the number of landmarks and k-NNs. In each instance, we processed queries within a fixed period (60 seconds). Increasing $L$ decreases~$|TP|$ because more landmarks allow fewer queries to be processed. This phenomenon is expected since large $L$ increase the running time of processing a single query. With more landmarks, there is a higher probability of finding matches for those queries due to the increasing accuracy of the embedding. In contrast, few landmarks allow more queries to be processed within a period since the running time (RT) for embedding a single query is small.
\autoref{fig:6} suggests that we only need 100 landmarks to detect the highest number of TP matches for the given data. The method has processed all the 500 queries within a minute, detecting 432 TP matches. The average time for processing a single query is 0.07 seconds.
Based on \autoref{fig:6}, we conclude that setting $k=150$ and $L={100}$ gives the best trade-off between the quality of the results and the RT. This result is consistent with existing real-time query matching techniques that process a query within a sub-second time~\cite{Liang}.
\begin{figure}
\begin{subfigure}{10cm}
\centering\includegraphics[width=7.8cm]{01_compare_datasets}
\caption{The trade-off between the number of TP and the number of landmarks for two different datasets.}
\end{subfigure}
\begin{subfigure}{10cm}
\centering\includegraphics[width=7.8cm]{03_compare_datasets_1.png}
\caption{The trade-off between the precision and the number landmarks for two different datasets.}
\end{subfigure}
\caption{The two figures (a) and (b) compares the Dataset-1 and Dataset-2 in terms of number of TP and Precision varying the $L$ The k-NNs are fixed to $k=150$ and $T=60$ seconds in each instance. The curves illustrate similar trends for similar parameter settings in (a). However, in (c), Dataset-2 exhibits low precision compared to Dataset-1.}
\label{fig:7}
\Description{Two line plots a and b displaying the precision and the number of true positives against different numbers of landmarks. The y-axis of plot a has a value range from 100 to 500. The y-axis of the second plot range from 0.8 to 1. Both plots have the same x-axis with a range of 100-3000 landmarks. Each plot has two curves in different colours.}
\end{figure}
To validate the robustness of the method, we compared the results of applying the EM-K indexing method to two reference databases and stream of queries derived from Dataset-1 and \mbox{Dataset-2}, respectively in \mbox{\autoref{fig:7}}. Both reference databases contained 5000 records. The two streams of queries are processed against the two reference databases separately with similar parameter settings. The query matching rate ($QMR$) is equal to 1 for both. Keeping the control parameters fixed at $K=7$, $B=150$ and $T=60$ seconds, we vary $L$ to compare different results.
\mbox{\autoref{fig:7}-(a)} compares the trade-off between the $|TP|$ per computational effort, and the number of landmarks. Increasing $L$ decreases the $|TP|$ for query matching in both datasets similar to \autoref{fig:6}. Few landmarks allow processing all the queries within 60 seconds, whereas many landmarks allow processing fewer queries. While both reference databases are similar in size, the total queries processed are different in size. A total of 500 queries are processed against Dataset-1, and 375 queries are processed against Dataset-2 in the experiment. Hence, we observe fewer TP matches for Dataset-2 than Dataset-1 in their highest performance. However, the curves illustrate similar trends for both datasets.
Based on the results produced by the previous experiment, \mbox{\autoref{fig:7}-(b)} compares the two datasets in terms of the precision (P) against varying $L$ In each instance, the queries are processed within T=60 seconds. Hence the P values are measured per computational effort. Dataset-2 has lower P values compared to Dataset-1. This behaviour is expected since we assume that the duplicate records in Dataset-2 contain a maximum of three edit distance errors. The pre-selected thresholds~($\theta_m$) are different for the two datasets. However, we do not have the exact details of error rates for this benchmark dataset. Hence, the method may allow more false positives in the final block of records retrieved for a given query, reducing the P.
However, the overall results suggest that the proposed method is robust over different datasets. The optimal parameter set that provides the best compromise for our data is $K=7$, $B$ (or $k$)$= 150$ and $L={100,300}$. We need a few landmarks to achieve the most number of TP matches for a stream of queries within a period. Few landmarks make the embedding efficient. As a result, a single query can be processed within less than a second against a reference database to find matches. Hence our method scales well for approximate query matching against a large-scale reference database for ER.
\section{Discussion}
The Em-K indexing methods embed a set of strings in a metric-space, particularly a lower-dimensional Euclidean space. There is a trade-off between the dimension and the accuracy of the embedding. Higher dimensions allow the embedding to be more accurate. However, it does not scale well for large datasets. In our experiments, we only selected two blocking variables for embedding and indexing. We can also use other blocking variables, such as addresses and gender. As a result, the dimension of the Euclidean space needs to extend accordingly to facilitate those.
The most costly part of the method is the amount of time to embed queries, which increases linearly with the number of landmarks. However, this approach is easily parallelizable since each query is processed separately. We can tune the parameters for a fast, less accurate query matching or a slow, more accurate method depending on the application.
Our method is designed to solve approximate query matching rather than exact query matching. It means we expect the queries to contain errors in their attribute values. We could easily perform an exact search based on lexicographical order considering the query and the reference dataset using a data structure such as a binary tree to find exact matches as a pre-filter for our method.
Our indexing method for query matching uses distance computations to embed the reference database in the Euclidean space. This embedding is a slow process that requires a minimum of $O(L^2)$ operations where $L$ is the number of landmarks. We consider this as the training phase, which only needs to be performed once. However, with the Kd-tree built, we can perform similarity queries with significantly fewer distance computations than a sequential scan of the entire dataset.
In some applications, large-scale reference databases grow with time. Hence the new entities need to be added incrementally. For example, some applications would require the addition of queries that have no matching records in the reference database already. To facilitate those, we need to extend the Kd-tree accordingly without repeating the embedding process that creates the initial tree. However, growing a Kd-tree can be a heuristic procedure where the tree could become unbalanced. Therefore, we can explore alternative tree structures such as R-tree that are robust against dynamic data.
\section{Related Work}
Indexing techniques are recognized as a crucial component for improving the efficiency and the scalability of the ER solutions. There exist a variety of indexing techniques which are also known as blocking or filtering. \mbox{Papadakis \textit{et al.}~\cite{sur2}} and Christen \textit{et al.}~\cite{sur1} presented surveys that include well-known indexing methods such as standard blocking, suffix array, q-gram blocking, sorted-neighbourhood, canopy clustering, and string-map based methods. We only survey a few closely related works since many of these existing methods are orthogonal to the focus of this paper.
Canopy clustering uses cheap comparison metrics to group similar records into overlapping canopies and creates blocks from records that share a common-canopy~\cite{MacCallum}. The method depends on global threshold values, and that reduces its flexibility. It also uses similarity measures such as TF-IDF and Cosine distance that can be computationally expensive~\cite{sur1}.
Mapping-based indexing methods map records to objects in a Euclidean space, preserving the original distances between them. Jin \textit{et al.}~\cite{Li-chen} proposed the Stringmap algorithm that maps records into a similarity preserving Euclidean space (with dimension between 15-20). Similar pairs are determined by building an R-tree. Stringmap has linear complexity, but it requires tuning several parameters. Moreover, the performance of such approaches tends to decrease with more than 20 dimensions. In contrast, our method operates in much smaller dimensions.
The Double Embedding scheme \cite{DES} uses two-dimensions, $K$ and $K'$ for embedding records such that $K'<K$. Similarity joins are performed in the metric-space using a Kd-tree and nearest-neighbour search to find candidate matches. The method is faster than the Stringmap algorithm. However, it attempts to keep the embedding contractive by increasing the distance computations and is not suitable for large-scale data.
Another metric-space indexing technique utilised an M-tree to produce complete and efficient ER results. The cost of the method and the quality of the results have remained similar to existing indexing techniques~\cite{Akgun}. However, it does not scale for large-scale data since it has a single step that combines the indexing, comparison, and classification steps.
In summary, the existing mapping-based indexing techniques have two components. The first is to map the records into a metric-space. The second is to perform similarity joins in the metric-space using a tree-like indexing structure.
These methods were developed offline by applying a mapping technique to map all the records into a multidimensional metric-space. The spatial mapping of records acts as a filtering step before the actual record matching. Hence, the focus is on matching similar records without grouping them into blocks. However, none of them handles new records by reusing the properties of an existing multidimensional metric-space. Therefore, query matching is not achievable. In this paper, similar ideas are significantly extended to accommodate the benefits of the Euclidean space for efficient query matching.
\section{Conclusions}
Indexing techniques reduce the pairwise comparisons in ER solutions.
Many existing mapping-based indexing techniques work in offline mode with fixed-size databases. Hence, these techniques are not suitable for applications that require real-time query matching, especially if it involves big, fast, or streaming data. Our method investigated the query matching problem in ER by using spatial mapping of records into a Euclidean space. We aimed to develop an indexing approach for a fast query process within a short time, returning as many as potential matches. The proposed method proved fast running time as well as scalability along with the data size. The use of vectors in the Euclidean space that represent records allowed fast query matching with comparable accuracy. Many directions are ahead for future work. First, we plan to extend the Em-K indexing method to be parallel. Second, the current Em-K indexing can be extended to other forms of ER problems, such as querying against a dynamic database or iterative ER~\cite{HARRA}.
\bibliographystyle{ACM-Reference-Format}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 713 |
# TODO Make it look like more ruby
(Dir[File.dirname(__FILE__) + "/core_ext/*"] + Dir[File.dirname(__FILE__) + "/rails_ext/active_record/*"]).each do |path|
require path
end
class ClassDiagramGenerator
cattr_accessor :associations_metadata
def initialize
@style = "nofunky"
@note= ""
@classes = []
@options = ClassDiagramGenerator::Options.new
ClassDiagramGenerator.associations_metadata = ActiveRecord::Metadata.new
end
# TODO Diagram style ?
OPTIONS_LIST = %w[attributes public_methods inheritance associations association_types debugging]
OPTIONS_LIST.each do |option|
define_method("show_#{option}") do
eval "@options.#{option} = true"
self
end
define_method("hide_#{option}") do
eval "@options.#{option} = false"
self
end
end
class Options
OPTIONS_LIST.each do |option|
attr_accessor option.to_sym
end
attr_accessor :classes
end
def with_note(note)
@note = "[note: #{note}]"
self
end
# Classes & introspection
def with_classes (*klasses)
@classes << klasses
@classes = @classes.flatten
self
end
def with_all_model_classes(pattern="app/models/**/*.rb", except_files=[])
files = Dir.glob(pattern) - except_files
# files += Dir.glob("vendor/plugins/**/app/models/*.rb") if @options.plugins_models
files.each do |file|
model_name = File.basename(file, '.rb').camelize
next if /_related/i =~ model_name
@classes << get_model_class(model_name)
end
self
end
def get_model_class(model_name)
begin
model_name.constantize
rescue LoadError
STDERR.print "\t#{model_name} raised LoadError.\n"
oldlen = model_path.length
model_path.gsub!(/.*[\/\\]/, '')
model_name = model_path.camelize
if oldlen > model_path.length
retry
end
STDERR.print "\tDone trying to remove slashes, skipping this model.\n"
rescue NameError
STDERR.print "\t#{model_name} raised NameError, skipping this model.\n"
end
end
private :get_model_class
def with_all_controller_classes(pattern="app/controllers/**/*_controller.rb", except_files=[])
begin
files = Dir.glob(pattern) - except_files
files.each { |file| @classes << get_controller_class(file) }
rescue LoadError
raise
end
self
end
def get_controller_class(file)
model = file.sub(/^.*app\/controllers\//, '').sub(/\.rb$/, '').camelize
parts = model.split('::')
begin
parts.inject(Object) {|klass, part| klass.const_get(part) }
rescue LoadError
Object.const_get(parts.last)
end
end
private :get_controller_class
# TODO Is it possible to move the generation code in another class?
def to_yumlme_dsl
# FIXME tmp FIX for modules
# @classes -= [ActsAsRateable, AuthenticatedBase, Geocodeable, HasStatus, XmasProjectMethods]
@options.classes = @classes
# puts @classes if @options.debugging
@yumlme_dsl = @note + @classes.map do |klass|
unless @options.inheritance
klass.to_yumlme_dsl(@options)
else
create_yuml_dsl_for_class_hierarchy(klass)
end
end.flatten.uniq.join(",")
end
def create_yuml_dsl_for_class_hierarchy(klass)
hierarchy = klass.hierarchy
hierarchy.delete(ActiveRecord::Base)
# FIXME write a test for this line
return klass.to_yumlme_dsl(@options) if hierarchy.size == 1
# TODO Make it look like ruby
dsl = []
last_item_index = hierarchy.size - 1
hierarchy.each_with_index do |klass, index|
if index < last_item_index
super_class_dsl = klass.to_yumlme_dsl(@options)
child_class_dsl = hierarchy[index+1].to_yumlme_dsl(@options)
dsl << "#{super_class_dsl}^-#{child_class_dsl}"
end
end
dsl
end
private :create_yuml_dsl_for_class_hierarchy
def to_png(file, diagram_options="")
output(file, "", diagram_options)
end
# http://yuml.me/diagram/class/[Customer]+->[Order].pdf
def to_pdf(file, diagram_options="")
self.output(file, ".pdf", diagram_options)
end
# def to_url( options, data, type ) #:nodoc:
# opts = options.clone
# diagram = opts.delete(:diagram)
# if type.nil?
# type = ""
# else
# type = ".#{type}" if type[0].chr != "."
# end
# "http://yuml.me/diagram/#{self.options_string(opts)}#{diagram}/#{data}#{type}"
# end
private
def output(file, type="", diagram_options="")
# http://yuml.me/diagram/nofunky;dir:TB;scale:120;/class/
# http://yuml.me/diagram/nofunky;dir:TB;scale:180;/class/
data = self.to_yumlme_dsl
uri = "/diagram/#{@style}#{diagram_options}/class/#{data}#{type}"
puts "*** #{uri}" if @options.debugging
writer = STDOUT
writer = open(file, "wb") unless file.nil?
res = Net::HTTP.start("yuml.me", 80) {|http|
http.get(URI.escape(uri))
}
writer.write(res.body)
writer.close
end
end | {
"redpajama_set_name": "RedPajamaGithub"
} | 483 |
\section*{}
\vspace{-1cm}
\footnotetext{\textit{$^{a}$~Institut f\"ur Experimentelle Physik, TU Bergakademie
Freiberg, Leipziger Stra{\ss}e 23, 09596 Freiberg, Germany; E-mail: roman.gumeniuk@physik.tu-freiberg.de}}
\footnotetext{\textit{$^b$~Max-Planck-Institut f\"ur Chemische Physik fester Stoffe,
N\"othnitzer Stra{\ss}e 40, 01187 Dresden, Germany}}
\footnotetext{\textit{$^{c}$~Ivan Franko National University of Lviv, Kyryla and Mefodiya Str. 6, UA-79005, Lviv, Ukraine}}
\footnotetext{\textit{$^{d}$~Swiss-Norwegian Beamlines at ESRF, CS 40220, 38043 Grenoble Cedex 9, France}}
\section{Introduction}
After their discovery in the early 1980\textit{s} by Remeika et al.\cite{Remeika1980}, a new series of primitive and face-centred cubic and tetragonal Sn-rich compounds containing rare-earth, alkaline-earth, actinide as well as transition 3$d$- and 4$d$-metals became immediately an object of numerous studies\cite{gum2018Rem}. Continuous interest in these phases is mainly due to the exciting superconducting properties and easy synthesis of nicely shaped large single crystals (i.e. by flux methods).
However, as it turned out in recent investigations, the determination of both the crystal structure and the chemical composition of such stannides is challenging. They reveal both concentration- and temperature-induced phase transitions. For example, La$_3$Rh$_4$Sn$_{13}$ crystallizes with space group (SG) $I4_132$, $a \approx 19$ \AA \cite{Bordet1991}, while La$_{3+x}$Rh$_4$Sn$_{13-x}$ ($x = 1$) is primitive cubic (SG $Pm\overline{3}m$, $a \approx 9$ \AA). \cite{Anand2011} On the other hand, body centred cubic (SG $I4_132$) Ce$_3$Rh$_4$Sn$_{13}$ is observed for $T < 350$\, K, while the primitive cubic modification (SG $Pm\overline{3}m$) exists at higher temperatures.\cite{kuo2018} Also, stannides with approximate composition $\sim$$M$\{Co, Rh\}$_{1.2}$Sn$_4$ ($M$ = Sc, Y, heavier rare-earth metal), which have initially been reported to be primitive tetragonal with unit cell parameters $a \approx 13$ \AA\, and $c \approx 9$ \AA\cite{Remeika1980,Espinosa1980,Cooper1980,Espinosa1982}, were later shown to possess composition 5:6:18 and to be body-centred tetragonal (SG $I$4$_1$/$acd$) with tripled lattice parameter $c$ (i.e. $a$\,$\approx$\,13\, \AA\, and $c$\,$\approx$\,27\,\AA).\cite{Chenavas1982}$^{,}$\cite{Miraglia1987} The first refinement of such a structural model was performed for Tb$_5$Rh$_6$Sn$_{18}$\cite{Miraglia1987} and revealed a statistical mixture of Sn and Tb to occupy one of the crystallographic sites [i.e. the (Sn$_{1-x}$Tb$_x$)Tb$_4$Rh$_6$Sn$_{18}$ composition]. Later studies showed this structure type to be characterized either by unphysical anisotropies of thermal displacements, as in the case of Sc$_5$Co$_6$Sn$_{18}$\cite{Kotur1999, Lei2009} or by numerous splits of crystallographic positions occupied by Sn-atoms (Sc$_5$Ir$_6$Sn$_{18}$\cite{Levy2019}). The crystal structure of Sc$_5$Rh$_6$Sn$_{18}$ has never been refined up to now. The unit cell parameters $a$\,$\approx$\,13.5\, \AA\, and $c$\,$\approx$\,27.1\,\AA\, and SG $I$4$_1$/$acd$ were reported in \cite{Chenavas1982} and authors of \cite{Kase2012}$^,$\cite{Bhattacha2018} assumed it to crystallize with the Tb$_5$Rh$_6$Sn$_{18}$ structure type.
Being a superconductor with $T_\mathrm{c}$ $\approx$~5~K and relatively high critical magnetic field ($B_\mathrm{c2}$ $\approx$~8~T), Sc$_5$Rh$_6$Sn$_{18}$ became an object of numerous investigations.\cite{Remeika1980,Espinosa1980,Espinosa1982,Venturini1986,Kase2012} A very recent muon-spin-rotation ($\mu$SR) study \cite{Bhattacha2018} performed on crystals of this stannide indicated a time-reversal symmetry (TRS) breaking of the superconducting state and thus, unconventional superconductivity with either a singlet d + id state with a line node or, alternatively, non unitary triplet pairing with point nodes \cite{Bhattacha2018}. This finding raises the question about the inversion center in the crystal structure of Sc$_5$Rh$_6$Sn$_{18}$. As it is well known, non-centrosymmetric superconductors (SC) are preferable candidates for unconventional superconducting properties, which includes TRS breaking as well.\cite{Bauer2012}
Taking all these observations into account as well as a strong structural disorder reported for isostructural Sc$_5$Ir$_6$Sn$_{18}$ superconductor \cite{Levy2019} we performed an extensive study of the crystal structure of Sc$_5$Rh$_6$Sn$_{18}$ applying temperature dependent high-resolution powder and single crystal X-ray diffraction as well as transmission electron microscopy. To understand the obtained structural model the analysis of the chemical bonding situation assuming different structural models is performed. These investigations and simulations revealed Sc$_5$Rh$_6$Sn$_{18}$ to crystallize with the centrosymmetric Sc$_5$Ir$_6$Sn$_{18}$ structure type. Also, we demonstrate the consequences of the structural disorder of the compound for its electrical and thermal transport properties.
\section{Experimental}
\subsection{Synthesis}
Samples with nominal composition Sc$_5$Rh$_6$Sn$_{18}$ and total mass of 2\,g were prepared from scandium ingots (Dr.\ Lamprecht, 99.5\,wt.\,\%), rhodium ingots (Chempur, 99.95\,wt.\,\%) and tin (Chempur 99.9999\,wt.\,\%) by arc melting (mass losses $<0.1$\,\%). Obtained buttons with 10 g of tin excess were placed in glassy carbon crucibles, sealed in a tantalum tube and enclosed in an evacuated silica ampoule. All manipulations were performed inside an argon-filled glove box [$p$(O$_2$,H$_2$O) $\leq$ 1 ppm]. To grow the crystals the samples were heated up to 1473\,K within 24\,h, held at this temperature for 48\,h and then cooled down to 673\,K within 1100\,h. The tin excess was removed from the samples by centrifugation at 873\,K. Obtained buttons were additionally washed in diluted (5\,\%) HCl acid. A typical Sc$_5$Rh$_6$Sn$_{18}$ crystal of a pyramidal shape with $\sim 1 \times 1 \times 1$ cm dimensions together with a well resolved Laue pattern, are shown in Fig.\ \ref{fig:cry}. Further, the crystals were oriented and bars parallel to $c$- and $ab$-directions were cut for the physical properties measurements.
\subsection{Differential thermal analysis}
Differential thermal analysis (DTA) was performed on a piece of single-crystal using a \textit{Netzsch DSC 404C}, under a steady argon flow at a heating rate of 5\,K\,min$^{-1}$. The melting point of Sc$_5$Rh$_6$Sn$_{18}$ was determined via the peak onset to be 1275\,K.
\subsection{X-ray diffraction}
Laboratory X-ray diffraction (XRD) of the powdered crystals was performed on a HUBER G670 imaging plate Guinier camera (Cu$K_{\alpha1}$ radiation, $\lambda$\,=\,1.54056 \r A). The \emph{WinXpow} program package was used for phase analysis \cite{STOE}.
Temperature-dependent synchrotron XRD was performed on powder at the Rossendorf beamline BM20 of the ESRF using synchrotron radiation of wavelength $\lambda$ = 0.45932\,\r A. Sieved, powdered samples (fraction \textless\,20\,$\mu$m) were measured in quartz glass capillaries in the temperature range from 100\,K to 300\,K using a nitrogen cryostream apparatus to cool the samples. Diffraction patterns were recorded from 5$^{\circ}$ to 26$^{\circ}$ 2$\theta$ with $\Delta$2$\theta$ = 0.002$^{\circ}$ steps and a counting time of 0.5\,s per step.
Single crystal XRD data were also collected with a Rigaku AFC7 diffractometer (Mo$K_{\alpha}$ radiation, $\lambda$\,=\,0.71073 \r A) equipped with Saturn724+ CCD detector.
Single crystal XRD data were collected in the temperature range 100\,K--300\,K (cooling with a nitrogen cryostream apparatus) at the BM01 beamline of the European Synchrotron Radiation Facility (ESRF, Grenoble) with $\lambda$ = 0.73331\,\AA. The images acquired on a Pilatus2M detector \cite{Eikenberry2003} were pre-processed with the \textit{SNBL toolbox} \cite{Dyadkin2016} and subsequently processed using the \textit{CrysAlis} \cite{Crysalis} software.
The lattice parameter refinement by least-squares fitting, Rietveld refinement as well as refinements of single crystal data have been done using the \emph{WinCSD} program package \cite{Akselrud2014}.
\begin{figure}
\centering
\includegraphics[width=6cm, height=6cm]{Fig01.eps}
\caption{Typical Laue pattern of a Sc$_5$Rh$_6$Sn$_{18}$ single crystal (shown in inset).}
\label{fig:cry}
\end{figure}
\subsection{Metallography}
For metallographic analysis the synthesized crystals were embedded in conductive resin and then ground, polished and finished with diamond abrasives. The metallographic microstructure was characterized by light-optical microscopy (Zeiss Axioplan 2) and by energy-dispersive X-ray spectroscopy (EDXS) on a JEOL JSM 6610 scanning electron microscope equipped with an UltraDry EDS detector. No additional phases were observed on the surface of the crystals and their composition on the basis of 8 independent measurements was found to be Sc$_{5.0(1)}$Rh$_{6.1(1)}$Sn$_{18.2(2)}$.
\subsection{Transmission Electron Microscopy}
For transmission electron microscopy (TEM) the powdered sample was dispersed in methanol. Several drops of the dispersion were loaded on a carbon coated copper grid and transferred to the microscope after complete dryness. The Quantifoil S7/2 (100-mesh hexagonal) copper grids were covered with 2 nm carbon film (Quantifoil Micro Tools, Jena, Germany). High-resolution TEM (HRTEM) imaging of the sample was performed on a FEI Tecnai F30 with a field-emission gun at an acceleration voltage of 300\,kV. The point resolution amounted to 1.9\,\r A, and the information
limit amounted to about 1.2\,\r A. The microscope was equipped with a wide-angle slow-scan CCD camera (MultiScan, 2k$\times$2k pixels; Gatan Inc., Pleasanton, CA, USA). The analysis of the TEM images was made with the \textit{Digital Micrograph} software (Gatan, USA).
\subsection{Electrical and thermal transport properties}
The Seebeck coefficient of the thermopower, the thermal conductivity as well as the resistivity were measured in a commercial system (PPMS, Quantum Design) on the TTO option.
\subsection{Theoretical calculations}
The electronic structure calculations were carried out within the local density approximation (LDA) to the density functional theory (DFT) using the all electron full-potential local orbital method (FPLO) \cite{Koepernik1999FPLO}. The Perdew-Wang parametrization for the exchange-correlation functional was applied \cite{perdew1992}. Sn 4$s$, 4$p$, 4$d$, Rh 4$s$, 4$p$ and Sc 3$s$, 3$p$ states were treated as semicore. The first Brillouin zone (BZ) was sampled with a 6\,$\times$6\,$\times$6 mesh and the linear tetrahedron method was employed for BZ integrations.
The position space chemical bonding analysis was based on the topological analysis of the electron density (ED) and electron localizability indicator (ELI) \cite{Kohout2004}. The ELI-D representation of ELI was used \cite{Kohout2006,Kohout2007}. Both ED and ELI-D were computed by a module implemented in the FPLO method \cite{Ormeci2006}, and their topological analysis was performed by the program DGrid \cite{DGrid}. The underlying theory of this approach is provided by Bader's quantum theory of atoms in molecules (QTAIM) \cite{Quantumtheory}. The basin intersection technique of Raub and Jansen was employed to determine the atoms participating in the bonds \cite{Raub2001}.
\section{Results and discussion}
\subsection{Crystal structure}
\subsubsection{Single crystal and powder X-ray diffraction}
All peaks collected during the room temperature single crystal diffraction measurement on Sc$_5$Rh$_6$Sn$_{18}$ at the BM01 beamline could be indexed with the unit cell parameters given in Table\ \ref{tbl:struct_rt}. No satellite reflections which could indicate either modulation or changes in unit cell dimensions were observed.
The analysis of the extinction conditions indicated the solely possible SG $I4_1/acd$. The positions of all atoms were successfully localized by the direct methods. They were the same as those reported for Tb$_5$Rh$_6$Sn$_{18}$ type.\cite{Miraglia1987} However, the refinement of atomic coordinates and isotropic displacement parameters with extinction correction converged with rather high reliability factors $R_F$ = 0.110; $R_W$ = 0.118 and residual electronic peaks of -8.1/10.4 e \r A\,$^{-3}$. Also, $B_\mathrm{iso}$ = 3.3(1) for the Sc1-atom was by a factor of $\sim$3 larger than that for all other atoms. By setting the occupation parameters of Sc1 to 0.92(1) and refining the anisotropic displacement parameters for all atoms (Table\ \ref{tbl:powder-nosplit} and \ref{tbl:bvalues}) we were able to obtain both low reliability factors and residual electronic density (Table\ \ref{tbl:struct_rt}). However, a closer look at the $B_\mathrm{anis}$ values (Table\ \ref{tbl:bvalues}) reveals a pronounced anisotropy for almost all Sn-atoms (e.g. $B_{33}\,\approx\,5B_{11}$ for Sn2; $B_{33}\,\approx\,6B_{11}$ for Sn3; $B_{22}\,\approx\,4B_{11}$ for Sn5 $etc$, see Fig. \ref{fig:Anis}b).
\begin{figure}[htb]
\includegraphics[width=\linewidth]{Fig02.eps}
\caption{(Color online) {Powder XRD pattern for Sc$_5$Rh$_6$Sn$_{18}$} at room temperature.}
\label{fig:diff}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{Fig03.eps}
\caption{Relative thermal expansion $\eta$ for Sc$_5$Rh$_6$Sn$_{18}$. The lines are guides for the eye.}
\label{fig:latticeparam}
\end{figure}
\begin{figure*}
\includegraphics[width=15cm, height=15cm]{Fig04.eps}
\centering
\caption{Corner sharing trigonal prisms arrays (tan) together with [\textit{M}Sn$_{12}$] cuboctahedra (grey, \textit{M} = Yb, Sc1) and icosahedra (a) or [Sn1Sc2Sn$_{15}$] polyhedra (c) in the structures of Yb$_3$Rh$_4$Sn$_{13}$ type and idealized (Tb$_5$Rh$_6$Sn$_{18}$ type\cite{Miraglia1987}) Sc$_5$Rh$_6$Sn$_{18}$, respectively. b) Condensed columns of cuboctahedra (transparent or grey) extending in $b$-direction with differently oriented trigonal prisms (tan or blue) in the structure of Yb$_3$Rh$_4$Sn$_{13}$ d) Separated cuboctahedra (transparent or grey) as well as 3 layers (tan, green and blue) of corner sharing trigonal prisms in Sc$_5$Rh$_6$Sn$_{18}$. (Sc - grey balls, Rh - red balls, Sn - pink balls).}
\label{fig:str}
\end{figure*}
Taking into account that the anisotropy is not systematic (which indicates the absorption correction to be performed properly) we assumed it to originate from mechanical stresses and strains appearing in the small crystals while crushing the initial sample. Therefore, in a further step prior to the single crystal diffraction measurement, stress annealing of the small crystals at 1070\, K for 2h was performed. However, further crystal structure refinement resulted in the same partial occupancy by Sc1 of its crystallographic site and unacceptable anisotropic displacement parameters (Table\ \ref{tbl:struct_rt}, \ref{tbl:powder-nosplit} and \ref{tbl:bvalues}). Similar effects were reported for refined structures of isostructural compounds Sc$_5$Co$_6$Sn$_{18}$ \cite{Kotur1999,Lei2009} and Sc$_5$Ir$_6$Sn$_{18}$\cite{Levy2019}.
Interestingly, refinement of the crystal structure of Sc$_5$Rh$_6$Sn$_{18}$ from high resolution synchrotron powder XRD (Table\ \ref{tbl:struct_rt}) (experimentally measured, theoretically calculated and differential profiles are given in Fig.\ \ref{fig:diff}) resulted in partial occupancies by Sn1, Sn4, Sn5 and Sn6 of their crystallographic positions (Table\ \ref{tbl:powder-nosplit}) (if one would set the occupational parameter $G = 1$ for mentioned atoms, this would lead to the increase of the $R_I$ by $\sim$2\, \%).
These observations prompted us to look for an alternative structural model. Taking the model with enhanced reliability factors and residual electronic density (see above) we performed differential Fourier syntheses to localize possible further atomic positions. However, the largest electronic density peaks were localized in close vicinity to Sn2 and Sn3 positions, which indicated them to be split. Other small peaks detected near Sn4-, Sn5- and Sn7-atoms could be implemented into the structural model by shifting these atoms off the centers (i.e. by doubling of the multiplicity of the corresponding Wyckoff sites and thus, $G\,\approx0.5$) (Table\ \ref{tbl:crys3}, \ref{tbl:crys4}). The comparison of the obtained model with that of the initial Tb$_5$Rh$_6$Sn$_{18}$ type is presented in Fig.\ \ref{fig:Baernighausen}.
To prove the correctness of such a refinement as well as to exclude any possible temperature induced structural changes we performed single crystal XRD down to 100 K. Atomic coordinates, equivalent and anisotropic displacement as well as occupational parameters for the crystal structures of Sc$_5$Rh$_6$Sn$_{18}$ at 100 K, 150 K, 200 K, 250 K and RT are given in Tables\ \ref{tbl:crys3}, \ref{tbl:crys4}. In the whole studied temperature range the same structural model, which is now characterized by reasonable thermal anisotropic displacements and stoichiometric 5:6:18 composition, was refined. The performed refinements confirm Sc$_5$Rh$_6$Sn$_{18}$ to crystallize with Sc$_5$Ir$_6$Sn$_{18}$ structure type\cite{Levy2019}. In agreement with this finding, all reflections in the powder XRD patterns of Sc$_5$Rh$_6$Sn$_{18}$ were indexed with the same structural model. The unit cell parameters smoothly increase with increasing temperature (Fig.\ \ref{fig:latticeparam}) and the relative thermal expansion $\eta\, \approx\, 0.5$\, \% in the 100-300\, K range is typical for intermetallic compounds\cite{Kittel86}.
Interatomic distances in the crystal structures of Sc$_5$Rh$_6$Sn$_{18}$ refined with both Tb$_5$Rh$_6$Sn$_{18}$ and Sc$_5$Ir$_6$Sn$_{18}$ types models are given in Table\ \ref{tbl:dist}. They agree mainly well with the sums of atomic radii of the elements ($r_\mathrm{Sc}$ = 1.61 \AA, $r_\mathrm{Rh}$ = 1.34 \AA, $r_\mathrm{Sn}$ = 1.41 \AA\cite{Emsley1998}). In both models Sc2-Rh and Sc2-Sn contacts are almost equal with the corresponding sums, while the Sn-Sn distances are by $\sim$1-2 \% longer. The shrinking of Rh-Sn bonds is of $\sim$4.8 \% or $\sim$5.4 \% in the Tb$_5$Rh$_6$Sn$_{18}$ and Sc$_5$Ir$_6$Sn$_{18}$ types models, respectively. Of special interest are Sc1-Sn and Sn1-Sn contacts: they exceed by $\sim$15 \% the corresponding sums of atomic radii in the Tb$_5$Rh$_6$Sn$_{18}$ type model (Tables\ \ref{tbl:dist}). These long distances could classify Sc$_5$Rh$_6$Sn$_{18}$ as a cage-compound\cite{Kase2012,Bhattacha2018}. However, applying the Sc$_5$Ir$_6$Sn$_{18}$ model Sc1-Sn and Sn1-Sn distances in this compound would exceed the corresponding sums by only $\sim$6 \% and $\sim$10 \%, respectively. On the other hand, the Sn1-Sc2 contact is shorter by $\sim$2~\%, and thus the Sn1 atom is involved in covalent bonding (see discussion below). Therefore, Sc$_5$Rh$_6$Sn$_{18}$ cannot be considered as a cage-compound.
Since the only difference between Tb$_5$Rh$_6$Sn$_{18}$ and Sc$_5$Ir$_6$Sn$_{18}$ types of arrangements is the split of some crystallographic sites (Fig.\ \ref{fig:Baernighausen}), we used the Tb$_5$Rh$_6$Sn$_{18}$ one for the further structural description. The close relationship of the Tb$_5$Rh$_6$Sn$_{18}$ type arrangement with the primitive cubic Remeika Yb$_3$Rh$_4$Sn$_{13}$ prototype (it is already reflected in the relations between the unit cell parameters: $a_\mathrm{tetr}\, \approx a_\mathrm{cub}\sqrt{2}$; $c_\mathrm{tetr}\, \approx 3a_\mathrm{cub}$) has been widely discussed in the literature\cite{Remeika1980,Miraglia1987,gum2018Rem}. Both structures are characterized by corner sharing [RnSn$_6$] trigonal prismatic arrays and reveal distorted [$M$Sn$_{12}$] cuboctahedra for $M$ = Yb, Sc1 atoms (Fig.\ \ref{fig:str}a-d).
The common blocks in both types are cuboctahedra (grey) condensed with 4 trigonal prisms (tan), which share their Sn-vertices (Fig.\ \ref{fig:str}a and c). However, whereas in Yb$_3$Rh$_4$Sn$_{13}$ the free space in-between the blocks is filled with ideal [Sn1Sn$_{12}$] icosahedra (red in Fig.\ \ref{fig:str}a) in Sc$_5$Rh$_6$Sn$_{18}$ these are now 16 vertices distorted Frank-Kasper\cite{Alvarez2005} [Sn1Sc2$_2$Sn$_{14}$] polyhedra (Fig. \ref{fig:Anis}c, Sn-Sn bonds are red and Sc-Sn - dark yellow in Fig.\ \ref{fig:str}c). Another crucial difference between these two types is that condensed cuboctahedra form in Yb$_3$Rh$_4$Sn$_{13}$ infinite columns along $b$-direction (grey and transparent in Fig.\ \ref{fig:str}b), while in the structure of Sc$_5$Rh$_6$Sn$_{18}$ they are separated from each other (grey and transparent in Fig.\ \ref{fig:str}d). One could also present the Remeika phase as alternating layers of blocks of cuboctahedra (transparent and grey) condensed with 4 differently oriented trigonal prisms (tan in the first layer and blue and tan in the second one) (Fig.\ \ref{fig:str}b). In contrary, three layers can be identified for Sc$_5$Rh$_6$Sn$_{18}$ (Tb$_5$Rh$_6$Sn$_{18}$-type) structural arrangement (Fig.\ \ref{fig:str}d). The first one (tan in Fig.\ \ref{fig:str}d) is identical with that of Yb$_3$Rh$_4$Sn$_{13}$, the second layer consists of distorted [RnSn$_6$] trigonal prisms (green in Fig.\ \ref{fig:str}d) which are separated from each other and condensed with cuboctahedra from the 1$^\mathrm{st}$ layer. Finally the third layer is again from Yb$_3$Rh$_4$Sn$_{13}$ type [contains trigonal prisms condensed with cuboctahedra (blue and grey, respectively in Fig.\ \ref{fig:str}d)]. However, it is shifted by 1/2 of the translation in a-direction compared to the 1st layer.
\subsubsection{Transmission electron microscopy}
\begin{figure}
\includegraphics[width=\linewidth]{Fig05}
\caption{TEM of the [100] zone in Sc$_5$Rh$_6$Sn$_{18}$. (a) Fourier transform of the high-resolution image. Spots marked with arrows indicate (0\,6\,$\bar{2}$) and (0\,4\,6) reflections which are lacking in the [001] zone. (b) Corresponding Fourier filtered high-resolution image. (c) Simulated diffraction pattern of the [100] zone. (d) Electron diffraction pattern of the [100] zone. Intensities appear equalized due to dynamic scattering compared to the kinematic simulation in (c).}
\label{fig:TEM100}
\end{figure}
Transmission electron microscopy (TEM) was performed on a single crystal of Sc$_5$Rh$_6$Sn$_{18}$ to verify the structure model derived from X-ray diffraction measurements and to check for the presence of the super structure reflections. A fast Fourier transform (FFT) of the [1\,0\,0] zone of the high-resolution image shows the expected, additional peaks e.g. (0\,6\,$\bar2$) or (0\,4\,6) which corroborate the tetragonal cell with $c$\,$\approx$\,$2a$ (Fig.\,\ref{fig:TEM100}). The electron diffraction pattern reveals rather equalized intensities of the reflections due to dynamic scattering (Fig.\,\ref{fig:TEM100}d). The (0\,6\,$\bar2$) or (0\,4\,6) reflections are also present in the diffraction pattern, but faintly.
In accordance with the structure model from X-ray diffraction no superstructure reflections or streaking effects as reported for other MR$_6$Sn$_{18}$ (M=Gd, Tb, Dy, R=Rh, Os) \cite{Miraglia1987,Hodeau1982} were observed in the zones [0\,0\,1] and [1\,1\,0]. The electron diffraction patterns display good agreement with simulation (Fig.\,\ref{fig:TEM001}, \ref{fig:TEM110}).
The situation, where XRD refinement of large complex crystal structures results in numerous split positions and TEM reveals no indication of any superstructure could indicate local deviations from the translational symmetry and thus, appearance of some domains without inversion center. Such a local disorder with domains breaking the average symmetry is recently reported for Ba$_{7.81}$Ge$_{40.67}$Au$_{5.33}$ clathrate \cite{Lory2017}, boron carbide \cite{Rasim2018}, $\beta$-Mg$_2$Al$_3$ \cite{Samson1965,Feuerbacher2007} and Ruthenium Zinc Antimonides \cite{Xiong2010}. These findings were confirmed in the mentioned compounds by combined ab initio DFT-calculations and high resolution TEM. A local disorder assuming the presence of some non-centrosymmetric domains in the crystal structure of Sc$_5$Rh$_6$Sn$_{18}$ could support unconventional superconductivity mechanisms. On the other hand the rather complicated structure of Sc$_5$Rh$_6$Sn$_{18}$ could be supposed to smooth such local effects of missing inversion symmetry \cite{Bauer2012}.
\subsubsection{Electronic structure and chemical bonding situation}
To shed light on the possible changes in the chemical bonding situation caused by the shortening of some contacts due to many split of Wyckoff positions, electronic structure calculations were performed using both the Tb$_5$Rh$_6$Sn$_{18}$-type and a fictitious model, which was created based on the Sc$_5$Ir$_6$Sn$_{18}$ type. To create it the initial symmetry was lowered from SG $I4_1/acd$ to $I4_1cd$ ($i.e.$ the number of crystallographic sites was doubled) (Fig.\ \ref{fig:Baernighausen}) and, then the Wyckoff positions with the shortest contacts (found in Table\ \ref{tbl:dist}), were selected to obtain the composition Sc$_5$Rh$_6$Sn$_{18}$ with no splits. However, electronic structure and chemical bonding analysis revealed negligible differences between the models. Consequently, the features of the ideal Tb$_5$Rh$_6$Sn$_{18}$-type model will be presented and discussed here.
\begin{figure}
\includegraphics[width=\linewidth]{Fig06.eps}
\caption{Electronic density of states (DOS) for Sc$_5$Rh$_6$Sn$_{18}$ assuming ideal Tb$_5$Rh$_6$Sn$_{18}$-type model. Inset: Total DOS in the Energy range from -0.1--0.1 eV.}
\label{fig:origDOS}
\end{figure}
The electronic density of states (DOS) for Sc$_5$Rh$_6$Sn$_{18}$ assuming the ideal Tb$_5$Rh$_6$Sn$_{18}$-type model is shown in Fig. \ref{fig:origDOS}. The value of the DOS at the Fermi energy (set to 0 eV) is 16.1 states eV$^{-1}$ prim.cell$^{-1}$. This indicates the theoretically calculated Sommerfeld coefficient of the electronic specific heat $\gamma_\mathrm{theor}$ = 38 mJ mol$^{-1}$ K$^{-2}$ to be by a factor of $\sim$2 smaller than the experimentally observed one \cite{Feig} and, thus an electronic instability of the Tb$_5$Rh$_6$Sn$_{18}$-type model. The states between -11.5 and -6 eV are heavily dominated by the 5$s$ contributions of the Sn atoms. The upper manifold of the valence region, [-5.25, 0] eV, consists mainly of Sn5$p$ and Rh$4d$ states. The $s$ and $p$ contributions from the Rh and Sc atoms are much smaller ($cf.$ bottom panel of Fig.\ \ref{fig:origDOS}). The lower part of the conduction band has largely Sc$3d$ and Sn$5p$ contributions. These observations are of general nature, and in order to have a deeper understanding of the bonding situation in Sc$_5$Rh$_6$Sn$_{18}$ position space chemical bonding analysis was carried out.
\begin{figure}
\includegraphics[width=\linewidth]{Fig07}
\caption{a: The ELI distribution around Sn1 at the center of the distorted Frank-Kasper polyhedron. The isosurface value is 1.0692. The atom types and bonds are identified. The thin orange lines show the boundaries of the box in which the ELI was computed. The location of the Sn1 atom is covered by the isosurface due to the Sn1 - Sc2 bond. The 3-center bond is formed by Sn1 and two Sn4 atoms. All Sn-Sn two-center bond types are shown. b: The structural arrangement in the same region as the ELI distribution illustrated by the [Sn1Sc2$_2$Sn$_{14}$] polyhedron. The distances between the respective Sn atoms are rounded to two digits and given in \AA. The bonds discussed in the text are colored in blue. The Sn1 and Sc2 atoms are illustrated as large brown or gray spheres, respectively. The smaller spheres represent Sn2-Sn4 (pink), Sn5 (yellow) and Sn6 (green).}
\label{fig:ELI}
\end{figure}
The topological analysis of the ED yields the atomic or QTAIM basins \cite{Quantumtheory}. Integration of the ED inside these basins gives the electron population of each atom, from which effective charges can be calculated. In Sc$_5$Rh$_6$Sn$_{18}$ as expected Sc (Rh) atoms are positively (negatively) charged, 1.3 (-1.2) (see Tables \ref{tbl:qtaim} and \ref{tbl:effchargecompar}). The results for the Sn atoms are mixed: the effective charge of Sn1 is -0.5, while all the other Sn atoms are positively charged. Note that the values for Sn2 (at \textit{16f}), Sn5 and Sn6 (both at the general position $32g$) are less than 0.1. The negative charge on Sn1 can be attributed to the absence of Rh atoms (most electronegative element in this compound) in its first coordination shell.
\begin{table}
\caption{Effective charges obtained through QTAIM analysis}
\centering
\label{tbl:qtaim}
\begin{tabular}{c@{\hspace{1cm}}c}
\hline
Wyckoff position & Effective Charge \\
\hline
Sc1 & +1.32 \\
Sc2 & +1.30 \\
Rh1 & -1.18 \\
Rh2 & -1.17 \\
Sn1 & -0.53 \\
Sn2 & -0.05 \\
Sn3 & +0.14 \\
Sn4 & +0.18 \\
Sn5 & +0.06 \\
Sn6 & +0.06 \\
\hline
\end{tabular}
\end{table}
The main features of the ELI-D topological analysis in Sc$_5$Rh$_6$Sn$_{18}$ can be summarized as one type of three-center Sn-only bonds and various two-center bonds formed by Sn-Sn, Sn-Sc, Sn-Rh and Rh-Sc pairs. In all cases the participating atoms are near neighbors. The two-center Sn-Sn bonds involve Sn5-Sn5 (1.98 e$^-$), Sn6-Sn6 (2.16 e$^-$) and Sn2-Sn3 (2.14 e$^-$) atoms having distances shorter than 2.91~\AA ~(see Table\ \ref{tbl:dist}). The three-center bond is formed by Sn1-Sn5-Sn5 and the Sn1-Sn5 distance, 3.244~\AA, is the fourth shortest Sn-Sn contact in this compound (Fig. \ref{fig:ELI}b). Sn1 contributes 70~\% of the total bond electrons, $\sim$1.95. Only one type of Sn-Sc contact is shorter than 3~\AA, between Sn1 and Sc2. Thus, these atoms form the only Sn-Sc bond with a high bond polarity, contribution of Sc2 is about 7~\% of the total bond electrons which is 1.9 (Fig. \ref{fig:ELI}a). The closest Sn neighbors of Rh1 are Sn4, Sn5 and Sn6. Electron populations of the corresponding bonds are between 1.72, 1.91 and 1.92, respectively. Rh1 atoms contribute $\sim$25~\% of them. The near neighbours of Rh2 atoms include all Sn atoms but Sn1. Bond electron populations vary between 1.7 and 2.0 with Rh2 accounting for 22 -- 26~\%. The Sc2-Rh1 and Sc2-Rh2 bonds are also polar with Sc2 contributions at the level of about 17~\%. However, the basins of these bonds contain fewer electrons, $\sim$0.3. The distances in the first cuboctahedral coordination shell of Sc1 are all longer than 3.34~\AA ~(see Table\ \ref{tbl:dist}), hence there are no two-center bonds involving Sc1 atoms as a major participant; Sc1 contributes to various two-center bonds only at few percent level. Therefore, Sc1 atoms' participation in atomic interactions is largely of ionic nature.
\begin{figure}
\includegraphics[width=\linewidth]{Fig08.eps}
\caption{Temperature dependence of the thermopower $S(T)$ for Sc$_5$Rh$_6$Sn$_{18}$. The violet line represents a linear fit of the data to $S = \alpha T$ in the temperature range 200\,K\,-\,350\,K.}
\label{fig:s}
\end{figure}
\begin{figure}[h]
\includegraphics[width=\linewidth]{Fig09.eps}
\caption{(Color online) Temperature dependence of thermal conductivity $\kappa$($T$) of Sc$_5$Rh$_6$Sn$_{18}$ with its electronic ($\kappa_{\mathrm{el}}$) and phonon ($\kappa_{\mathrm{ph}}$) contributions measured in different directions. In the high-temperature range above 200 K, ($\kappa_{\mathrm{exp}}$) and ($\kappa_{\mathrm{el}}$) for Sc$_5$Rh$_6$Sn$_{18}$ are biased due to radiation heat losses which follow roughly a $\propto T^3$ law. The expected real conductivities are shown by dotted lines.}
\label{fig:kappa}
\end{figure}
\subsection{Electrical and Thermal transport properties}\label{transport}
The electrical resistivity of Sc$_5$Rh$_6$Sn$_{18}$ reveals an anisotropic bad metallic behavior above the superconducting transition (inset to Fig. \ref{fig:s}). It increases with increasing temperature in the direction parallel to \textit{ab} and almost linearly decreases for $\rho \parallel c$. Such a temperature dependence of $\rho$(\textit{T}) could be ascribed to the strong structural disorder and low charge carrier concentration observed in Sc$_5$Rh$_6$Sn$_{18}$ (for more details see discussion in \cite{Feig})
The temperature dependence of the thermopower $S(T)$ of Sc$_5$Rh$_6$Sn$_{18}$ measured along $ab$ ($S \parallel ab$)- and $c$ ($S \parallel c$)-directions is shown in Fig.\,\ref{fig:s}. No anisotropy is observed. $S$($T$) is negative in the whole measured temperature range (1.8\,K--400\,K) and indicates the dominance of electron-like charge carriers in agreement with reported Hall-effect studies \cite{scte2013}. The thermopower in the temperature range 200\,K $\leq\ T \leq$ 350\,K is described by $S = \alpha T$ with a small slope $\alpha$\,=\,$-0.08$\,$\mu$V K$^{-2}$. At lower temperatures the experimental curves deviate slightly from the linear behavior due to the presence of additional scattering mechanisms. Such behavior of $S$($T$) resembles a typical metal-like behavior. The dimensionless parameter $q = N_Ae\alpha/\gamma$ (Faraday constant $N_A \,e \approx 96485$ C\,mol$^{-1}$), which characterizes the thermoelectric material in terms of an effective charge carrier
concentration per f.u.\ (or the Fermi volume $V_\mathrm{F}$ of the
charge carriers) \cite{Zlatic2007}, is estimated to be 0.11 for
Sc$_5$Rh$_6$Sn$_{18}$. For Fermi liquids the value of $q$ is expected to be approximately 1.
The temperature dependence of the thermal conductivity $\kappa$($T$) for Sc$_5$Rh$_6$Sn$_{18}$ with the heat flow parallel to the $ab$ ($\kappa \parallel ab$)- and $c$ ($\kappa \parallel c$)-directions is plotted in Fig.\ \ref{fig:kappa}. With decreasing temperature $\kappa(T)$ decreases in the whole measured temperature range showing a small kink at $T_\mathrm{kink}$ = 75\,K. No anisotropic behavior of the thermal conductivity within the standard deviation was observed. The total $\kappa$($T$) is small over the whole measured temperature range, which is obviously due to the strong structural disorder in Sc$_5$Rh$_6$Sn$_{18}$.
For metallic compounds $\kappa(T)$ can be decomposed into
$\kappa_\mathrm{el}$ and $\kappa_\mathrm{ph}$. The electronic contribution $\kappa_\mathrm{el}$ may be estimated from the experimental resistivity data by applying the Wiedemann-Franz law $\kappa_{\mathrm{el}}(T) = L_0T/\rho(T)$, where $L_0$ is the Lorenz number $2.44 \times 10^{-8}$ W\,$\Omega$\,K$^{-2}$. The phonon contribution $\kappa_\mathrm{ph}$ for Sc$_5$Rh$_6$Sn$_{18}$ was estimated by subtracting $\kappa_\mathrm{el}$ from the measured value of $\kappa$. Due to the anisotropy in the electrical resistivity $\kappa_\mathrm{el}^{\parallel ab}$ is dominating the thermal conductivity for $T > 100$\, K, while for the $c$-direction the main contribution is from $\kappa_\mathrm{ph}^{\parallel c}$.
\section{Conclusions}
Crystal structure refinements performed for Sc$_5$Rh$_6$Sn$_{18}$ in the temperature range of 100-300\, K from both high-resolution synchrotron powder and single crystal X-ray diffraction methods revealed it to crystallize with the centrosymmetric Sc$_5$Ir$_6$Sn$_{18}$ structure type (a split variant of the Tb$_5$Rh$_6$Sn$_{18}$ prototype). Such a structural model assumes shortening of Sc-Sn contacts and does not allow to classify Sc$_5$Rh$_6$Sn$_{18}$ as a cage-compound. Further TEM characterization confirmed the absence of any satellites (i.e. possible modulation) or superstructure reflections (i.e. changes in unit cell dimensions). Splits of numerous Sn-positions together with the absence of any signatures of superstructure can however indicate a local disorder and thus, the possible absence of an inversion center in some domains. They can be indicative of packing incompatibilities of the large polyhedra. Both Sc$_5$Ir$_6$Sn$_{18}$ and Tb$_5$Rh$_6$Sn$_{18}$ types show close relationships with the primitive cubic Yb$_3$Rh$_4$Sn$_{13}$ Remeika phase revealing a similar array of corner sharing [RhSn$_6$] trigonal prisms condensed with distorted [$M$Sn$_{12}$] ($M$ = Yb or Sc) cuboctahedra.
The performed chemical bonding situation analysis revealed highly polar character for Sc2-Sn1, Sn-Rh and Sc2-Rh bonds as well as two- and three-center bonds involving Sn-atoms. Sc1 atoms' participation in atomic interactions is mainly of ionic nature. No differences in bonding situations between the idealized Tb$_5$Rh$_6$Sn$_{18}$-like model and a fictitious structure, which was created based on the Sc$_5$Ir$_6$Sn$_{18}$ type and included only the shortest contacts, was observed.
Thus, both crystal structure refinements as well as analysis of chemical bonding situation confirm the average crystal structure of Sc$_5$Rh$_6$Sn$_{18}$ to be centrosymmetric, which would be rather incompatible with unconventional superconductivity (SC) with a non unitary triplet electron-phonon pairing with point nodes in the gap as recently proposed \cite{Bhattacha2018}. On the other hand the local disorder could support such a scenario. This prompted us to perform an additional study of SC properties of Sc$_5$Rh$_6$Sn$_{18}$, which revealed it to be a conventional SC with stronger electron-phonon coupling and isotropic $s$-wave gap \cite{Feig}.
The anisotropic behavior of temperature dependence of electrical resistivity in Sc$_5$Rh$_6$Sn$_{18}$ is in details discussed in \cite{Feig}. The thermopower $S(T)$ of the stannide reveals no anisotropy and is small and negative in the whole measured temperature range, which indicates dominance of electron-like charge carriers in this metallic system.
The thermal conductivity $\kappa$($T$) for Sc$_5$Rh$_6$Sn$_{18}$ is isotropic as well and smaller than for a metal. This can be explained with the intrinsic structural disorder in the studied compound. In accordance with the anisotropy in the electrical resistivity $\kappa_\mathrm{el}^{\parallel ab}$ is dominating thermal conductivity for $T > 100$\, K, while for the $c$-direction the main contribution is from $\kappa_\mathrm{ph}^{\parallel c}$.
\section*{Acknowledgments}
This work is performed within the DFG (Deutsche Forschungsgemeinschaft) grant 325295543. The authors are grateful to J.\ Grin for his interest and steady support. The authors thank U.\ Burkhardt, P.\ Scheppan and S.\ Kostmann for the metallographical analysis, J.\ Chang for his contribution in the initial phase of this project. We acknowledge D.\ Sokolov for providing a high quality Laue pattern of the crystal. H.\ Borrmann contributed single crystal XRD measurements at the initial stage of this research. The authors are indebted C.\ Hennig for his support during the adjustment of BM20 beamline at ESRF.
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,630 |
Oliver Alexander Reinhard Petszokat (born 10 August 1978), better known by his stage name Oli.P, is a German singer, actor and television presenter.
Biography
Petszokat, whose father Reinhard Petszokat was a policeman, began ballroom dancing at the age of ten. His biggest dance success was in 1995 as a participant in the winning team for the Youth Cup team of the Dance Sport Federation of North Rhine-Westphalia eV.
In Germany, Petszokat became famous as pop singer Oli.P. On German broadcaster RTL, he played character Ricky Marquart in the television series Gute Zeiten, schlechte Zeiten. In several television shows he worked as television presenter (Big Brother, ...).
In 1999, Petszokat married German actress Tatiani Katrantzi. Together they have one child. In 2007, the couple separated.
Awards
1998: Bravo Otto
1999: Bravo Otto
2000: Echo
Discography
(as Oli.P)
Albums
Studio albums
1998: Mein Tag
1999: o.ton
2001: P.ulsschlag
2002: Startzeit
2004: Freier Fall
2016: Wie früher
2019: Alles Gute!
Compilation albums
2002: Lebenslauf – Gold & Platin 98-01
Singles
1997: "Liebe machen"
1998: "Flugzeuge im Bauch"
1998: "I Wish"
1999: "Der 7. Sinn"
1999: "So bist du (und wenn du gehst...)"
2000: "Niemals mehr"
2000: "Plötzlich stand sie da"
2001: "Girl You Know It's True"
2001: "When You Are Here"
2002: "Das erste Mal tat's noch weh"
2002: "Nothing's Gonna Change My Love for You"
2003: "Alles ändert sich (alles oder nichts)"
2003: "Neugeboren"
2004: "Engel" / "Unsterblich"
2016: "Wie früher"
2016: "Wohin gehst du"
2019: "Flugzeuge im Bauch (2K19)"
2019: "Lieb mich ein letztes Mal"
2019: "Hallo Schatz"
Filmography
TV series
1996–1997: Alle zusammen – jeder für sich (230 episodes)
1998–1999: Gute Zeiten, schlechte Zeiten (480 episodes)
1999: Hinter Gittern – Der Frauenknast (4 episodes)
2002: Der kleine Mönch (7 episodes)
2004: Wie erziehe ich meine Eltern? (1 episode)
2004: Im Namen des Gesetzes (1 episode)
2005: Axel! will's wissen (1 episode)
2006–2007: Gott sei dank … dass Sie da sind! (6 episodes)
2010–2011: Hand aufs Herz (234 episodes)
2015: Die tierischen 10 (VOX)
2016: Rote Rosen (ARD)
Films
2000: Wie angelt man sich seinen Chef?
2001: Girl
2003: Baltic Storm
2003: Motown
2005: Ein Hund, zwei Koffer und die ganz große Liebe
2007: Die ProSieben Märchenstunde: Dörnröschen – ab durch die Hecke
2008: Funny Movie – Dörtes Dancing
References
External links
German television presenters
1978 births
Living people
German male film actors
German male television actors
21st-century German male singers
RTL Group people
Mass media people from Berlin | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 500 |
\section{Introduction}
Dicrete integrable systems represent a few decades old field of study that excites both mathematics and physics communities.
Mathematical areas of differential equations, dynamical systems, geometry, algebraic geometry are all intertwined in that field, which, on the other hand, has applications in various physics theories.
For detailed expostions and references, see for example \cites{GrammaticosDIS, BS2008book, DuistermaatBOOK} and referenced therein.
This paper is focused to discrete systems arising from the dynamics of billiards within confocal quadrics.
A class of such systems, \emph{double reflection nets}, is introduced and studied in \cites{DragRadn2012jnmp,DragRadn2014bul,DragRadn2015umn}.
Double reflection nets are \emph{discrete line congruences}, that is maps that assigns a line to each vertex of the lattice $\mathbf{Z}^m$, such that neighbouring lines always intersect.
In this work, we present and study a novel class of discrete systems arising from billiards within confocal quadrics.
Unlike double reflection nets, these new systems are defined on a non-cubic lattice and they assign hyperplanes of a given projective space to the verteces of the lattice.
This paper is organised as follows.
Section \ref{sec:drn} contains overview of confocal quadrics, related biliards, and double reflection nets.
In Section \ref{sec:hyper}, we introduce new discrete systems that are naturally related to the double reflection nets.
We derive their basic properties and discuss the cases when $m=2$ and $m=3$.
\section{Double reflection nets}
\label{sec:drn}
In this section, we give first a brief overview of most important notions regarding confocal families of quadrics and double reflection nets.
For more details on billiards within quadrics and confocal families see \cite{DragRadn2011book} and references therein.
On double reflection nets, see \cites{DragRadn2012jnmp,DragRadn2014bul,DragRadn2015umn}.
A \emph{pencil of quadrics} in the $d$-dimensional projective space $\mathbf{P}^d$ is a $1$-parameter family of quadrics:
\begin{equation}\label{eq:pencil}
\mathcal{Q}_{\lambda}\ :\ \big((A-\lambda I)x,x\big)=0,
\end{equation}
where $A$ is a symmetric non-degenerate operator and $I$ the identity operator.
A \emph{confocal family of quadrics} is a family dual to a pencil of quadrics:
\begin{equation}\label{eq:confocal}
\mathcal{Q}_{\lambda}^*\ :\ \big((A-\lambda I)^{-1}x,x\big)=0.
\end{equation}
When a confocal family of quadrics is given in the projective space, it is possible to define the billiard reflection off the quadrics from the family, see \cite{CCS1993}.
That definition is consistent with the billiard reflection in the Euclidean space.
By Chasles' theorem, each line in the space is touching exactly $d-1$ quadrics from (\ref{eq:confocal}) and these quadrics are preserved by reflections off quadrics from the confocal family.
Thus, a billiard trajectory within confocal quadrics always have $d-1$ \emph{caustics}.
Lines $\ell$, $\ell_1$, $\ell_2$, $\ell_{12}$ represent a \emph{double reflection configuration} if there are quadrics $\mathcal{Q}_{\alpha}$ and $\mathcal{Q}_{\beta}$ in (\ref{eq:confocal}) such that:
\begin{itemize}
\item
pairs $\ell$, $\ell_1$ and $\ell_2$, $\ell_{12}$ and satisfy reflection law off $\mathcal{Q}_{\alpha}^*$;
\item
pairs $\ell$, $\ell_2$ and $\ell_1$, $\ell_{12}$ and satisfy reflection law off $\mathcal{Q}_{\beta}^*$;
\item
four tangent planes at the reflection points are in a pencil.
\end{itemize}
A \emph{double reflection net} is a map $\varphi\ :\ \mathbf{Z}^m \to \mathcal{L}$, with $\mathcal{L}$ be the set of all lines in the projective space,
such that there exist $m$ quadrics $\mathcal{Q}_1^*$, \dots , $\mathcal{Q}_m^*$ from the confocal pencil, satisfying the following conditions:
\begin{itemize}
\item
the sequence $\{\varphi(\mathbf{n_0} + i\mathbf{e}_j)\}_{i\in\mathbf{Z}}$ represents a billiard trajectory within $\mathcal{Q}_j^*$,
for each $j \in \{1,\dots,m\}$ and $\mathbf{n_0} \in\mathbf{Z}_m$;
\item
the lines $\varphi(\mathbf{n_0})$, $\varphi(\mathbf{n_0} + \mathbf{e}_i)$, $\varphi(\mathbf{n_0} + \mathbf{e}_j)$, $\varphi(\mathbf{n_0} + \mathbf{e}_i+\mathbf{e}_j)$ form a double reflection configuration, for all $i,j \in \{1,...,m\}$, $i \neq j$ and $\mathbf{n_0} \in \mathbf{Z}^m$.
\end{itemize}
In other words, for each edge in $\mathbf{Z}^m$ of direction $\mathbf{e}_i$, the lines corresponding to its vertices meet at $\mathcal{Q}_i^*$, while the four tangent planes at the intersection points, associated to an elementary quadrilateral, belong to a pencil.
The integrability of double reflection nets follows from the Six-pointed star theorem \cite{DragRadn2008}, which states that tere exist configurations consisting of twelve planes with the following properties:
\begin{itemize}
\item
The planes may be organized in eight triplets, such that each plane in a triplet is tangent to a different quadric from (\ref{eq:confocal}) and the three touching points are collinear.
Every plane in the configuration is a member of two triplets.
\item
The planes may be organized in six quadruplets, such that the planes in each quadruplet belong to a pencil and are tangent to two different quadrics from (\ref{eq:confocal}).
Every plane in the configuration is a member of two quadruplets.
\end{itemize}
Moreover, such a configuration is determined by three planes tangent to three different quadrics from (\ref{eq:confocal}), with collinear touching points.
Such a configuration of planes is shown in Figure \ref{fig:cubo-oct}: each plane corresponds to a vertex of the cuboctahedron.
\begin{figure}[h]
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\caption{A configuration of planes from the Six-pointed star theorem.}\label{fig:cubo-oct}
\end{figure}
These cuboctahedra will represent basic building blocks for lattices of hyper-planes that we introduce in the next section.
\section{Hyper-plane billiard nets}
\label{sec:hyper}
Consider a lattice $\mathbf{Z}^m$ in $\mathbf{R}^m$.
That lattice generates a \emph{honeycomb}, that is a filling of the space by polytopes \cite{Coxeter}.
In this case, it is a regular honeycomb consisting of $m$-cubes.
The set of all midpoints of the edges of the cubes is
$$
\mathcal{M}^m=\bigcup_{1\le i\le m}\big(\mathbf{Z}^m+\frac12\mathbf{e}_i\big).
$$
The lattice $\mathcal{M}^m$ determines a honeycomb containing two types of polytopes (see \cite{Coxeter}):
\begin{itemize}
\item
\emph{rectified $m$-cubes} -- the verteces of each polytope of this kind are midpoints of the edges of an $m$-cube in the lattice $\mathbf{Z}^m$;
\item
\emph{cross polytopes} -- the verteces of each cross polytope are midpoints of all edges with a common endpoint in $\mathbf{Z}^m$.
\end{itemize}
This is an example of a convex uniform honeycomb \cite{Wells}.
For $m=3$, it is shown in Figure \ref{fig:honey3}, and for $m=2$ in Figure \ref{fig:tiling2}.
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action=draw**]
(-1.5,1.5,1.5)
\end{pspicture*}
\caption{Honeycomb consisting of cuboctahedra and octahedra.}\label{fig:honey3}
\end{figure}
Integrable systems on such lattices were studied in \cites{KingS2003,KingS2006}
Assume that $\varphi \ :\ \mathbf{Z}^m\to\mathcal{L}$ is a given double reflection net.
For each $\mathbf{n_0}\in\mathbf{Z}^m$ and $i\in\{1,\dots,m\}$, the lines $\varphi(\mathbf{n_0})$ and $\varphi(\mathbf{n_0}+\mathbf{e}_i)$ are reflected to each other off the quadric $\mathcal{Q}_i^*$ from the confocal family.
We define the map
$$
\mathcal{H}\ :\ \mathcal{M}^m\to \mathbf{P}^{d*},
$$
such that it assigns to the midpoint of the edge $(\mathbf{n_0}, \mathbf{n_0}+\mathbf{e}_i)$ the tangent plane to $\mathcal{Q}_i^*$ at the point of reflection.
We introduce also the map
$$
\mathcal{P}\ :\ \mathcal{M}^m\to \mathbf{P}^{d*},
$$
which assigns the intersection point $\varphi(\mathbf{n_0})\cup\varphi(\mathbf{n_0}+\mathbf{e}_i)$ to the the midpoint of the edge $(\mathbf{n_0}, \mathbf{n_0}+\mathbf{e}_i)$.
Since each hyperplane of the space, except of a subset of measure $0$, is touching exactly one quadric from the given confocal family.
Thus, the touching point is uniquely determined and map $\mathcal{P}$ is uniquely determined by $\mathcal{H}$.
The inverse, to determine $\mathcal{H}$ when $\mathcal{P}$ is given, is not straightforward, since each point of the $d$-dimensiona space belongs to $d$ confocal quadrics.
From the construction, $\mathcal{H}$ and $\mathcal{P}$ have the following properties:
\begin{prop}
\begin{itemize}
\item
For each cross polytope of the honeycomb, $\mathcal{P}$ assigns to all its vertices collinear points.
Moreover, the points joined to the opposite vertices are on the same quadric from the confocal family.
\item
For each square $2$-face of any rectified $m$-cube, $\mathcal{H}$ assignes to its vertices hyperplanes that belong to one pencil and form a harmonic quadruple.
The hyperplanes corresponding to the opposite vertices are tangent to the same quadric from the confocal family.
\item
The hyperplanes assigned to any two adjacent verteces of a square $2$-face uniquely determine the hyperplanes assigned to the other two verteces.
\end{itemize}
\end{prop}
\begin{rem}
For $m=2$, $\mathcal{M}^2$ determines a regular tesselation by squares, see Figure \ref{fig:tiling2}.
\begin{figure}[h]
\centering
\psset{unit=1.5}
\begin{pspicture}(-0.5,-0.5)(3.5,3.5)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(0,0.5)(.5,1)(0,1.5)(-0.5,1)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(0,1.5)(.5,2)(0,2.5)(-0.5,2)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(1,-0.5)(1.5,0)(1,.5)(0.5,0)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(1,0.5)(1.5,1)(1,1.5)(0.5,1)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(1,1.5)(1.5,2)(1,2.5)(0.5,2)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(1,2.5)(1.5,3)(1,3.5)(0.5,3)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(3,0.5)(3.5,1)(3,1.5)(2.5,1)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(3,1.5)(3.5,2)(3,2.5)(2.5,2)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(2,-0.5)(2.5,0)(2,0.5)(1.5,0)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(2,0.5)(2.5,1)(2,1.5)(1.5,1)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(2,1.5)(2.5,2)(2,2.5)(1.5,2)
\psset{fillstyle=solid, linecolor=gray!50, fillcolor=gray!50}
\psline(2,2.5)(2.5,3)(2,3.5)(1.5,3)
\psset{linecolor=gray,linewidth=0.02}
\psline(-0.5,0)(3.5,0)
\psline(-0.5,1)(3.5,1)
\psline(-0.5,2)(3.5,2)
\psline(-0.5,3)(3.5,3)
\psline(0,-0.5)(0,3.5)
\psline(1,-0.5)(1,3.5)
\psline(2,-0.5)(2,3.5)
\psline(3,-0.5)(3,3.5)
\psset{linecolor=black,linewidth=0.05}
\psline(0.5,0)(0,0.5)
\psline(1.5,0)(0,1.5)
\psline(2.5,0)(0,2.5)
\psline(3,0.5)(0.5,3)
\psline(3,1.5)(1.5,3)
\psline(3,2.5)(2.5,3)
\psline(0,0.5)(2.5,3)
\psline(0,1.5)(1.5,3)
\psline(0,2.5)(0.5,3)
\psline(0.5,0)(3,2.5)
\psline(1.5,0)(3,1.5)
\psline(2.5,0)(3,0.5)
\end{pspicture}
\caption{The lattice $\mathcal{M}^2$.}\label{fig:tiling2}
\end{figure}
For each square, if the values of $\mathcal{H}$ are given at two neighbouring vertices, it is possible to uniquely determine the values at the remaining two vertices.
The hyperplanes joined to the opposite points of a square are always tangent to the same quadric from the confocal pencil.
However, the discrete dynamics depends on the type of each square:
\begin{itemize}
\item
For the squares with vertices of the form
$$
\mathbf{n_0}+\dfrac{\mathbf{e_1}}{2},
\quad
\mathbf{n_0}+\dfrac{\mathbf{e_2}}{2},
\quad
\mathbf{n_0}+\mathbf{e_1}+\dfrac{\mathbf{e_2}}2,
\quad
\mathbf{n_0}+\dfrac{\mathbf{e_1}}2+\mathbf{e_2},
$$
the corresponding hyperplanes are in a pencil and harmonically conjugated.
Such squares are white in Figure \ref{fig:tiling2}.
\item
For the squares with vertices of the form
$$
\mathbf{n_0}+\dfrac{\mathbf{e_1}}{2},
\quad
\mathbf{n_0}+\dfrac{\mathbf{e_2}}{2},
\quad
\mathbf{n_0}-\dfrac{\mathbf{e_1}}{2},
\quad
\mathbf{n_0}-\dfrac{\mathbf{e_2}}2,
$$
the corresponding touching points are collinear, and in general not harmonically conjugated.
Such squares are gray coloured in Figure \ref{fig:tiling2}.
\end{itemize}
We denoted by $\mathbf{n_0}$ a point of $\mathbf{Z}^2$ and by $\mathbf{e_1}$, $\mathbf{e_2}$ unit coordinate vectors.
\end{rem}
It would be interested to write explicitely recursive relations that maps $\mathcal{H}$, $\mathcal{P}$ and double reflection nets satisfy on the lattice $\mathcal{M}^m$ and to see how they fit in the classification from \cite{ABS2003}.
\begin{bibdiv}
\begin{biblist}
\bib{ABS2003}{article}{
author={Adler, V. E.},
author={Bobenko, A. I.},
author={Suris, Yu. B.},
title={Classification of integrable equations on quad-graphs. The consistency approach},
journal={Comm. Math. Phys.},
volume={233},
date={2003},
number={3},
pages={513--543},
}
\bib{BS2008book}{book}{
author={Bobenko, Alexander I.},
author={Suris, Yuri B.},
title={Discrete differential geometry: Integrable structure},
series={Graduate Studies in Mathematics},
volume={98},
publisher={American Mathematical Society},
place={Providence, RI},
date={2008},
pages={xxiv+404},
}
\bib{CCS1993}{article}{
author={Chang, Shau-Jin},
author={Crespi, Bruno},
author={Shi, Kang-Jie},
title={Elliptical billiard systems and the full Poncelet's theorem in $n$ dimensions},
journal={J. Math. Phys.},
volume={34},
number={6},
date={1993},
pages={2242--2256},
}
\bib{Coxeter}{book}{
author={Coxeter, H. S. M.},
title={Regular polytopes},
edition={3},
publisher={Dover Publications, Inc., New York},
date={1973},
pages={xiv+321},
}
\bib{DragRadn2008}{article}{
author={Dragovi\'c, Vladimir},
author={Radnovi\'c, Milena},
title={Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms},
journal={Adv. Math.},
volume={219},
date={2008},
number={5},
pages={1577--1607},
}
\bib{DragRadn2011book}{book}{
author={Dragovi\'c, Vladimir},
author={Radnovi\'c, Milena},
title={Poncelet Porisms and Beyond},
publisher={Springer Birkhauser},
date={2011},
place={Basel},
}
\bib{DragRadn2012jnmp}{article}{
author={Dragovi\'c, Vladimir},
author={Radnovi\'c, Milena},
title={Billiard algebra, integrable line congruences, and double reflection nets},
journal={Journal of Nonlinear Mathematical Physics},
volume={19},
number={3},
pages={1250019},
date={2012},
}
\bib{DragRadn2014bul}{article}{
author={Dragovi\'c, Vladimir},
author={Radnovi\'c, Milena},
title={Bicentennial of the great Poncelet theorem (1813--2013): current advances},
journal={Bull. Amer. Math. Soc. (N.S.)},
volume={51},
date={2014},
number={3},
pages={373--445},
}
\bib{DragRadn2015umn}{article}{
author={Dragovi{\'c}, V. I.},
author={Radnovi{\'c}, M.},
title={Pseudo-integrable billiards and double reflection nets},
language={Russian},
journal={Uspekhi Mat. Nauk},
volume={70},
date={2015},
number={1(421)},
pages={3--34},
translation={ journal={Russian Math. Surveys}, volume={70}, date={2015}, number={1}, pages={1-31}, },
}
\bib{DuistermaatBOOK}{book}{
author={Duistermaat, Johannes J.},
title={Discrete integrable systems: QRT maps and elliptic surfaces},
series={Springer Monographs in Mathematics},
publisher={Springer},
place={New York},
date={2010},
pages={xxii+627},
isbn={978-1-4419-7116-6},
}
\bib{GrammaticosDIS}{collection}{
title={Discrete integrable systems},
series={Lecture Notes in Physics},
volume={644},
booktitle={Proceedings of the International School held in Pondicherry, February 2--14, 2003},
editor={Grammaticos, B.},
editor={Kosmann-Schwarzbach, Y.},
editor={Tamizhmani, T.},
publisher={Springer-Verlag, Berlin},
date={2004},
pages={xviii+439},
isbn={3-540-21425-9},
}
\bib{KingS2003}{article}{
author={King, A. D.},
author={Schief, W. K.},
title={Tetrahedra, octahedra and cubo-octahedra: integrable geometry of multi-ratios},
journal={J. Phys. A},
volume={36},
date={2003},
number={3},
pages={785--802},
}
\bib{KingS2006}{article}{
author={King, A. D.},
author={Schief, W. K.},
title={Application of an incidence theorem for conics: Cauchy problem and integrability of the dCKP equation},
journal={J. Phys. A},
volume={39},
date={2006},
number={8},
pages={1899--1913},
}
\bib{Wells}{book}{
author={Wells, A. F.},
title={Three-dimensional nets and polyhedra},
note={Wiley Monographs in Crystallography},
publisher={Wiley-Interscience [John Wiley \& Sons], New York-London-Sydney},
date={1977},
pages={xii+268},
}
\end{biblist}
\end{bibdiv}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,128 |
See Sending G-code for more information on how to send the commands below.
Rebooting the printer may not be necessary but it is recommended.
M501 ; Restores the setting currently stored in EEPROM. This is basically the same as turning the printer off and on.
M502 resets all settings/parameters to the default values of the currently installed Motion Controller firmware. These settings are sometimes altered by the user or slicing software although I suspect the latter is less likely to have occurred.
Most often the two settings altered by the users are the PID values and the Steps per mm for the extruder stepper motor.
; Please add a blank line to the end of the file while editing. DokuWiki Code Blocks removes blank lines at the end of the block but g-code files should end with a single blank line at the end. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,833 |
It is a statistical fact that you are more likely to die while horseback riding (1 serious adverse event every ~350 exposures) than from taking Ecstasy (1 serious adverse event every ~10,000 exposures). Yet, in 2009, the scientist who said this was fired from his position as the chairman of the UK's Advisory Council on the Misuse of Drugs. Professor David Nutt's remit was to make scientific recommendations to government ministers on the classification of illegal drugs based on the harm they can cause. He was dismissed because his statement highlighted how the UK Government's policies on narcotics are at odds with scientific evidence. Today, the medical use of drugs such as cannabis remains technically illegal.
Such incidents of silencing are sadly commonplace when it comes to politically controversial scientific topics. The US Government muzzled climate scientists in a similar manner in 2007, when it was reported that 46% of 1600 surveyed scientists were warned against using terms like "global warming" and 43% said their published work had been revised in ways that altered their conclusions. US preparations for oncoming climate change were checked as a result, a failing that persists today. Going back further, the story of Nikolai Vavilov is chilling. Vavilov was a plant geneticist in the Soviet Union under Joseph Stalin. He was jailed in 1940 for criticizing the pseudo-scientific views of Trofim Lysenko, a protégé of Stalin. Vavilov died of starvation in prison a few years later; scientific dissent from Lysenko's "theories" of Lamarkian inheritance was outlawed in 1948. Soviet agriculture languished for decades because of Lysenkoism; meanwhile famine decimated the population.
The scientific method is defined by the Oxford English Dictionary as "a method or procedure...consisting in systematic observation, measurement, and experiment, and the formulation, testing, and modification of hypotheses". It is our finest instrument for unearthing the truth. Applied correctly it is blind to and corrects for our inherent biases. Scientists are trained to wield this formidable tool in their quest to understand the universe around us. The truths they uncover can be at odds with our current beliefs; but when the facts (based on evidence and arrived at through rigorous testing) change, minds also need to change.
This is untenable. Scientists have a moral obligation to engage with the public about their findings; to advise and speak out on policy, and to critique its consequent implementation. Science impacts on the life of every single species on our planet. It is ludicrous that the very people who discover the facts are not part of any subsequent policy-making dialogue. Science needs to be an essential component of the public discourse; currently it is not. The consequences of that disconnect can be dire, as evinced by the criminalization of drugs that can provide relief to sufferers of chronic pain, troubling delays in programs of vital national importance, and the famine that slaughtered millions of Soviet citizens under Stalin's regime.
By passionately advocating for evidence-based policy scientists will expand scientific research, reversing the trend of recent years; and by thus visibly working for the common weal scientists will earn the public's trust, protecting long-term investigations from short-sighted cuts. Scientific advancement is utterly dependent on public funding and public backing. The Space Race, the Human Genome Project, the search for the Higg's Boson and the Mars Curiosity Rover Mission were all enthusiastically embraced by the public. The progress of science demands that scientists engage the public. But for that to happen the notion that a scientist should stay hidden away in a laboratory needs to be retired. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,580 |
using Microsoft.WindowsAzure.Commands.Utilities.Common;
using Microsoft.Azure.Management.RemoteApp.Models;
using Microsoft.Azure.Commands.RemoteApp.Common;
using System.Management.Automation;
using System;
using Hyak.Common;
namespace Microsoft.Azure.Commands.RemoteApp.Cmdlet
{
public abstract partial class RemoteAppArmCmdletBase : AzurePSCmdlet
{
public RemoteAppArmCmdletBase()
{
}
private RemoteAppManagementClientWrapper _RemoteAppClient = null;
public RemoteAppManagementClientWrapper RemoteAppClient
{
get
{
if (_RemoteAppClient == null)
{
_RemoteAppClient = new RemoteAppManagementClientWrapper(Profile, Profile.Context.Subscription);
}
return _RemoteAppClient;
}
set
{
// for testing purpose only
_RemoteAppClient = value;
}
}
private void HandleCloudException(object targetObject, CloudException e)
{
CloudRecordState cloudRecord = RemoteAppCollectionErrorState.CreateErrorStateFromCloudException(e, String.Empty, targetObject);
if (cloudRecord.state.type == ExceptionType.NonTerminating)
{
WriteError(cloudRecord.er);
}
else
{
ThrowTerminatingError(cloudRecord.er);
}
}
public Collection FindCollection(string ResourceGroupName, string collectionName)
{
Collection response = null;
response = RemoteAppClient.Get(ResourceGroupName, collectionName);
if (response == null)
{
WriteErrorWithTimestamp("Collection " + collectionName + " not found in resource group " + ResourceGroupName);
}
return response;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,727 |
\subsection*{Introduction}
Over the past few years the Einstein--$SU(2)\!-\!\sigma$ model has
attracted a great deal of attention \cite{w1, bw1, ch, w2, bw2, bsw,
s}. This model is interesting because its rich phenomenology is
sensitive to the value of a dimensionless parameter $\eta$
which leads to various bifurcation
phenomena. The most interesting bifurcation was found by Lechner et
al. \cite{w2} who showed that the critical behavior in gravitational
collapse changes character from continuous to discrete
self-similarity when the coupling constant $\eta$ increases above a
critical value $\eta_c$. This phenomenon was interpreted in terms of
dynamical systems theory as the homoclinic loop bifurcation where
the two critical solutions, continously self-similar (CSS) and
discretely self-similar (DSS), merge in phase space. Since the
echoing period $\Delta$ of the DSS solution diverges as
$\eta\rightarrow \eta_c$, the numerical analysis of this bifurcation
is extremely difficult and for this reason some of the aspects of
critical behavior near the bifurcation point, in particular the
black hole mass scaling law, were left open in \cite{w2}.
Below, using high precision numerical methods, we confirm the main
findings of \cite{w2}. In addition, we find that just above the
bifurcation point the marginally supercritical side of the
transition between dispersion and black holes exhibits a fine
structure which is due to the competition between two coexisting
critical solutions, the DSS one and the CSS one. The description of
this phenomenon and its interpretation is the main purpose of this
paper. The rest of the paper is organized as follows. For readers'
convenience, in section~2 we first briefly repeat the basic setting
of the model and then we summarize what is known about it. In
section~3 we present numerical results and finally, in section~4, we
interpret them.
\subsection*{The model}
The spherically symmetric Einstein-$SU(2)-\sigma$ system is
parametrized by three functions: the metric coefficients $A(t,r)$,
$\delta(t,r)$ and the $\sigma$-field $F(t,r)$, which satisfy the
following system of equations
\begin{equation}
\Box_g F - \frac{\sin(2 F)}{r^2} = 0,\quad \Box_g = -e^{\delta}
\partial_t(e^{\delta} A^{-1} \partial_t) +\frac{e^{\delta}}{r^2}
\partial_r(r^2 e^{-\delta} A\:
\partial_r),
\end{equation}
\begin{eqnarray}
{\partial_t A} &=& -2 \eta\: r A (\partial_t F) (\partial_r F),
\\
{\partial_r \delta} &=& -\eta\: r \left((\partial_r F)^2 + A^{-2}
e^{2\delta} (\partial_t F)^2 \right),
\\
\partial_r A &=& \frac{1-A}{r} - \eta\: r \left( A (\partial_r F)^2 + A^{-1}
e^{2\delta} (\partial_t F)^2 + 2 \:\frac{\sin^2{\!F}}{r^2}\right),
\end{eqnarray}
where $\eta$ is a dimensionless coupling constant. For $\eta=0$ this
system reduces to the $\sigma$ model in Minkowski spacetime. The
initial value problem for this system was studied by Bizo\'n et al.
\cite{bct} for $\eta=0$ and by Husa et al. \cite{w1} for $\eta>0$.
In these studies an important role is played by self-similar
solutions. A countable family of continuously self-similar (CSS)
solutions, herefater denoted by $CSS_n$ ($n=0,1,...$), was shown to
exist for $0\leq\eta<0.5$ in \cite{b1,bw1,bw2}.
These solutions are regular within the past light cone of the
singularity, however they have a spacelike hypersurface of
marginally trapped surfaces, i.e. an apparent horizon outside the
past light cone if $\eta
> \eta_{n}$, where $\eta_n$ is an increasing sequence ($\eta_{0} = 0.08$, $\eta_{1} = 0.152$, etc.).
Linear stability analysis
shows that the "ground state" $CSS_{0}$ is stable while the
excitations $CSS_{n}$ have exactly $n$ unstable modes.
Besides the CSS solutions, the system (1-4) has also a discretely self-similar (DSS)
solution for $\eta\geq \eta_c\approx 0.17$.
This solution was constructed by Lechner \cite{ch}
via a pseudospectral method following the lines of Gundlach
\cite{g}.
Next, we summarize what is known about the critical behavior in
gravitational collapse in this model. The first numerical studies of
this problem, reported in \cite{w1}, focused on relatively large
coupling constants $\eta>0.2$. In this range a "clean" type II
critical DSS behavior was observed, however the attempts to resolve
critical evolutions for lower values of $\eta$ encountered numerical
difficulties and for $0.18<\eta<0.2$ only an approximate DSS
behavior was observed. Furthermore the echoing period $\Delta$ was
found to increase sharply as the coupling constant decreases from
0.5 to 0.18. The critical behavior for smaller couplings
$0.1<\eta<0.2$ was studied in \cite{w2} (still smaller couplings are
less interesting because then the model admits naked singularities).
In the range $0.1<\eta<0.14$ a "clean" CSS critical behavior was
observed, thus it became clear that somewhere in the interval
$0.14<\eta<0.2$ there must be a transition between CSS and DSS
critical solutions. The detailed studies of this transition
\cite{w2} led to a conjecture that there exist a critical value of
the coupling constant $\eta_{c}\approx 0.17$ for which the system
exhibits the homoclinic loop bifurcation, i.e. the CSS saddle merges
with the DSS limit cycle in the phase space. These results left open
the question which of the two solutions in the transition region
acts as the critical solution at the threshold of black hole
formation. In particular, near the bifurcation point the black-hole
mass scaling could not be properly resolved.
\subsection*{Numerical results}
We have solved equations (1)-(4) for marginally critical initial
data fine tuned to the DSS solution for coupling constants close to
the critical value $\eta_c=0.17$. Since the echoing period $\Delta$
increases sharply as the coupling constant tends to its critical
value from above, it becomes more and more difficult to follow the
evolution over a large number of DSS cycles\footnote{ By an
elementary dimensional analysis the number of cycles scales as
$n\sim -(1/\lambda \Delta) \ln|p-p^*|$, where $\Delta$ is the
echoing period and $\lambda$ is the eigenvalue of the growing mode
of the DSS solution.}. We used the fully constrained implicit
evolution scheme based on the Newton-Raphson iteration. In order to
resolve the singular behavior near the origin we used the grid which
is uniform in $\ln(r)$. To get several cycles of the DSS attractor
near the bifurcation point we had to fine tune parameters of initial
data with precision of $70$ digits - this was achieved with the help
of the arprec library \cite{arp}
Actually, it was not our aim to determine $\Delta$ with high
precision, but rather to show that there exists a $\Delta$ to which
the evolution converges. To this end, we determine $\Delta$ as a
function of time (cycles) as the evolution approaches the limit
cycle i.e. the DSS solution.
For a marginally critical
solution we plot
the function $F$ versus
$\ln(r)$ for some late time $t_1$ and
superimpose the profile of the first echo at time $t_2$ shifted by
$\ln(r) \rightarrow \ln(r)+\Delta_r$. The time $t_2$ and the radial
echoing period $\Delta_r$ are chosen to minimize the discrepancy
between two profiles. We also define the temporal echoing period
$\Delta_t$ by the formula $t_2=t^*(1-e^{-\Delta_t})+e^{-\Delta_t}
t_1$, where $t^*$ is the accumulation time. Repeating this
calculation for a sequence of pairs $(t_n,t_{n+1})$, we get a
sequence of values $\Delta_r$ and $\Delta_t$.
Of
course, if the evolution converges to the DSS solution, both
$\Delta_r$ and $\Delta_t$ should converge to the same constant. In
Fig.~1
we show the convergence of
$\Delta_{r}$ and $\Delta_{t}$ during a critical evolution for the
coupling constant $\eta = 0.1725$. Note that the curve levels off,
thus signaling the closeness of the evolution to the limit cycle. As
the coupling constant decreases, $\Delta$ grows and the approach to
the DSS solution becomes slower.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth,angle=-90]{Fig2.eps}
\caption{\small{For $\eta=0.1725$ we show $\Delta_r$ and $\Delta_t$
as the functions of the cycle number $N$. Fitting the curve
$\Delta+c e^{-N}$ to the data we obtain $\Delta\approx 1.803$.}
}\label{fig2}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth,angle=-90]{Fig3.eps}
\caption{\small{Fitting the echoing period $\Delta$ (determined as
in Fig.~1) to the analytic prediction $\Delta = a \ln|\eta-\eta_c| +
const$ we get $\eta_c=0.1701$ and the slope $a=-0.389$ which is in
very good agreement with the analytic prediction
$a=-2/\lambda_{CSS}$ and the linear perturbation result \cite{ch}
$\lambda_{CSS}(\eta_c)\approx 5.14$. } }\label{fig2}
\end{figure}
Let $p^*$ be a critical parameter value which separates dispersion
from black holes (this value can be found by standard bisection). In
agreement with \cite{w1} we find that for $\eta>0.17$ the solution
corresponding to $p^*$ is DSS, in particular for $p$ slightly below
$p^*$ we observe DSS subcriticality, i.e. the solution approaches
the DSS solution and then disperses. Looking at the maximum value of
the spatial derivative of the scalar field at the origin as the
function of $p$, we find a typical subcritical scaling law (see
Fig.~3)
\begin{equation}\label{subscal}
\max |\partial_r F(t,0)| \sim (p^*-p)^{-\gamma_{DSS}}.
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth,angle=-90]{Fig4.eps}
\caption{\small{The subcritical scaling (5) for the coupling
constant $\eta=0.19$. The slope of the linear fit is approximately
equal to $-0.109$. The wiggles, which are imprints of discrete
self-similarity, have the period $\approx 4.8$.} }\label{fig3}
\end{figure}
For $p > p^*$ black holes are formed, however this happens in a
rather unusual manner. This is shown in Fig.~4 where the metric
function $A$ is seen to develop two minima very close to zero which
signals an almost simultaneous formation of a small and a large
black hole.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth,angle=-90]{Fig5_bis.eps}
\caption{\small{The series of snapshots of the metric function
$A(t,r)$ from the evolution of marginally supercritical initial data
for the coupling constant $\eta=0.19$. At the last frame one can see
the formation of two apparent horizons.} }\label{fig4}
\end{figure}
Let us denote their apparent horizon radii by $r_{in}$ and
$r_{out}$, respectively. We find that the outer radius exhibits the
standard DSS supercritical scaling (see Fig.~5a)
\begin{equation}\label{superscal}
r_{outer} \sim (p-p^*)^{\gamma_{DSS}},
\end{equation}
but the inner radius does not seem to scale. The latter fact was
already mentioned in \cite{w2}. The corresponding graph shows a
sea-saw structure, i.e. short straight lines with jump
discontinuities at certain values of the parameter $p$ (see
Fig.~5b).
\begin{figure}[h]
\centering \subfigure[]{
\includegraphics[width=0.34\textwidth,angle=-90]{Fig6.eps}}
\subfigure[]{
\includegraphics[width=0.34\textwidth,angle=-90]{Fig7.eps}}
\caption{\small{$\eta=0.19$. (a) The locus of the outer apparent
horizon is shown to satisfy the power law (6) with the slope
$\gamma_{DSS}=0.109$. The wiggles superimposed on the linear fit
have the period $4.8$ which agrees with the analytic prediction
$\Delta/2\gamma_{DSS}$. (b) The locus of the inner horizon is shown
not to scale. The jump discontinuities are periodic with the period
$4.8$.} }\label{fig6}
\end{figure}
In order to understand this strange behavior we looked in more
detail at the evolution of initial data fine-tuned to the location
of these jumps.
With the help of high resolution numerical methods we found the
following remarkable structure: for a given family of initial data
there is a sequence of discrete parameter values $p_{1}> p_{2}
> p _{3}...> p_{n}$ such that a solution with $p\in(p_n,p_{n+1})$
approaches to the $CSS_{1}$ solution $n$ times, i.e. the
solution comes close to the $CSS_{1}$ solution, turns away and
returns $n$ times before leading to black hole formation.
Multiple
approaches to the $CSS_1$ solution were already noticed in \cite{w2}
where they were called episodic CSS, however the corresponding
fine-structure in the parameter space was not seen there. The
sequence $\{p_n\}$ with $n\leq 5$ is listed in Table~1.
\begin{table} [h]
\begin{center}
\begin{tabular}{l||c|c|c|c|c|c}
\hline \hline
$n$ & 1 & 2 & 3 & 4 & 5 & $\infty$ \\
$p_n$ & $0.529001923689295$ & $0.528771570563995$ & $0.528769577618968$
& $0.528769560376615$ & $0.528769560227152$ & $0.528769560226138$\\
\hline \hline
\end{tabular}
\caption{The first five critical parameter values $p_n$ for the
coupling constant $\eta=0.19$.} \label{tab:pn}
\end{center}
\end{table}
Because of numerical limitations we were not able to
resolve higher $p_n$, however the data shown in Table~1 seem to
indicate that the sequence $p_n$ converges to $p^*$ as $n$ tends to
infinity. Actually, we find that the two consecutive parameters
$p_n$ satisfy the scaling law
\begin{equation}
\frac{p_{n}-p^*}{p_{n+1}-p^*} \approx
\exp\left(\frac{\Delta}{2\gamma_{DSS}}\right).
\end{equation}
Now we return to the problem of scaling of the inner radius
$r_{in}$.
For $p = p_{n} + \varepsilon$, i.e. for $p$ just above one of the
$p_{n}$'s we see a clear CSS scaling (see Fig.~6a)
\begin{equation}\label{subscal}
r_{in} \sim (p-p_n)^{\gamma_{CSS}}.
\end{equation}
For $p = p_{n} - \varepsilon$ the solution displays a kind of
pseudo-dispersion after its last CSS-episode. This
pseudo-dispersion manifests itself as follows: after leaving the CSS
solution, the maximum of the function $F$ decreases, the inner
minimum of $A$ disappears and later a spike develops which leads to
the formation of an apparent horizon at $r_{outer}$. In this range
of $p$ we see the subcritical CSS scaling (see Fig.~6b)
\begin{equation}\label{subscal2}
\max |\partial_r F(t,0)| \sim (p_{n}-p)^{-\gamma_{CSS}},
\end{equation}
however the masses of black holes formed in such evolutions are
"large" and do not scale. We remark that since the solutions on
both sides of $p_n$ form black holes, the bisection which gives
critical parameter values $p_{n}$ has to be performed in a sense
"by hand".
\begin{figure}[h]
\centering\centering \subfigure[]{
\includegraphics[width=0.34\textwidth,angle=-90]{Fig10.eps}}
\centering \subfigure[]{
\includegraphics[width=0.34\textwidth,angle=-90]{Fig11.eps}}
\caption{\small{$\eta=0.19$. (a) The supercritical (8) and (b)
subcritical (9) scalings around $p_n$ for $n=2$ (we get the same
picture for each $n$). The slopes of the linear fits are equal to
$\pm 0.195$ which agrees with the analytic prediction $\pm
1/\lambda_{CSS}$ where $\lambda_{CSS}(\eta=0.19)=5.1$ was obtained
independently from the linear perturbation theory by Lechner
\cite{ch}.} }\label{fig6}
\end{figure}
\subsection*{Interpretation of numerical results}
The results presented above confirm and extend the findings of
Lechner et al. \cite{w2}. Probing the bifurcation point $\eta_c$
with higher accuracy we improved the evidence that $\Delta$ diverges
as $\eta$ tends to the critical value $\eta_{c} = 0.17$ from above,
which in turn confirms the picture that the DSS cycle merges with
the CSS solution at the critical coupling constant $\eta_{c}$.
A natural question is:
what is the meaning of the series of critical parameter values
$p_{n}$ within this picture?
We conjecture that our system shows a so called Shil'nikov
bifurcation \cite{k}. In his classification of loop bifurcations
for three dimensional systems, Shil'nikov
considered a system with a saddle point together with
a homoclinic orbit which bifurcates for some value of a parameter.
Assuming that the eigenvalues of the saddle point are real and
satisfy the following conditions: $\lambda_{1}
> 0
> \lambda_{2}>\lambda_{3}$ and $\lambda_{1} + \lambda_{2}> 0$ (plus some less important technical conditions), Shil'nikov showed
that a saddle limit cycle bifurcates and the phase space picture
looks qualitatively as in Fig.~7a.
\begin{figure}[h]
\centering \subfigure[]{
\includegraphics[width=0.35\textwidth,angle=-90]{Fig13.eps}}
\subfigure[]{
\includegraphics[width=0.35\textwidth,angle=-90]{Fig12.eps}}
\caption{\small{(a) Shil'nikov bifurcation. (b) The conjectured phase space picture.} }\label{fig6}
\end{figure}
Of course, our system is infinite dimensional and the Shil'nikov
theorem cannot be applied directly. Nevertheless, it is expected
that a similar picture to Fig.~7a will be valid for higher
dimensional systems as long as only a few largest eigenvalues of
the perturbation matter. Recently, Donninger \cite{d} has studied
linear perturbations around the CSS solution and found that for
coupling constants around the critical value $\eta_c$ the first
three largest eigenvalues do in fact satisfy the above stated
Shil'nikov conditions. Combining this property with the fact that
the bifurcating DSS solution is a saddle limit cycle, we conjecture
that the (one-dimensional) unstable manifold of the DSS solution
lies on the stable manifold of the neighboring CSS solution. This is
sketched in Fig.~7b.
More precisely, the DSS unstable manifold winds around the limit
cycle (infinitely many times) and eventually runs into the CSS
saddle. Suppose that a curve of initial data intersects this spiral
manifold at values $p_{n}$, with $\lim p_{n}= p^*$, where $p^*$
corresponds to the intersection with the limit cycle. Then, the
dynamical behavior will have exactly the form we observed above: for
$p$ equal to one of the $p_{n}$'s, the solution spirals $n$ times
around the limit cycle each time coming closer to the CSS--saddle
before hitting it. For $p = p_{n} \pm \varepsilon$, the behavior is
similar, except that the solution does not hit the CSS--saddle but
escapes along its unstable manifold. This is the reason why one
observes the CSS scaling around $p_n$ with an exponent related to
the unstable eigenvalue of the CSS solution. Note that the scaling
law (7) follows immediately from the picture shown in Fig.~7b
because during one cycle of evolution the distance from the DSS
limit cycle increases by the factor $e^{\lambda_{DSS} \Delta}$ (and
$\gamma_{DSS}=1/\lambda_{DSS}$).
\subsection*{Acknowledgments} This research was supported in part by
the Austrian Fonds zur F\"orderung der wissenschaftlichen Forschung
(FWF) Project P15738-PHY and in part by the Polish Ministry of
Science grant no. 1PO3B01229.
| {
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Q: PHPMyAdmin with php < 8 So I am installing PHPMyAdmin for months now on my php 8 machine by using apt install phpmyadmin, which always worked fine, it always ran at php 7.2, the webserver at least, not sure if phpmyadmin it self also used php 7.2, anyway, it did not install or use php 8.1
But today, after I tried to install phpmyadmin on 3 different machines I noticed that when using apt install phpmyadmin it was installing php 8.1, which is conflicting my Laravel projects which need php artisan functions, which only seem to work with php 8.
So my question is, is there any way to not install php 8.1 but just use 8 or 7.2?
| {
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package de.iritgo.aktario.core.network;
import de.iritgo.aktario.core.base.Transceiver;
/**
* BroadcastTransceiver, nutzen ActionProcessor um an alle Clients Actions zu schicken!
*/
public class BroadcastTransceiver implements Transceiver
{
/**
* Clone Transceiver
*
* @return Transceiver.
*/
public Transceiver cloneTransceiver()
{
return new BroadcastTransceiver();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,150 |
Q: Why am I getting HDD total space less than 931 GiB? I've been trying for a few days already, but still cannot figure it out how to get the proper size of my HDD drive with a python script.
My HDD is 1Tb. As I know in Gb it is 1000Gb, and in GiB it is 931GiB roughly.
When I type in the terminal lsblk it shows this:
NAME MAJ:MIN RM SIZE RO TYPE MOUNTPOINT
sda 8:0 0 931,5G 0 disk
├─sda1 8:1 0 512M 0 part /boot/efi
└─sda2 8:2 0 931G 0 part /
Ok.Then I try lshw --class disk it shows 931GiB as well:
*-disk
description: ATA Disk
product: ST1000LM035-1RK1
physical id: 0.0.0
bus info: scsi@0:0.0.0
logical name: /dev/sda
version: SDM2
serial: WDEWEKZF
size: 931GiB (1TB)
capabilities: gpt-1.00 partitioned partitioned:gpt
configuration: ansiversion=5 guid=f166251c-436c-421f-aba8-9910d76f9fab logicalsectorsize=512 sectorsize=4096
Then I try to get the size through a python script:
total, used, free, percent = disk_usage('/')
print(f"Total: {total}")
print(f"Used: {used}")
print(f"Free: {free}")
total2, used2, free2, percent2 = disk_usage('/boot/efi')
print(f"Total: {total2}")
print(f"Used: {used2}")
print(f"Free: {free2}")
output:
Total: 982900588544
Used: 118413897728
Free: 814486605824
Total: 535805952
Used: 5484544
Free: 530321408
982900588544 / 1024 / 1024 / 1024 = 915 GiB.
535805952 = 500 MiB.
df command shows this:
Filesystem 1K-blocks Used Available Use% Mounted on
udev 8092080 0 8092080 0% /dev
tmpfs 1627768 1712 1626056 1% /run
/dev/sda2 959863856 115646148 795389500 13% /
tmpfs 8138832 12368 8126464 1% /dev/shm
tmpfs 5120 4 5116 1% /run/lock
tmpfs 8138832 0 8138832 0% /sys/fs/cgroup
/dev/sda1 523248 5356 517892 2% /boot/efi
tmpfs 1627764 24 1627740 1% /run/user/1000
The sum of all 1K-blocks gives 1Tb.
So, where is another 931 - 915 = 16 GiB of HDD space?
And how to get the size in a correct way?
Linux Mint 20.1 x64
Thanks.
A: If that's ext4, it's the size that is lost to filesystem metadata, mainly inode tables. As an example, a /home partition here.
*
*Partition is 751619276800 bytes (sudo /sbin/blockdev --getsize64 /dev/mapper/Watt-home)
*"df" size is 739691814912 (df --block-size=1 /home)
*Inode count is 45875200 (df -i /home)
*Inodes on ext4 are 256 bytes.
So if you do the math, (751619276800-739691814912-(45875200*256))/1024^2 ≈ 175MiB. That's the rest of the filesystem metadata (superblocks, etc.).
To make sure this is right, compare to a filesystem that was initialized with a lower inode ratio — one way is with the -T largefile or -T largefile4 option (see /etc/mke2fs.conf for the possibilities). I have one here:
*
*Partition size: 429496729600
*df size: 429229522944
*inodes: 409600
Note how much closer the df size is to the partition size (over 99.9%). That's because there are far fewer inodes. And if you do that math again, (429496729600-429229522944-(409600*256))/1024^2 ≈ 155MiB.
Keep in mind that on ext4, the number of inodes is a hard limit on the number of files you can have. It (or rather the ratio of 1 inode per N blocks) is also set once at mkfs and can not be changed. But if you have a filesystem that you know will only be used to store large files, you can save some space by having fewer of inodes, as I did on my second filesystem.
You can see the overhead subtracted out in the kernel source code: https://elixir.bootlin.com/linux/latest/source/fs/ext4/super.c#L6095 — and also the existence of a minixdf mount option that will stop it from doing so, and maybe do more weirdness too. I didn't check, and the only documentation I found about it was them trying to remove it but keeping it when people complained.
BTW: In addition to this overhead for inode tables, etc., there is often 5% of space reserved, typically for root. That doesn't subtract from the total size, but will subtract from the available space. You can change this amount tune2fs -m; other options let you specify by block count instead (-r) and change which user (-u) or group (-g) can use the reserved space. One benefit is even if users fill a partition, the sysadmin has some space to use for recovery.
Note: ext2/ext3 used 128-byte inodes, half the size. Small filesystems still do. You can actually set it a mkfs time with the -I option; see the mkfs.ext4 manpage for caveats (I would not recommend changing to 128).
A: It is lost to hidden internal data used by the file system itself to bookkeep all the file structure. It is not displayed by tools you are using, and thus appears as missing.
| {
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{"url":"https:\/\/www.physicsforums.com\/threads\/exponential-decay-growth.442654\/","text":"# Exponential decay\/growth\n\n1. Oct 29, 2010\n\n### h00zah\n\nif i have a set of data, where as time increases, so does y, but is bounded by a number say y=c, how do i formulate my equation? how do i find the constants?\n\ni have y=c-a-x*b\n\nthis is basically a transformation of y=b*ax+c\n\n2. Oct 29, 2010\n\n### fzero\n\nIf you have some reason to believe that\n\n$$y = c - b a^x$$\n\nis the function that fits the (x,y) data, you can compute\n\n$$\\ln (c-y) = (\\ln a) x + \\ln b.$$\n\nIf you can find c from the asymptotic form of y, then a plot of $$\\ln (c-y)$$ vs x will be linear. You can read $$\\ln a$$ from the slope and $$\\ln b$$ from the intercept. You can read up on linear regression theory if you want to determine the accuracy of the fit.\n\n3. Oct 29, 2010\n\n### fzero\n\nIf you have some reason to believe that\n\n$$y = c - b a^x$$\n\nis the function that fits the (x,y) data, you can compute\n\n$$\\ln (c-y) = (\\ln a) x + \\ln b.$$\n\nIf you can find c from the asymptotic form of y, then a plot of $$\\ln (c-y)$$ vs x will be linear. You can read $$\\ln a$$ from the slope and $$\\ln b$$ from the intercept. You can read up on linear regression theory if you want to determine the accuracy of the fit.","date":"2019-03-19 08:07:03","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\": 0.36671146750450134, \"perplexity\": 302.33796266089803}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912201922.85\/warc\/CC-MAIN-20190319073140-20190319095140-00400.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{T}{here} is an exponential growth in the number of digital images being produced and circulated through different media sources day by day. This includes natural images of the real-world scenes taken by a camera and computer-generated images. The computer-generated images include images that are generated by different graphics and rendering software and also those generated using the latest deep learning algorithms called Generative Adversarial Networks (GAN). The process of computer generation of images tends to be such realistic nowadays that it is impossible to differentiate natural images from computer-generated images. Figure \ref{fig:sample} shows some sample images where, we can observe the extend of photorealism attained in GAN generated (left image\footnote{\url{https://github.com/NVlabs/stylegan2}}) and computer graphics generated (middle image\footnote{\url{https://cgsociety.org/c/featured/1f9s/the-forever}}) images, making it hard to classify them as computer-generated images and for the natural image (right image, from the Computer Graphics versus Photographs dataset \cite{tokuda2013computer}) at first glance, we might have an impression that it is a computer graphics generated image. Hence, images that reach us always have a question of authenticity, that is, whether the image is a projection of a real-world event taken by a camera or is it computer-generated content. Even though computer-generated images are mostly seen produced for creative art, entertainment, advertisement, joke, or satirical purposes, they have high potential to easily propagate through social media causing misinformation particularly, when presented with fake stories or fake news \cite{anoop2019leveraging}. They also have much darker sides like the earlier incidents of claiming pornographic images of children as computer-generated graphics images to escape from legal actions\footnote{\url{www.sciencedaily.com/releases/2016/02/160218144928.htm}}, to the recent incidents of creating nude photographs of people from their original photographs through GAN algorithms\footnote{\url{www.technologyreview.com/2020/10/20/1010789/ai-deepfake-bot-undresses-women-and-underage-girls/}}. The deficiencies in human perception to distinguish natural and computer-generated images without the assistance of any additional tools \cite{schetinger2017humans} highly demands and points out the necessity of computational algorithms in digital image forensics to investigate images since the authenticity of an image legally depends on whether it is a natural image or computer-generated. Distinguishing natural from computer-generated images has thus become one of the fundamental and most actively researched problems in digital image forensics.
\begin{figure}[!t]
\centering
\subfloat{\includegraphics[height=2.35cm]{IEEEtran/figures/gan_sample.jpg}}\hspace{0.8pt}%
\label{fig:gan}
\hfil
\subfloat{\includegraphics[height=2.35cm]{IEEEtran/figures/graphics_sample.jpg}}\hspace{0.8pt}%
\label{fig:graphic}
\hfil
\subfloat{\includegraphics[height=2.35cm]{IEEEtran/figures/real_sample.jpg}}%
\label{fig:real}
\caption{A sample of GAN generated (left image), Computer graphics generated (middle image) and natural (right image) images}
\label{fig:sample}
\end{figure}
The previous works distinguishing natural images from photo-realistic computer-generated ones either addresses the \textit{natural images versus computer graphics} problem or the \textit{natural images versus GAN images} problem, at a time. For the \textit{natural image versus computer graphics} problem, when an image is not computer graphics it shall fall to the natural image category, but it may sometimes actually belong to the GAN category of computer-generated images which is not considered for the task; similar issue may also occur for the \textit{natural images versus GAN images} problem. Therefore, in a real-world scenario to provide a complete forensic solution to distinguish natural images from computer-generated images, since the image generation is unknown, it is highly essential to consider all categories of images generation, including natural images taken by a camera, computer graphics, and GAN images. We, to the best of our knowledge, for the first time attempt to address this gap of a generalized algorithm in digital image forensics to distinguish natural images from photo-realistic computer-generated images including both computer graphics and GAN images as a three-class classification task by proposing a Multi-Colorspace fused EfficientNet model.
The major contributions of our work include:-
\begin{itemize}[leftmargin=5mm]
\item We introduce a deep learning based image forensic solution to identify natural images and photo-realistic computer-generated images including both the computer graphics and GAN images
%
\item We propose a Multi-Colorspace fused EfficientNet model build by parallelly fusing three EfficientNet networks that follow transfer learning methodology where each network operates in different colorspaces, RGB, LCH, and HSV, chosen after analyzing the efficacy of colorspace transformations in this forensic problem
%
\item Our Multi-Colorspace fused EfficientNet model obtains good forensic performance, outperforming the baselines in terms of accuracy, robustness towards post-processing, and generalizability towards other datasets
%
\item We also conduct psychophysics experiments to assess the capability of humans to classify natural images and photo-realistic computer-generated images including computer graphics and GAN images
%
\item We analyze the behavior of our model through visual explanations to understand the salient regions that contribute to model's decision making and compare with manual explanations from the psychophysics experiments provided by the human participants in the form of region markings
\end{itemize}
The rest of this paper is organized as follows. Section \ref{sec:related} introduces some of the relevant related works and demarcates our work from the related works. Section \ref{sec:methodology} discusses our methodology with the motivation and detailed description of our proposed work. Section \ref{sec:empirical} presents the empirical study with details of dataset, experimental settings, results and discussion including the statistical significance, robustness, generalizability, feature visualization and psychophysics experiments, and model behavior analysis using activation maps. Finally, section \ref{sec:conclusion} draws the conclusions.
\section{Related Work}
\label{sec:related}
Since the previous image forensic works distinguishing natural images from computer-generated images either addresses only the \textit{natural images versus computer graphics} or the \textit{natural images versus GAN images} problem, in this section we present a brief review of these two categories separately.
\subsection{Natural Images versus Computer Graphics Images}
The works distinguishing natural images and computer graphics have been reported since 1990s, which are mostly based on color features \cite{athitsos1997distinguishing,smith1997multi}. The differences in generation of natural images and computer graphics, camera or device properties, etc., are considered for the traditional feature based classification works \cite{ng2005physics,swaminathan2008digital,gallagher2008image}. Many other feature based works using color, texture, and shape based statistical features followed this area of image forensics study \cite{li2012distinguishing,wang2014statistical,peng2017discrimination}. Leveraging features from various image transformation domains like wavelet \cite{farid2003higher}, contourlet \cite{ozparlak2011differentiating}, quaternion wavelet \cite{wang2017forensics}, etc., is another approach that can be seen in the traditional feature based works. Later, with revolutionary progress in the area of neural networks, contributions can be seen using Convolutional Neural Networks (CNN) which avoids the burdensome process of finding out discriminative features and exhibit comparatively higher performance than the traditional feature based classification approaches \cite{cui2018identifying,nguyen2019capsule}. A few deep learning based works that employ the transfer learning methodology by using pre-trained off-the-shelf networks \cite{de2018exposing,long2021identifying,meena2021distinguishing} can also be seen in the literature. Apart from considering the full-sized images, some works crop images into fixed size patches and derive results of the full-sized images from these image patches \cite{rahmouni2017distinguishing,quan2018distinguishing,he2018computer}. Besides the aforesaid objective studies there are also a very few number of subjective studies that involve humans to distinguish natural images and graphics images using certain psychophysics experiments \cite{farid2012perceptual,mader2017identifying,fan2017image}.
\subsection{Natural Images versus GAN Images}
The works in this category are comparatively much recent ensuing the advent of new and powerful class of deep learning algorithms called Generative Adversarial Networks. Unlike previous problem, very few works employ the traditional hand-crafted feature based classification \cite{marra2018detection,li2018detection}, and majority of the works in literature approach this problem using deep learning algorithms \cite{nataraj2019detecting,marra2019incremental,hsu2020deep}. Besides the works targeting to detect images generated from a single GAN algorithm \cite{marra2018detection,mo2018fake,tariq2018detecting}, there are also works for the attribution of known GANs which are used to generate the fake images \cite{yu2019attributing,joslin2020attributing,goebel2020detection}. Many works in this problem are seen to specifically work over the GAN generated human faces rather than considering heterogeneous image contents \cite{barni2020cnn,hulzebosch2020detecting,hu2021exposing}.
\subsection{Our work in context}
In the literature, works either address only the \textit{natural images versus computer graphics} problem or \textit{natural images versus GAN images} problem, at a time. However, such a closed set will not suit the real-world scenario that requires a single forensic system to authenticate an image by investigating multiple types of image generations, where in most cases the image generation is unknown. Therefore, unlike the previous image forensic works that had always been dealt with as a two-class classification problem, we for the first time, to the best of our knowledge, attempt the image forensic task of distinguishing natural images from computer-generated images as a three-class classification problem classifying natural images, computer graphics, and GAN images. We perform a deep neural network based classification with transfer learning methodology that avoids the burdensome process of feature extraction and feature selection as in the case of conventional feature extraction based approaches.
Different from the transfer learning based work to distinguish natural images from computer graphics proposed by Rezende et al. \cite{de2018exposing} using the ResNet architecture with 25.6M parameters, our choice of network, the EfficientNet, is 4.9 times smaller, with just 5.3M parameters. When compared to the deep learning based works proposed by Cui et al. \cite{cui2018identifying}, and Quan et al. \cite{quan2018distinguishing}, that utilizes an earlier
Columbia
dataset \cite{ng2005columbia} with considerably less amount of images for a deep learning task (800 images per class), the choice of our dataset is more challenging in the real world forensic scenario by maintaining heterogeneity in each of the three classes so that to build a generalized robust model that is unbiassed towards any particular image category, origin or generating algorithm, without compromising the number of images in dataset (4000 images per class). Also, we avoid patch based implementation in our deep learning approach because, firstly such patch based approaches are computationally very expensive than taking full-sized images for checking whether an image is fully computer-generated or is it taken by a camera (e.g., \cite{quan2018distinguishing} extracts 200 patches from a single image), and moreover, such patch based implementations might be more suitable for image forgery problems to detect the manipulated image regions.
Apart from \cite{he2019detection,li2020identification} that choose certain colorspace transformations in their work to distinguish natural images from computer-generated images, we examine in detail which colorspaces provide high classification accuracies for the task of distinguishing natural images from computer-generated images including computer graphics and GAN images, and also the chances of improvement in accuracy by fusing the networks operating in different colorspaces. Among the various works in literature to distinguish natural images from computer-generated images, no much works are seen to discuss the interpretability or behavior of model; a work in this regard would be \cite{quan2018distinguishing}, that tries to understand what the model learns to differentiate natural and computer graphics images. Whereas, in our three-class classification work for natural, graphics and GAN images, besides visualizing the explanations of correct and wrong predictions for model behavior analysis, we also compare the visual explanations of our model with the human explanations labeled as region markings during the psychophysics experiments, to look for any similarities between the model and human explanations, and to understand whether our model is predicting the decisions meaningfully.
\section{Methodology}
\label{sec:methodology}
We formulate the image forensic task of distinguishing natural images from computer-generated images as a three-class classification task with the classes being Real, GAN, and Graphics, where the class Real indicates natural images, and the classes GAN and Graphics indicates computer-generated images. Even though both GAN and Graphics images are computer-generated, we maintain them as separate classes, since they follow entirely different process of image generations. Accordingly, we put up an amendment to the depiction of general framework followed for the \textit{natural images versus computer-generated images} problem as outlined by Quan et al. \cite{quan2018distinguishing}, by incorporating our three-class classification approach, in figure \ref{fig:framework}. Framework A indicates conventional feature based classification that finds a mapping $y = \text{\textit{clf}}(\text{\textit{ftr}}(x))$ between the training data $x$ and corresponding label $y$ using a good choice of feature set (\textit{ftr}) and classifier (\textit{clf}) combination. Whereas, framework B indicates deep neural network based classification that avoids tiresome process of hand-crafted feature extraction and feature selection. In this work, we follow the framework B of deep neural network based classification, where we aim to find the best-fit mapping function $M: y= M(x)$ for the training data ${(x_1,y_1),(x_2,y_2),...,(x_n,y_n)}$, where $x_i$ indicates $i^{th}$ image in the training set and $y_i$ indicates the corresponding image label, denoted as 0 for the class GAN, 1 for Graphics and 2 for Real. Deep neural networks also allow the option of transfer learning where a network pre-trained over very large datasets for some \textit{n}-class classification task can be utilized for another \textit{m}-class classification task even with less number of training data. Such a knowledge transfer from the source network helps to obtain high accuracy in the target network of different tasks. We incorporate transfer learning methodology which helps to transfer information from an object classification network trained on the huge ImageNet \cite{russakovsky2015imagenet} dataset with 1000 classes, to our three-class forensic task.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{IEEEtran/figures/framework.png}
\caption{The general machine learning frameworks for image forensic problem}
\label{fig:framework}
\end{figure}
\subsection{Motivation}
Some state-of-the-art works for distinguishing natural images from computer-generated images discuss the future scope of fusing deep learning models to create ensembled architectures that can improve classification accuracies \cite{de2018exposing,quan2018distinguishing}. Our concept of network fusion for distinguishing natural from computer-generated images is motivated from ColorNet
\cite{gowda2018colornet}, where the authors' demonstrate that colorspace transformations can significantly effect classification accuracies and observe that there is no hundred percentage correlation between different colorspace transformed images.
Our choice of base network for network fusion was centered on the motivation that it should have less number of parameters so that to reduce network complexity but without compromising on classification accuracy. This might make easier the chances of network fusion by not much shooting up the complexity of fused network. Hence, among the wide range of deep neural network architectures we choose one of the latest networks, EfficientNet \cite{tan2019efficientnet}, as the base network for our study that shows high performance in ImageNet recognition challenge \cite{russakovsky2015imagenet}. The classification based on transfer learning methodology using a pre-trained EfficientNetB0 model helps to reduce training complexity, by keeping the number of trainable parameters of a single EfficientNetB0 network to a very short number of only 3843.
\subsection{Single Colorspace EfficientNet Network (SC-EffNet)}
We perform classification based on transfer learning methodology using a pre-trained EfficientNetB0 model by removing its top dense layer with 1000 neurons and instead, fitting a fully connected dense layer with 3 neurons and softmax activation for our three-class classification task. All other layers in the EfficientNetB0 network are kept frozen while training and validating our task. The initial phase of classification
was performed on the dataset by considering input images without any color conversion, i.e., in the RGB colorspace itself (\textit{SC-EffNet\textsubscript{RGB}}). EfficientNetB0 network can intake input images within the data range 0-255 since data normalization is included as a part of its architecture. Hence, while implementing an EfficientNetB0 model for RGB images, the input images are not rescaled to the range 0-1 as like the normal procedure of multiplication with 1./255, which is most commonly performed while implementing many off-the-shelf deep neural network architectures.
In our image forensic task of classifying GAN, Graphics, and Real images we were curious to know the colorspaces that significantly effect classification accuracies. Hence, next we perform classifications using the EfficientNetB0 model over colorspace transformed images. The colorspaces chosen for our set of experiments include HLS, HSV, LAB, LCH, XYZ, YCbCr, YDbDr, YIQ, YPbPr, and YUV, which are the most commonly known and used color spaces. Except HLS and YDbDr all other colorspaces we have chosen are also experimented in ColorNet \cite{gowda2018colornet}, because of their easiness in transformation from RGB.
For compiling the model we use categorical cross-entropy as the loss function, Adam optimizer with learning rate 0.001, batch size of 256 and 100 epochs.
In case of colorspaces other than RGB, we perform an additional rescaling procedure over the transformed images as their intensities do not follow the range 0-255 for being admitted to the EfficientNetB0 model. The rescaling procedure we have followed in the proposed work is given in algorithm \ref{alg:rescale}. We were curious whether rescaling colorspace transformed images would in any case reduce the classification accuracies. But we could find that the classification accuracies instead improved on rescaling colorspace transformed images to the range 0-255 for every colorspace transformation, particularly for LAB and LCH colorspaces. Also, we were able to implement classification in HLS colorspace only after rescaling. The test accuracies of classification for RGB images, and for the colorspace transformed images without rescaling and after rescaling is shown in table \ref{tab:rescaling_acc}
\SetAlFnt{\small}
\SetAlCapFnt{\small}
\SetAlCapNameFnt{\small}
\begin{algorithm}[!t]
\DontPrintSemicolon
\KwIn{Colorspace transformed image $T_{img}$ with color channels $[ch_1, ch_2, ch_3]$}
\KwOut{Rescaled image $R_{img}$}
Initialize $R_{img}$ to be empty\\
\For{$i \gets 1$ \textbf{to} 3}
{
$min(i) = min(T_{img}[ch_i])$\\
$max(i) = max(T_{img}[ch_i])$\\
$ R_{img} \left[ch_{i}\right] = round \left( \frac {T_{img}\left[ch_{i}\right] - min(i)} {max(i) - min(i)} \times 255 \right) $
}
\Return{$R_{img}$}
\caption{Rescale image to the data range 0-255}
\label{alg:rescale}
\end{algorithm}
\begin{table}[!t]
\centering
\caption{Accuracy of \textit{SC-EffNet} for different colorspaces in percentage
(The highest accuracy is given in boldface)
}
\label{tab:rescaling_acc}
\begin{tabular}{lcc}
\hline
Colorspace & Without Rescaling & After Rescaling \\
\hline
RGB & \textbf{82.13} & - \\
HLS & - & 77.79 \\
HSV & 77.96 & 80.38 \\
LAB & 40.29 & 77.42 \\
LCH & 36.33 & 80.52 \\
XYZ & 80.00 & 80.26 \\
YCbCr & 74.66 & 75.75 \\
YDbDr & 75.13 & 75.58 \\
YIQ & 74.83 & 76.54 \\
YPbPr & 74.17 & 75.92 \\
YUV & 74.38 & 75.79\\
\hline
\end{tabular}
\end{table}
Our three-class forensic task of distinguishing GAN, Graphics, and Real images obtains a highest accuracy of 82.13 percentage when images are in the RGB colorspace itself, as against ColorNet \cite{gowda2018colornet} performed over the object classification task which obtains highest accuracy in the LAB colorspace. Also, we observe that three other colorspaces HSV, LCH, and XYZ have their accuracies near to RGB after rescaling, unlike ColorNet where except LAB every other colorspace have similar values of accuracies. A much more detailed view of the classification results showing accuracies of each class separately is shown in table \ref{tab:class_acc}.
\begin{table}[!t]
\centering
\caption{Class accuracy and total accuracy of \textit{SC-EffNet} for different colorspaces in percentage (The highest two accuracies in each class and total accuracy are given in boldface)}
\label{tab:class_acc}
\begin{tabular}{lcccc}
\hline
Colorspace & GAN & Graphics & Real & Total Accuracy \\
\hline
RGB & 88.75 & \textbf{79.88} & \textbf{77.75} & \textbf{82.13
} \\
HLS & 90.13 & 70.75 & 72.50 & 77.79 \\
HSV & \textbf{93.88} & \textbf{75.63} & 71.63 & 80.38 \\
LAB & 88.13 & 70.38 & 73.75 & 77.42 \\
LCH & \textbf{92.87} & 74.88 & 73.82 & \textbf{80.52} \\
XYZ & 90.75 & 74.61 & \textbf{75.42} & 80.26 \\
YCbCr & 84.38 & 68.88 & 74.00 & 75.75 \\
YDbDr & 87.38 & 66.25 & 73.13 & 75.58 \\
YIQ & 86.38 & 69.88 & 73.37 & 76.54 \\
YPbPr & 84.13 & 70.88 & 72.75 & 75.92 \\
YUV & 81.00 & 71.00 & 75.38 & 75.79 \\
\hline
\end{tabular}
\end{table}
We can observe that the accuracy of each class varies highly with the colorspaces. The accuracy of class GAN is comparatively higher than the other two classes for all the colorspaces. The highest accuracy for class GAN is observed in HSV colorspace and for the classes Graphics and Real is observed in RGB colorspace. LCH and XYZ are the other colorspaces that shows nearest higher accuracies for these classes. Since the highest accuracies for each of the classes when viewed individually are obtained in different colorspaces, there is a scope of increasing the total accuracy of our task by combining these colorspaces.
We try to combine the \textit{SC-EffNet} networks of the colorspaces which shows highest accuracy for each class when treated individually and also the colorspaces which shows highest overall accuracy, i.e., the combinations of RGB, HSV, LCH, and XYZ, to form a Multi-Colorspace fused EfficientNet.
\subsection{Multi-Colorspace fused EfficientNet Network (MC-EffNet)}
For combining the networks of colorspaces, each colorspace except RGB is rescaled and passed through a separate EfficientNetB0 model pre-trained over the ImageNet dataset. The top dense layer of each EfficientNetB0 with 1000 neurons is removed and all its layers are kept frozen for the training phase, similar to the \textit{SC-EffNet} based classification. The EfficientNetB0 networks without top dense layer now return a feature vector of size 1280, for each colorspace network. We construct a parallelly fused model where outputs of all colorspace networks used for fusion are concatenated and provided to a dense layer with three neurons suitable for our classification task. The test accuracies of the fused models formed from RGB, HSV, LCH, and XYZ colorspace networks are shown in table \ref{tab:fused_acc}. Our fusion technique shows increase in the overall accuracy, especially the combination of three colorspace networks RGB, LCH and HSV that produces a high accuracy of 87.96 percentage, an increase of 5.83 percentage points from the \textit{SC-EffNet\textsubscript{RGB}} model. But, the addition of XYZ colorspace network again to this fused model is seen to slightly degrade the accuracy. Hence, we adhere to the three colorspaces RGB, LCH and HSV to build our Multi-Colorspace fused EfficientNet model, \textit{MC-EffNet-1}.
\begin{table}[!t]
\centering
\caption{Accuracy of \textit{MC-EffNet} for colorspace network combinations in percentage (The highest accuracy is given in boldface)}
\label{tab:fused_acc}
\begin{tabular}{lc}
\hline
Colorspace network combination & Accuracy \\
\hline
RGB + HSV & 86.04 \\
RGB + LCH & 86.63 \\
RGB + XYZ & 82.63 \\
RGB + LCH + HSV & \textbf{87.96} \\
RGB + LCH + HSV + XYZ & 86.83\\
\hline
\end{tabular}
\end{table}
Since the image forensic computation models, apart from providing high accuracies, should also show good amount of robustness towards post-processed images, we tested \textit{MC-EffNet-1} over JPEG compressed images. We observe that, even though \textit{MC-EffNet-1} gives high classification accuracy for original images without any post-processing, the model accuracy decreases highly for JPEG compressed images, even for a quality factor of 90 (shown in table \ref{tab:mc1_compression}). Interestingly, the decrease in test accuracy for JPEG compressed images is comparatively higher for class GAN than Graphics and Real, when observed class wise. We also provide accuracies of the base \textit{SC-EffNet} networks over JPEG compressed images in table \ref{tab:mc1_compression}. For \textit{SC-EffNet\textsubscript{RGB}}, we can observe that with an increase in compression (or decrease in quality factor), there exists a decrease in classification accuracy, but the rate of decrease is not as high as for \textit{MC-EffNet-1}. But when we check \textit{SC-EffNet\textsubscript{LCH}} and \textit{SC-EffNet\textsubscript{HSV}}, we can observe a very quick decay in their accuracies for compressed images. This helps to finalize that even though LCH and HSV colorspace transformations can highly increase the classification accuracies of images without any post-processing, they do not behave well with JPEG compressed images.
\begin{table}[!t]
\centering
\caption{Model accuracy over original images and JPEG compressed images in percentage for different quality factors ($\mathrm{qf}$)}
\label{tab:mc1_compression}
\begin{tabular}{ccccccc}
\hline
\multirow{2}{*}{Model} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Original\\images\end{tabular}} & \multicolumn{5}{c}{JPEG compressed images}\\
& & qf=90 & qf=80 & qf=70 & qf=60 & qf=50
\\ \hline
\textit{MC-EffNet-1} & 87.96 & 74.79 & 68.20 & 64.75 & 63.37 & 62.16
\\
\textit{SC-EffNet\textsubscript{RGB}} & 82.13 & 80.20 & 77.04 & 77.63 & 78.00 & 78.13
\\
\textit{SC-EffNet\textsubscript{LCH}} & 80.79 & 62.96 & 59.29 & 56.00 & 54.17 & 54.46
\\
\textit{SC-EffNet\textsubscript{HSV}} & 80.38 & 67.58 & 62.08 & 59.63 & 58.54 & 57.25
\\ \hline
\end{tabular}
\end{table}
On further investigation we observe blocking artifacts in the compressed images, particularly in the compressed GAN images, which on colorspace transformations becomes very much visible as blocks with uniform intensity values, replacing original intensities and region or shape information in that area of compressed images.
This creates a difference in image content between the same image moving through the RGB pipeline and the LCH or HSV pipeline of \textit{MC-EffNet-1} for the compressed images, which might be the major cause for such degradation in model accuracy for compressed images. This difference in image content between the compressed images passed through the RGB pipeline and the LCH or HSV pipeline can be seen increasing while quality factor decreases.
To maintain the advantage of high model accuracy provided by LCH and HSV colorspace transformations and to eliminate the negative effect of blocking artifacts when dealing with compressed images, we attach an additional pre-processing block to the LCH and HSV pipeline, that employs a laplacian of gaussian filter over the images, and add the residuals to corresponding images to avoid loss of information. The advantage of passing an image through different filters before computations are well explored in many fields of image forensics \cite{cozzolino2017recasting,cozzolino2014image,fridrich2012rich}. Such pre-processing operations can very well study hidden data representations to understand natural image statistics or any deviations from these statistics. Inspired by such image forensics works, our usage of laplacian of gaussian filter to pre-process colorspace transformed images refines \textit{MC-EffNet-1} model to a more robust \textit{MC-EffNet-2} model that achieves improved robustness towards JPEG compressions. Also, with the addition of laplacian of gaussian pre-processing block in \textit{MC-EffNet-2}, the overall test accuracy improves to 89.38 percentage, an increase of 1.42 percentage points from \textit{MC-EffNet-1}. The overall architecture of our Multi-Colorspace fused EfficientNet model, \textit{MC-EffNet-2} is given in figure \ref{fig:mc-effnet2}.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.8\linewidth]{IEEEtran/figures/mc-effnet2.png}
\caption{ The overall architecture of our Multi-Colorspace fused EfficientNet model, \textit{MC-EffNet-2}}
\label{fig:mc-effnet2}
\end{figure*}
\section{Empirical study}
\label{sec:empirical}
\subsection{Dataset}
In this study, we utilize a total of 12000 images, where GAN, Graphics and Real classes contain 4000 images each. For Graphics and Real classes, images are collected from the Computer Graphics versus Photographs dataset \cite{tokuda2013computer}, which is a challenging dataset with diversity in image category, origin, quality and content. The class Graphics of the dataset include photorealistic images which are not easily manually predictable as computer graphics images and excludes graphical icons. Similarly, to incorporate heterogeneity and to avoid bias in the class GAN, we collect images from four different GAN algorithms, ProgressiveGAN \cite{karras2017progressive}, StyleGAN \cite{karras2019style}, StyleGAN2 \cite{karras2020analyzing} and StyleGAN2-ADA \cite{Karras2020ada}, considering their excellent performances to generate high quality realistic images. The entire dataset maintains heterogeneity in every class, with several different categories like outdoor and indoor scenes, objects, animals, characters, landscapes, architectures, etc. The entire dataset is split to the ratio 60:20:20 to form the train, validation and test sets, where images belonging to different categories are split proportionally across each set.
\subsection{Experimental Settings}
In our \textit{MC-EffNet-2} model, with the use of transfer learning methodology, the concatenation of feature outputs from the three colorspace pipelines (RGB, LCH and HSV) produces a feature vector of length 3840, which is then provided to a dense layer with 3 neurons and softmax activation making the total trainable parameters of the model to be 11523. The model is compiled with categorical cross-entropy as loss function, Adam optimizer with learning rate 0.001, batch size of 256 and 100 epochs. We compare the performance of our \textit{MC-EffNet-2} model with a set of baselines discussed below.
\subsubsection*{Baselines}
Since our work is the first of its kind considering the task of distinguishing natural images from photo-realistic computer-generated images including, both computer graphics and GAN images, as a three-class classification task, we perform baseline comparison for our work by implementing state-of-the-art works belonging to the categories, \textit{natural images versus computer graphics}, \textit{natural images versus GAN images}, and one another off-the-shelf deep neural network architecture, as three-class classification tasks.
\begin{itemize}[leftmargin=5mm]
\item Quan et al. \cite{quan2018distinguishing} (\textit{natural images versus computer graphics}): A CNN based work that proposes a local-to-global strategy for predicting the classification results of local patches after which the global classification results of the full-sized images is derived by majority voting. They compare their work with another patch-based CNN approach \cite{rahmouni2017distinguishing}, and four other state-of-the-art feature based works \cite{pevny2010steganalysis,ng2005physics,peng2017discrimination,zhang2011distinguishing}, where their work obtains higher accuracies and robustness. Hence we choose this work as a baseline to compare our work, by replacing the final dense layer of two neurons in their CNN model with three neurons to suit our three-class classification task.
%
\item Rezende et al. \cite{de2018exposing} (\textit{natural images versus computer graphics}): A work that uses ResNet-50 for classifying natural images and computer-generated images. They perform a result comparison between their 7 deep learning experimental settings and the 17 approaches implemented in \cite{tokuda2013computer}. Among all the 24 results, their experimental setting of transfer learning combined with a shallow classifier SVM with RBF kernel obtains highest accuracy than the other deep learning settings and feature based approaches. Hence, we choose their high accuracy experimental setting as one of the baselines for comparing our work by replacing the top layer to suit our three-class classification task.
%
\item Nataraj et al. \cite{nataraj2019detecting} (\textit{natural images versus GAN images}): A CNN based work to detect GAN images by using the co-occurrence matrix of the RGB channels. Their work obtains higher accuracy when compared against three state-of-the-art works, first based on steganalysis features \cite{fridrich2012rich,cozzolino2014image}, second, deep learning work extracting residual features \cite{cozzolino2017recasting}, and third, fine-tuning generic deep learning architecture of XceptionNet \cite{chollet2017xception} pre-trained on ImageNet. Hence, we choose this work as a baseline to compare our proposed work, by replacing their final sigmoid layer with our dense layer of 3 neurons with sofmax activation.
%
\item InceptionResNet \cite{szegedy2017inception} (Off-the-shelf deep neural network): A model that shows high classification accuracy for ImageNet classification task with almost 55.8M parameters. We attempt transfer learning methodology on InceptionResNetV2 pre-trained over ImageNet dataset by freezing all its layers during the training phase and replacing the final prediction layer of 1000 neurons with three neurons. Other hyperparameters include batch size 256, Adam optimizer with learning rate 0.01 and 100 epochs.
\end{itemize}
\subsection{Results and Discussions}
Our \textit{MC-EffNet-2} model achieves a test accuracy of 89.38 percentage, a gain of 1.42 percentage points when compared to \textit{MC-EffNet-1}, and a gain of 7.25 percentage points when compared to \textit{SC-EffNet\textsubscript{RGB}} that achieves highest accuracy among single colorspace models. Figure \ref{fig:mc_effnet2_cm} shows the confusion matrix of test result for our \textit{MC-EffNet-2} model. Class GAN obtains higher accuracy than the other two classes Graphics and Real. For class GAN, \textit{MC-EffNet-2} obtains an accuracy of 96.75 percentage, i.e., a gain of 2.87 percentage points when compared to \textit{SC-EffNet\textsubscript{HSV}} that obtains the highest accuracy of 93.88 percentage for class GAN among single colorspace models. Similarly, class Graphics achieves an accuracy of 83.75 percentage, i.e., a gain of 3.87 percentage points when compared to the highest accuracy of 79.88 percentage for class Graphics obtained for \textit{SC-EffNet\textsubscript{RGB}} among single colorspace models. The class Real seems to be most advantaged of the fusion technique which achieves 87.63 percentage accuracy, a high gain of 9.88 percentage points when compared to the highest accuracy of 77.75 percentage obtained for class Real of \textit{SC-EffNet\textsubscript{RGB}} among single colorspace models.
\begin{figure}[!t]
\centering
\includegraphics[width=0.33\textwidth]{IEEEtran/figures/conf.png}%
\caption{Confusion matrix of \textit{MC-EffNet-2}}
\label{fig:mc_effnet2_cm}
\end{figure}
Table \ref{tab:baseline_comparison} presents a comparison of model performances in terms of test accuracies of individual classes and total accuracy for our \textit{MC-EffNet-2} model against the chosen baselines. The results indicate that our model obtains the highest overall accuracy, and highest accuracies even for the individual classes. Among the baselines, the next higher accuracy shown by InceptionResnet is less than our model by 9.92 percentage points. All the models including those proposed for \textit{natural images versus computer graphics} problem, except the model proposed by Quan et al. \cite{quan2018distinguishing}, show a similar trend of higher accuracy for class GAN followed by class Real and then Graphics. Whereas, the model proposed by Quan et al. \cite {quan2018distinguishing} shows higher accuracy for the class Graphics for which the model was originally proposed, but not higher than the accuracy for class Graphics achieved by our \textit{MC-EffNet-2}.
\setlength{\tabcolsep}{5pt}
\begin{table}[!t]
\centering
\caption{Comparison of model performance accuracies in percentage \\(The highest accuracy is given in boldface)}
\label{tab:baseline_comparison}
\begin{tabular}{lcccc}
\hline
\multicolumn{1}{c}{Model} & GAN & Graphics & Real & Total Accuracy \\
\hline
Quan et al. \cite{quan2018distinguishing} & 56.08 & 72.88 & 69.79 & 66.25 \\
Rezende et al. \cite{de2018exposing} & 59.40 & 52.83 & 57.40 & 56.54 \\
Nataraj et al. \cite{nataraj2019detecting} & 76.00 & 48.25 & 57.13 & 60.46 \\
InceptionResNet \cite{szegedy2017inception} & 85.25 & 73.50 & 79.63 & 79.46 \\
\textit{MC-EffNet-2 (Our model)} & \textbf{96.75} & \textbf{83.75} & \textbf{87.63} & \textbf{89.38}\\
\hline
\end{tabular}
\end{table}
\newcommand{\protect\includegraphics[height=1.8ex,keepaspectratio]{IEEEtran/figures/Rlogo.png}}{\protect\includegraphics[height=1.8ex,keepaspectratio]{IEEEtran/figures/Rlogo.png}}
\subsubsection{Statistical significance}
In addition to the significant gains achieved by our \textit{MC-EffNet-2} model in terms of accuracy over various baselines, we conduct statistical significance test between our model and the baselines. We perform the Stuart-Maxwell\footnote{\url{http://www.john-uebersax.com/stat/mcnemar.htm\#stuart}} test
with conventional significance level, i.e., a p-value of 0.05. We obtain a p-value of 0.00187 between our \textit{MC-EffNet-2} model and the model that obtained highest accuracy among the baselines (InceptionResNet), and a p-value of 0.01036 between our \textit{MC-EffNet-2} model and the model that obtained highest accuracy among the Single Colorspace EfficientNet Networks (\textit{SC-EffNet\textsubscript{RGB}}), which provides enough evidence to conclude that the results of our \textit{MC-EffNet-2} model are statistically significant over the best baselines.
\subsubsection{Robustness against Post-processing}
Post-processing operations are quite common when uploading images to the web or social media. Therefore, apart from producing good accuracies on original images in the dataset, an effective algorithm for image forensics should also be robust over post-processing operations. We evaluate robustness of our model and baselines towards the typical post-processing operation of JPEG compression, where the models trained on original data are tested over ten different JPEG compression quality factors within the range 100 to 10, in steps of 10. The result of our robustness test is shown in figure \ref{fig:robustness}, where we can observe that even though accuracy of our \textit{MC-EffNet-2} model drops with the decrease of quality factor, it always achieves better performance than the baselines for all the quality factors. Also, we can observe that our \textit{MC-EffNet-2} model attains highly improved robustness than \textit{MC-EffNet-1} with the inclusion of pre-processing block to the colorspace transformations. As like the classification results of original images without compression here also InceptionResnet is the baseline that shows next higher results for different quality factors. The entire results over different compression quality factors indicate that our \textit{MC-EffNet-2} model achieves better robustness towards post-processing based on JPEG compression than the baselines.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{IEEEtran/figures/robust.png}
\caption{Classification accuracies for various JPEG compression quality factors}
\label{fig:robustness}
\end{figure}
\subsubsection{Generalizability}
We analyze generalizability of our \textit{MC-EffNet-2} model and the baselines by testing over three dataset combinations that are unseen during the training phase. In the first dataset, images for Real and Graphics classes are collected from PIM-Google (photographic images from Google Image Search) and PRCG (photo-realistic computer graphics images) sets of Columbia
dataset \cite{ng2005columbia}, respectively, and images for class GAN is collected from the generated images of PG\textsuperscript{2} (Pose Guided Person Generation \cite{ma2017pose}) GAN algorithm that produces high quality and realistic person images. The Columbia
dataset \cite{ng2005columbia} contains large diversity in image content. Many previous state-of-the-art works utilize this dataset for the two-class \textit{natural images versus computer graphics} problem \cite{ng2005physics,quan2018distinguishing,cui2018identifying,ni2019evaluation}. From this dataset with 800 images per class, we collect only twenty percent of the data, i.e., 160 images per class randomly (without considering the five images which were associated with incorrect labels in PIM-Google class as per the findings in \cite{quan2018distinguishing}), since it is only for testing model generalizability. To build a balanced dataset we follow the same number of 160 random images from PG\textsuperscript{2} for class GAN. Apart from PIM-Google and PRCG, Columbia
dataset also consists of another set, PIM-Personal (photographic images from the authors' personal collections), with 800 natural images which constitutes the second dataset for generalizability test, by replacing class Real of the first dataset with 160 images randomly collected from PIM-Personal class.
Another dataset combination mostly used in many previous state-of-the-art works \cite{rahmouni2017distinguishing,quan2018distinguishing,ni2019evaluation} for the \textit{natural images versus computer graphics} problem is the RAISE \cite{dang2015raise} versus Level-Design Reference Database \cite{piaskiewicz2017level}. We collected 160 images randomly from RAISE dataset that consists of high-resolution raw and uncompressed images specifically for image forensic research investigations and converted them directly to JPEG format to form class Real of the third dataset. The Level-Design Reference Database consisting of screenshots from various video games can be seen utilized in \cite{rahmouni2017distinguishing} by selecting only those screenshots which seem to be photo-realistic followed by cropping them to remove the gaming information like dialogues, text bars, etc. From these images provided by \cite{rahmouni2017distinguishing}, we collect 160 images to form class Graphics of the third dataset. To form class GAN we collect images generated by CycleGAN \cite{zhu2017unpaired} that generates high-quality realistic images, utilized in many state-of-the-art works to detect GAN images \cite{nataraj2019detecting,marra2019incremental,goebel2020detection}. For CycleGAN images, instead of choosing a single category of images from various unpaired image-to-image translation categories of objects and scenes, we collect 160 images randomly from the horses-to-zebra, zebra-to-horse, apple-to-orange and orange-to-apple categories. Table \ref{tab:generalizability} shows the results of generalizability tests for our model and baselines when tested over the three datasets, where we can clearly observe that our model outperforms all the baselines. Higher accuracies obtained for our model over the datasets on which the model was not originally trained, indicates the promising nature of our approach for tackling future challenges in computer-generated images. Also, we can observe that even though our model is trained on the category of GAN algorithms that generate whole new images, such as StyleGAN, it could perform well on CycleGAN which belongs to the attribute transfer category of GAN algorithms.
\setlength{\tabcolsep}{4.5pt}
\begin{table}[!t]
\centering
\caption{Model generalizability over different datasets in percentage (The highest accuracy is given in boldface)}
\label{tab:generalizability}
\begin{tabular}{lccc}
\hline
\multicolumn{1}{c}{Model} &
\begin{tabular}[c]{@{}c@{}}PG\textsuperscript{2}$\times$\\PRCG$\times$\\ PIM-Google\end{tabular} & \begin{tabular}[c]{@{}c@{}}PG\textsuperscript{2}$\times$\\PRCG$\times$\\ PIM-Personal\end{tabular} &
\begin{tabular}[c]{@{}c@{}}Cycle GAN$\times$\\Raise$\times$\\Level-Design\end{tabular} \\ \hline
Quan et al. \cite{quan2018distinguishing} & 54.22 & 56.27 & 60.01 \\
Rezende et al. \cite{de2018exposing} & 51.25 & 51.21 & 50.92 \\
Nataraj et al. \cite{nataraj2019detecting} & 49.17 & 53.63 & 48.75 \\
InceptionResNet \cite{szegedy2017inception} & 62.08 & 67.16 & 71.74 \\
\textit{MC-EffNet-2 (Our model)} & \textbf{81.04} & \textbf{85.21} & \textbf{84.79} \\
\hline
\end{tabular}
\end{table}
\subsubsection{Feature Visualization}
Our \textit{MC-EffNet-2} model, projects raw pixels of input images with dimension 224$\times$224$\times$3, or feature vector of size 150528 to a lower dimension feature vector of size 3840, with an intention to provide good amount of separability between the three classes so that the top classifier layer attains a high classification accuracy. To understand the separability of features projected from our model we implement a technique for dimensionality reduction called t-Distributed Stochastic Neighbor Embedding (t-SNE) \cite{van2008visualizing} that can visualize high dimensional features into a two-dimensional plane. We project both the raw image features, and the output features from \textit{MC-EffNet-2} into two-dimensional plots, with three different colors indicating three different classes. The plots are given in figure \ref{fig:tsne} where green circles represent the class GAN, pink diamonds represent the class Real and blue squares represent the class Graphics. As can be seen from the t-SNE visualizations, the raw image features are more clustered particularly towards the center of the plot, whereas, the output features from our \textit{MC-EffNet-2} model are seen to be more separated. Thus, the t-SNE visualizations prove that our \textit{MC-EffNet-2} model suits the forensic task of classifying natural images from computer-generated images including both computer graphics and GAN images by projecting the raw image pixels to a much better and separable feature space.
\begin{figure*}[!t]
\centering
\fbox{\subfloat[Raw image features]{\includegraphics[width=0.48\textwidth]{IEEEtran/figures/tsne_input.jpg}}%
\label{fig:rgb_viz}}
\hfil
\fbox{\subfloat[\textit{MC-EffNet-2} features]{\includegraphics[width=0.48\textwidth]{IEEEtran/figures/tsne_output.jpg}}%
\label{fig:mc_effnet2_viz}}
\caption{t-SNE visualizations of the feature vectors. (\tikzsymbol{fill=cadmiumgreen} indicates GAN, \tikzsymbol[rectangle]{minimum width=6pt,minimum height=6pt,blue,fill=white} indicates Graphics and, \tikzsymbol[diamond]{pastelpink,fill=mediumcandyapplered,} indicates Real images)}
\label{fig:tsne}
\end{figure*}
\subsubsection{Psychophysics Experiments}
We were also curious about how accurately humans can distinguish natural images from photo-realistic computer-generated images including computer graphics and GAN images. Hence, in this study, we also perform a manual classification test for GAN, Graphics and Real class of images by gathering information from eleven human participants on a set of 330 images randomly selected from the test data used in our study. The participants belong to the age group 22 to 40 years with normal or corrected-to-normal visual acuity and normal color vision. We utilize an annotation tool VIA \cite{dutta2016via}, that helps participants to label each image as GAN, Graphics or Real, and also to mark parts/regions in the image that explains their decisions. This way of asking human participants to provide evidence/explanation in the form of region markings allows to omit the chances of lucky guesses and moreover provides insight to what the participants perceive as a suspect and/or evidence in the images.
In VIA the participants can zoom and or zoom out images for better analysis and no other constraints like viewing distance, time for observation, etc. are imposed. Each participant is asked to label thirty images randomly chosen and assigned to them and each image is annotated only once. For a participant, the entire experiment on thirty images including marking their explanations in the images takes nearly 45 minutes.
We also compute the accuracy of our \textit{MC-EffNet-2} model over the same set of 330 images selected for psychophysics experiments, for comparison. Figure \ref{fig:psycho_acc} shows the confusion matrices of manual classification performed by human participants and that of our \textit{MC-EffNet-2} model over 330 images. For manual classification, we obtain a total accuracy of 62.42 percentage, whereas for the same set of images our \textit{MC-EffNet-2} model obtains a higher accuracy of 85.15 percentage, i.e., a very high gain of 22.73 percentage points. We can observe that the ability of humans to classify Real images is almost near to our \textit{MC-EffNet-2} model with a decrease of 5 percentage points for manual classification, but for Graphics images \textit{MC-EffNet2} highly outperforms manual classification. Similarly, in case of GAN images, manual classification accuracy is almost half of \textit{MC-EffNet-2}. The overall results indicate that the ability of humans to identify photo-realistic computer-generated images is very low and hence there is a high necessity of image forensic algorithms that can computationally aid to distinguish natural images and photo-realistic computer-generated images. Our \textit{MC-EffNet-2} model with a high classification accuracy, especially for the photo-realistic computer-generated images, is thus a better solution for the forensic task of distinguishing natural images from photo-realistic computer-generated images. We also compute the Stuart-Maxwell statistical significance test between our model and manual classifcation, where we obatin a p-value of 2.481E-06 indicating the significance of our \textit{MC-EffNet-2} model over manual classification.
\begin{figure}[!t]
\centering
\subfloat[Manual classification]{\includegraphics[width=0.33\textwidth]{IEEEtran/figures/psycho_manual.png}%
\label{fig:psycho_manual}}
\hfil
\subfloat[\textit{MC-EffNet-2}]{\includegraphics[width=0.33\textwidth]{IEEEtran/figures/psycho_mceffnet2.png}%
\label{fig:psycho_mceffnet2}}
\caption{Confusion matrices of classification performed by human participants and our \textit{MC-EffNet-2} model}
\label{fig:psycho_acc}
\end{figure}
\subsection{Understanding the Explanations}
Apart from analyzing model performance, we also investigate the behavior of our \textit{MC-EffNet-2} model so that we can trust the predictions of our deep learning model. We utilize Gradient-weighted Class Activation Mapping (Grad-CAM) \cite{selvaraju2017grad} that makes use of class-specific gradient information to make a deep learning model more transparent through visual explanations. To obtain visual explanations we employ Grad-CAM at the penultimate layer of our fully trained and saved\textit{ MC-EffNet-2} model, which constructs coarse localization maps of the salient regions in input images that are significantly important for the predictions.
Since our task is formulated as an image classification problem build using transfer learning methodology over the source networks of pre-trained EfficientNets, originally meant for object classification task, first, we try to identify whether our fused model is still looking for objects while making the decisions, or is it looking for the regions significant for classifying natural images and computer-generated ones in a forensic perspective. To address this question we take the Grad-CAM explanations of our \textit{MC-EffNet-2} model and also the base network, EfficientNetB0 which is pre-trained on the ImageNet data, on a set of images in our dataset. The Grad-CAM explanations from both the models for GAN, Graphics and Real images are shown in table \ref{tab:gradecam_comparison}. The EfficientNetB0 network which is pre-trained on the ImageNet data, as obvious, mainly highlights the objects present in images. But, we can observe that, even though our entire fused \textit{MC-EffNet-2} model is built over the base network of EfficientNet which was originally designed for an object classification task, after applying transfer learning towards our forensic task, does not primarily give importance to the objects as in the source task, rather highlights the regions that are significant in classifying them as GAN, Graphics or Real in a forensic perspective. This indicates the fitness of our \textit{MC-EffNet-2} model as a forensic solution to classify GAN, Graphics and Real images.
\setlength{\tabcolsep}{1pt}
\begin{table*}[!t]
\centering
\caption{Grad-CAM explanations from the base network EfficientNetB0 and our \textit{MC-EffNet-2} model for GAN, Graphics and Real images}
\label{tab:gradecam_comparison}
\begin{tabular}{lll||lll||lll}
\hline
\multicolumn{3}{c}{GAN} & \multicolumn{3}{c}{Graphics} & \multicolumn{3}{c}{Real}
\\ \hline
\multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Original \\ image\end{tabular}}} & \multicolumn{2}{c||}{Grad-CAM explainations} & \multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Original \\ image\end{tabular}}} & \multicolumn{2}{c||}{Grad-CAM explainations} & \multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Original \\ image\end{tabular}}} & \multicolumn{2}{c}{Grad-CAM explainations} \\ \cline{2-3} \cline{5-6} \cline{8-9}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{EfficientNetB0} & \multicolumn{1}{c||}{\textit{MC-EffNet-2}} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{EfficientNetB0} & \multicolumn{1}{c||}{\textit{MC-EffNet-2}} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{EfficientNetB0} & \multicolumn{1}{c}{\textit{MC-EffNet-2}} \\ \hline
\includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/pggan_boat_74.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/pggan_boat_74_eff_schooner.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/pggan_boat_74_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/3dcommunity_luxary.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/3dcommunity_luxary_eff_hometheatre.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/3dcommunity_luxary_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/freeFoto_01_21_55_prev.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/freeFoto_01_21_55_prev_eff_oldenglishsheepdog.png} & \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/freeFoto_01_21_55_prev_cam.jpg} \\
\includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/pggan_chair_69.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/pggan_chair_69_eff_studiocouch.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/pggan_chair_69_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/3dshop_L104287.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/3dshop_L104287_eff_hometheatre.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/3dshop_L104287_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/freeFoto_16_06_10_prev.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/freeFoto_16_06_10_prev_eff_jeep.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/effnet_vs_mceffnet2/freeFoto_16_06_10_prev_cam.jpg} \\
\hline
\end{tabular}
\end{table*}
Accordingly, we try to understand what makes our \textit{MC-EffNet-2} model label an image as GAN, Graphics or Real in context of image forensics. In this stage, we also utilize the manual explanations in the form of region markings (yellow bounding boxes\footnote{The bounding box boundaries are thickened for better visibility}) from the human participants of our psychophysics experiments, to compare whether the Grad-CAM explanations given by our model have any similarities with manual explanations. We first take the case of images for which our model and human participants provide correct predictions and analyze the Grad-CAM and manual explanations of these images. Table \ref{tab:correct_explanations} shows the examples of this case, where each set of three columns indicates the GAN, Graphics and Real classes, respectively, and for each class, we provide three sample images with their corresponding Grad-CAM and manual explanations. In the first image of a sheep among the GAN generated ones, we can observe that the image is not completely formed and there are missing regions like legs of the sheep. When we look at the Grad-CAM explanations provided by our \textit{MC-EffNet-2} model, we can see that the model captures image regions of legs and also the slight difference in fur color and texture near the neck region. For the same image, human participant has marked the leg region as an explanation for their decision of labeling that image as GAN image. Similarly in the second GAN image of a bird where the image is not completely formed at the head region and the texture of bird is misplaced towards bottom of the image, Grad-CAM captures both these regions along with the tail region of bird and also some part of the surroundings. The human participant has manually marked the region at head and misplaced texture at the bottom of the image. In next GAN image of a human face where no misformations or misplacements can be seen evidently, Grad-CAM mainly highlights texture of hair, some regions on face and also some part of the surroundings. In this case, human participant has highlighted the hair region as an explanation. Hence, in the GAN image examples, we can see that there are similarities between Grad-CAM explanations provided by our model and manual explanations.
Next, we analyze Grad-CAM and manual explanations of Graphics images in the second set of columns. For all the correct predictions of Graphics images Grad-CAM explanations are most commonly seen to highlight uneven illuminations or illuminated regions in the images. Human participants are also seen to mark such regions of uneven illuminations. In the third set of Real images, Grad-CAM explanations are commonly seen to be centered on the surroundings, focussing on complex variabilities in the background regions. In the first Real image, along with the surrounding regions a major significance can be seen given to the shadow of swan in water, by Grad-CAM as well as the human participant. Similarly, the feather regions in second image and the clouds in third image can be seen highlighted in both Grad-CAM and manual explanations. From the overall results, we can sum up that the explanations of our \textit{MC-EffNet-2} model are mostly similar to manual explanations, and also our model is able to identify more number of salient regions than human participants, to distinguish natural and computer-generated images. The explanations demonstrate the powerful nature of our \textit{MC-EffNet-2} model to take the decisions meaningfully.
\setlength{\tabcolsep}{1pt}
\begin{table*}[!t]
\centering
\caption{Explanations of images for which our \textit{MC-EffNet-2} model and human participants both produces correct predictions}
\label{tab:correct_explanations}
\begin{tabular}{ccc||ccc||ccc}
\hline
\multicolumn{3}{c}{GAN} & \multicolumn{3}{c}{Graphics} & \multicolumn{3}{c}{Real} \\
\hline
\begin{tabular}[c]{@{}c@{}}Original\\image\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Grad-CAM\\explaination\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Manual\\explanation\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Original\\image\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Grad-CAM\\explaination\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Manual\\explanation\end{tabular} &\begin{tabular}[c]{@{}c@{}}Original\\image\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Grad-CAM\\explaination\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Manual\\explanation\end{tabular}
\\ \hline \vspace{2pt}
\includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_sheep_67.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_sheep_67_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_sheep_67_anno.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dcommunity_1_5204.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dcommunity_1_5204_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dcommunity_1_5204_anno.JPG}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_01_19_30_prev.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_01_19_30_prev_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_01_19_30_prev_anno.JPG}
\\ \vspace{2pt}
\includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_bird_70.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_bird_70_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_bird_70_anno.JPG}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dcommunity_1_7334.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dcommunity_1_7334_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dcommunity_1_7334_anno.JPG}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_01_45_74_prev.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_01_45_74_prev_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_01_45_74_prev_anno.JPG}
\\ \vspace{2pt}
\includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_celebahq_72.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_celebahq_72_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/pggan_celebahq_72_anno.JPG}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dshop_L104131.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dshop_L104131_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/3dshop_L104131_anno.JPG}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_04_23_75_prev.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_04_23_75_prev_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/correct_exp/freeFoto_04_23_75_prev_anno.jpg}
\\
\hline
\end{tabular}
\end{table*}
We also try to get insights on wrong predictions by examining images for which the human participants provide correct predictions but our \textit{MC-EffNet-2} model produces wrong predictions. A few examples of this case are given in table \ref{tab:wrong_explanations}. The first image (a) is GAN which is manually predicted as GAN itself but \textit{MC-EffNet-2} misclassifies it as a Real. The Grad-CAM explanation from \textit{MC-EffNet-2} shows that the decision is produced from the head region, rock at bottom of the image and the surroundings but not mainly from the misformed regions like the leg region. Similarly in (d), Graphics is misclassified as Real taking into account the region of mountains and some parts of the background rather than considering the regions of illuminations in the image as usually seen in case of graphics images. For Graphics image (c) which is manually predicted as a Graphics itself, \textit{MC-EffNet-2} misclassifies it as a GAN. The Grad-CAM explanation shows that it produces decision from only the object region of one of the photographs in the image and not the regions of illuminations as usually seen in case of Graphics images. Similarly, in Real image (e), the object regions of sheeps are highlighted by \textit{MC-EffNet-2} than the meaningful explanations like shadows or surroundings as usually seen for the Real images, which might be the reason for its misclassification as GAN. Whereas, in the images (b,f) \textit{MC-EffNet-2} highlights the uneven illuminations which might be the reason for its misclassification as Graphics image.
\setlength{\tabcolsep}{1pt}
\begin{table*}[!t]
\centering
\caption{Explanations of images for which human participants provide correct predictions but \textit{MC-EffNet-2} produces wrong predictions}
\label{tab:wrong_explanations}
\begin{tabular}{ccc||ccc||ccc}
\hline
\multicolumn{3}{c}{GAN} & \multicolumn{3}{c}{Graphics} & \multicolumn{3}{c}{Real} \\
\hline
\begin{tabular}[c]{@{}c@{}}Original\\image\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Grad-CAM\\explaination\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Manual\\explanation\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Original\\image\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Grad-CAM\\explaination\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Manual\\explanation\end{tabular} &\begin{tabular}[c]{@{}c@{}}Original\\image\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Grad-CAM\\explaination\end{tabular}
&\begin{tabular}[c]{@{}c@{}}Manual\\explanation\end{tabular}
\\ \hline \vspace{2pt}
\includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/pggan_bird_61.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/pggan_bird_61_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/pggan_bird_61_anno.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/3dshop_L103998.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/3dshop_L103998_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/3dshop_L103998_anno.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/freeFoto_01_54_13_prev.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/freeFoto_01_54_13_prev_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/freeFoto_01_54_13_prev_anno.JPG}
\\
\multicolumn{3}{c||}{(a) \textit{MC-EffNet-2} miclassifies GAN as Real}
& \multicolumn{3}{c||}{(c) \textit{MC-EffNet-2} miclassifies Graphics as GAN}
& \multicolumn{3}{c}{(e) \textit{MC-EffNet-2} miclassifies Real as GAN}
\\ \hline
\includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/stylegan_car_89.png}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/stylegan_car_89_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/stylegan_car_89_anno.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/3dcommunity_Nexus.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/3dcommunity_Nexus_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/3dcommunity_Nexus_anno.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/freeFoto_01_14_2_prev.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/freeFoto_01_14_2_prev_cam.jpg}
& \includegraphics[width=19mm, height=19mm, valign=c]{IEEEtran/figures/wrong_exp/freeFoto_01_14_2_prev_anno.JPG}
\\
\multicolumn{3}{c||}{(b) \textit{MC-EffNet-2} miclassifies GAN as Graphics}
& \multicolumn{3}{c||}{(d) \textit{MC-EffNet-2} miclassifies Graphics as Real}
& \multicolumn{3}{c}{(f) \textit{MC-EffNet-2} miclassifies Real as Graphics}
\\ \hline
\end{tabular}
\end{table*}
\section{Conclusion}
\label{sec:conclusion}
In this work, we proposed deep learning based Multi-Colorspace fused EfficientNet model to classify natural images and photo-realistic computer-generated images including both computer graphics and GAN images, as against the state-of-the-art works that have always discussed either \textit{natural images versus computer graphics} or \textit{natural images versus GAN images} problem, at a time. We compared our model with state-of-the-art methods where our model outperforms all the baselines in terms of performance accuracy, robustness against typical post-processing operation of JPEG compression, and generalizability towards other datasets, which demonstrates the utility of our model in real-world forensic applications. We also conducted psychophysics experiments to realize how capable humans are in classifying natural images and photo-realistic computer-generated images, where, the results of manual classification accuracy was lower than our model accuracy, particularly in classifying the photo-realistic computer-generated images, indicating the necessity and usefulness of our computational model for the task. We also analyzed the behavior of our model by visualizing the salient regions in the images that are responsible for classification decisions. We compared these visual explanations of our model with the explanations manually labeled by the human participants for their correct predictions, where we could observe similarities between the explanations of our model and manual explanations, indicating that our model takes decisions meaningfully. To our best knowledge, such a comparison of visual explanations to understand whether our model behaves alike human explanations to produce the decisions meaningfuly is a new attempt that might even be useful in other digital image forensics or multimedia security tasks. To aid future research, these manual classifications along with the manually labelled visual explanations, and other relevant materials, including the source code will be made publicly available at \url{https://github.com/manjaryp/GANvsGraphicsvsReal} and \url{https://dcs.uoc.ac.in/cida/projects/dif/mceffnet.html} along with the publication. In the future, we are considering to extend our work to identify other forensic attacks like the recaptured images. We are also considering to extend our work to classify natural and computer generated videos.
\section*{Acknowledgment}
The authors would like to thank Prof. Dr. Anderson Rocha from University of Campinas for sharing their dataset in \cite{tokuda2013computer}, the authors of \cite{quan2018distinguishing,de2018exposing,ni2019evaluation} for making thier source codes publicly available, and the authors of \cite{ng2005columbia,rahmouni2017distinguishing} for making thier datasets publicly available,. The authors would also like to thank the Master students (CS2019-21) and staff at the Department of Computer Science, University of Calicut for their involvement and co-operation to conduct the psychophysics experiments in this work.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
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{"url":"http:\/\/www.math-only-math.com\/formulas-of-profit-and-loss.html","text":"# Formulas of Profit and Loss\n\nFormulas of profit and loss are given below.\n\nFormulas of profit and loss are given below.\n\nWhen the Selling Price (SP) is greater than Cost Price (CP) the man makes a Profit or Gain.\n\nSelling Price (SP) > Cost Price (CP) \u2192 Profit or Gain\n\nProfit = Selling Price (SP) \u2013 Cost Price (CP)\n\nIf profit % is required to find then,\n\nProfit % = (Profit\/Cost Price) \u00d7 100\n\nWhen the Selling Price (SP) is less than Cost Price (CP) the man suffers a Loss.\n\nSelling Price (SP) < Cost Price (CP) \u2192 Loss\n\nLoss = Cost Price (CP) - Selling Price (SP)\n\nIf loss % is required to find then,\n\nLoss % = (Loss\/CP) \u00d7 100\n\nDepending on the formulas of profit and loss For let us consider some examples:\n\n1. Mr. Smith bought a book for $85 and sold it for sold it for$ 115. Find his profit or loss percent.\n\nSolution:\n\nCost Price (CP) = $85; Selling Price (SP) =$ 115\n\nSince SP > CP,\n\nTherefore, Mr. Smith makes a profit.\n\nProfit = Selling Price (SP) \u2013 Cost Price (CP)\n\n= 115 \u2013 85\n\n= $30 Therefore, profit % = (Profit\/Cost Price) \u00d7 100 = (30\/85) x 100 = 35.29 % Answers: 35.29 % 2. Mr. Brown bought a TV for$ 5800 and sold it for sold it for $7000. Find his profit or loss percent. Solution: Cost Price (CP) =$ 5800;\n\nSelling Price (SP) = $7000 Since SP > CP, Therefore, Mr. Brown makes a profit. Profit = Selling Price (SP) \u2013 Cost Price (CP) = 7000 \u2013 5800 =$ 1200\n\nTherefore, profit % = (Profit\/Cost Price) \u00d7 100\n\n= (1200\/5800) x 100\n\n= 20.69 %\n\nAnswers: 20.69 %\n\n3. Robert bought pencils for $150.As they were of bad quality, he had to sell them for$ 127. Find his loss or gain percent.\n\nSolution:\n\nCost Price (CP) = $150, Selling Price (SP) =$ 127\n\nSince SP < CP,\n\nTherefore, Robert suffers a loss.\n\nLoss = Cost Price (CP) \u2013 Selling Price (SP)\n\n= 150 \u2013 127\n\n= $23 Therefore, loss % = (Loss\/CP) \u00d7 100 = (23\/150) \u00d7 100 = 15.33% Answers: 15.33 % 4. Jack bought a pairs of shirt for$ 125 and sold them for $108. Find his loss or gain percent. Solution: Cost Price (CP) =$ 125,\n\nSelling Price (SP) = $108 Since SP < CP, Therefore, Jack suffers a loss. Loss = Cost Price (CP) \u2013 Selling Price (SP) = 125 \u2013 108 =$ 17\n\nTherefore, loss % = (Loss\/CP) \u00d7 100\n\n= (17\/125) \u00d7 100\n\n= 13.6 %\n\nAnswers: 13.6 %\n\nFormulas of Profit and Loss.\n\nTo find Cost Price or Selling Price when Profit or Loss is given.\n\nWorksheet on Profit and Loss.\n\n### New! Comments\n\nHave your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.\n\nDidn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.","date":"2017-06-24 07:10:36","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\": 0.622744083404541, \"perplexity\": 7088.019478234617}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-26\/segments\/1498128320227.27\/warc\/CC-MAIN-20170624064634-20170624084634-00509.warc.gz\"}"} | null | null |
Seek an internship with a difference
08/09/2021 | 3 mins
How does helping out your community while gaining skills and experience for your future career sound? Pretty great, right? How about also scoring credit for your time? It's a no-brainer.
The UWA McCusker Centre for Citizenship offers hundreds of internship opportunities every year to UWA students looking to lend a hand and develop key employability skills at the same time.
Internships focus on creating caring, connected and socially engaged students who enjoy actively contributing to the world around them.
You could be placed with one of 300+ partners, spanning not-for-profit, community and government organisations, and undertake a structured, hands-on internship for credit.
Over 100 hours, you could gain insight into areas like social justice, community development, environmental conservation, the provision of care, human rights, advocacy, community capacity building and activism.
What can I take away from an internship?
Professional experience in a community, not-for-profit or government environment
Valuable transferable skills
A deeper awareness of critical social issues challenging society.
A strong understanding of professional responsibility and active citizenship
Connections and professional networks within the community
An understanding of how to apply your knowledge and skills post-degree
What do other students say?
"I would strongly encourage other students to do a McCusker Centre for Citizenship internship. You get the opportunity to meet new people and establish a professional network. What you learn during an internship are a variety of transferable practical skills, which are invaluable for your future career."
Nicole (pictured), Political Science and Marketing student and intern at the Department of Transport
"I would 100% recommend it to other students. The experience in the workplace and ability to see your degree put into action is such a worthwhile experience. Being able to make worthwhile change while still studying is empowering."
Gabby, Biomedical Science student and intern at the Salvation Army
What impact could I make?
Master of Professional Engineering student Alexandra Lyons (pictured) conducted an environmental sustainability audit for Perth Festival, providing estimations of the Festival's carbon footprint and recommendations on how they could reduce emissions associated with the Festival. She also produced an emissions calculator to help future Perth Festival programmers understand the carbon footprint of various decisions and operation choices.
"By understanding where the bulk of our emissions are coming from, we can make informed actions to start reducing them," says Alexandra.
Software Engineering student Alex Arnold (pictured) developed a smartphone app for the Men's Resource Centre in Albany, aimed at supporting men's physical and mental health. Alex continued to work on the project voluntarily for the next 18 months after he finished his internship. The app is now widely available on the Apple App Store and Google Play, and was launched by WA Health Minister, the Honourable Roger Cook.
"The internship is one of the best learning experiences you'll ever have while at university. It's not like anything else you'll do," says Alex.
Ready to make a difference?
You can undertake an internship by enrolling in the UWA McCusker Centre for Citizenship Internship unit. Internships are very popular, so your first step is to check with your Student Advising Office for more information. Interns are selected on a competitive basis and then matched to suitable roles. To find out more and to check your eligibility, visit the Internships page on the McCusker Centre for Citizenship website. You can also hear from other interns and read their stories.
Updated:04/07/2022 12:45 PM (this date excludes nested assets)
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Q: Qt5 QTreeView editable with virtual keyboard Working on a touch-screen kiosk system that has a virtual keyboard widget. In all dialogs, the virtual keyboard is the only method of non-touch input. I am now working on a QTreeView for displaying file names using QFileSystemModel, and I have added a column called New Name (all other columns other than Name are hidden). I have overridden createEditor() and destroyEditor() for the delegate I assigned to the QTreeView, and have overridden the data() and setData() for the model. The createEditor() override returns a pointer to a local QLineEdit object so that I can control the results of the edit (plugging them in to my model's data object for that new column).
With my physical keyboard I am able to make changes to the New Name cell for the given row, but I cannot figure out how to type on my virtual keyboard without losing focus and calling destroyEditor() on the item I'm editing. Any ideas as to how this can be done? I've dug into the Qt code, but no luck so far.
A: The virtual keyboard must not have any focusable widgets. It should be focus-neutral. It won't be stealing the focus, then. This answer has a working example that doesn't steal focus and synthesizes keypress events that get posted to the focused widget.
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Thirsty Classics: "Vampyros Lesbos" Invites Us Into the Male Gaze
By Drew Gregory
Thirsty Classics is a nine-week miniseries celebrating lesbian cinema from before 1980. We often talk about these films like homework or mere stepping stones, but Drew is here to share how they can be fun… if you're horny enough. This week: Jesús Franco's Vampyros Lesbos starring Ewa Strömberg and Soledad Miranda.
Every year my high school peer counseling program hosted "gender meetings." All the boys would sit in the middle of a circle and the girls got to ask questions. The next week, we switched.
One year a girl asked us why men watched lesbian porn. The room broke out in uncomfortable giggles. "Men are imaginative creatures," one of the boys said. "So when we watch we imagine we're in the middle of them." More giggles. "What? It's the truth."
At the time, I was completely unaware of my transness. I was just a 16-year-old boy who exclusively watched lesbian porn. "What an idiot," I thought. "We imagine ourselves as one of the women. Obviously."
I'm the one on the left
Whether it's porn or Blue is the Warmest Color or Jesús Franco's exploitation classic Vampyros Lesbos, a lot of the lesbian sex that has been recorded on film and video was done so for cis straight men. And if we're going to trust the wisdom of that teenage boy, the point is for men to imagine themselves between the two women. That is how it's staged, that is how it's shot, that is how it's cast.
But what does that mean? And what does it mean if, I, a lesbian, still enjoy it? As a trans woman, it would be validating to say I only watch gay male porn because lesbian porn is too fake. Or, even, that The L Word is boring because they aren't real lesbians. I can say neither of these things.
Vampyros Lesbos is not my fantasy. But it is a fantasy. And, while this fills me with some guilt, I do, in fact, enjoy it. Some of it, anyway.
The opening credits combine ship imagery, the very 60s psychedelic score, and our first glimpse of Countess Nadine Carody. She's on her back, reaching towards the camera, her red scarf billows in the manufactured wind. Then, we watch a striptease.
A woman with a bob stands completely naked. She is the cliché male fantasy: big boobs, thin, surprisingly hairless for 1971, blank facial expression, completely motionless. You know, the dream. (insert several eye roll emojis)
The countess enters. She's wearing a black bra and underwear set visible underneath her black lace nighty. She has another long red scarf around her neck and she dances into frame holding a candelabra. She approaches a mirror. She dances with herself, she kisses herself.
Countess Carody, played by frequent Franco collaborator Soledad Miranda is also thin. She also has perfectly symmetrical boobs. And she also has a blank expression. But she is not a doll. Throughout this dance, and throughout the whole film, she will lead with her sexuality, aggressively seeking what, and who, she wants.
The true pleasure of this scene, at least for me, are the cutaways to estate lawyer Linda Westinghouse (Ewa Strömberg). She is at the show with her boyfriend, Omar, but during this five-minute sequence he has ceased to exist. She watches the dance, getting more and more flustered. She watches as the countess removes her lingerie and places it on the real life mannequin. Carody touches her. Kisses her. All the while Linda is biting her lip, her blue eyes sparkling brighter than her matching blue eye shadow.
"You're very excited. What's up?" Omar asks. "Oh nothing," she replies with a smile.
Period sex
It turns out Linda has seen the Countess Carody before. In her dreams. We see one of these dreams: island scenery, a moth in a net, a kite, a scorpion, blood dripping down a window, Carody's freckled face. She calls to her. Linda. Linda. Lindaaa.
Linda tells her therapist, "The strange thing is that the dream arouses me. More than once I've reached orgasm." He tells her she's just not having good enough sex with Omar. She should take a male lover. He says this as he hovers over her body.
Luckily for Linda, she is called to the Kadidados Islands, to meet with Countess Carody and settle Count Dracula's estate. When she first arrives she spots blood dripping down a window just like in her dream. But it's soon forgotten when she spies Carody lounging in a white bathing suit and huge black glasses. "You must be tired," the countess says. "Would you like a swim before we get down to business?"
When a lawyer is invited to swim with her client who is definitely a vampire, the problem of clothing may arise. The lawyer, of course, did not plan to swim. Fortunately, skinny dipping is always an option. Cue 60s music. Cue Linda frolicking into the water naked.
The movie is filled with moments like this. The score maintains a certain level of absurdity, as does the constant zooms, and the bright red blood, and the costumes, and the dialogue. Linda's business suit looks like it's mid-aughts Juicy Couture and the characters regularly say things like, "I lost myself completely in her. She was me, and I was her."
Whether or not the movie turns you on matters less if you enjoy semi-accidental camp. There are plenty of moments to laugh with and at the movie, especially once Linda gets taken to Vampire Hospital, or whatever we want to call the place where women who have gone mad with their love of the countess are housed and studied.
A very watery Scorpio
But before that we have a sex scene. Linda has passed out due to Carody's spiked wine and she wakes up still wearing her suit. The countess appears in the doorway, blood dripping down her mouth. Her lips are soft and wet and her eyes drip with lust. They meet in the middle of the room. Carody's hand moves its way down Linda's body, the back of her nails pressing into her skin. She removes Linda's clothes. She kisses her sunburnt shoulder.
They lie on the floor. Carody maintains control, Linda accepting everything wide-eyed. Carody's straight brown hair covers the right side of Linda's face as she kisses down the left. Another zoom. And she bites. She lifts her head up and the blood drips out the side of her mouth like spit or cum. She takes another bite. Linda moans.
"A hickey from Kenickie is like a hallmark card."
If you lived exclusively on the internet, it would seem that straight men and queer women are into very different people. It would appear straight men are interested in fucking Scarlett Johansson, while queer women are more interested in having Rachel Weisz run us over with a truck.
One of the best parts of being queer is an expanded idea of what's attractive and what's sexy. But no one, queer or not, is free from the ways our larger society influences our judgments. Let's be real, Rachel Weisz still fits just about every "normative" box. And were ScarJo to walk into the Cubby Hole, someone would probably tease her for being the worst before happily taking her home.
Attraction should be examined, but it's also difficult to resist or force. Vampire stories have always been about the temptation of sexuality. An innocent protagonist drawn into a world of lust, one they can either embrace or defeat. It's why it fits so well with queerness. Linda is drawn to Carody and must decide whether or not she will transgress the laws of heterosexuality. I'm drawn to Carody and must decide whether or not I'm going to let cis straight men continue to decide what gets me off.
Heterosexuality
Maybe the issue is less about the individuals themselves and more about what they're doing. After all, queer women do not look any one specific way. Hell, I have no idea if Soledad Miranda was queer. All I know is she had a husband and that doesn't mean anything. She was at least queer enough (or talented enough) to believably portray lust towards other women.
This goes back to my high school classmate. We were likely watching the same content: YouTube searches of "girls kissing" before graduating to the top videos on free porn sites. Yet, he watched and imagined himself in between. I watched and imagined myself as one of the women.
Here I cast myself as Linda. I am not a busty blonde with blue eyes. But I would like to be pursued by a dangerous countess. The endless shots of boobs are fine, but the shots of Carody's face, determined, dominating, lips and eyes always wet, are when the film is sexiest to me. Linda affects Carody unlike any other, and by the time they have sex a second time, the dynamic has changed. This time Linda does the topping and Carody is at her mercy. This is maybe even hotter than the first scene, because I am so deeply desired. I mean, Linda is so deeply desired.
The actual money shot
Linda ultimately decides to murder Carody and return to heterosexuality. This film was, of course, made by a man. Jesús Franco might enjoy the idea of lesbianism, but it must never intrude on his longterm hetero plans.
It's a relief that since 1971, more and more lesbian films have been directed by actual lesbians. And not just indie dramedies. Every genre of film has had more women and more queer women making work, including softcore and hardcore porn.
The ways we actually look, the ways we actually have sex, our diverse queer community, is appearing on screen. Instead of finding a punctum within the male gaze, we get to embrace our own gaze. Our own gazes.
But part of that gaze, an important part, is sometimes it's not that deep. Sometimes, we too, just want to see tits.
Related:thirsty classics
Drew Gregory
Drew is an LA-based writer, filmmaker, and theatremaker. Her writing can be found at Bright Wall/Dark Room, Cosmopolitan UK, Thrillist, I Heart Female Directors, and, of course, Autostraddle. She is currently working on a million film and TV projects mostly about trans lesbians. Find her on Twitter and Instagram @draw_gregory.
Drew has written 222 articles for us.
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I'm not personally interested in 70s lesbian vampire movies, but I'd happily read anything Drew writes; it's all so good.
RMarlene
That was indeed the money shot! I wish I could wear gloss like that without looking like I just drooled all over my face.
Ok after having this on my mind for a few more minutes, I keep thinking about Lair of the White Worm. I know it's a few years later than this series is running, and the ending isn't any better, but when Amanda Donohoe rises up out of the basket with that little curl on her forehead I mean DAMN. I was down to join the snake party right there.
I totally get the confusion over how men watch lesbian porn. I mean, if you want a male stand-in, why not watch straight porn? Two girls and a guy? It just seems… weird to have two people having sex that's performed for a non-participatory third perspective be the standard format, especially since many of the performers aren't really into women, making it seem extra fake and…. weird. In order to derive undiminished pleasure from that you'd have to be some kind of hyper privileged asshole who assumes all women exist to cater to your desires and….. ohhhhhhhhh I get it.
I've never seen Lair of the White Worm! I will absolutely check it out.
And yeahhhhh I think you figured out the porn question! lol
Oh god yes watch that movie! It's based on the story by Stoker, and has an early version of Hugh Grant being himself. Favorite line is "Scotch is a drink" lol my dad's family was Scottish and he always says he's Scotch. Anyway yeah it's a cheesy 80's B-movie with some seriously trippy scenes and a decent dose of humor and a murderous strap-on sacrifice and a ferret tucked up a kilt. I had it on VHS back in the day and tried to get my friends to watch it at parties but most people balked at having to see Hugh Grant being himself.
This sounds like something I'd very much enjoy. Thank you!
Please let me know if you ever get around to watching this, I'll be very curious to hear what you think.
This trashy gem is one of my all time favoritest movies. I saw it back when it actually ran at the cinema.
Fun fact- the kilt wearing friend-of-Hugh is the 12th doctor, Peter Capaldi in his splendid youth.
Wiki says: "Russell originally wanted to cast Tilda Swinton, but she turned down the role, and Amanda Donohoe was cast instead" My head just exploded imagining Tilda Swinton in the role.
Whoooooaaaa I never heard that! I woulda LOVED Tilda in this!
I think it's also some straight boy nonsense about "more is better", whether that's beer or women or….
Okay, so 1) this article was great and 2) it reminded me of my enduring quest to figure out what the name of the first porny thing I ever saw: a lesbian vampire movie that came on late night tv at a sleepover in middle school that everyone else ignored except me, and of course I just found it fascinating (so strange! so enthralling! How can anyone keep their eyes off the screen? I am clearly very straight and chill and normal but this is just so interesting for no particular reason! Anyway I came out ten years later).
It was really stylized, like maybe 70s art house? A lot of weird lighting, red and black and yellow tones with heavy dark shadows. Set in Europe somewhere, maybe multiple places? And it was in a foreign language (not Spanish), but set in (I think) Spain? With English subtitles. The lady vampires where more similar looking, I think everyone had shiny long black hair? Maybe it WAS Vampyros Lesbos and i'm just misremembering it in a pubescent haze colored by my love for doppelbanging?
What was this weird sexy lesbian vampire porn movie? Help me solve this mysteryyyy!!!
This is so tricky! Nothing immediately comes to mind, specifically because of the shiny long black hair! So many of the lesbian vampires are blondes. lol
Jean Rollin made so many lesbian vampire films I feel like maybe it was one of his?
Or Daughters of Darkness which has two blonde vampires that look similar if the hair is flexible.
So I'm scrolling through old threads but maybe
The Shiver of the Vampires or Blood and Roses?
amidola
"I am not a busty blonde with blue eyes.But I would like to be pursued by a dangerous countess."
As a busty blonde, I just might make the latter part of this statement my tinder tagline.
You know, I'm a total wuz. I couldn't make it past the second season of True Blood because it was just too damn bloody. I'm not going to watch this movie, despite the hot countess because of the very fake blood. It's a bit embarrassing,actually.
Still, I'm very grateful for the archetype of the lesbian vampire. Because since 1936, they've managed to unapologetically portray female desire and be absolutely drool worthy, while being independent and bad ass, when not a lot of women were allowed to be.
I raise my boring chai latte to Lesbian vampires, the heroes we needed when it was very dark, indeed.
And I also raise it to you, Drew, because I have always wondered and heard about this movie, but never dared to watch it.
I'm gonna hide this here so @amidola doesn't get freaked out by the blood – Don't look! But this is an old promo I did for a Halloween show. I was the terrifying vampire mistress with the huge heels and knife. I was dangerous! My seductive maidens were busy elsewhere at the time though.
OMG! You look amazing! Gotta keep track of your seductive maidens though.
Thanks Drew!
Oh, I knew where those maidens were…. just sitting around, just sitting around.
Why, here's two of them now!
Well, thankfully, this wasn't disturbing at all.😄
Ha yay! I'm gonna recommend Lair of the White Worm to you as well, as I think the gore is…. minimal? And amusing? And there are snakes/ladders.
Yesss Dracula's Daughter! There are so many classic movies like that one that have these really great gay moments even if the films as a whole didn't warrant a full write up.
And hey we all have our gore tolerance! Glad I could give you a window into this movie without you having to torture yourself. haha | {
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} | 7,600 |
Fitzpatrick, Patrick Vincent
McCabe, Desmond
Fitzpatrick, Patrick Vincent (1792–1865), political and financial manager to Daniel O'Connell (qv), was born 19 July 1792 at 2 Ormond Quay, Dublin, eldest son of Hugh Fitzpatrick (qv), printer, engraver, and publisher, and his wife Jane Fitzpatrick (d. 1814). Boarding first at the Drogheda classical academy under the Rev. James Keelan, he next entered the catholic seminary, St Patrick's college, Maynooth, as a lay scholar for two years, before educating himself at home; he also studied briefly in TCD c.1811/12. His affluent family circumstances altered drastically in 1813 on the state prosecution and conviction of his father for the publication of seditious libel. Shortly after his father's release from prison in October 1814, his mother died. Patrick continued the family book trade for several years. Then, radicalised in political sentiment, he appears to have qualified in law c.1820 (probably in London), and was practising as a barrister on the home circuit by the mid 1820s, when he joined the Catholic Association. In 1828 he published a discursive twelve-part poem, Thaumaturgis; though he wrote poetry throughout his life, his verse was never collected.
These interests were decisively overtaken from the later 1820s by his intimate involvement in politics. On 22 June 1828, taking up the casual suggestion of Sir David Roose (d. 1836), he excitedly urged Daniel O'Connell to contest the seat for Co. Clare in the ensuing general election, promising to raise the campaign expenses. O'Connell yielded to his persuasion by 24 June, and within ten days Fitzpatrick had accumulated £2,800 towards the election fund. During early 1829 he came up with the notion of a national Irish testimonial to O'Connell when it became clear that there was unbending establishment resistance to the promotion of O'Connell in the legal profession. He took day-to-day charge of the project, impressing O'Connell with his zeal, good humour, and organisational discipline. O'Connell's personal and political finances were in a parlous state for much of his life; the transformation necessary for his continued full-time representation of the Irish nationalist cause was contrived, from the summer of 1830, when Fitzpatrick brought into being an annual O'Connell tribute. His discreet management of clerical contacts thawed initial suspicions in the diocese of Waterford and elsewhere, making the Sunday collection a considerable success. By late 1830 he had made O'Connell aware that his own legal business was suffering, and he was soon given formal employment as manager of the tribute, on a percentage commission. By mid 1831 he supplemented his exertions as fund manager with miscellaneous assistance in gathering political information and rumour. In July and August that year he was conveying O'Connell's views on the reform bill to interested parties in Dublin. By 1832 he was warmly acknowledged by O'Connell as indispensable confidant and adviser.
Close to one-quarter of all surviving correspondence to and from O'Connell during 1833–45 was received by or issued from Fitzpatrick. His duties were extraordinarily various, ranging from rummaging about in O'Connell's house in Merrion Square, Dublin, for deeds or books, to dispatching newspapers and journals to London or Derrynane when required, superintending the O'Connell trust fund established with the annual tribute, and the constant communication of sagacious advice and commentary on the Irish political situation, delivered with nicely balanced respect and sardonic humour. O'Connell made it his habit to unburden his mind almost daily to Fitzpatrick, making him privy to the fluctuations in his political thinking, in confidence that his views would be sensitively and accurately interpreted for others in Dublin. Because of his confidential relationship with O'Connell, he was seen as a source of special information by influential supporters whose goodwill needed to be cultivated and maintained. On the rare occasions when he could not conceal low spirits, he transmitted that mood to O'Connell – testimony to how much O'Connell depended for emotional support on Fitzpatrick's sanguine temperament, particularly after the death of his wife in 1836. Throughout numerous crises Fitzpatrick managed to disentangle O'Connell's fraught finances and reassure him about the future.
Though becoming partner, with Morgan O'Connell (qv), in a Cork brewery during the early 1830s, Fitzpatrick was, for nearly twenty years, totally occupied as self-sacrificing lieutenant in the campaigns of O'Connell. Remigius Sheehan (qv), a tenacious political opponent, correctly depicted him as the foundation on which Irish agitation was based. As O'Connell sank into ill health in late 1846, he secured Fitzpatrick a sinecure as assistant registrar of deeds. Having resolved O'Connell's financial affairs in January 1847, Fitzpatrick used his clerical influences to secure the release of the Rev. John Miley (qv) to accompany O'Connell on his pilgrimage to Rome. O'Connell finally parted from Fitzpatrick at Hastings in March, two months before his death.
Fitzpatrick's retirement from politics was comfortable and reportedly convivial. He died unmarried at 26/7 Eccles St., Dublin on 25 September 1865, and was buried in Glasnevin cemetery. His papers were inherited by his nephew, W. J. Fitzpatrick (qv).
Ryrie Bjolla Padraig (P. V. Fitzpatrick), Thaurnaturgus (1828); Freeman's Journal, 26 Sept. 1865; W. J. Fitzpatrick (ed.), Correspondence of Daniel O'Connell, the Liberator (2 vols, 1888); id., History of the Dublin catholic cemeteries (1900); O'Donoghue; R. J. O'Duffy, Historic graves in Glasnevin cemetery (1915); IBL, vii (1915); O'Connell, Corr., iv–viii (1978–80); Oliver MacDonagh, O'Connell: the life of Daniel O'Connell, 1775–1847 (1991)
Forename: Patrick, Vincent
Surname: Fitzpatrick
Career: Politics, Business and Finance
Born 19 July 1792 in Co. Dublin
Died 25 September 1865 in Co. Dublin | {
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} | 2,487 |
\section{INTRODUCTION}
For a generic description of the very early universe governed by high energy physics
where effects of quantum gravity can occur,
Weinberg \cite{PRD08Weinberg} suggested the most general corrections with four spacetime derivatives,
$\Delta L$, \bea
&& \Delta L = \sqrt{-g} [ f_1(\phi) (\phi^{,c} \phi_{,c})^2
+ f_2(\phi) \phi^{,c} \phi_{,c} \Box \phi
+ f_3(\phi)(\Box\phi)^2
- f_4(\phi) R^{ab} \phi_{,a} \phi_{,b}
{}
\nonumber\\
&& \quad\;\, {} - f_5(\phi) R \phi^{,c} \phi_{,c} - f_6(\phi)R\Box\phi
+ f_7(\phi) R^2
+ f_8(\phi)R^{ab}R_{ab}
{}
\nonumber\\
&& \quad\;\, {}
+ f_9(\phi) C^{abcd}C_{abcd} ]
+ f_{10}(\phi) \eta^{abcd} C_{ab}^{\;\;\; ef} C_{cdef} \label{Weinberg Lagrangian} . \eea
Here, $R_{ab}$ is the Ricci tensor, $R$ is the Ricci scalar,
$\eta^{abcd}$ is a totally antisymmetric Levi-Civita tensor density,
and $C_{abcd}$ is the Weyl tensor.
This $\Delta L$ is added to
the standard Lagrangian of the minimally coupled scalar field (MSF), $L_{0}$,
describing the universe filled with scalar field \cite{85Mukhanov, 05Mukhanov, book08Weinberg},
given by
\bea
&& L_{0} = \sqrt{-g} \Big[ {1 \over 16\pi G} R -{1 \over 2} \phi^{,c} \phi_{,c} -V(\phi) \Big] \label{L0} ,
\eea
where
$G$ is Newton's constant and
$V(\phi)$ is a potential as a function of single scalar field $\phi$.
These correction terms with just four spacetime derivatives have been previously discussed by Elizalde {\em et al}. \cite{95Elizalde et al} in a different context.
$R^2$ or $R^{ab}R_{ab}$ terms were studied
by DeWitt (1967) searching for quantum theory of gravity
and by Birrell and Davis studying on quantum fields in curved space \cite{DeWitt B and D}.
In earlier times, Weyl, Pauli, and Eddington suggested a simpler version of the additional term(s) \cite{WPE}. Especially, the term proportional to $R^2 $, in a pure gravity theory without scalar field, has been discussed by Starobinsky \cite{80 Starobinsky}, a special example of general $f(R)$ gravity \cite{10 Sotiriou and Faraoni, Nojiri:2017ncd, Nojiri:2010wj} which substitutes the standard Einstein-Hilbert action. An inflation model based on Starobinsky gravity as well as non-minimally coupled scalar field theory \cite{13 Kallosh and Linde,11Linde et al, 09Bezrukov and Shaposhnikov, 98 H and N nonMSF} well explains the observational results pictured in the $n_s$(spectral index)-r(tensor-to-scalar ratio) plane, and these are preferred among other inflationary models by Planck Collaboration \cite{Planck2015} who measures the cosmic microwave background (CMB) anisotropy. In addition to inflation, dark energy related scenarios are well accommodated by theories of modified gravity and scalar field \cite{01 H and N, 15Vagnozzi et al, A and T, 06Copeland et al}. Weinberg in his 2008 paper derived the tensor mode equation for only $f_{10}$ correction \cite{PRD08Weinberg}.
Noh and Hwang considered $f_7$ and $f_8$ as constants without other correction terms and aimed at the explanation of cosmological gravitational wave \cite{97 N and H}.
Here, we mainly generalize this theory such that $f_7$ and $f_8$ are small corrections as functions of a scalar field.
In section II, we derive gravitational field equations and scalar field equation of motion. In section III, we apply the standard cosmological metric to the equations derived in section II. In section IV, we use a perturbative approximation and obtain solutions under the condition of large scale limit; these are our main results.
In section V, we briefly discuss our results.
We take the convention of Hawking and Ellis \cite{Hawking and Ellis} and the notation of Hwang and Noh \cite{05 H and N}. Here, $c \equiv 1 \equiv \hbar$ .
\section{EINSTEIN EQUATIONS AND EQUATION OF MOTION WITH TWO CORRECTION TERMS }
The action considered here is
\bea
&& S = \int d^4x \sqrt{-g} \Big[ {1 \over 16\pi G} R - {1 \over 2} \phi^{,c} \phi_{,c} - V(\phi)
+ f_1(\phi) R^2 + f_2(\phi)R^{ab}R_{ab}
\Big] \label{action} ,
\eea
where $f_1(\phi)$ and $f_2(\phi)$ are the dimensionless functions corresponding to $f_7$ and $f_8$ respectively in Eq. (\ref{Weinberg Lagrangian}).
Varying the action (\ref{action}) with respect to the metric and the scalar field \cite{09Yun, 72Weinberg, book08Weinberg, 83Barth}
yields the gravitational field equations (GFE) and equation of motion (EOM):
\bea
&& R_{ab} - {1 \over 2} g_{ab} R - 8\pi G ( T^{(f_1)}_{ab} + T^{(f_2)}_{ab} ) = 8\pi G T^{(MSF)}_{ab}
\label{GFE} ,
\eea
where \bea
&& T^{(MSF)}_{ab} = \phi_{,a} \phi_{,b} - \Big( {1 \over 2} \phi^{,c} \phi_{,c} + V \Big) g_{ab} ,
\label{EM tensor MSF}
\\ && T^{(f_1)}_{ab} \equiv 2 f_1 \Big( {1 \over 2} R^{2} g_{ab} - 2R R_{ab} - 2 g_{ab} \Box R + 2 R_{;ab} \Big)
\nonumber \\ && {} - 8 f_{1,c} R^{;c} g_{ab} + 8 f_{1,(a} R_{,b)}
+ 4 f_{1;ab} R - 4 \Box f_1 R g_{ab} ,
\\ && T^{(f_2)}_{ab} \equiv f_2 g_{ab} R^{cd} R_{cd} - 2g_{ab} (f_2 R^{cd}) _{;cd}
+ 4 ( f_2 {R_{(a}}^c ) _{;b)c} - 2 \Box (f_2 R_{ab}) - 4 f_2 {R_a}^c R_{bc}
\nonumber \\ && {} = 2 f_2 \Big( {1 \over 2} R^{cd} R_{cd} g_{ab} + R_{;ab} - 2 R^{cd} R_{acbd} - {1 \over 2} g_{ab} \Box R - \Box R_{ab} \Big) {}
\nonumber \\ && {} + 2 ( -g_{ab} f_{2,c} R^{;c} - 2 f_{2,c} R_{ab;d} g^{cd} + 2 f_{2,c} R^{c}_{(a;b)} + f_{2,(a} R_{,b)} ) {}
\nonumber \\ && {} + 2 ( - f_{2;cd} R^{cd} g_{ab} - \Box f_2 R_{ab} + 2 f_{2;c(a} R_{b)}^{c} ) \label{EM tensor f2} ,
\eea
and
\bea && \Box \phi = V_{,\phi } - f_{1,\phi} R^2 - f_{2,\phi} R^{ab} R_{ab} \label{EOM} .
\eea
Here,
semicolons denote covariant derivatives, symmetrization of a tensor is defined as $T_{(ab)} \equiv {1 \over 2} (T_{ab} + T_{ba})$ ,
d'Alembertian of $\phi$ is written as $ \Box \phi \equiv g^{ab} \phi_{,a;b} $ ,
$V_{,\phi} \equiv {\partial V \over \partial \phi} $, and $ \dot \phi \equiv {\partial \phi \over \partial t} $ .
In Eq. (\ref{EM tensor f2}), the Bianchi identities \cite{72Weinberg} have been used in order to specify each component of the energy-momentum tensor conveniently.
If $f_1$ and $f_2$ are constants, then the GFE are in agreement with the previous results by Noh and Hwang \cite{97 N and H} .
\section{EVOLUTION OF BACKGROUND UNIVERSE AND GRAVITATIONAL WAVE}
We assume a homogenous, isotropic, and spatially flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric \cite{72Weinberg}
for the description of the background universe and consider tensor-type linear perturbation:
\bea
&& ds^2 = a^2 \big[- d \eta^2 + ( \delta_{\alpha \beta} + 2 C_{\alpha\beta} ) dx^{\alpha} dx^{\beta} \big] \label{metric ten PT} .
\eea
Here, $a(t)$ is the cosmic scale factor, $x^0 \equiv\eta $, and $ dt \equiv a d\eta $ .
According to the notation of Hwang and Noh \cite{05 H and N}
who have formulated cosmological linear perturbation theory in various generalized gravity including scalar- and tensor-type perturbation,
$ C^{(t)}_{\alpha\beta}$ should be used instead of $C_{\alpha\beta}$ to indicate the tensor mode.
However, the superscript (t) is omitted in this paper, since we deal with only gravitational wave.
$ C_{\alpha\beta}({\mathbf x} ,t)$ is tracefree and transverse with respect to
the flat three-dimensional metric $\delta_{\alpha\beta}$ ,
$C^{\alpha}_{\alpha} \equiv 0 \equiv C^{\alpha}_{\beta,\alpha}$ .
$C_{\alpha\beta}$ is also invariant under a gauge transformation
\cite{80Bardeen, 92Mukhanov, 05Mukhanov, 05 H and N, 11H, 84Kodama}.
Useful quantities calculated from the metric (\ref{metric ten PT}), are listed in the appendices of Noh and Hwang \cite{97 N and H}.
They include $G^{a}_{b}, \Box R$, etc.
By substituting the metric (\ref{metric ten PT}) into GFE (\ref{GFE}) and EOM (\ref{EOM}),
we obtain
\bea
&& 8\pi G T^{0(MSF)}_{0}
\nonumber \\ && = -3 H^2
- 96 \pi G \big[ (3f_1 + f_2) ( 2 H \ddot H - \dot{H}^2 + 6 H^2 \dot H )
+ \dot{f}_1 H R + \dot{f}_2 ( 3 H^3 + 2 H \dot H ) \big] \label{Einstein eq 00-component} ,
\\ && T^{0(MSF)}_{\alpha} = T^{\alpha(MSF)}_{0} = 0 ,
\\ && 8\pi G T^{\alpha (MSF)}_{\beta}
\nonumber \\ && = - ( 2 \dot H + 3 H^2 ) \delta^{\alpha}_{\beta} + D^{\alpha}_{\beta}
\nonumber \\ && - 8 \pi G \Big\{ 4 ( 3 f_1 + f_2 ) \delta^{\alpha}_{\beta} ( 2 \dddot H +12 H \ddot{H} + 9 \dot{H}^2 + 18 H^2 \dot{H} )
+ 8 \dot{f}_1 \delta^{\alpha}_{\beta} ( \dot{R} + HR )
+ 4 \ddot{f}_1 R \delta^{\alpha}_{\beta}
\nonumber \\ && {} - 4 f_1 ( R D^{\alpha}_{\beta} + \dot{R} \dot{C}^{\alpha}_{\beta} )
- 4 \dot{f}_1 R \dot{C}^{\alpha}_{\beta}
+ 2 \dot{f}_2 \delta^{\alpha}_{\beta} ( 8 \ddot{H} + 36 H \dot{H} + 12 H^3 )
+ 4 \ddot{f}_2 \delta^{\alpha}_{\beta} ( 2 \dot{H} + 3 H^2 )
\nonumber \\ && {} + 2 f_2 [ \ddot{D^{\alpha}_{\beta}} + 3 H \dot{D^{\alpha}_{\beta}}
- 6 ( \dot{H} + H^2 ) D^{\alpha}_{\beta} - { \Delta \over a^2 } D^{\alpha}_{\beta}
- 6 ( \ddot{H} + 2 H \dot{H} ) \dot{C}^{\alpha}_{ \beta } - 4 \dot{H} { \Delta \over a^2 } C^{\alpha}_{ \beta } ]
\nonumber \\ && {} + 2 \dot{f_2} [ 2 \dot{D^{\alpha}_{\beta} } + 3H D^{\alpha}_{\beta} - 6 ( \dot{H} + H^2 ) \dot{C}^{\alpha}_{ \beta } ]
+ 2 \ddot{f_2} D^{\alpha}_{\beta} \Big\} \label{Einstein eq spatial component} , \eea
and
\bea
&& \ddot\phi + 3 H \dot\phi + V_{,\phi}
- 36 f_{1,\phi} \big( \dot{H}^2 + 4 \dot H H^2 + 4 H^4 \big)
-12 f_{2,\phi} \big( \dot H^2 + 3 \dot H H^2 + 3 H^4 \big)
= 0 \label{EOM BG} ,
\eea
where
the Hubble parameter, $H \equiv {\dot{a} / a}$, the Ricci scalar, $R = 6 ( \dot H + 2 H^2 )$,
and \bea
&& D^{\alpha}_{\beta} \equiv \ddot{C}^{\alpha}_{\beta} + 3H \dot{C}^{\alpha}_{\beta} - { \Delta \over a^2 } C^{\alpha}_{\beta} \label{D def} .
\eea
Putting $f_1$ and $f_2$ to be constant and removing the $\phi$-dependent terms, we get the results which agree with those of Noh and Hwang \cite{97 N and H}.
Therefore, their remarks on the qualitative sameness of the background contribution from $R^2$ and $R^{ab}R_{ab}$ theories
also hold in this case.
We can split the energy momentum tensor into the background part (function of only time)
and the small perturbed part (function of both time and space)
in the cosmological linear perturbation theory based on the typical FLRW model \cite{11H, 05Mukhanov, book08Weinberg},
$T^{a}_{b}({\mathbf x} ,t) = \overline{T^{a}_{b}}(t) + \delta T^{a}_{b}({\mathbf x} ,t) $.
The background parts are easily read off from the Eqs. (\ref{Einstein eq 00-component}) and (\ref{Einstein eq spatial component}):
\bea
&& H^2
+ 32 \pi G \Big[ (3 f_1 + f_2 ) ( 2 H \ddot{H} - \dot H^2 + 6 H^2 \dot H )
+ 6 \dot f_1 ( 2 H^3 + H \dot H ) + \dot f_2 ( 3 H^3 + 2 H \dot H ) \Big]
\nonumber \\ && {} = - { 8 \pi G \over 3 } T^{0 (MSF)}_{0}
= { 8 \pi G \over 3 } \mu^{(MSF)} = { 8 \pi G \over 3 } \Big( { \dot\phi^2 \over 2 } + V \Big) ,
\nonumber \\ && \dot H
+ 16 \pi G \Big[ 2 (3 f_1 + f_2 ) ( \dddot H + 3 H \ddot H + 6 \dot H^2 )
\nonumber \\ && {} + 6 \dot f_1 ( 2 \ddot H + 7 H \dot H - 2 H^3 )
+ \dot f_2 ( 4 \ddot H + 12 H \dot H - 3 H^3 )
+ 6 \ddot f_1 ( \dot H + 2 H^2 ) + \ddot f_2 ( 2 \dot H + 3 H^2 ) \Big]
\nonumber \\ && {} = 4 \pi G \Big( T^{0(MSF)}_0 - { 1 \over 3 } \overline{T^{\alpha}_{\alpha}}^{(MSF)} \Big)
= -4 \pi G \dot\phi^2
\label{Friedmann eq} .
\eea
The second equation can also be checked by diffentiating the first one and by using the EOM (\ref{EOM BG}).
The perturbed part of Eq. (\ref{Einstein eq spatial component}) is
\bea
&& D^{\alpha}_{\beta} + 8\pi G \Big\{ 4 f_1 ( R D^{\alpha}_{\beta} + \dot R \dot C^{\alpha}_{\beta} ) + 4 \dot{f}_1 R \dot{C}^{\alpha}_{\beta}
\nonumber \\ && {} - 2 f_2 [ \ddot{D^{\alpha}_{\beta}} + 3 H \dot{D^{\alpha}_{\beta}}
- 6 ( \dot{H} + H^2 ) D^{\alpha}_{\beta} - { \Delta \over a^2 } D^{\alpha}_{\beta}
- 6 ( \ddot{H} + 2 H \dot{H} ) \dot{C}^{\alpha }_{ \beta } - 4 \dot{H} { \Delta \over a^2 } C^{\alpha}_{ \beta } ]
\nonumber \\ && {} - 2 \dot{f_2} [ 2 \dot{D^{\alpha}_{\beta} } + 3H D^{\alpha}_{\beta} - 6 ( \dot{H} + H^2 ) \dot{C}^{\alpha }_{ \beta } ]
- 2 \ddot{f_2} D^{\alpha}_{\beta} \Big\}
= 0 \label{GW eq} .
\eea
Eq. (\ref{GW eq}) is a fourth order differential equation for $C^{\alpha}_{\beta}({\mathbf x} ,t)$ .
Thus, it is theoretically hard to deal with because more initial conditions are required for numerical analysis and these equations allow unnecessary unphysical solutions.
With this concern for the problems of higher-derivative theories,
the research on a perturbative method for reducing the order of derivatives has been done by Simon {\em et al}. \cite{89Simon, 90Simon, 93Parker, 18Solomon}.
\section{second order differentional equations after feedback}
Considering the quantum corrections are small and neglecting $f_n^2$ terms allow the order reduction of the differential Eqs. (\ref{Friedmann eq}, \ref{GW eq}):
\bea
&& H^2 = 8 \pi G \Big\{ {1 \over 3 } \mu^{(MSF)}
+ 8 \pi G ( 3 f_1 + f_2 ) \big[ 8\pi G \big( \mu^{(MSF)} + p^{(MSF)} )^2 + 4 H \dot{p}^{(MSF)} \big]
\nonumber \\ && {} + 32\pi G H \big[ \dot f_1 \big( 3 p^{(MSF)} - \mu^{(MSF)} \big) + \dot f_2 p^{(MSF)} \big] \Big\}
\nonumber \\ && {} = 8 \pi G \Big\{ { 1 \over 3 } \Big( { \dot\phi^2 \over 2 } + V \Big)
- 64\pi G ( 3 f_1 + f_2 ) \Big[ 4\pi G \dot\phi^2 \Big( { \dot\phi^2 \over 4 } + V \Big) + H \dot\phi V_{,\phi} \Big]
\nonumber \\ && {} + 32\pi G H \Big[ \dot{f}_1 \big( \dot\phi^2 - 4V \big) + \dot{f}_2 \Big( { \dot\phi^2 \over 2 } - V \Big) \Big]
\Big\}
\label{Friedmann eq feedback}
\eea
and \bea
&& D^{\alpha}_{\beta} + 32\pi G \Big\{ f_1 \dot R \dot C^{\alpha}_{\beta} + \dot{f}_1 R \dot{C}^{\alpha}_{\beta}
\nonumber \\ && {} + f_2 [ 3 ( \ddot{H} + 2 H \dot{H} ) \dot{C}^{\alpha }_{ \beta } + 2 \dot{H} { \Delta \over a^2 } C^{\alpha}_{ \beta } ]
+ 3 \dot{f_2} ( \dot{H} + H^2 ) \dot{C}^{\alpha }_{ \beta }
\Big\} = 0 . \label{GW eq feedback}
\eea
A much simplified second order differential equation (\ref{GW eq feedback}) for $C^{\alpha}_{\beta}$ is obtained
by a feedback method: inserting $D^{\alpha}_{\beta} = \mathcal{O} (f^1_n) $ from Eq. (\ref{GW eq})
into the big curly brackets in Eq. (\ref{GW eq}) itself and neglecting very small $ \mathcal{O} (f^2_n)$ terms.
Likewise, using Eq. (\ref{Friedmann eq}) and Eq. (\ref{EOM BG}), we derived a modified Friedmann Eq. (\ref{Friedmann eq feedback})
in which the curly brackets may be regarded as ${1 \over 3}$ of the effective energy density in this model.
Meanwhile, it is allowed to add a term of $f_n^2$-order, $ 96\pi G f_2 (\dot H + H^2) D^{\alpha}_{\beta} $ , to Eq. (\ref{GW eq feedback}) and to recover the $f_1$ gravity terms before the feedback:
\bea
&& D^{\alpha}_{\beta} + 32 \pi G \Big\{ \big( f_1 R \big) \dot{\phantom{i}} \dot C^{\alpha}_{\beta} + f_1 R D^{\alpha}_{\beta}
+ 3 \big[ f_2 ( \dot H + H^2 ) \big] \dot{\phantom{i}} \dot{C}^{\alpha }_{ \beta }
+ 3 f_2 ( \dot H + H^2 ) D^{\alpha}_{\beta} + 2 f_2 \dot{H} { \Delta \over a^2 } C^{\alpha}_{ \beta } \Big\}
\nonumber \\ && {} = F D^{\alpha}_{\beta} + \dot F \dot{C}^{\alpha}_{\beta} + 64\pi G f_2 \dot H { \Delta \over a^2 } C^{\alpha}_{ \beta }
= 0 \label{GW eq feedback 2} ,
\eea
where \bea
F \equiv 1 + 32\pi G [ f_1 R + 3 f_2 ( \dot H + H^2 ) ] . \eea
Dividing Eq. (\ref{GW eq feedback 2}) by $F$ and using the definition of $D^{\alpha}_{\beta}$ in Eq. (\ref{D def})
lead to an equation for the tensor mode in the compact form:
\bea
&& {1 \over a^3 F } \big( a^3 F \dot C^{\alpha}_{\beta} \big)\dot{\phantom{i}}
- \big( 1- 64\pi G f_2 \dot H \big) { \Delta \over a^2 } C^{\alpha}_{ \beta }
\nonumber \\ && {} = { 1 \over {a^2 z} } \Big[ { v^{\alpha}_{\beta}}''
- \Big( { z'' \over z } + c^2_T \Delta \Big) v^{\alpha}_{\beta} \Big] = 0 \label{MS eq} ,
\\ && {} v^{\alpha}_{\beta} \equiv z C^{\alpha}_{\beta} ,
\quad z \equiv a \sqrt{F} , \eea
and \bea
&& {} c^2_T \equiv 1- 64\pi G f_2 \dot H .
\eea
Here, $ ' \equiv {\partial \over \partial\eta} $.
Eq. (\ref{MS eq}) is often called Mukhanov-Sasaki equation \cite{85Mukhanov, 92Mukhanov, 86Sasaki}.
If $c_T $ is the gravitational wave propagation speed,
then it is affected not by the general function $f_1(\phi)$, but by the small $f_2$ correction term depending on time.
Moreover, $c_T$ should be less than the speed of light, thus the constraint that $ f_2 \dot H > 0 $ is required.
In the large scale limit, a general integral form solution is obtained:
\bea
C^{\alpha}_{\beta}({\mathbf x} ,t) = c^{\alpha}_{\beta}({\mathbf x}) + d^{\alpha}_{\beta}({\mathbf x}) \int^{t} {dt \over {a^3 F}}
\label{LS sol} ,
\eea
where $ c^{\alpha}_{\beta}(\mathbf{x}) $
and $ d^{\alpha}_{\beta}(\mathbf{x}) $
are the time-independent integration constants.
Ignoring the decaying transient $d$-solution in an expanding universe,
we note that
the evolution of tensor type perturbation in the large scale limit
is described by the conserved quantity $ c^{\alpha}_{\beta}(\mathbf{x}) $.
\section{DISCUSSIONS}
We have derived complicated fourth order differential equations of the gravitational wave as well as the background evolution in the inflationary universe implemented with the additional two modified gravity theories including a scalar field. Reducing the order by the perturbative approximation yields the more tractable equation and its solutions in the large scale limit. With model-dependent variables $F, z$, or $c_T$ \cite{13 Yunes and Siemens, 10Garfinkle et al, 17 Creminelli and Vernizzi} , the form of Eq. (\ref{MS eq}) is maintained in various generalized gravity theories such as a model motivated by string theory.
Those variables have been tabulated in Ref. \cite{05 H and N} .
If $f_1$ and $f_2$ are constants,
Einstein gravity and Starobinsky gravity correspond to a limit of $F =1$ and $ F = 1 + 32\pi G f_1 R$ respectively .
It would be more appropriate to call Eq. (\ref{MS eq}) Field-Shepley \cite{68Field} equation if the priority were concerned.
According to Weinberg \cite{PRD08Weinberg} , if the field equations derived from the MSF Lagrangian (\ref{L0}) are used in the correction Lagrangian (\ref{Weinberg Lagrangian}) and $\phi$ and $V(\phi)$ are suitably redefined, then Eq. (\ref{Weinberg Lagrangian}) can be simplified to have only three terms, $f_1, f_9$, and $f_{10}$ . In other words, the ten terms in Eq. (\ref{Weinberg Lagrangian}) are not independent to one another if the perturbative method at the action level and the redefinition approach are applied.
We suggest an interpretation of the logic behind his argument that is simpler than our approach to the full Lagrangian as follows. Assuming that $\Delta L$ (\ref{Weinberg Lagrangian}) is much smaller than $L_0$ (\ref{L0}), Einstein's equation (we set $8 \pi G \equiv 1$ in this section only)
\bea
R_{ab} = \phi_{,a} \phi_{,b} + g_{ab} V
\eea
derived from $L_0$ (\ref{L0})
and its trace equation
\bea
R = 2 ( X + 2 V )
\eea
with a convenient definition $X \equiv {1 \over 2} g^{ab} \phi_{,a} \phi_{,b} $
can be put into $\Delta L$ (\ref{Weinberg Lagrangian}).
Assuming that $ f_8 = -4 f_7 $,
\bea
f_7 R^2 + f_8 R^{ab} R_{ab} = - 12 f_7 X^2 \equiv 4 f_1 X^2 .
\eea
Thus, the $f_1$-gravity form
\cite{99 AP D and M}
is obtained from the seventh and eighth terms in $\Delta L$ (\ref{Weinberg Lagrangian}) with the abovementioned assumptions.
Our approach in a different context results in a modified propagation speed of gravitational wave that is measurable in principle.
We selected and considered only two terms, $f_7$ and $f_8$ in the correction Lagrangian (\ref{Weinberg Lagrangian})
and directly analyzed the action without any redefinitions and simplification, while we and Weinberg share the same assumption that the correction Lagrangian is small.
We used the approximation at the wave equation (\ref{GW eq}), while he did the approximation at the action level.
Comparison between two methods may be another issue.
There are several future investigations about this research.
Firstly, quantizing Eq. (\ref{MS eq}) from the action level is straightforward
by following the known prescriptions \cite{05 H and N, 92Mukhanov} .
The unitarity shall be considered during quantization of the theories here to preserve the inner product of quantum states; however, the unitarity-violating term is encountered in a study of quantum cosmology \cite{12 Kiefer and Kraemer} . Indeed, quantizing gravity is an abstruse issue for the very early universe.
More fundamentally, various generalized gravity theories with higher-derivative expansion are motivated by string theory
\cite{12EMM, 07Gasperini, 03 Gasperini and Veneziano, 87GSW} .
Secondly, if the Riemann-tensor-squared Lagrangian is studied,
then the tensor mode equations in this paper will be able to transform into Weinberg's counterpart \cite{PRD08Weinberg}.
Thirdly, a heavy numerical analysis may allow a comparison of the exact equations and the approximate equations.
\begin{acknowledgments}
The author is grateful
to Prof. Jai-chan Hwang for his teachings on cosmology
and
to Prof. Sang Gyu Jo for his thoughtful advice and critical review.
The author also thanks Prof. Chan-Gyung Park for his much help in {\em Mathematica} usage.
\end{acknowledgments}
| {
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The 2005 Radivoj Korać Cup was the third season of the Serbian-Montenegrin men's national basketball cup tournament. The Žućko's left trophy awarded to the winner Reflex from Belgrade.
Venue
Qualified teams
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Qualifications
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\section{Introduction}
\subsection{Instanton-dyons and confinement}
At high temperatures QCD matter is in form of quark-gluon plasma (QGP) state, which is weakly coupled
because of the asymptotic freedom phenomenon. The topological solitons to be discussed below
have large action $S=O(1/\alpha_s)\gg1 $ and are therefore strongly suppressed, $\sim exp(-S)$.
However, as $T$ decreases toward the deconfinement transition, the coupling grows and such objects become
important.
The non-trivial configurations of interest are Instantons \citep{Belavin:1975fg},
which in the Euclidean finite temperature formulation are known as
Calorons. Such solutions have been generalized to the case of non-zero expectation value of the Polyakov loop
by Lee-Li-Kraan-van Baal in refs \cite{Kraan:1998sn,Lee:1998bb} and are known as LLKvB calorons.
An important novel feature of these solution was the realization of instanton substructure:
each LLKvB caloron
consists of $N_c$ objects, known as instanton-dyons (or instanton-monopoles).
Color confinement phenomenon has many manifestations, and thus many definitions. In this series of papers
we focus on one particular aspect of it, namely on the shift of the vacuum expectation value of the Polyakov loop
from its ``trivial value" $<P>\approx 1$ at high $T$ to small $<P>\approx 0$ at $T < T_c$.
Multiple numerical simulations in the framework of lattice gauge theory have documented such shift,
as well as modification of the effective potential $V(P,T)$ with $T$ leading to it. Since a contribution of the quarks (and non-diagonal gluons) to thermodynamical quantities is proportional to (powers) of $<P>$, vanishing of it,
effectively switches off quark-gluon plasma contributions. So,
in papers of this series we focus on the calculation of this effective potential
and on the deconfinement phase transition phenomenon.
Another manifestation of confinement is a disordering of large Wilson loops. It has been argued in \cite{Gerhold:2006sk} that an ensemble of instanton-
dyons can generate the expected area law. However, this issue is rather subtle and depends on the
infrared tails of the soliton fields, which are modified by screening effects and thus are not robust
enough to be conclusive.
One more
approach to the confinement issue is reached via the static quark potentials, which do exist
at any $T$ and were extensively studied on the lattice. We intend to calculate those in our approach later.
Finally, a classic formulation of confinement includes absence of color degrees of freedom from vacuum spectra, at $T=0$. Addressing it directly is not possible for the type of models we discuss, since the calorons and instanton-dyons themselves become difficult to use at sufficiently low $T$.
The idea that the effective potential of the Polyakov loop $P$ is due to the back reaction of the instanton-dyons
goes back to Diakonov and collaborators \cite{Diakonov:2009ln}, who provided the first estimates indicating how this may happen, but were unable to prove it. Using the so called ``double-trace
deformation of Yang-Mills theory", at large $N$ on $S^1\times R^3$,
Unsal and Yaffe~\cite{Unsal:2008ch} argued that
there can be confining behavior, with unbroken center symmetry, even in weak coupling.
This construction was extended by Unsal and collaborators \cite{Poppitz:2011wy,Poppitz:2012sw,Poppitz:2012nz}
to a class of deformed supersymmetric theories with soft supersymmetry breaking. In such a
setting , with weak coupling and
an exponentially $small$ density of the dyons, the minimum of the potential is at the confining
value of $P$ induced by the repulsive interaction in the dyon-anti-dyon pairs (called
$bions$ by the authors). (The supersymmetry was needed to cancel the perturbative Gross-Pisarski-Yaffe-Weiss (GPYW) holonomy potential , which otherwise favors trivial value $<P>=1$.
Sulejmanpasic and one of us \cite{Shuryak:2013tka} have proposed a simple model, with ``repulsive cores" in the dyon-antidyon channel,
which can generate confining $V(P)$ at certain temperature $T_c$ in pure gauge theory.
To evaluate the free energy of the instanton-dyon ensemble we
performed numerical simulations for pure gauge $SU(2)$ theory, in the first paper of this series \cite{Larsen:2015vaa},
to be below referred as I. The essential element was inclusion of dyon-antidyon
interactions, determined in the previous work \cite{Larsen:2014yya} using a gradient flow method.
A similar conclusion has been recently reached by Liu, Shuryak and Zahed
\cite{Liu:2015ufa} using analytic mean field theory. It however uses the mean field approximation
which is only applicable
for high enough dyon density, or $T<T_c$.
\subsection{Quarks in the instanton-dyon ensemble}
In
this paper we include quarks, fermions in the fundamental color representation, to the instanton-dyon ensemble.
Those will be
referred to as ``dynamical quarks", since the so called fermionic determinant will be
included in the ensemble measure.
The topological objects, instantons and instanton-dyons, have a certain number of 4-dimensional
zero modes prescribed by
the topological index theorems. Topology ensures that any smooth deformation of the objects themselves does not
shift fermionic eigenvalues from zero.
When the ensemble of topological solitons is dense enough, the fermionic zero modes can
collectivize and produce the so called Zero Mode Zone (ZMZ).
For an ensemble of instantons this phenomenon has been studied in
great detail in the 1980's and 1990's, for a review see \cite{Schafer:1996wv}.
The main physical phenomenon associated with ZMZ is the spontaneous breaking of the $SU(N_f)$
chiral symmetry, ``chiral breaking" for short.
\begin{table}[h]
\begin{tabular}{ l | c | c | c | r }
\hline
& $M$ & $\bar{M}$ & $L$ & $\bar{L}$ \\
\hline
e & 1 & 1 & -1 & -1 \\
m & 1 & -1 & -1 & 1 \\
\hline
\end{tabular}
\caption{Quantum numbers of the four different kinds of the instanton-dyons of the SU(2) gauge theory.
The two rows are electric and magnetic charges.
}
\label{tab1}
\end{table}
In the case of $SU(2)$ gauge group there are only two types of instanton-dyons, called $M$ and $L$ types
(also known as BPS and ``twisted" or KK ones), their electric and magnetic charges are given in Table \ref{tab1} .
Physical (antiperiodic in time direction) fermions have zero modes on the $L$ dyons.
The zero modes produce the simplest effect
of the dynamical fermions - binding of the $\bar L L$ dyon pairs into ``molecules" , studied
by Shuryak and Sulejmanpasic
\cite{Shuryak:2012aa}. The first numerical simulations with fermions were done by
Faccioli and Shuryak \cite{Faccioli:2013ja}, who studied 1, 2 and 4 flavor theory with the SU(2) color:
they found chiral symmetry breaking in the first two cases, but the last one, $N_f=4$ appeared marginal.
Many technical aspects of our work follows their setting.
Recent work by Liu, Shuryak and Zahed \cite{Liu:2015jsa} was also devoted to the role of quarks
in the dense confining instanton-dyon ensemble. Their basic conclusion is that in this regime
the quark condensate, signaling chiral symmetry breaking, satisfies certain
universal gap equation, which has non-zero solutions provided the number of quark flavors $N_f<2 N_c$.
So, the border case for 2 colors is $N_f=4$, which is also a near-critical one according to Ref. \cite{Faccioli:2013ja}.
In the present work we focus on the simplest case with the spontaneous breaking of chiral symmetry, with only
two quark flavors $N_f=2$. The central issue addressed is interrelation between confinement and chiral symmetry breaking.
The paper is structured as follows: in section \ref{Zero}
we describe the physics of the fermionic zero modes and the technical tool -- the hopping matrix --
used to evaluate the determinant. We then explain the general setting of the interactions in section \ref{Setting}. After that we show how the chiral condensate is obtained from the eigenvalue distribution in section \ref{EigenDestribution} and the mass gap is discussed in section \ref{GapWay}. The data sets used and how they were analyzed is explained in section \ref{SectionData}. We end with the physical results in section \ref{Results}, where we show, among other, the Polyakov loop and the chiral condensate's dependence on temperature.
\section{The Zero Mode Zone }\label{Zero}
The term ``dynamical quarks" in the title implies inclusion of the
fermionic determinant in the measure for gauge field configurations.
The main approximation made by us -- similar to what was done in the instanton ensemble --
is that the set of all fermionic states is translated to the subspace spanned by zero modes.
This determinant can be viewed as a sum of closed fermionic loops with ``hopping amplitudes"
between dyons and antidyons. Since sectors that are self-dual or anti-self-dual have its eigenvalues protected, then the overlap of $L$ and $L$ dyons or $\bar{L}$ and $\bar{L}$ dyons have to be zero. The resulting form
of the ``hopping
matrix" is
\begin{eqnarray}
{\bf \hat T}\equiv \left(\begin{array}{cc}
0&{\bf T}_{ij}\\
-{\bf T}_{ji}&0
\end{array}\right)
\label{T12}
\end{eqnarray}
Each of the entries in ${\bf T}_{ij}$ is a ``hopping amplitude" for a fermion between
the i-th L-dyon and the j-th $\bar{\rm L}$-antidyon. The diagonal matrix elements are zero,
and therefore a single or many infinitely-separated dyons will have zero determinant and ``veto"
such configurations. However, nonzero non-diagonal hopping matrix elements make
the determinant nonzero.
The only modification of the partition function used in this work relative to that in I is the fermionic factor
\begin{eqnarray} \left( det({\bf \hat T}) \right)^{N_f}
\end{eqnarray}
Basically, $det({\bf \hat T})$ can be seen as a set of loop diagrams, connecting all L-dyons and antidyons of the ensemble. It can
either be dominated by short loops, including small number (2,..) dyons, to be referred to as a ``molecular regime",
or by very long loops, including finite fraction of the ensemble (``collectivized regime"). The former has unbroken
and the latter broken chiral symmetry. It is the purpose of our simulations to determine,
as a function of the dyon density, the weights of such short and long loops.
We define the individual hopping amplitude as the matrix element of the Dirac operator between different zero-mode eigenfunctions
\begin{eqnarray}
T_{ij} &=& < i | D\hspace{-1.6ex}/\hspace{0.6ex} | j > \label{eqn_hop}
\end{eqnarray}
where $i$ and $j$ are zero-modes belonging to i-th $L$ and j-th $\bar{L}$ dyons.
If the gauge field in the Dirac operator is a sum of two solitons, using the equations of motion for
both zero modes, one can reduce the covariant derivative to the ordinary derivative.
Including a mass term, changes the hopping matrix by a constant $m$ times the identity matrix.
\section{The general setting} \label{Setting}
The setup is almost the same as in our paper I \cite{Larsen:2015vaa}, with the difference being the inclusion of
the fermionic determinant in the zero-modes approximation. This factor creates an additional fermion-induced interaction between the $L$ type dyons.
The dimensionless holonomy $\nu = v/(2\pi T)$ is related to the expectation value of the Polyakov loop through the ($SU(2)$) relation
\begin{eqnarray}
P &=& \cos (\pi \nu)
\end{eqnarray}
We seek to minimize the free energy
\begin{eqnarray}
f &=& \frac{4 \pi^2}{3}\nu ^2 \bar{\nu}^2 -2n_M\ln\left[\frac{d_\nu e }{n_M}\right] -2n_L\ln\left[\frac{d_{\bar{\nu}} e }{n_L}\right] \nonumber\\
& &+\Delta f \end{eqnarray}
where the first term is the perturbative Gross-Pisarski-Yaffe-Weiss holonomy potential, the next terms
contain semiclassical independent dyon contributions, with
\begin{eqnarray} d_\nu &=& \Lambda \left( \frac{8\pi ^2}{g^2} \right)^2 e^{-\frac{\nu 8\pi ^2}{g^2}} \nu ^{\frac{8\nu}{3}-1}/(4\pi)
\end{eqnarray}
and $\Delta f \equiv -\log (Z_{changed})/\tilde{V_3} $ is defined via the partition function studied numerically
\begin{eqnarray} Z_{changed}&=& \frac{1}{\tilde V_3^{2(N_L+N_M)}}\int D ^3x \det (G) \exp( -\Delta D_{DD} (x) ) \nonumber \\
& &\times \prod _i \lambda _i ^{N_f}
\end{eqnarray}
The last factor is the fermionic determinant, now written as
the product of all eigenvalues of the hopping matrix $T_{ij}$.
Further explanation of $G$ and $\Delta D_{DD}$ can be found in \cite{Larsen:2015vaa}, and we therefore just present their
expressions here without too many comments.
\begin{eqnarray}
G &=& \delta _{mn} \delta _{ij} ( 4\pi \nu_m-2\sum _{k\neq i}\frac{e^{-M_D T |x_{i,m}-x_{k,m}|}}{T|x_{i,m}-x_{k,m}|} \\
& & +2\sum _{k}\frac{e^{-M_D T |x_{i,m}-x_{k,p\neq m}|}}{T|x_{i,m}-x_{k,p\neq m}|} ) \nonumber \\
& & +2\delta_{mn}\frac{e^{-M_D T |x_{i,m}-x_{j,n}|}}{T|x_{i,m}-x_{j,n}|}-2\delta_{m\neq n}\frac{e^{-M_D T |x_{i,m}-x_{j,n}|}}{T|x_{i,m}-x_{j,n}|} \nonumber
\end{eqnarray}
Dyon 2-point interactions $\Delta D_{DD}$ is a sum over all the different dyon to dyon combinations
\begin{eqnarray}
\Delta D_{DD} &=& \sum _{j >i} \Delta S_{D_i D _j}
\end{eqnarray}
where $\Delta S_{D_i D _j}$ is the correction to the action between dyon i and dyon j. If the two dyons are a dyon and its anti-dyon, we have for distances larger than $x_0$
\begin{eqnarray}
\Delta S_{D\bar{D}} &=& -2\frac{8\pi^2 \nu}{g^2}(\frac{1}{x}-1.632e^{-0.704x})e^{-M_D r T} \nonumber \\
x &=& 2\pi \nu r T
\end{eqnarray}
For the rest of the combinations we have
\begin{eqnarray}
\Delta S_{DD} &=& \frac{8\pi^2 \nu }{g^2}\left( -e_1e_2\frac{1}{x}+m_1m_2\frac{1}{x}\right)e^{-M_D r T} \nonumber \\
x &=& 2\pi \nu rT
\end{eqnarray}
where the charge is given by table \ref{tab1}. For distances smaller than $x_0$ we have a core between dyon pairs of the types $LL$, $MM$, $\bar{L}\bar{L}$, $\bar{M}\bar{M}$, $L\bar{L}$ and $M\bar{M}$
\begin{eqnarray}
\Delta S_{DD} &=& \frac{\nu V_0}{1+\exp\left[\sigma T(x-x_0)\right]}\\
x &=& 2\pi \nu rT
\end{eqnarray}
where $x_0$ is the size of the dyons core. In this paper we work with $x_0 =2$, just as in our earlier paper I. It is important to note that for $M$ type dyons one has to use $\nu$ and for $L$ type dyons one has to use $\bar{\nu}=1-\nu$.
\section{Eigenvalue distributions and the chiral condensate}\label{EigenDestribution}
The Banks-Casher relation for the chiral condensate tells us that, in the infinite volume limit, the chiral condensate for massless fermions is proportional to
the density of eigenvalues at zero value
\begin{eqnarray}
|<\bar{\psi}\psi >| & = & \pi \rho (\lambda)_{\lambda \to 0,m \to 0,V \to \infty}
\end{eqnarray}
For any system with a finite volume, the typical size of small eigenvalues is of size $1/V$
and the density will always be $0$ at $\lambda =0$ and $m=0$. We see this behavior in our ensemble as seen for zero mass in Fig. \ref{Eigenvaluesm0Dense} and \ref{Eigenvaluesm0}. We also find that a finite mass as in Fig. \ref{Eigenvaluesm001Dense} and \ref{Eigenvaluesm001} has the effect of allowing eigenvalues around zero, and if the mass is large enough, smooth the maximum of the eigenvalue distribution into the region around $\lambda =0$.
To understand
finite volume effects on the distribution, one may study those using chiral random matrix theory,
for review see
\cite{Verbaarschot:2009jz}.
In principle,
using expressions obtained in this framework one can recover the value of the chiral condensate in the infinite volume case.
\begin{figure}[h]
\includegraphics[scale=0.48]{eigenvaluesM0p0y6.jpg}
\caption{Eigenvalue distribution for $n_M=n_L=0.47$, $N_F=2$ massless fermions at $S=7$.}
\label{Eigenvaluesm0Dense}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.48]{eigenvaluesM0p19y6.jpg}
\caption{Eigenvalue distribution for $n_M=n_L=0.08$, $N_F=2$ massless fermions at $S=7$.}
\label{Eigenvaluesm0}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.48]{eigenvaluesM001p0y6.jpg}
\caption{Eigenvalue distribution for $n_M=n_L=0.47$, $N_F=2$ $m=0.01$ fermions at $S=7$.}
\label{Eigenvaluesm001Dense}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.48]{eigenvaluesM001p19y6.jpg}
\caption{Eigenvalue distribution for $n_M=n_L=0.08$, $N_F=2$ $m=0.01$ fermions at $S=7$.}
\label{Eigenvaluesm001}
\end{figure}
We will be determining the chiral condensate by two different methods:
(i)
The first one is based on the part of the eigenvalue distributions with the smallest $\lambda$. It
requires an understanding of both the {\em finite volume} and {\em quark mass} effects on the
distribution. This understanding we obtain from analytic random matrix results. We explain this approach in section \ref{SizeEffects}.
Vanishing of the condensate is used to define the ensemble parameters corresponding
to chiral symmetry breaking transition, $T_{\bar{\psi}\psi}$.
The second strategy (ii) we will use, is based on the
determination of
the so called {\em gap width} in the distribution, near $\lambda =0$: we will refer to it as $T_{gap}$. This approach is explained in section \ref{GapWay}.
Ideally, both critical temperatures should coincide, defining the location of the chiral symmetry breaking
$T_\chi$.
\subsection{The finite size effects}\label{SizeEffects}
To understand the scaling of the finite volume effects we performed
simulations for $64$ and $128$ dyons, at the same density.
(The volume of the sphere with $128$ dyons being 2 times larger than the sphere of the $64$ ones.)
The quark mass in both simulations were set to zero.
The resulting eigenvalue distributions are shown in Fig. \ref{fig_finite_volume}.
\begin{figure}[h]
\includegraphics[scale=0.46]{eigenvaluesfitComparison.jpg}
\caption{(Color Online) The points are the eigenvalue distribution for 64 (blue circles) and 128 (red squares) dyons at $S=8$ and
density of dyons $n_M=0.33$, $n_L =0.20$, $N_F=2$. The curves are the fit with eq. (\ref{analytic_f}) with $\Sigma _{2,64}= 1.30 \pm 0.06$ and $\Sigma _{2,128} = 1.28 \pm 0.06$ and the scaling as $\Sigma _{1,64}=0.79 \pm 0.05$ and $\Sigma _{1,128}=0.51\pm 0.04$ for these two cases, respectively. The lower purple line is the difference between the two fits. Eq. (\ref{SigmaEq}) gives $\Sigma = 0.38\pm 0.13$, while the maximum of the difference between the two curves gives $\Sigma =0.3$ after normalizing the difference (note: This approach of using the maximum of the difference between the two volumes, has not been used to analyze the data, but is simple used here to visualize the effect).}
\label{fig_finite_volume}
\end{figure}
We fit the distribution of the eigenvalues with the form taken from random-matrix theory \cite{Verbaarschot:2009jz} for $SU(2)$ gauge group for massless fermions given by
\begin{eqnarray}
\rho(x) &=& V\Sigma _2[\frac{x}{2}(J_2( x)^2 - J_1( x)J_3( x)) \nonumber
\\ & &+
\frac{1}{2} J_2(x)(1 - \int _0 ^x dt J_2(t)) ]\label{analytic_f}
\end{eqnarray}
where $x=\lambda V \Sigma _1$ and $J_n$ is the Bessel function. Both the scaling factor $V \Sigma _1$ and the overall factor $V\Sigma_2$ should be proportional to the value of the chiral condensate $\Sigma$.
In the limit $V\to \infty $ the formula gives $\rho(0) \propto V\Sigma _2$ as required.
Ideally, the parameter values for two different volumes should agree.
When the fits for different volumes were done, we found that the values for parameter $\Sigma _2$ agree very well indeed.
(This is related to the fact that the height of the distributions at the r.h.s. of Fig. \ref{fig_finite_volume}
do agree.)
Note that the main difference between the two distributions is a shift to the left for bigger volume.
This is expected in larger volume clusters of a condensate inside which quark propagation gets larger,
and the eigenvalues smaller. The formula, from random matrix theory, prescribes a particular
``mesoscopic" scaling with the volume.
However, the fit by this formula produces values of $\Sigma _1$ which are not the same. This indicates that, at least our smaller volume, is not yet
in the range in which the expected large volume scaling applies.
The physics behind this behavior is as follows: there are basically two components of the ensemble, generating
two different dependencies on the volume. As we already mentioned in the introduction, there is collectivized
dyons, producing the condensate, and dyon-antidyon pairs. The former
component produces eigenvalue distribution shifting with the volume, while the latter
contribution is volume-independent .
The existence of two components lead us to construct a value of $\Sigma$ out of all four parameters of the fit given by
\begin{eqnarray}
\Sigma &=& \Sigma _{2}(2\Sigma _{1}^{128}/\Sigma _{1}^{64}-1) \label{SigmaEq}
\end{eqnarray}
In the case of only almost zero-modes, from the collectivized
dyons, doubling
the volume should double $V \Sigma _1$. In the opposite case of only dyon-antidyon pairs, $V \Sigma _1$ should be unchanged. As can be seen in Fig. \ref{fig_finite_volume} the situation is sometimes in between the two extremes.
The expression (\ref{SigmaEq}) is an interpolation between the two regimes.
This resulting value of $\Sigma$ will be used in the plots to follow, such as showing the temperature dependence
of the condensate. We show $\Sigma_2$, $2\Sigma_1^{128}$/$\Sigma_1^{64}-1$ and $\Sigma$ for the results in section \ref{A} in Fig. \ref{Sigmacomp}.
\begin{figure}[h]
\includegraphics[scale=0.54]{Sigma_Comparison_Square_fitValues.jpg}
\caption{(Color Online) $\Sigma_2$ (blue circle), $2\Sigma_1^{128}$/$\Sigma_1^{64}-1$ (red square) and $\Sigma$ (purple triangle) as a function of input action $S=8\pi^2/g^2$ for the results in section \ref{A}. It is observed how the rise in $\Sigma_2$ and $2\Sigma_1^{128}$/$\Sigma_1^{64}-1$ are correlated, while, $2\Sigma_1^{128}/\Sigma_1^{64}-1$ goes to zero for higher $S$ while $\Sigma_2$ does not.}
\label{Sigmacomp}
\end{figure}
As the density increases, it is seen how the scaling becomes closer and closer to that of the volume, as expected from Eq. (\ref{analytic_f}), such that the limit to infinite volume gives the chiral condensate as $\rho (0)$.
\subsection{The effect of the quark mass}
Nonzero quark mass moderates the distribution of the smaller eigenvalues. Furthermore, for $\lambda <m $
the fermions are effectively decoupled, and thus the distributions should be the same as for a quenched (no
dynamical quarks) theory. The latter is known to produce a singularity at $\lambda\rightarrow 0$ observed in
the instanton liquid simulations and on the lattice already in the mid-1990's.
Our simulations with the mass $0.01$ produce eigenvalue distributions shown in Fig. \ref{Eigenvaluesm001Dense} and \ref{Eigenvaluesm001}. Note that, in contrast to the zero mass case, one finds a peak
near zero eigenvalue. Eigenvalues outside of the range of the mass, $\lambda>m$ behave as in the massless case, as can be seen by comparing to Fig. \ref{Eigenvaluesm0Dense} and \ref{Eigenvaluesm0}. In the range of $\lambda = m$ the distribution is smoothed due to the singularity at $\lambda\rightarrow 0$. The same behavior is seen on the lattice \cite{Dick:2015twa}, even when a gap appears.
\subsection{Gaps of the eigenvalue distribution}\label{GapWay}
At high temperatures --or very dilute dyon ensembles, in our model -- the chiral symmetry
remains unbroken. As it has been shown in multiple lattice simulations, in this case
the Dirac eigenvalue distribution develops a finite
gap, between $\lambda=0$ and the point where the eigenvalue distribution starts to rise.
Vanishing of this gap therefore provides another way of observing the location of the chiral symmetry breaking. Not to confuse it with the critical temperature obtained from the other method,
we call this temperature for $T_{gap}$.
The procedure used is explained by
an example shown in Fig. \ref{fig_gap}:
we fit the distribution by a straight line, and use its intersection with the x-axis as the measure for the gap.
The fact that a gap appears, means that the lowest excitations are not massless.
\begin{figure}[h]
\includegraphics[scale=0.5]{LineGap_fitValues.jpg}
\caption{The eigenvalue distribution for 64 dyons at $S=7.5$, $\nu=0.434$, $N_F=2$, $n_M=0.43$ and $n_L =0.22$. A straight line has been fitted through point 3 to 6 from the left. The gap size is defined as the cross point with the x-axis. }
\label{fig_gap}
\end{figure}
\section{Data and analysis}\label{SectionData}
The setting has already been explained above.
An ``update cycle" is defined as a sequence of Metropolis updates of all coordinates of of all dyons.
Each ``run" consisted of 4000 such ``update cycles", out of which the typical thermal relaxation time was of the order of 500 cycles. The ``useful data" selected were the mean action values collected for the last 1000 cycles.
The free energy of the model, depending on its parameters, is determined from
the integrated expectation value of the action $<S(\lambda)>$, following a standard approach
\begin{eqnarray}
e^{-F(\lambda)} &=& \int D x e^{-\lambda S}\\
F(1) &=& \int _0 ^1 <S(\lambda)> d\lambda +F(0)
\end{eqnarray}
An example of the lambda dependence is illustrated in Fig. \ref{fig_lambda}.
The quick descent in the expectation value of the action at small $\lambda$
required more measurement points in the range $\lambda=0..1$. Therefore we had
a step size of $1/90$ until $\lambda=0.1$, while for larger lambda the step size is increased to 0.1.
These values, shown in the upper two rows of the Table \ref{tab2}, constitute 19 runs.
The next three rows of the Table \ref{tab2} correspond to three parameters of the model
used for free energy minimization. (Those are the value of the holonomy $\nu$, the radius of the system
defining the total dyon density and the number of $M$-type dyons $N_M$.)
This three-dimensional space was canned systematically, in a lattice form defined by min and max values and
a step defined in the Table. This was done for all values of two remaining ``input parameters",
the Debye mass $M_d$ and classical action $S$. This gives 67200 different combinations.
\begin{table}[h]
\begin{tabular}{ l | c | c | r }
\hline
& Min & Max & Step size \\
\hline
$\lambda$ & 0 & 0.1 & 1/90 \\
$\lambda$ & 0.1 & 1.0 & 0.1 \\
$\nu$ & 0.175 & 0.525 & 0.025 \\
$r$ & 1.05 & 2.00 & 0.05 \\
$N_M$ & 16 & 26 & 2 \\
$M_d$ & 3 & 6 & 1 \\
$S$ & 5 & 9.5 & 0.5 \\
\hline
\end{tabular}
\caption{The input parameters used for the final run.
}
\label{tab2}
\end{table}
\begin{figure}[h]
\includegraphics[scale=0.5]{Lambda_dependence.jpg}
\caption{A typical example of the expectation values of the action $<S>$ obtained from the simulation as a function of $\lambda$. Contribution to the free energy from the overall constant $F(0)$ is not included. }
\label{fig_lambda}
\end{figure}
\subsection{Data Analysis}
After the integration over lambda is done, the values of the free energy for each combination of parameters
are determined. The main part of the data analysis is the fit, defining dependence of the free energy
in the 3-dimensional space (of two dyon densities and holonomy) near its minimum. We therefore fit this
set of data with a 3-dimensional parabola
\begin{eqnarray}
f &=& (v-v_0)M(v-v_0)+f_0
\end{eqnarray}
which has 10 variables. $v$ and $v_0$ are 3D vectors with $v$ containing the variables holonomy $\nu$, radius $r$, and number of M dyons $N_M$ and $v_0$ describing the correction to the point that were the minimum. $M$ is a 3 times 3 matrix with $M=M^T$ containing the coefficients for the fit.
This expression was fitted to
free energy values of $5^3=125$ points from a cube, containing
5 points around the minimum in each direction. The resulting values of the $10$ parameters fitted
are used as follows:(i) $v_0$ and its uncertainties give the values of densities and holonomy
at the minimum, plotted as results below; (ii) the diagonal component of $M$ in
the holonomy direction was converted into the value of the Debye mass $M_d$.
An additional requirement of the procedure, to make the ensemble approximately self-consistent, is that the Debye mass
from the fit should be within
$\pm 0.5$ of the used input Debye mass value.
To obtain the
chiral properties -- such as the Dirac eigenvalue distributions and its dependence on dyon number and volume --
we only used the ``dominant" configurations for each action S,
defined as follows. Since $N_M$ is always an integer, we use the value closest to that obtained from the fit.
The eigenvalue distributions is then analyzed as explained in section \ref{SizeEffects}.
\section{Physical Results}\label{Results}
Accurate gauge-independent
determination of the hopping matrix element Eq. (\ref{eqn_hop}) is, in general, not a trivial
procedure. While zero modes for a single dyon are well known, combining a pair of $L$ and $\bar{L}$ dyons
is not as simple as it is for instantons: the complication is caused by magnetic charges and the Dirac
strings associated with them, transporting singular magnetic flux
to their centers. Ideally those are invisible pure-gauge artifacts, whose direction
is irrelevant: but it is not so for simple configurations like the sum ansatz. ``Combing gauge factors", which appear
in the zero mode wave function, complicate the
calculation, although numerically their effect is relatively small: see more in Appendix A of \cite{Liu:2015jsa}.
Currently we are working on solving
the Dirac equation for ``streamline"
configurations defined in \cite{Larsen:2014yya}, but this work is not yet finished.
As a temporal solution, we use two paramaterizations of the hopping matrix element.
We perform simulations with both sets.
The parameterizations themselves are explained in the Appendix.
The physical results are, respectively, split up into two sections, one for each choice of $T_{ij}$. Since the overall constant $c'$ is unknown, values of $c'$ have been chosen, such that the transition happens around $S=7.5$. We are actively trying to obtain $c'$ from numerical simulations. While the different $T_{ij}$'s behave similar for large distances, the behavior is different around zero. This also means that the constant $c'$ can be different in the two cases. For these results $c'$ was chosen such that the density of L dyons didn't become too small, while having a smooth Polyakov loop that went to zero in the range of $S=5-10$.
The plots below have two scales, on their bottom and top. The former one shows the ``instanton action" parameter $S$,
one of the major parameters of the model controlling the diluteness of the ensemble. We also indicate at the top the corresponding temperature, relative to the critical temperature $T_c$, chosen as $S=7.5$. It should be noted that this is a choice, and is done in order to set a scale. The real input is the action $S$ or the coupling constant $g$. The temperature is found from the running coupling constant.
\begin{eqnarray} S(T)={8\pi^2 \over g^2(T)}=b \cdot ln\left({T\over \Lambda}\right), \,\, b={11\over 3} N_c -\frac{2}{3}N_F,\end{eqnarray}
This top temperature scale is
approximate and should only be used for qualitative comparison to other models and lattice data.
\subsection{Parameterization A for $T_{ij}$}\label{A}
The results in this subsection are for
\begin{eqnarray}
T_{ij} & =& \bar{v} c'\exp{\left(-\sqrt{11.2+(\bar{v}r/2)^2}\right)}
\end{eqnarray}
Minimizing the free energy gives the dominating parameters for a specific action S or Temperature T. This is done for $\Lambda =4$ and $-Log(c')=-2.60$. This gives the holonomy, the density, Fig. \ref{Density_lamb28c2}, and Debye mass, Fig. \ref{DM_lamb28c2}. The dominating configurations have been analyzed using the methods described in section \ref{EigenDestribution} in order to obtain the chiral condensate, which is shown together with the Polyakov loop in Fig. \ref{PandCC_lamb28c2} and is also compared to the gap in Fig. \ref{gab4}.
We observe a smooth transition towards zero expectation value of the Polyakov loop $P$ as temperature decreases. We also observe a non-zero value of the Chiral condensate as temperature decreases. This is a more abrupt change, though in some way still smooth. Its inflection point (change of curvature) is found around $S=7.5$, though the transition happens between $S=6.5-8$. Below $S=7$ the results fluctuate around a constant.
The chiral symmetry breaking can also be observed through the shrinking of the gap around zero as shown together with the chiral condensate in Fig. \ref{gab4}. Again, thinking of the inflection points of the two curves, we conclude from it
that the critical temperature for chiral condensate and the gap do coincide within errors,
at the same $S=6.5-8$ point.
Confinement and chiral symmetry are therefore different phenomena, but are both triggered by the increase in the density of dyons.
The Debye mass, Fig. \ref{DM_lamb28c2}, as compared to lattice results \cite{Kaczmarek:2005ui}, is seen to be around $66\%$ too large. This could be due to the choice of working with a hard core, or it could signal that the correct value for the size of the core is slightly larger.
\begin{figure}[h]
\includegraphics[scale=0.50]{Fit_density_square.jpg}
\caption{(Color online) Parameterization A: The density of the $M$ (blue circles) and $L$ (red squares) dyons as a function of action $S=8\pi^2/g^2$ or temperature $T/T_c$.}
\label{Density_lamb28c2}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.5]{Fit_PandCC_square_fitValues.jpg}
\caption{(Color online) Parameterization A: The Polyakov loop $P$ (blue circles) and the chiral condensate $\Sigma$ (red squares) as a function of action $S=8\pi^2/g^2$ or temperature $T/T_c$. A clear rise is seen around $S=7.5$ for the chiral condensate. $\Sigma$ is scaled by 0.2. The black constant line corresponds to the upper limit of $\Sigma$ under the assumption that the entire eigenvalue distribution belongs to the almost-zero-mode zone, i.e. the maximum of $\Sigma _2$. }
\label{PandCC_lamb28c2}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.5]{Fit_GAPandCC_square_fitValues.jpg}
\caption{(Color online) Parameterization A: The gap scaled up 15 times (blue circles) and the chiral condensate $\Sigma$ (red squares) as a function of action $S=8\pi^2/g^2$ or temperature $T/T_c$. A clear rise/fall is seen around $S=7-7.5$. We get a critical temperature from $S=6.5-8$ for the condensate and $S=6.5-8$ for the gap. $\Sigma$ is scaled by 0.2. The black constant line is defined in the caption of Fig. \ref{PandCC_lamb28c2}.}
\label{gab4}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.50]{Fit_Md_square.jpg}
\caption{Parameterization A: Debye mass $M_d$ as a function of action $S=8\pi^2/g^2$ or temperature $T/T_c$.}
\label{DM_lamb28c2}
\end{figure}
\subsection{Parameterization B for $T_{ij}$ }\label{B}
The results in this subsection are for
\begin{eqnarray}
T_{ij} & =& \bar{v} c'\frac{e^{-\bar{v}r/2}}{\sqrt{1+\bar{v}r/2}} \label{T2}
\end{eqnarray}
with $-\log (c') = -0.388$ and $\Lambda = 3.2$.
Just as for the other choice of $T_{ij}$ discussed in the previous subsection, we obtain the parameters of density, Fig. \ref{Density_lamb32c1_Norm}, holonomy (Polyakov loop Fig. \ref{PandCC_lamb32c1_Norm}), and Debye mass, Fig. \ref{DM_lamb32c1}, as a function of temperature by minimizing the free energy. The chiral condensate Fig. \ref{PandCC_lamb32c1_Norm} and \ref{gab32}, and gap width Fig. \ref{gab32}, have been obtained from configurations with the parameters obtained by minimizing the free energy. The main difference between the two choices of $T_{ij}$ comes from the behavior around $r=0$.
The almost exponential behavior as shown in Eq. (\ref{T2}), means that $L$ dyons become more likely at high densities. The other thing is that it is harder to make the different elements in $T_{ij}$ of similar size, which results in a scaling behavior of the chiral condensate that only becomes around $37\% \pm 10\%$ of the volume, and not $100\%$ as with the other choice of $T_{ij}$. This does not mean that the chiral condensate which we show in Fig. \ref{PandCC_lamb32c1_Norm} does not exist, but it does mean that we need a larger volume in this case to obtain a cleaner result.
It also means that the overlap between almost-zero-modes and dyon-antidyon pairs was larger.
\begin{figure}[h]
\includegraphics[scale=0.50]{Fit_density_Normal.jpg}
\caption{(Color online) Parameterization B: The density of the $M$ (blue circles) and $L$ (red squares) dyons as a function of action $S=8\pi^2/g^2$ or temperature $T/T_c$.}
\label{Density_lamb32c1_Norm}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.50]{Fit_PandCC_Normal_fitValues.jpg}
\caption{(Color online) Parameterization B: The Polyakov loop $P$ (blue circles) and the chiral condensate $\Sigma$ (red squares) as a function of action $S=8\pi^2/g^2$ or temperature $T/T_c$. $\Sigma$ is scaled by 0.1. The black constant line is defined in the caption of Fig. \ref{PandCC_lamb28c2}.}
\label{PandCC_lamb32c1_Norm}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.50]{Fit_GAPandCC_Normal_fitValues.jpg}
\caption{(Color online) Parameterization B: The gap scaled up 20 times (blue circles) and the chiral condensate $\Sigma$ (red squares) as a function of action $S=8\pi^2/g^2$ or temperature $T/T_c$. A fall is seen around $S=7$ for the gap, while it goes close to zero around $S=5-6.5$. At $S=5-6$ the chiral condensate starts to consistently become different from zero.
It should be noted in this case that $2\Sigma _1^{128}/\Sigma _1^{64}-1$ never becomes larger than $37\% \pm 10\%$. $\Sigma$ is scaled by 0.1. The black constant line is defined in the caption of Fig. \ref{PandCC_lamb28c2}.}
\label{gab32}
\end{figure}
\begin{figure}[h]
\includegraphics[scale=0.50]{Fit_Md_Normal.jpg}
\caption{Parameterization B: Debye mass $M_d$ as a function of action $S=8\pi^2/g^2$ or temperature $T/T_c$.}
\label{DM_lamb32c1}
\end{figure}
\section{Conclusion}
We have performed simulations for ensembles of instanton-dyons
for the setting with two colors $N_c =2$ and two quark flavors $N_f=2$, with variable temperature (coupling constant). We have simulated the partition function for 64 and 128 dyons, calculated the free energy, and derived the values of the
Polyakov loop, the chiral condensate and the gaps in the Dirac eigenvalue distributions
at the free energy minimum, for each value of the main external parameter $S$ defining the dyon density.
We also observe gaps in the eigenvalue distribution which goes close to zero in the same interval as the inflection point for the chiral transition.
We find that the required condition for both the chiral symmetry breaking and confinement
is
basically sufficiently high
density of the dyons.
Furthermore, unlike in the case of pure gauge theory without quarks studied in the previous paper,
the holonomy dependence on the density is smoother. We don't observe holonomy vanishing, and also
the densities of the $M$ and $L$ type dyons does not become equal, even at the lowest $T$ we studied.
All of these features make exact determination of $T_c$ difficult and definition-dependent.
It is important to note, that
the repulsive core between the dyons of the same type is essential for these results. For the Polyakov loop expectations value, the core ensures that the holonomy is pushed towards smaller $M$ dyons as density increases, thus making the Polyakov loop expectation value smaller, instead of creating a clump of only $M$ dyons. For the chiral condensate it is important to obtain configurations where the separation from $L$ to $\bar{L}$ dyons are of the same size between the closest dyons, such that the determinant goes from being diagonal dominated between dyon-antidyon pairs to a collective liquid instead.
While the model itself can definitely be improved -- especially the hopping matrix elements can be defined more accurately --
the overall mechanism for obtaining confinement and chiral symmetry breaking appears to be very solid, and should not be qualitatively affected by small changes in the interactions.
The extensions of the model to other values of $N_c,N_f$ are straightforward,
and we expect to be able to do so in the near future. Another obvious direction of improvement
is larger systems, better statistical accuracy and better control over large volume and quark mass
extrapolations.
\vskip .25cm \textbf{Acknowledgments.} \vskip .2cm
This work was supported in part by the U.S. Department of Energy, Office of Science under Contract No. DE-FG-88ER40388.
\vspace{0.5cm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,147 |
{"url":"http:\/\/msp.org\/involve\/2011\/4-4\/p02.xhtml","text":"Vol. 4, No. 4, 2011\n\n Recent Issues\n The Journal Cover Page Editorial Board Editors\u2019 Addresses Editors\u2019 Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Ethics Statement Subscriptions Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print)\nThe family of ternary cyclotomic polynomials with one free prime\n\nYves Gallot, Pieter Moree and Robert Wilms\n\nVol. 4 (2011), No. 4, 317\u2013341\nAbstract\n\nA cyclotomic polynomial ${\\Phi }_{n}\\left(x\\right)$ is said to be ternary if $n=pqr$, with $p$, $q$ and $r$ distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behavior of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family ${\\Phi }_{pqr}\\left(x\\right)$ with $p, $p$ and $q$ fixed and $r$ a free prime.\n\nKeywords\nternary cyclotomic polynomial, coefficient\nPrimary: 11C08\nSecondary: 11B83\nMilestones\nReceived: 21 July 2010\nRevised: 19 July 2011\nAccepted: 15 August 2011\nPublished: 21 March 2012\n\nCommunicated by Kenneth S. Berenhaut\nAuthors\n Yves Gallot 12 bis rue Perrey 31400 Toulouse France Pieter Moree Max-Planck-Institut f\u00fcr Mathematik Vivatsgasse 7 D-53111 Bonn Germany Robert Wilms Sterbeckerstrasse 21 D-58579 Schalksm\u00fchle Germany","date":"2017-05-28 08:47:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 10, \"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\": 0.2726561427116394, \"perplexity\": 6118.153474987382}, \"config\": {\"markdown_headings\": false, \"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-2017-22\/segments\/1495463609610.87\/warc\/CC-MAIN-20170528082102-20170528102102-00052.warc.gz\"}"} | null | null |
package es.um.unosql.subtypes.outliers.impl;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.stream.Collectors;
import java.util.stream.Stream;
import es.um.unosql.subtypes.outliers.OutlierDetector;
import es.um.unosql.uNoSQLSchema.SchemaType;
import es.um.unosql.uNoSQLSchema.StructuralVariation;
import es.um.unosql.uNoSQLSchema.uNoSQLSchema;
public class EpsilonOutlierDetector implements OutlierDetector
{
private double threshold;
private Map<SchemaType, List<StructuralVariation>> outliers;
public EpsilonOutlierDetector(double threshold)
{
this.threshold = threshold;
this.outliers = new HashMap<SchemaType, List<StructuralVariation>>();
}
@Override
public void setFactor(double factor)
{
this.threshold = factor;
}
@Override
public double getFactor()
{
return this.threshold;
}
@Override
public List<StructuralVariation> removeOutliers(SchemaType schemaType)
{
if (threshold < 0.0)
throw new IllegalArgumentException("Epsilon value must be greater than 0");
long numObjects = schemaType.getVariations().stream().mapToLong(var -> var.getCount()).sum();
double countThreshold = Math.round(numObjects * threshold);
List<StructuralVariation> variationsToRemove = new ArrayList<StructuralVariation>();
variationsToRemove.addAll(schemaType.getVariations().stream().filter(var -> var.getCount() < countThreshold).collect(Collectors.toList()));
schemaType.getVariations().removeAll(variationsToRemove);
return variationsToRemove;
}
@Override
public void removeOutliers(uNoSQLSchema schema)
{
for (SchemaType schemaType : Stream.concat(schema.getEntities().stream(), schema.getRelationships().stream()).collect(Collectors.toList()))
outliers.put(schemaType, removeOutliers(schemaType));
}
@Override
public Map<SchemaType, List<StructuralVariation>> getOutliers()
{
return outliers;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,337 |
Q: Why Does the Kafka Producer Close for Each Commit in Flink's `EXACTLY_ONCE` Mode I use the flink-connector-kafka in my Flink application. The semantic is set to EXACTLY_ONCE. I see the following log that indicates that the Kafka producer has been closed and reconnected:
Closing the Kafka producer with timeoutMillis = 0 ms.
Proceeding to force close the producer since pending requests could not be completed within timeout 0 ms.
Looking at the source code, I found the close call from the producer commit function. The commit function calls the recycleTransactionalProducer in finally block, and the recycleTransactionalProducer funcation call the close function, which prints the log. Why is/does the Kafka producer has been close for each commit?
Source code from package:
org.apache.flink.streaming.connectors.kafka;
org.apache.kafka.clients.producer;
@Override
protected void commit(FlinkKafkaProducer.KafkaTransactionState transaction) {
if (transaction.isTransactional()) {
try {
transaction.producer.commitTransaction();
} finally {
recycleTransactionalProducer(transaction.producer);
}
}
}
private void recycleTransactionalProducer(FlinkKafkaInternalProducer<byte[], byte[]> producer) {
availableTransactionalIds.add(producer.getTransactionalId());
producer.flush();
producer.close(Duration.ofSeconds(0));
}
private void close(Duration timeout, boolean swallowException) {
long timeoutMs = timeout.toMillis();
if (timeoutMs < 0)
throw new IllegalArgumentException("The timeout cannot be negative.");
log.info("Closing the Kafka producer with timeoutMillis = {} ms.", timeoutMs);
A: Quoting from http://apache-flink-user-mailing-list-archive.2336050.n4.nabble.com/Problems-with-FlinkKafkaProducer-closing-after-timeoutMillis-9223372036854775807-ms-td39488.html :
... when using exactly-once semantics for the FlinkKafkaProducer, there is a fixed-sized pool of short-living Kafka producers that are created for each concurrent checkpoint.
When a checkpoint begins, the FlinkKafkaProducer creates a new producer for that checkpoint. Once said checkpoint completes, the producer for that checkpoint is attempted to be closed and recycled.
So, it is normal to see logs of Kafka producers being closed if you're using an exactly-once transactional FlinkKafkaProducer.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,835 |
Q: With Rails Minitest, how do tests pass but fail when retested? I'm using transactional fixtures in Minitest, my tests run successfully (and pass) when I first run them:
rake test test/models/number_test.rb
Run options: --seed 31462
# Running:
..
Finished in 0.271344s, 7.3707 runs/s, 7.3707 assertions/s.
2 runs, 2 assertions, 0 failures, 0 errors, 0 skips
However, when I immediately run the again, they fail:
rake test test/models/number_test.rb
Run options: --seed 22968
# Running:
EE
Finished in 0.058652s, 34.0997 runs/s, 0.0000 assertions/s.
1) Error:
NumberTest#test_to_param_is_number:
ActiveRecord::InvalidForeignKey: PG::ForeignKeyViolation: ERROR: insert or update on table "calls" violates foreign key constraint "fk_calls_extension_id"
DETAIL: Key (extension_id)=(760421015) is not present in table "extensions".
: COMMIT
2) Error:
NumberTest#test_twilio's_API_is_configured_to_come_to_this_number's_URL:
ActiveRecord::InvalidForeignKey: PG::ForeignKeyViolation: ERROR: insert or update on table "calls" violates foreign key constraint "fk_calls_extension_id"
DETAIL: Key (extension_id)=(760421015) is not present in table "extensions".
: COMMIT
2 runs, 0 assertions, 0 failures, 2 errors, 0 skips
I'm using schema_plus gem to add foreign keys to my tables.
Because fixtures are loaded in alphabetical order, I'm using the deferrable: :initially_deferred option which only does the referential integrity check at the end of the transaction so all the data is loaded to all tables before the checks. This is what's made the first run of the tests to work… however I'm not sure why it's any different for the second run.
When running the retest, shouldn't all the database tables be emptied and fixtures reloaded using the deferrable option too? It's like deferrable isn't honoured after the first time.
To get it to work, I always have to run rake db:reset between running tests which seems crazy.
Update 1: If I comment out all the fixtures for calls (which actually have nothing to do with any test in number_test.rb), all works fine… I can rerun number tests as often as I like and they still pass. So, it does seem to be something with the deferral.
A: It was a genuine referential integrity problem. numbers and calls both eventually link back to users.
Turns out, the users fixture didn't exist on the test server. That'd do it.
A: Rails will try to disable constraint triggers in the DB to clean it up. You need to have the superuser rol to accomplish this. so do:
I enter sudo -u postgres psql and type:
ALTER USER yourdbusername SUPERUSER;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,920 |
Home Tablets
Android tablets still unwanted? Honeycomb makes up just 1% of Android usage, study finds
By Zach Epstein
Android Honeycomb, first announced by Google at the Consumer Electronics Show in January 2011, is now widely available across dozens of Android tablet models. As Google's first publicly available operating system developed specifically for media tablets, Honeycomb was hotly anticipated ahead of its unveiling, but end users have seemingly not been impressed by the OS or the slates that have emerged carrying the platform. According to a recent study, Honeycomb tablets account for just 1% of Android usage. Read on for more.
Ad network Chitika analyzed hundreds of millions of impressions served to Android devices during the course of one week, and found that Gingerbread accounted for two-thirds of all Android devices. Froyo powered 28% of Android devices and Donut and Cupcake each accounted for 2% of usage. Honeycomb, which is found on all modern Android tablets, was running on just 1% of Android devices drawing traffic from Chitika's network.
"We found that versions, Gingerbread (2.3) and Froyo (2.2) dominate all other Android versions. Together, they make up nearly 95% of total Android traffic in our network,"Chitika's Ryan Cavanagh wrote in a post on the company's blog. "Gingerbread is the real standout dominating the market with a 67% share. This makes sense because Gingerbread is the most recent version on mobile devices. Froyo, the version released just prior to Gingerbread owns 28% of the Android market. Although Honeycomb is a more current version than Gingerbread, it's only available on smart tablets. Our servers see a much higher share of mobile phone traffic than we do smart tablets, (though they are gaining)."
For the time being, Google's next-generation operating system — Android 4.0 Ice Cream Sandwich — is found on only one device. We reviewed the Samsung Galaxy Nexus last month and found the software to be much more cohesive than earlier versions of the Android platform. As Android 4.0 makes its way to tablets in 2012, perhaps Android tablet adoption will be accelerated. In the meantime however, tablets running Honeycomb are seemingly not in high demand.
An analysis of Google's Android platform version tracker suggests that Honeycomb tablets account for just 4.8 million of the 200 million Android devices that have been activated to date.
Android Google honeycomb market share Tablets usage
Zach Epstein, Executive Editor
Zach Epstein has been the Executive Editor at BGR for more than 10 years. He manages BGR's editorial team and ensures that best practices are adhered to. He also oversees the Ecommerce team and directs the daily flow of all content.
Zach first joined BGR in 2007 as a Staff Writer covering business, technology, and entertainment. His work has been quoted by countless top news organizations, and he was recently named one of the world's top 10 "power mobile influencers" by Forbes. Prior to BGR, Zach worked as an executive in marketing and business development with two private telcos.
Zach Epstein's latest stories
Nolah Evolution 15 mattress review
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,592 |
Church, Culture & Religion, Friends, Holidays, Inspired, Lessons Learned, Lifelong Learning, Remembering, Uncategorized, Vacations
Tourist in My Own Town – First Visit to the State Capitol & the Church Beside – Richmond, Virginia
Photo Credit: Wikimedia.org
I love history but am not a very serious student of it. Wanna be, but truth be told, not so much. Children's picture books with real (not revisionist) history as text are about my speed. Seriously, I do appreciate context and seeing puzzle pieces of our stories fit together. What a gift to have someone else bring me along with their children on a history field trip, sort of. That was my yesterday.
A friend, new to Richmond, Virginia, invited me to join her for a trip to the State Capitol downtown. I agreed to drive since I knew my city so well. [Ha!] It was a hot humid day which made it perfect to be inside an air-conditioned government building.
We headed into Richmond via Monument Avenue. A really gorgeous, tree-lined street with huge houses on each side. It's called Monument Avenue because of all the monuments . Most are of Confederate generals atop their horses. I told my passengers that there's some sort of code about the hewn statues – denoting, by the position of horse and rider, whether the generals survived the waror not. Well, it turns out that's a myth. Strike one for the city "insider".
We missed our turn into the city on purpose to drive across the James River by way of the Belvidere (Robert E. Lee) Bridge. Richmond is a striking city with the James running through it. On the U-turn back toward the city, I pointed out the Virginia War Monument and then what I thought was the Capitol Building, right behind, on a grassy knoll that slopes right down to the River. Wrong! Strike two. [I still don't know what that great white columned building is. Anybody?]
Strike three for me was assuming there would be parking attached to the Capitol building. I circled and circled and circled. We finally called the Capitol information line and found that St. Paul's Episcopal Church a couple of blocks away offers their parking lot for $5/hour. That's a deal in downtown Richmond, if you can't get a metered space.
The Virginia State Capitol was designed by Thomas Jefferson. It is magnificent. There are free guided tours or we could meander around on our own. The state legislators were not in session, but the halls themselves made us feel welcome. As did the lovely lady at the information table in the rotunda.
I am not going to do a guided tour – you, like me, have your own level of interest in history. I have just captioned a few of the pictures I took. You should visit your state capitol. I came away with a much greater appreciation of the cost of liberty and the processes of state government.
"Brothers" statue depicted the poignancy of reuniting after fighting on opposite sides of the Civil War.
The stairwells and marble floors had the look of a grand hotel.
President George Washington – the only statue he posed for, they say; life-sized rendering. [Let me know if that's a myth or not. The statue was definitely life-sized. That I could tell.]
After exiting the Capitol building, we made our way around the grounds to the Governor's Mansion (which was open to the public).
The old meets the new in the Governor's Mansion. Period antiques throughout the main floor and lacrosse sticks belonging to the Governor's children at the front door.
So many fascinating persons from our history displayed in portraits, statues, and busts. Many were of Confederate generals, US political figures, and foreign dignitaries. Then there were others of great and different import – civil rights champion Oliver W. Hill, Jr., and Pocahontas in pearls.
We covered the Capitol Building, Governor's Mansion, and grounds in 1 1/2 hours. That was fast. So if you're visiting your state capitol, you might want to take more time. On our way back to the car, we stopped inside St. Paul's Episcopal Church (on the advisement of our new friend in the Capitol rotunda.
Photo Credit: St. Paul's Episcopal Church
The sanctuary of the church was massive with stunning stained glass windows. The sun was pouring in and it was like a gallery of art pieces depicting the life of Christ. While we walked the perimeter of the church, the organist was at the keyboard of the pipe organ housed in the balcony of the church. Maybe he was practicing. For us it was like a private concert.
Our young newcomers who had stayed tuned in to our self-guided tour were done…as were their Mom and I.
Leaving downtown, we scooted around Virginia Commonwealth University to my favorite pizza joint there – Piccola Italy on Main.
Now, you can take a morning to see Virginia's State Capitol…or your own. I only visited the U.S. Capitol once, and never visited my home state's Capitol in Atlanta, Georgia. Hopefully you can avoid getting lost and hit a homerun your first time out. It was a win for us, in the end, for sure.
Virginiacapitol.gov
Virginia Capitol Tourists' Guide
A Self-Guided Tour of the Virginia State Capitol (pdf)
TripAdvisor – Virginia State Capitol Building
10 Buildings that Changed America
St. Paul's Episcopal Church, Richmond, Virginia
Piccola Italy Pizza and Subs
Civil RightsCivil WarConfederacycontextfield tripgovernmenthistoryHometownJames Riverlife of JesusOliver Hills JrPipe organPocahontasRobert E. LeeSt. Paul's Episcopal Churchstained glass windowsstaycationTouristVirginia State Capitol
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2 thoughts on "Tourist in My Own Town – First Visit to the State Capitol & the Church Beside – Richmond, Virginia"
That was wonderful Debbie! VA is my home state and I had numerous historical school field trips, but as a kid they always bored me. I was able to see a few things as an adult and appreciated it more. I'm heart broken to learn that the Monument Ave statue poses are a myth! 🙁 Glad you all had a good time.
admindeb says:
Thanks, Sharon, for commenting. There are so many beautiful, fascinating sights in Richmond. History has its darker side, and Richmond certainly has more than its share of that kind of history. I don't believe we should get rid of Civil War monuments, but to keep them before us, remembering that grievous time and the great internal cost of that war…and what must still be fought for. Anyway…long lament on the statue pose myth. That was a bummer for me, too. Blessings! | {
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\section{Introduction}
Fifth force experiments such as the Cassini satellite experiment
put stringent bounds on the gravitational coupling of nearly
massless scalar particles. Future satellite tests of fifth forces
and putative violations of the equivalence principle will even
lead to stronger constraints. As no scalar field has ever been
observed, these bounds would not be so dramatic if the existence
of nearly massless fields was not suggested by the late time
acceleration of the universe expansion\cite{supernovae1,supernovae2}. In fact, these constraints
have fundamental consequences for models of dark energy. Indeed,
models of dark energy known as quintessence\cite{wetterich,ratra,stein,quintreview}
require the existence of a runaway scalar field with a tiny mass now,
of order $H_0\sim 10^{-43}$ GeV. The range of the interactions mediated
by the quintessence scalar field is of order of the Hubble horizon size.
Hence, unless the quintessence field has a very small
gravitational interaction with ordinary matter, fifth force
experiments are not compatible with a quintessence scenario. For
instance, embedding quintessence models in spontaneously broken
supergravity proves to be extremely difficult as the gravitational
couplings are generically large\cite{braxmartin}. Within string
theory, the dilaton has been argued to be a quintessence candidate
provided the coupling to matter is universal and possesses a
minimum playing the role of an attractor where all gravitational
problems are evaded\cite{damour}. Of course, it would be extremely
interesting to confirm this possibility explicitly. String moduli
fields are also natural candidates for quintessence.
Unfortunately, their gravitational coupling is generically of
order one. On the other hand, there exists a well-motivated scalar
field with a small gravitational field: the radion measuring the
inter-brane distance in Randall-Sundrum scenarios. In this case,
the gravitational coupling of the radion to matter on a warped
brane is suppressed by the warp factor and becomes very small for
a large radion.
Chameleon field theory combine both a quintessence-like behaviour
leading to dark energy at late time and a gravitational coupling
to matter which can be large\cite{cham1,cham2}.
So how come they are not definitely
ruled out by fifth force experiments? In fact, it is useful to
draw an analogy with photons. In some circumstances, photons do
get a mass which alters their properties. This is notoriously the
case in superconductors where the Meissner effect (the fact that
the magnetic field is expelled from a superconductor) can be seen
as the result of the Higgs mechanism with a mass given to the
photons\cite{weinberg2}. In less extreme situations, like in a crystal, photons
are slowed down when interacting with matter. Similar phenomena
can occur for scalar fields. Typically, scalar particles have an
effective potential obtained as a combination of the bare
potential appearing in the Lagrangian and a term proportional to
the matter density. This effective potential may have a density
dependent minimum. In this case, we call the field a chameleon as
its mass depends on the environment.
Chameleon fields are generically more massive in a dense
environment. This is enough to evade the gravitational bounds in
most cases. Indeed, the range of the chameleon mediated force
becomes too small to be detected. Even when this is not the case,
chameleon theories may enjoy another non-trivial property: the
existence of thin shells. More precisely, the field created by a
massive object may be essentially trapped inside the massive body.
In this case, the interaction between massive bodies is
essentially non-existent. Combining these two effects, one can
build satisfying examples of chameleon theories. We will review
their main properties here.
Recently, the PVLAS experiment has measured the dichroism of light
propagating through a magnetic field\cite{PVLAS}. This can be understood by
coupling a scalar field to photons. In this case, one can use the
environment dependent mass of chameleon fields to generate a large
mass for the chameleon in the sun. Therefore, chameleons would not
be produced by the Primakov effect and therefore the CAST
experiment\cite{CAST} would not see the photon regeneration by inverse
Primakov effect. We will present some of these ideas very briefly.
\section{Scalar-Tensor Theories}
\subsection{Coupling to matter}
Chameleon fields appear in scalar--tensor theories of gravity\cite{scalartensor}. We
start with a discussion of these theories.
We consider theories where a scalar field $\phi$ couples both to
gravity and matter, generating a potential fifth force. The
Lagrangian of such scalar--tensor theories reads
\begin{equation}
S = \frac{1}{2\kappa_4^2}\int d^4 x\sqrt{-g} (R- (\partial \phi)^2
-2\kappa_4^2 V(\phi))
\end{equation}
Matter couples to both gravity and the scalar field
according to
\begin{equation}
S_m(\psi, A^2(\phi)g_{\mu\nu}),
\end{equation}
where $\psi$ is a matter field and $A$ is an arbitrary function of
$\phi$. The Klein--Gordon equation can be written in terms of an
effective potential
\begin{equation}\label{Veff}
V_{\rm eff}(\phi)=V(\phi) +\rho_m A(\phi).
\end{equation}
The effective potential depends on the environment through the
matter energy density $\rho_m$. We will assume that $V(\phi)$ is
a runaway potential and for the models we consider $A(\phi)$
increases with $\phi$. In that case the potential has a minimum
whose location depends on $\rho_m$, i.e. on the environment. Such
a field has been called a chameleon field.
The field $\phi$ acts
on all types of matter and, in the Einstein frame, there is a new
force associated with the scalar field
\begin{equation}\label{fifthforce}
F_{\phi} = - \kappa_4 m \alpha_\phi\frac{\partial \phi}{\partial
x_\mu},
\end{equation}
where $m$ is the mass of the test particle in the Einstein frame
and
\begin{equation}
\alpha_\phi = \frac{\partial \ln A}{\partial \kappa_4 \phi}
\end{equation}
The force $F_{\phi}$ cannot be too large, otherwise
experiments would have already detected it.
For massless fields $V(\phi)\equiv 0$ and a point-like matter
source, the Klein-Gordon equation becomes
\begin{equation}
\Delta \phi= -\kappa_4 m \alpha_\phi \vert_{r=0}\delta^{(3)}(r)
\end{equation}
where $m= A(\phi)\vert_{r=0} m_0$ is the Einstein frame mass and
$m_0$ the bare mass of the source. The resulting field $\phi=
-\kappa_4 \alpha_\phi\vert_{r=0} /4\pi r$ leads to a force between
bodies $F_\phi= 2G_N \alpha_{\phi\vert_{r=r_1}}
\alpha_{\phi\vert_{r=r_2}} m_1 m_2 /r_{12}$ where $\kappa_4^2=8\pi
G_N$. This produces a fifth force where
\begin{equation}
F_\phi= 2\alpha_1\alpha_2 F_{\rm Newton}
\end{equation}
and $\alpha_1=\alpha_{\phi\vert_{r=r_1}}$. The Cassini experiments
impose that $\alpha_\phi^2\le 5. 10^{-5}$ for a constant coupling.
Hence massless particles (or nearly massless particles with a mass
less than $10^{-3}$ eV) must have a very small coupling to
gravity. Chameleon field theories enable to overcome this
obstacle.
\subsection{The radion}
A simple and interesting example of non-trivial coupling to
gravity is provided by the Randall-Sundrum scenario where matter
is confined on 4d hyperplanes embedded in an $AdS_5$
vacuum\cite{randallsundrum}. The two boundaries of space-time are
called the UV and the IR brane reflecting the fact that the metric
is warped. Distances on the IR branes are warped down compared to
scales on the UV. Consider now matter on the UV brane of positive
tension. The coupling of matter to gravity depends on the radion
field $\phi$ (for a derivation of the following equations and
references, see e.g. the review\cite{branereview})
\begin{equation}
A(\phi)=\cosh \frac{\kappa_4 \phi}{\sqrt 6}
\end{equation}
where the inter-brane distance is
\begin{equation}
d=-l \ln (\tanh \frac{\kappa_4\phi}{\sqrt 6})
\end{equation}
For small distances compared to the AdS curvature $l$, the
coupling becomes
\begin{equation}
A(\phi)= \frac{1}{2} e^{\kappa_4 \phi/\sqrt 6}
\end{equation}
The gravitational coupling is constant
\begin{equation}
\alpha_\phi= \frac{1}{\sqrt 6}
\end{equation}
Of course, this is too big for the Cassini bound. However, in
this case and in the large interbrane-distance limit, the chameleon mechanism
can be applied to hide the interaction mediated by the radion\cite{chamrad}
by introducing a bare potential for the radion field.
\subsection{Chameleon Cosmology}
We concentrate on a particular model where $A(\phi)=
e^{\beta\phi}$ and $\beta=O(1)$.
We consider the family of potential
\begin{equation}
V=M^4 f((\frac{M}{ \phi})^n)
\end{equation}
where $f$ leads to ordinary quintessence with a long time tracking
solution. A typical example is provided by $f(x)=e^x$. As $\phi
\gg M$ now, the potential is nothing but
\begin{equation}
V=M^4 + \frac{M^{4+n}}{\phi^n}
\end{equation}
Cosmologically, it mimics a cosmological constant. For
gravitational tests, only the Ratra--Peebles part of the potential
matters.
This model satisfies the chameleon property of having a
$\rho$--dependent minimum. As $\beta=0(1)$, the coupling of matter
to the chameleon field is large and may be in conflict with
experiments. We will study the gravitational aspects in the next
section. Here we concentrate on cosmological properties.
In a Friedmann--Robertson--Walker Universe, the (non)-conservation
of matter equation reads
\begin{equation}
\dot \rho +3H \rho =\alpha_\phi \dot \phi \rho.
\end{equation}
leading to
\begin{equation}
\rho=A(\phi) \rho_m,\ \rho_m=\frac{\rho_0}{a^{3(1+w_m)}}
\end{equation}
while the Klein--Gordon equation can be written in terms of an
effective potential
\begin{equation}\label{effpot}
V_{\rm eff}(\phi)=V(\phi) +\rho_m(1-3w_m) A(\phi).
\end{equation}
Let us now go through the different cosmological eras\cite{cham2}. During
inflation the chameleon potential has an effective minimum which
is time-independent. Moreover, as the mass of the chameleon field
at the minimum is $m\gg H$, the field oscillates rapidly and
converges to the minimum extremely rapidly, behaving like a dust
component. By the end of inflation, the field is stuck at the
minimum. As inflation stops and the radiation era starts, the
minimum is pushed far away (as it depends only of non-relativistic
matter). The field is therefore in an overshooting regime where it
becomes kinetically dominated, being far away to the right of the
potential. The field overshoots before stopping at $\phi_{stop}=
\phi_{in}+ \sqrt{6\Omega_\phi^i}m_{\rm Pl}$ where $\Omega^i_\phi$
is the initial chameleon fraction density. After stopping the
field is in an undershooting position. In that case, the field
would remain still until either being caught up by the minimum or
the beginning of the matter era. When caught by the minimum the
field oscillate and converges to the minimum, which is a tracker
solution. This follows from the fact that $m\gg H$ at the minimum
throughout the history of the Universe. The field converges to the
minimum faster than $a^{-3}$ due to the time variation of the
mass at the minimum.
In fact if the field is far away from the minimum after
overshooting, it is sensitive to short bursts when relativistic
particles become non--relativistic
\begin{equation}
\ddot\phi +3H\dot\phi =\frac{\beta}{m_{\rm Pl}}T^\mu_\mu
\end{equation}
as, during such periods, the trace of the energy momentum tensor
$T^\mu_\mu$ of the decoupling species is temporarily
non-vanishing, resulting in a kick\cite{dn} of order of a fraction of
$\beta$. Taking into account all these kicks, the field decreases
by about $\Delta\phi \sim -\beta m_{\rm Pl}$. By BBN, either the
field is close to the minimum, in which case the electron kick
which occurs during BBN does not lead to a large variation of
$\phi$ during BBN, or the field is still far away from the minimum
in which case the electron kick leads to large variations of
$\phi$ and therefore of masses
\begin{equation}
\vert \frac{\Delta m}{m}\vert =\beta \vert \frac{\Delta
\phi}{m_{\rm Pl}}\vert
\end{equation}
the latter case being excluded. As a result, the
initial value of $\phi$ cannot be larger that one and
$\Omega_\phi^i\le 1/6$, a weaker bound than in quintessence. Once
at the minimum by BBN, the field follows the attractor in the
matter era. Once the vacuum energy dominates, the matter density
decreases extremely fast. The chameleon field follows the minimum
until $m\sim H$ where it starts lagging behind eventually having
the same evolution as a quintessence field with no coupling to
matter.
\section{Gravitational Tests}
\subsection{The massive chameleon}
The effective potential with $f(x)=e^x$ leads to a stabilisation
of the scalar field for
\begin{equation}\label{minimum}
\phi= \left(\frac{n \Lambda^{4+n}M}{ \rho}\right)^{1/(n+1)},
\end{equation}
where $\rho$ is the matter energy density . The mass at the bottom
of the potential is given by
\begin{equation}
m^2= n(n+1) \frac{\Lambda^{n+4}}{\phi^{n+2}}
\end{equation}
In the atmosphere, the mass of chameleons is larger than $10^{-3}$
eV implying no consequence for Galileo's Pisa experiment and
similar tests.
\subsection{The thin shell}
Let us now consider a situation where the gravitational
experiments are performed on a body embedded in a surrounding
medium. The body could be a small ball of metal in the atmosphere
or a planet in the inter-planetary vacuum. The effective
potential~(\ref{effpot}) is not the same inside the body and
outside because $\rho _{\rm m }$ is different. The effective
potential can be approximated by
\begin{equation}
\label{approxVeff}
V_{\rm eff}\simeq \frac12 m_\phi^2(\phi-\phi_{\rm min})^2\, ,
\end{equation}
As already
mentioned the minimum and the mass are different inside and
outside the body. We denote by $\phi_{\rm b}$ and $m_{\rm b}$ the
minimum and the mass in the body and by $\phi_{\infty}$ and
$m_{\infty}$ the minimum and the mass of the effective potential
outside the body. Then, the Klein-Gordon equation reads
\begin{equation}
\label{radialKG} \frac{{\rm d}^2\phi}{{\rm
d}r^2}+\frac{2}{r}\frac{{\rm d}\phi}{{\rm d}r}= \frac{\partial
V_{\rm eff}}{\partial \phi}\, ,
\end{equation}
where $r$ is a radial coordinate. Requiring that $q$ remains
bounded inside and outside the body and joining the interior and
exterior solutions, one can determine the complete profile which
can be expressed as
\begin{eqnarray}
\phi_{<}\left(r\right) &=& \phi_{\rm b}+\frac{R_{\rm
b}\left(\phi_{\infty}-\phi_{\rm b}\right)\left(1+m_{\infty }R_{\rm
b}\right)}{\sinh \left(m_{\rm b}R_{\rm b}\right)\left[m_{\infty
}R_{\rm b}+m_{\rm b}R_{\rm b} \coth \left(m_{\rm b}R_{\rm
b}\right)\right]}\frac{\sinh \left(m_{\rm b}r\right)}{r}\, ,
\qquad r\le R_{\rm b}\, , \nonumber \\ \phi_{>}\left(r\right) &=&
\phi_{\infty}+R_{\rm b}\left(\phi_{\rm b}-\phi_{\infty}\right)
\frac{m_{\rm b}R_{\rm b}\coth\left(m_{\rm b}R_{\rm
b}\right)-1}{\left[m_{\infty}R_{\rm b}+m_{\rm b}R_{\rm
b}\coth\left(m_{\rm b}R_{\rm b}\right)\right]}\frac{{\rm
e}^{-m_{\infty}\left(r-R_{\rm b}\right)}}{r} \, , \qquad r\ge
R_{\rm b}\, \nonumber \\
\end{eqnarray}
Assuming, as it is always the case in practise, that $m_{\rm b}\gg
m_{\infty}$, $m_{\rm b}R_{\rm b}\gg 1$, one has
\begin{equation}
\frac{\partial \phi_>(r)}{\partial r}\simeq -\frac{R_{\rm b}}{r^2}
\left(\phi_{\infty}-\phi_{\rm b}\right)\, ,
\end{equation}
from which we deduce that the acceleration felt by a test particle
is given by
\begin{equation}
a=\frac{G_Nm_{\rm b}}{r^2}\left[1+\frac{\alpha
_\phi\left(\phi_{\infty}-\phi_{\rm b}\right)}{\Phi _{_{\rm
N}}}\right]\, ,
\end{equation}
where $\Phi _{_{\rm N}}=G_Nm_{\rm b}/R_{\rm b}$ is the Newtonian
potential at the surface of the body. Therefore, the theory is
compatible with gravity tests if
\begin{equation}
\frac{\alpha _\phi\left(\phi_{\infty}-\phi_{\rm b}\right)}{\Phi
_{_{\rm N}}}\ll 1\, .
\end{equation}
Large compact bodies have a thin shell implying that no distortion
of solar system planetary orbits are predicted. Lunar ranging
experiments are not affected either.
\subsection{Chameleon in a cavity}
Gravitational experiments on earth and future satellite
experiments involve vacuum chambers which can be modelled out as
spherical cavities of radius $R$. Solving the chameleon equations in this
situation, following the same method as in the previous
subsection, we find that the mass of the chameleon field inside
the cavity is determined by the resonance equation
\begin{equation}
\frac{\sinh m_0 R}{m_0R} = n+2
\end{equation}
Having determined $m_0$, one can deduce the value of the field
$\phi_0$ inside the cavity. Notice that for most values of $n$ we
have
\begin{equation}
m_0 R= O(1)
\end{equation}
When $\beta= O(1)$, the mass of the chameleon in gravitational
experiments on earth is of order $1/R$ and is too large to evade
gravitational tests (the range is given by $R\sim 1 $ m).
Fortunately, typical test bodies on earth have a thin shell
implying no deviation from Newton's law for Eotvos or Eotwash experiments\cite{eric}. Future satellite
experiments are such that test bodies do not have a thin shell.
Hence large deviations from Newton's law are predicted. When
$\beta \gg 1$, tests bodies have a thin shell and satellite
experiments would not see any deviation. (For a discussion
of the case $\beta \gg 1$, see reference\cite{shaw}).
\section{PVLAS vs CAST}
Recently, the coupling of a scalar field to photons have been
invoked in order to explain the PVLAS results on dichroism\cite{PVLAS}.
The scalar field is required to have a mass of order
$10^{-12}$ GeV and a coupling strength suppressed by a scale of
order $M=10^6$ GeV. The coupling to photons is given by
\begin{equation}
-\frac{1}{4} \int d^4 x e^{\phi/M} F_{\mu\nu}F^{\mu\nu}
\end{equation}
The results of the PVLAS collaboration are in conflict with
astrophysical bounds such as CAST\cite{CAST}, which for the same
mass for the scalar field, require much smaller couplings
($M>10^{10}$GeV).
The chameleon mechanism can help in explaining the PVLAS results
and, at the same time, be in agreement with astrophysical bounds\cite{PVLASus}.
Our model is of the
scalar-tensor type
\begin{eqnarray}
S&=&\int d^4x \sqrt{-g}\left(\frac{1}{2\kappa_4^2}R-
g^{\mu\nu}\partial_\mu\phi \partial_\nu \phi -V(\phi)
-\frac{e^{\phi/M}}{4} F^2\right)\nonumber \\ &+& S_m( e^{\phi/M}
g_{\mu\nu},\psi_m)
\end{eqnarray}
where $S_m$ is the matter action and the fields $\psi_m$ are the
matter fields. The effective gravitational coupling is given by
\begin{equation}
\beta= \frac{m_{\rm Pl}}{M},
\end{equation}
and therefore very large ($\beta = 10^{13}$) when assuming the
results from the PLVAS experiment ($M=10^6$~GeV). To prevent large
deviations from Newton's law one must envisage non--linear effects
shielding massive bodies from the scalar field. One natural
possibility is that the scalar field $\phi$ coupled to photons has
a runaway (quintessence)--potential leading to the chameleon
effect. For exponential couplings, this is realised when
\begin{equation}\label{poti}
V(\phi)= \Lambda^4\exp (\Lambda^n/\phi^n) \approx \Lambda^4 +
\frac{\Lambda^{4+n}}{\phi^n}
\end{equation}
In the presence of matter, the dynamics of the scalar field is
determined by an effective potential
\begin{equation}\label{effpot1}
V_{\rm eff }(\phi)=\Lambda^4\exp (\Lambda^n/\phi^n)+
e^{\phi/M}(\rho +\frac{{\bf B}^2}{2})
\end{equation}
where $\rho$ is the energy density of non-relativistic matter.
As already mentioned, the PVLAS experiment is in conflict with the
CAST experiment on the detection of scalar particles emanating
from the sun, as it requires $M\ge 10^{10}$ GeV. However, this
bound does not apply when the mass of the scalar field in the sun
exceeds $10^{-5} \rm GeV$. Let us evaluate the mass of the
chameleon field inside the sun. Furthermore, from the effective
potential one obtains
\begin{equation}\label{relati}
m_{\rm sun}= m_{\rm lab} \left(\frac{\rho_{\rm sun}}{\rho_{\rm
lab}}\right)^{(n+2)/2(n+1)}.
\end{equation}
Now $\rho_{\rm sun}/\rho_{\rm lab}\approx 10^{14}$ and, with
$n=0(1)$, one finds
\begin{equation}
m_{\rm sun} \sim 10^{-2} {\rm GeV} \gg 10^{-5} {\rm GeV}
\end{equation}
implying no production of chameleons in the sun. Hence, the CAST
experiment is in agreement with the chameleon model due to the
fact that the chameleon field is very massive in the sun.
\section{Conclusion}
We have given an brief overview of chameleon field theories. They
provide exciting new mechanisms for both gravitation and cosmology.
A scalar field coupled to matter can be problematic, since it
mediates a new force. But if the field self-interacts in a non-linear way,
as it is the case in chameleon field theories, the effect of the
field can be hidden from current experiments. As we pointed out,
future experiments will be able to search for such chameleon fields.
We have speculated that the PVLAS anomaly finds a natural interpretation
within these theories.
\section*{Acknowledgments}
We would like to thank out friends and collaborators on various
aspects of chameleon theories: A. Green, J. Khoury, D. Mota, D.
Shaw and A. Weltman.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,288 |
America's scenic byways offer the finest roadside panoramas
Try not to let the fall colors along the Kancamagus Scenic Byway's route through the White Mountain National Forest distract you from keeping your eyes on the road. | Photo by Erin Paul Donovan/Alamy Stock Photo
By Diane Bair and Pamela Wright | Spring 2021 issue
With international travel restricted over much of the last year and fewer travelers boarding planes, many Americans seeking fresh scenery have rediscovered a love for hitting the road. Among sought-after routes are the 150 scenic byways designated in 46 states.
Congress established America's National Scenic Byways program 30 years ago to recognize, preserve, and enhance select roads throughout the United States based on one or more archaeological, cultural, historic, natural, recreational, and scenic qualities.
"Scenic byways are paths to adventure and meaningful experiences for all of us," writes National Park Foundation President and CEO Will Shafroth. "They offer opportunities to explore national parks, admire stunning geologic features, connect with our shared heritage, and #RecreateResponsibly."
Those opportunities are likely to grow. The Reviving America's Scenic Byways Act of 2019 sought nominations for new roads to be designated, and 63 proposals were under review at press time.
In the meantime, pack the car and hit the road. Here are four beauties: scenic routes where the journey is the destination. Visit fhwa.dot.gov/byways for more information.
New Hampshire: Kancamagus Scenic Byway
One-way distance: 34.5 miles
Locally dubbed the "Kanc," this east-west route slices through the heart of the White Mountain National Forest in northern New Hampshire. The road climbs through a mountain pass and twists and turns as it follows the banks of the Swift and Pemigewasset rivers. Several trails lead to waterfalls or mountaintops. Pullouts and picnic areas are plentiful.
Don't miss: The short walk to Sabbaday Falls, with three tiers of tumbling cascades.
New Mexico: Billy the Kid Trail
Total distance: 84 miles
This historic loop road, flanked by the sprawling Lincoln National Forest in southeastern New Mexico, travels through Old West towns, once the stomping grounds of notorious outlaws and lawmen. Acres of grassy plains and rolling ranchlands bump up against the lofty peaks of the Sierra Blanca and Capitan mountain ranges, which offer expansive vistas.
Don't miss: A visit to Lincoln Historic Site, including the courthouse where Billy the Kid made his most famous escape.
Alabama: Coastal Connection
One-way distance: 130 miles
Slow down and savor the water views (and fresh seafood!) on this southern meander along the Gulf Coast. There are plenty of places to pause, including small fishing villages, like Bayou La Batre, known as the Seafood Capital of Alabama. Still backwaters and wildlife preserves offer opportunities to see nesting sea turtles and hundreds of species of migratory birds.
Don't miss: Fort Morgan State Historic Site, with a museum, nature trails, beaches, and picnic areas.
Kansas: Flint Hills Scenic Byway
One-way distance: 48 miles
Welcome to the Heartland. This ribbon of road in central Kansas conjures up images of early settlers riding wagons westward. You'll have many of the same views: miles and miles of rolling native grasses and wildflowers, comprising North America's largest remaining tallgrass prairie. Break up the trip with stops in historic towns like Council Grove, home to the Kaw Mission State Historic Site and Museum.
Don't miss: Tallgrass Prairie National Preserve, with hiking and biking trails, and ranger-led tours of the Spring Hill Ranch.
Scenic state routes
Some states have established their own byway programs to highlight gorgeous drives that are worth exploring. Here are a few of our favorites that aren't part of the National Scenic Byways program.
Hawai'i: Holo Holo Kōloa
Hear the whispers of ancient ancestors on this south shore drive on Kaua'i, with archaeological, natural, historic, and cultural points of interest. And, yes, scenic ocean views!
Don't miss: McBryde Garden, with the largest ex situ collection of native Hawaiian flora in the world.
Louisiana: Toledo Bend Forest Byway
Skirt the shores of the South's largest man-made body of water, lined with towering stands of stately pines.
Don't miss: South Toledo Bend State Park, a nesting ground for bald eagles, with nature trails and an observation deck.
Washington: North Cascades Highway
Part of the North Cascades Highway in Washington runs through North Cascades National Park. | Photo by James Schwabel/Alamy Stock Photo
Jagged mountain peaks, plunging waterfalls, deep valleys, and pristine lakes provide the backdrop on this sinewy, up-and-down ride through an area dubbed the "North American Alps."
Don't miss: Diablo Lake Overlook with views of impossibly blue waters rimmed by towering mountain peaks.
New England–based writers Diane Bair and Pamela Wright cover food and travel for several publications and are frequent Boston Globe contributors. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,509 |
\section{Introduction}
Class II methanol masers are well established as tracers of an early stage of massive star formation \citep[][and references therein]{Ellingsen2006}. The most prevalent and strongest methanol maser line is the $5_1-6_0A^+$ transition at 6.7 GHz, first discovered by \citet{Menten1991}. To date, over 1000 6.7 GHz methanol masers are known \citep{Pestalozzi2005,Green2009}. The $2_0-3_{-1}E$ transition at 12.2 GHz is not as common or as intense but is generally closely associated with the 6.7 GHz masers \citep{Breen2011a}. \citet{Breen2011a} speculate that sources that have both 6.7 and 12.2 GHz methanol masers may be slightly more evolved than those with only 6.7 GHz masers. The location of the masers in relation to the newly formed star is still not well understood. High resolution observations have shown masers to have a variety of morphologies \citep{Norris1993,Phillips1998,Walsh1998,Minier2002,Dodson2004}, including linear and curved structures and even a perfect ring shape \citep{Bartkiewicz2005}. Some of these structures have been explained as parts of edge-on or inclined disks \citep{Norris1993,Minier2000,Pestalozzi2004}, in outflows \citep{Minier2000} or associated with shocks \citep{Walsh1998,Dodson2004}.
The high intensity of the methanol masers enables them to be easily monitored with smaller radio-telescope dishes, which typically have more time available for long-term monitoring programmes. Masers are expected to be extremely sensitive to changes in their environment, including local conditions in the volume of masing gas (which may affect the maser path length), as well as the radiation field. Class II methanol masers are believed to be pumped by infrared radiation from warm dust \citep{Sobolev1997,Cragg2002,Cragg2005} and the masers amplify the continuum emission from background HII regions. Thus monitoring flux density variations in masers could lead to new insights into massive star formation but interpretation of the variation is complicated by the dependence of the maser intensity on many external factors.
Variability in class II methanol masers was first identified by \citet{Caswell1995}. \citet{Goedhart2004} conducted an extensive monitoring program for four years finding several sources that exhibited periodic or quasi-periodic variations. The source G9.62+0.20E was the first confirmed periodic maser \citep{Goedhart2003} exhibiting simultaneous flares at 6.7, 12.2 and 107 GHz \citep{VanderWalt2009}. The source G12.89+0.49 was found to exhibit rapid variations with a period of 29.5 days. Over 100 cycles were observed in the Hartebeesthoek monitoring program and the data for this source were published separately \citep{Goedhart2009}. To date, only two other periodically variable sources have been detected: IRAS 18566+0408 which also shows correlated variability in a formaldehyde maser with a period of 237 days \citep{Araya2010} and G22.357+0.066 which has a period of 179 days \citep{Szymczak2011}.
The range of periods found (between 29.5 to 509 days) is inconsistent with stellar pulsations of main sequence stars, for which periods range between a few hours to 3 days at the most \citep{Moffat2012}. Likewise, stellar rotation rates for massive protostars range from a few days up to 16 days \cite{Nordhagen2006}. Recently, \citet{Inayoshi2013} proposed that periods in the observed range could be explained by pulsation of massive protostars undergoing rapid mass accretion. However, their model does not explain the asymetric flare profiles seen in several of the masers. On the other hand, binary systems could also explain the range of periods detected. A large fraction (69\%) of main sequence binary stars are found in binary systems and this is expected to be a good representation of their properties at birth \citep{Sana2012}. \citet{Apai2007} surveyed the radial velocities of 16 embedded massive stars and found that 20\% of their targets were close binaries. There are a number of ways in which a binary system could modulate maser emission. For example, \citet{Muzerolle2013} have detected periodic variations in the infrared luminosity of a young protostar due to pulsed accretion, modulated by a binary companion. On the other hand, \citet{VanderWalt2011} suggests that a colliding-wind binary (CWB) system can explain both the range of periods and the flare profiles of most of the masers. In the CWB model, the maser flares are due to changes in the background free-free emission caused by a pulse of ionizing radiation passing through partially ionized gas. The pulse of radiation is produced by shocked gas from stellar winds when the stars are at periastron. This model can explain the raipid rise of maser intensity as well as the slower decay when the hydrogen recombines.
In this paper we present the results of approximately 10 years of monitoring of six longer-period sources which have been confirmed to be (quasi-)periodic, from the inception of the monitoring programme up to the failure of the main polar shaft bearing in October 2008.
\section{Observations}
All observations were made with the Hartebeesthoek Radio Astronomy Observatory (HartRAO) 26 m telescope. Observing took place at one to two week intervals, with observations at two to three day intervals if a source was seen to be flaring. The antenna surface at this time consisted of perforated aluminium panels. The surface was upgraded to solid panels during 2003--2004, resulting in increased efficiency at higher frequencies. However, the telescope pointing was affected until the telescope was rebalanced in 2005 April and a new pointing model was derived and implemented in 2005 September. Pointing checks during observations were done by observing offset by half a beam-width at the cardinal points and fitting a two-dimensional Gaussian beam model to the observed peak intensities. The sources were observed with hour angle less than 2.3 h to minimize pointing errors. Pointing corrections after the implementation of the new pointing model were typically around 4 to 10 per cent at 12.2 GHz and 1 to 5 per cent at 6.7 GHz. Any observations with pointing corrections greater than 30 per cent were excluded from the final data set since it was found that these were invariably outliers in the time series.
Amplitude calibration was based on regular drift scan measurements of 3C123, 3C218 and Virgo A (which is bright but partly resolved), using the flux scale of Ott et al. (1994). Pointing errors in the north-south direction were measured via drift scans at the beam half power points and the on-source amplitude was corrected using the Gaussian beam model.
During 2003 the receivers were upgraded to dual polarization, and observations were switched to a new control system and a new spectrometer. Frequency-switching mode was used for all observations.
The observing parameters of the monitoring programme are given in Table 1. Prior to 2003 only left circular polarization was recorded and two different bandwidths were used at 6.7 GHz, depending on the target source's velocity range.
The strong methanol maser source G351.42+0.64 was used as a comparison to identify potential periods induced by the telescope systems. No evidence of periodicity was found. Detailed analysis of this source was presented in \citet{Goedhart2009} and will not be repeated in this paper.
\begin{table}
\begin{center}
\caption{Observing parameters. Average system temperatures and rms noise are given.}
\label{tab:instr}
\begin{tabular}{lrrrrr}
\hline
Observation dates & BW & chan. & vel. res & T$_{sys}$ & rms \\
& (MHz) & & km s$^{-1}$ & (K) & (Jy) \\
\hline
\multicolumn{6}{c}{Rest frequency: 6.668518 GHz}\\
1999/01/17 -- 2003/04/03 & 0.64 & 256 & 0.112 &51 &0.5 \\
& 0.32 & 256 & 0.056 & 51 & 0.5 \\
2003/07/04 -- 2008/09/30 & 1.00 & 1024 &0.044 &70 &0.4 \\
\\
\multicolumn{6}{c}{Rest frequency: 12.178593 GHz}\\
2000/01/30 -- 2003/04/07 & 0.64 & 256 &0.062 &139 &2.0 \\
2003/08/25 -- 2008/08/06 & 1.00 & 1024 &0.048 &99 &0.3 \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Period search methods}
There are many methods of searching for periodicities in unevenly sampled astronomical data, most of which are variations on Fourier transforms \citep[eg][]{Scargle1989} or folding data by a trial period and measuring the dispersion of the data points through a test statistic \citep[eg][]{Stellingwerf1978}. We have tried a number different methods of detecting periodicity and found the Lomb-Scargle periodogram, using the fast algorithm of Press and Rybicki (1989), to be the most sensitive. However, this method can give rise to false detections. We follow the method recommended by \citet{Frescura2008} to determine the false alarm probability function. A cumulative density function (CDF) was constructed from 100000 Monte Carlo simulations of time series of gaussian noise, from which the maximum power in the Lomb-Scargle periodogram was found. The mean and variance of the noise was estimated from emission-free channels in the spectra and the same timestamps as the observations were used. CDFs were calculated individually for each source and transition. In the cases of G9.62+0.20 and G328.24-0.55 the two observational phases had different noise distributions. This was reflected in the synthetic time-series. The effect of oversampling was also investigated, using a conservative factor of 4, and grossly oversampling at a factor of 20. The CDFs are very similar in all cases. Frequencies with false alarm probability $\leq$1e-4 were considered to be significant. Figure~\ref{fig:CDF} shows the CDF for G9.62+0.20 at 6.7 GHz. We conclude from this that a power level of 16.7 corresponds to a false-alarm probability of 0.01\%.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{cdf.eps}}
\caption{The cumulative density function for the maximum power found for periodograms of synthetic noise for G9.62+0.20 at 6.7 GHz. }
\label{fig:CDF}
\end{figure}
Data were typically detrended using an unconstrained first or second-order polynomial prior to period search. Despite this, low-frequency compononents were still found in the data, particularly for some of the maser peaks in G9.62+0.20. \citet{Kidger1992} discuss the problem of confirming periodicity with the intent to predict future behaviour. They state that the data sample should be of very long duration, covering at least six cycles, and the variations should be of large amplitude. More cycles are necessary to confirm periodicity in the case of objects with low amplitude. This is consistent with our own observations. \citet{Goedhart2004} identified a number of sources which were potentially periodic, but had observed only three cycles for some objects. Continued monitoring of G196.45-1.68 (which had shown two and a half sinusoidal cycles) and G316.64-0.09 (three regularly-spaced flares of decreasing amplitude) showed no further evidence of periodic variations.
Epoch-folding was used to verify the frequencies found from the Lomb-Scargle periodogram. It was found that the standard epoch-folding method of finding the maximum $\chi^2$ was not very sensitive to flares of varying amplitude. The Davies L-statistic \citep{Davies1990} appears to be the most sensitive for the sort of time-series that we have observed. In the case of sources which may not be strictly periodic, it is useful to have an estimate of the uncertainty in the periods. We investigated methods recommended by \citet{Leahy1987} and \citet{Larsson1996} but these are dependent on being able to accurately model the wave-form of the pulse. Some sources show significant variation in the pulse shape from one cycle to another. All of the sources appear to have varying amplitudes in each cycle, which adds an additional free parameter. The width of the peak in the epoch-folded periodogram appears to be the most robust way to reflect the uncertainty in the periods. In the case of quasi-periodic sources, this is not due to measurement errors, but is a reflection of the spread in times at which the flares peak.
\section{Results}
\textbf{\textit{The Lomb-Scargle periodograms will be available online as supplementary material.}}
\subsection{G9.62+0.19E}
G9.62+0.20E is the brightest 6.7 GHz methanol maser known and was observed to reach a peak flux density of 7344 Jy on 24 June 2006. It has been found to be associated with the HII region E in the massive star forming complex G9.62+0.20 \cite{Garay1993} and was classified as a hypercompact HII region by \citet{Kurtz2002}. The distance to this region from trigonometrical parallax of the 12.2 GHz masers is 5.2 kpc \citep{Sanna2009}. Infall motions with HCN (4--3) and CS (7--6) lines were found towards the submilliter core in region E with an infall rate of $4.3\times10^{-3} M_\odot$ yr$^{-1}$ \cite{Liu2011}.
The range of variation for all spectral channels for G9.62+0.19E at 6.7- and 12.2 GHz is shown in Figures~\ref{fig:g0096_67_spectra} and \ref{fig:g0096_122_spectra}. We calculate the maximum, mean and minimum over time for each channel. The time series for the peak channels at 6.7 GHz is shown in Figure~\ref{fig:g0096_67_ts} and at 12.2 GHz in Figure~\ref{fig:g0096_122_ts}. At 6.7 GHz, the components at -0.837, -0.222, 1.227 and 5.266 km s$^{-1}$ have been steadily increasing in mean intensity, while the components at 3.027 and 3.729 km s$^{-1}$ have been decreasing. The mean brightness of the components at 12.2 GHz have also been increasing slowly but most notable is the increase in the amplitude of the flares in the 1.250 and 1.635 km s$^{-1}$ components.
Table~\ref{tab:g962-freqs} summarises the periods found from the Lomb-Scargle periodogram and epoch-folding. The time-series were detrended using a third-order polynomial. Several low-frequency components were found with high power, corresponding to long-term trends in the data. Following the criteria of \citet{Kidger1992} we consider a period to be confirmed only if at least six cycles have been observed. The period with the highest power, for any of the spectral features at 6.7 GHz, is 244.4 days. Clear periodicity is seen in the features at 1.227, 1.841, 2.237 and 3.027 km s$^{-1}$ . A harmonic series is seen in the periodograms for 1.841 and 2.237 km s$^{-1}$ . At 12.2 GHz the period with the highest power is 244.0 days. Clear periodicity is seen in the same velocity range as for the 6.7 GHz transition and several features show a harmonic series. The large number of significant frequencies found seem unlikely to be real detections. The periods were investigated further using epoch-folding. Clear peaks are only seen for periods corresponding to $\sim$ 243 days and multiples thereof. Similar results are seen for the 12.2 GHz data. Taking the weighted mean of the epoch-folded periods we find a period of 243.3$\pm$2.1 days. The time-series were folded modulo the other periods derived from the Lomb-Scargle periodogram but no clear waveform was seen.
Delays in flaring between features were estimated using the z-transformed discrete correlation function of \citet{Alexander1997}. We use the 1.227 km s$^{-1}$ feature as the reference. At 6.7 GHz there appears to be a lag of 34 days between the the dominant peak and the -0.222 km s$^{-1}$ feature (Figure~\ref{fig:g0096_67_dcf}). There may be an 8 day lag between 1.183 and 1.841 km s$^{-1}$ . There are smaller lags between the main peak and 2.2 and 3.0 km s$^{-1}$ but these may not be significant given the noise in the correlation function. No lags are found between the features in the 12.2 GHz time series and there is no lag between the same velocity features at 12.2 and 6.7 GHz.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g0096_67_spectra.eps}}
\caption{Range of variation across all spectral channels for G9.62+0.19E at 6.7 GHz during 2003--2008.}
\label{fig:g0096_67_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g0096_122_spectra.eps}}
\caption{Range of variation across all spectral channels for G9.62+0.19E at 12.2 GHz during 2003--2008.}
\label{fig:g0096_122_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g0096_67_ts.eps}}
\caption{6.7 GHz time series for peak velocity channels in G9.62+0.19E.}
\label{fig:g0096_67_ts}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g0096_122_ts.eps}}
\caption{12.2 GHz time series for peak velocity channels in G9.62+0.19E.}
\label{fig:g0096_122_ts}
\end{figure}
\begin{table*}
\caption{Periods found from Lomb-Scargle periodogram and epoch-folding for G9.62+0.19E. Identified harmonic series are indicated in bold text.}
\label{tab:g962-freqs}
\begin{center}
\begin{tabular}{lllllll}
\hline
Velocity & mean & mean & S/N & L-S & E-F period & E-F \\
& flux & rms & & Significant periods & & HWHM \\
& density & noise & & & & \\
km s$^{-1}$ & Jy & Jy & & Days & Days & Days \\
\hline
\multicolumn{7}{c}{6.7 GHz}\\
-1.276 & 30.1 & 1.9 & 15 & 1575, 945.2, 683.2, 187.8 & -- & -- \\
-0.837 & 65.7 & 2.2 & 30 & 1013 & -- & --\\
-0.222 & 564.5 & 8.4 & 67 & 821.9, 443.0, 241.3, 142.1, 131.0 & -- & --\\
1.227 & 5079 & 75 & 637 & 1620, 756.1, 446.5, 268.8, \textbf{244.4,127.4, 81.0, 61.0} & 243.8 & 4.1\\
1.841 & 151.6 & 3.4 & 45 & 450.1, 272.6, \textbf{244.4}, 167.8, 128.0, \textbf{121.7, 81.2, 61.0 } & 243.6 & 3.8\\
2.237 & 45.6 & 2.1 & 22 & 1890, \textbf{243.4}, 128.0, \textbf{122.0, 81.2, 61.0} & 243.7 & 3.4 \\
3.027 &52.0 & 2.5 & 23 & \textbf{246.6}, 225.0, \textbf{122.2, 81.7} & 242.6 & 5.5 \\
3.729 & 17.0 & 1.9 & 9 & 1829 & -- & -- \\
5.266 & 72.0 & 1.4 & 51 & 394.7 & -- & --\\
6.363 & 12.6 & 1.1 & 11 & -- & -- & --\\
\multicolumn{7}{c}{12.2 GHz}\\
-1.010 & 26.6 & 0.9 & 30 & 1071 & -- & -- \\
-0.192 & 28.9 & 0.9 & 32 & 997.5 & -- & --\\
1.250 & 277.5 & 1.8 & 124 & \textbf{244.0, 121.7, 81.1, 60.9, 48.7, 40.6} & 243.2 & 3.6\\
1.635 & 79.2 & 1.3 & 60 & \textbf{244.0, 121.7, 81.1, 60.9, 48.7, 40.6} & 243.5 & 2.2\\
2.115 & 4.7 & 0.9 & 5 & \textbf{245.2, 120.0, 81.3, 60.6} & 243.3 & 6.5\\
3.173 & 12.18 & 0.9 & 14 & \textbf{80.4} & 241.6 & 5.5\\
3.798 & 21.5 & 0.9 & 24 & 803.4, 140.4 & -- & -- \\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{G009.62+0.20_67_full_zdcf.eps}}
\caption{Discrete correlation function between pairs of features at 6.7 GHz for G9.62+0.19E.}
\label{fig:g0096_67_dcf}
\end{figure}
\subsection{G188.95+0.89}
These masers are situated in the star forming region AFGL 5180 or S252. VLBI observations at 6.7 and 12.2 GHz show a linear distribution, with the two transitions spatially co-located \citep{Minier2000}. No continuum radio source has been detected, but the masers are projected on a bright mm source with an estimated mass of 50 M$_\odot$ \citep[MM1 in][]{Minier2005}. \citet{Longmore2006} estimate the mass of the mid-infrared core associated with the methanol masers to be 7 M$_\odot$ based on its luminosity of 8.4 L$_\odot$. Parallax measurements of water masers associated with the star forming region place it at a distance of 2.02 kpc \citep{Niinuma2011}.
Figures~\ref{fig:g1889_67_spectra} and \ref{fig:g1889_122_spectra} show the range of variation in the spectra at 6.7 and 12.2 GHz, respectively. The time series are shown in Figures~\ref{fig:g1889_67_ts} and \ref{fig:g1889_122_ts}. As with G9.62+0.20, the variations are much more pronounced at 12.2 GHz. The first five cycles at 6.7 GHz show a sinusoidal waveform, but the last four cycles show more of a sawtooth pattern. The feature at 11.361 km s$^{-1}$ has been steadily weakening since early 2004 but weak periodic variations can be seen. \cite{VanderWalt2011} explains this decay by recombination of the ionized gas in part of the HII region along the line of sight to this maser.
Two separate polynomials were used to detrend the time series for the feature at 11.3 km s$^{-1}$ prior to period search. The periods above the significance threshold for the Lomb-Scargle periodogram and the periods found from epoch-folding are summarised in Table~\ref{tab:g1889-freqs}. The epoch-fold peaks at 12.2 GHz were asymmetrical and noisy, so they have not been fitted. Using the 6.7 GHz values only we find a weighted mean period of 395$\pm$8 days. No phase lags between features were found.
\begin{table*}
\caption{Periods from Lomb-Scargle periodogram and epoch-folding for G188.95+0.89.}
\label{tab:g1889-freqs}
\begin{center}
\begin{tabular}{lllllll}
\hline
Velocity & mean & mean & S/N & L-S & E-F period & E-F \\
& flux & rms & & Significant periods & & HWHM \\
& density & noise & & & & \\
km s$^{-1}$ & Jy & Jy & & Days & Days & Days \\
\hline
\multicolumn{7}{c}{6.7 GHz}\\
8.420 & 9.8 & 1.1 & 9 & 399.6 & 393.9 & 31.0 \\
9.649 & 29.7 & 1.7 & 25 & 1669, 468.9, 394.0 & 396.0 & 16.4 \\
10.439 & 490.0 & 9.6 & 51 & 1830, 886.4, 472.8, 396.8 & 393.9 & 15.8 \\
10.702 & 520.0 & 9.5 & 55 & 1773.1, 915.2, 472.8, 396.8 & 396.3 & 14.5\\
11.361 & 71.1 & 2.1 & 33 & 1830, 411.2 & 399.6, 395.9, 21.8 \\
\multicolumn{7}{c}{12.2 GHz}\\
10.337 & 127.4 & 1.7 & 76 & 1418, 394 & -- & --\\
10.721 & 160.0 & 1.8 & 89 & 1418, 813, 391 & -- & -- \\
11.010 & 157.7 & 1.6 & 70 & 1418, 827, 387 & --&--\\
11.394 & 19.6 & 1.4 & 14 & 1504 & -- &--\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g1889_67_spectra.eps}}
\caption{Range of variation across all spectral channels for G188.95 at 6.7 GHz during 2003--2008.}
\label{fig:g1889_67_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g1889_122_spectra.eps}}
\caption{Range of variation across all spectral channels for G188.95+0.89 at 12.2 GHz during 2003--2008.}
\label{fig:g1889_122_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g1889_67_ts.eps}}
\caption{6.7 GHz time series for peak velocity channels in G188.95+0.89.}
\label{fig:g1889_67_ts}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g1889_122_ts.eps}}
\caption{12.2 GHz time series for peak velocity channels in G188.95+0.89.}
\label{fig:g1889_122_ts}
\end{figure}
\subsection{G328.24-0.55}
There are two maser groups in the beam, viz. G328.236-0.547 and G328.254-0.532 \citep{Phillips1998}. G328.236-0.547 is broken up into two groups with velocity ranges -46 to -42 km s$^{-1}$ and -37 to -31 km s$^{-1}$ . G328.254-0.547 has features at -50 km s$^{-1}$ and at -40 to -36 km s$^{-1}$ . The latter group is unfortunately blended with G328.254-0.532 and the spectra are probably dominated by the emission from this group. The two velocity groups in G328.236-0.547 are separated by 83 arcsec and lie on either side of an unresolved HII region. \citet{Phillips1998} offer three explanations for this morphology: shock fronts in a bipolar outflow, two clusters on either side of a thick disk, or two separate sources in a binary system. \citet{Dodson2004} imaged this region with the LBA and find that the maser spots may have a linear distribution. However, they find no reason to think that the two regions could be related since there is no alignment in position angle of the maser spots. These regions are estimated to be at the near kinematic distance of 3 kpc.
Figures~\ref{fig:g3282_67_spectra} and \ref{fig:g3282_122_spectra} show the range of variation in the spectra at 6.7 and 12.2 GHz respectively. There is only one spectral feature at 12.2 GHz strong enough for analysis. We see correlated periodic variations across several peaks corresponding to the velocity range covered exclusively by G328.236-0.547. The overall intensity of the peak at -44.751 km s$^{-1}$ has increased almost threefold since the start of the monitoring programme. The average intensity of the features corresponding to G328.254-0.532 do not show the same trend. The 6.7 GHz periodograms are dominated by long-term trends. Epochfolding after a second-order detrend finds a weighted mean period of 220.5$\pm$1.0 days for five features -- -45.410, -44.751, -44.268, -43.259 and -36.410 km s$^{-1}$ . The significant periods found in the Lomb-Scargle periodogram and best-fit periods from epoch-folding are summarised in Table ~\ref{tab:g3282-freqs}. The highest spectral power is found for a period of 220.3 days. The 12.2 GHz epoch folded periodogram is not as sharply delineated as at 6.7 GHz because of the shorter time series. We adopt a weighted mean period of 220.5$\pm$1.0 days. The changing amplitudes of the flares leave the peaks weakly defined. The features corresponding to G328.254-0.547 show correlated, regular variations with a characteristic period close to 300 days but the epoch-fold does not show peaks corresponding to those found by the Lomb-Scargle periodogram and no clear waveform is seen when the time series are folded modulo 300 days. The variations show no correlation with G328.236-0.547. No time-delays were found.
\begin{table*}
\caption{Periods from Lomb-Scargle periodogram and epoch-folding for G328.24-0.55}
\label{tab:g3282-freqs}
\begin{center}
\begin{tabular}{lllllll}
\hline
Velocity & mean & mean & S/N & L-S & E-F period & E-F \\
& flux & rms & & Significant periods & & HWHM \\
& density & noise & & & & \\
km s$^{-1}$ & Jy & Jy & & Days & Days & Days \\
\hline
\multicolumn{7}{c}{6.7 GHz}\\
-49.581 & 41.3 & 2.1 & 20 & 1548, 311 & -- & --\\
-45.410 &55.8 & 2.2 & 25 & 1639, 978, 220.3 & 220.0 & 5.2\\
-44.751 & 805.4 & 12.4 & 65 & 1689, 221.1 & 221.5 & 3.5 \\
-44.268 & 222.0 & 4.0 & 55 & 220.3, 189.6 & 220.0, 5.7\\
-43.259 & 106.8 & 2.7 & 40 & 1798, 220.3 & 219.9 & 5.3 \\
-38.649 & 56.1 & 2.2 & 25 & 1639, 395, 309.7 & -- & --\\
-37.990 & 145.4 & 3.3 & 44 & 1548.3, 392.5, 352.8, 294.9 & -- & --\\
-37.420 & 308 & 5.7 & 55 & 1506, 389.8, 302.9 & -- & --\\
-37.068 & 185.9 & 4.0 & 47 & 1429, 290.3 & -- & --\\
-36.410 & 43.8 & 2.1 & 21 & 1394, 381.8, 218.6 & 220.8 & 8.4\\
\multicolumn{7}{c}{12.2 GHz}\\
-44.798 & 21.2 & 1.1 & 20 & 477.6, 216.3 & -- & --\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3282_67_spectra.eps}}
\caption{Range of variation across all spectral channels for G328.24-0.55 at 6.7 GHz during 2003--2008.}
\label{fig:g3282_67_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3282_122_spectra.eps}}
\caption{Range of variation across all spectral channels for G328.24-0.55 at 12.2 GHz.}
\label{fig:g3282_122_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3282_67_ts.eps}}
\caption{6.7 GHz time series for peak velocity channels in G328.24-0.55.}
\label{fig:g3282_67_ts}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3282_122_ts.eps}}
\caption{12.2 GHz time series for the peak velocity channel in G328.24-0.55.}
\label{fig:g3282_122_ts}
\end{figure}
\subsection{G331.13-0.24}
The maser spots in this source have a linear distribution with a velocity gradient, except for a single component \citep{Phillips1998}. The masers lie at the edge of an extended HII region with an irregular morphology. \citet{Phillips1998} speculate that there may be more than one star embedded in the UC HII region because the masers are offset from the centre. \citet{DeBuizer2009} included this region in their sample to search for outflows associated with linear maser structures. Using ATCA, they find SiO emission centred on the maser location and distributed at a very similar angle to the masers although the outflow may be oriented along the line of sight. There is a 3mm continuum source which overlaps with the cm continuum source but is slightly offset.
Figure~\ref{fig:g3311_67_spectra} shows the spectra at 6.7 GHz with the spots mapped by \citet{Phillips1998} indicated. While there are 12.2 GHz masers associated with this source, they lie below the detection threshold of the 26-m telescope. The 6.7 GHz time-series are plotted in Figure~\ref{fig:g3311_67_ts}. All peaks show repeated flares with a delay between the two velocity groupings. The last flare failed to manifest at the expected time in the second grouping. The second grouping is also showing a gradual increase in the base level intensity. The Lomb-Scargle periodograms show a spread in periods of about 7 days and Table~\ref{tab:g3311_67-freqs}). The spread in derived periods may be due to the flares not repeating very well. The epoch-folded peaks are correspondingly very broad. We adopt a weighted mean period of 509$\pm$10 days from the epoch-folded periodograms. In Figure~\ref{g3311_67_fold} we show the time-series folded modulo 509 days. The delays between the peak of the flares can be clearly seen. It is also clear that the the second group of masers (B, C and D) do not flare as regularly as the first group (H and I). Figure~\ref{fig:g3311_67_dcf} shows the discrete correlation function between the reference feature at -91.505 km s$^{-1}$ (spot I) and the other features. The magnitude of the delay is estimated by fitting a second-order polynomial to the peak of the correlation function but the peaks are quite broad and the derived delays should be treated with care. The changing pulse shapes make it very difficult to identify the exact magnitude of the delay. The feature at H starts to flare at the same time as I, but it appears to peak about 19 days earlier. Feature D flares about 49 days after I, and features C and B flare 59 and 53 days later. Broadly outlined, this implies that the flare propagates from the south-west (closer to the peak of the HII region) to the north-east. The resolution of the SiO maps from ATCA is unfortunately not good enough to determine if the flaring is related to the outflow. We would also need to be able to know the absolute positions of the masers relative to the outflow.
\begin{table*}
\caption{Periods from Lomb-Scargle periodogram and epoch-folding for G331.13-0.24 at 6.7 GHz}
\label{tab:g3311_67-freqs}
\begin{center}
\begin{tabular}{lllllll}
\hline
Velocity & mean & mean & S/N & L-S & E-F period & E-F \\
& flux & rms & & Significant periods & & FWHM \\
& density & noise & & & & \\
km s$^{-1}$ & Jy & Jy & & Days & Days & Days \\
\hline
-91.505 & 4.0 & 0.9 & 5 & 507, 370, 255 & 507.5 & 19.3 \\
-90.802 & 6.5 & 0.9 & 8 & 507, 368, 255 & 507.6 & 15.8\\
-85.578 & 14.3 & 0.9 & 16 & 517, 254 & 512.8 & 28.1 \\
-84.920 & 11.6 & 0.9 & 13 & 816, 503 & 508.2 & 27.7\\
-84.305 & 22.0 & 0.9 & 23 & 512 & 511.9 & 28.5\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3311_67_spectra.eps}}
\caption{Range of variation across all spectral channels for G331.13-0.24 at 6.7 GHz during 2003--2008.}
\label{fig:g3311_67_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3311_67_ts.eps}}
\caption{6.7 GHz time series for peak velocity channels in G331.13-0.24.}
\label{fig:g3311_67_ts}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3311_67_fold.eps}}
\caption{6.7 GHz time series for G331.13-0.24 folded modulo 510 days. The data have been detrended with a first-order polynomial.}
\label{g3311_67_fold}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{G331.13-0.24_67_bb_zdcf.eps}}
\caption{Discrete correlation function between pairs of features for G331.13-0.24.}
\label{fig:g3311_67_dcf}
\end{figure}
\subsection{G338.93-0.06}
This source does not appear to have been mapped at cm wavelengths.
Figures~\ref{fig:g3389_67_spectra} and \ref{fig:g3389_122_spectra} show the spectra at 6.7 and 12.2 GHz. The time series at 6.7 and 12.2 GHz are shown in Figures~\ref{fig:g3389_67_ts} and \ref{fig:g3389_122_ts}. Two of the peaks at 6.7 GHz show correlated periodic variability. The third peak at -41.376 shows variability and also shows significant power in the Lomb-Scargle periodogram. The periods found are summarised in Table~\ref{tab:g3389-freqs}. The 12.2 GHz peak shows the same periodicity as the 6.7 GHz peaks at -42.166 and -41.946 km s$^{-1}$ but is not as well sampled as at 6.7 GHz. The epochfolded periodograms also indicate a remarkable level of periodicity. We derive a weighted mean period using the longer time-series at 6.7 GHz of 132.8$\pm$0.8 days. The wave form of this source is different from the other periodic sources since it shows a very sharply defined minimum and does not have a quiescent period. No discernable delays were found between the two periodic features.
\begin{table*}
\caption{Periods from Lomb-Scargle periodogram and epoch-folding for G338.93-0.06}
\label{tab:g3389-freqs}
\begin{center}
\begin{tabular}{lllllll}
\hline
Velocity & mean & mean & S/N & L-S & E-F period & E-F \\
& flux & rms & & Significant periods & & FWHM \\
& density & noise & & & & \\
km s$^{-1}$ & Jy & Jy & & Days & Days & Days \\
\hline
\multicolumn{7}{c}{6.7 GHz}\\
-42.166 & 24.0 & 0.9 & 26 & 1158, 132.6, 87.4, 84.9, 66.4 & 132.8 & 1.4\\
-41.946 & 31.2 & 1.0 & 32 & 132.6, 66.4 & 132.9 & 1.1\\
-41.376 & 10.9 & 0.8 & 13 & 2837, 1320, 396.8, 297.1, 235.4, 201.9, 86.8 & -- & -- \\
\multicolumn{7}{c}{12.2 GHz}\\
-42.183 & 6.43 & 1.0 & 6 & 132.4 & 132.6 & 4.4 \\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3389_67_spectra.eps}}
\caption{Range of variation across all spectral channels for G338.93-0.06 at 6.7 GHz during 2003--2008.}
\label{fig:g3389_67_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3389_122_spectra.eps}}
\caption{Range of variation across all spectral channels for G338.93-0.06 at 12.2 GHz.}
\label{fig:g3389_122_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3389_67_ts.eps}}
\caption{6.7 GHz time series for peak velocity channels in G338.93-0.06.}
\label{fig:g3389_67_ts}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3389_122_ts.eps}}
\caption{12.2 GHz time series for the peak velocity channel in G338.93-0.06.}
\label{fig:g3389_122_ts}
\end{figure}
\subsection{G339.62-0.12}
The 6.7 GHz spectra are shown in Figure~\ref{fig:g3396_67_spectra} with maser features mapped by \citet{Walsh1998} indicated. The time-series for the peak velocity channels is shown in Figure~\ref{fig:g3396_67_ts}. The features at -37.661, -37.310, -36.959, and -32.92 km s$^{-1}$ show correlated periodic variations. We plotted the maser spots listed in table 2 of \citet{Walsh1998} in Figure~\ref{fig:g3396_spotmap} since it is necessary to understand the variability of the different features. The feature at -30.549 km s$^{-1}$ was not detected by \citet{Walsh1998} but has been steadily increasing in intensity. Table ~\ref{tab:g3396_67-freqs} summarises the peaks in the Lomb-Scargle and epoch-folding periodograms. Each individual feature shows its own uncorrelated variability, in addition to the shared periodic flares. Spots A, B and C form a tight cluster to the north-east of the region and spot D seems to have vanished. Spots E, F and G are slightly dispersed but all show a common periodicity. The same periodicity is probably also present in spot B. We adopt a weighted-mean period of 200.3$\pm$1.1 days. The uncorrelated variability as well as the destruction and formation of new maser spots seems to point to volatile local conditions. No discernable delays were found.
\begin{table*}
\caption{Periods from Lomb-Scargle periodogram and epoch-folding for G339.62-0.12 at 6.7 GHz}
\label{tab:g3396_67-freqs}
\begin{center}
\begin{tabular}{lllllll}
\hline
Velocity & mean & mean & S/N & L-S & E-F period & E-F \\
& flux & rms & & Significant periods & & HWHM \\
& density & noise & & & & \\
km s$^{-1}$ & Jy & Jy & & Days & Days & Days \\
\hline
-37.661 & 39.3 & 1.2 & 32 & 685.5, 201 & 200.9 & 6.5 \\
-37.310 & 23.4 & 1.1 & 22 & 1606, 1222, 892, 703, 557, 199 & 198.7 & 4.6 \\
-36.959 & 25.1 & 1.1 & 23 & 1249, 200 & 200.6 & 2.7 \\
-35.729 & 106.4 & 2.0 & 54 & 2677, 1196, 446, 201 & 200.7 & 8.5 \\
-33.490 & 38.4 & 1.3 & 31 & 1479 & -- & --\\
-32.920 & 83.4 & 1.7 & 50 & 1653, 936, 202& 201.0 , 5.1 \\
-32.261 & 37.3 & 1.2 & 31 & 1653, 892 & -- & --\\
-30.549 & 7.3 & 1.0 & 7 & 2555 & -- & --\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3396_67_spectra.eps}}
\caption{Range of variation across all spectral channels for G339.62-0.12 at 6.7 GHz during 2003--2008.}
\label{fig:g3396_67_spectra}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3396_67_ts.eps}}
\caption{6.7 GHz time series for peak velocity channels in G339.62-0.12.}
\label{fig:g3396_67_ts}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{g3396_spotmap.eps}}
\caption{Distribution of maser spots for G339.62-0.12 from table 2 of \citet{Walsh1998}}
\label{fig:g3396_spotmap}
\end{figure}
\section{Discussion}
We summarise the confirmed periods in Table~\ref{tab:vital_stats}, including the number of observations, the time-span covered and the number of cycles observed. The observations cover a sufficiently long time-span to confirm true periodicity in these sources. The range of periods observed (including the 29.5 day period reported for G12.89+0.49) can only be readily explained by orbital motions, most probably due to a binary system.
\begin{table}
\caption{Summary of periods and the observation statistics}
\label{tab:vital_stats}
\begin{center}
\begin{tabular}{lrrrrr}
\hline
Source(transition) & Period & $\sigma$P & num. & time span & num. \\
& (days) & (days) & obs &(days ) & cycles\\
\hline
G9.62+0.19E (6.7 GHz) & 243.3 & 2.1 & 708 & 3544 & 14.5 \\
G9.62+0.19E (12.2 GHz) & & & 617 & 3111 & 12.7 \\
G188.95+0.89 (6.7 GHz) & 395 & 8 & 488 & 3544 & 9.1 \\
G188.95+0.89 (12.2 GHz) & & & 373 & 3102 & 7.9 \\
G328.24-0.55 (6.7 GHz) & 220.5 & 1.0 & 320 & 3483 & 15.8 \\
G328.24-0.55 (12.2 GHz) & & & 112 & 1432 & 6.5 \\
G331.13-0.24 (6.7 GHz) & 509 & 10 & 318 & 3519 & 6.9 \\
G338.93-0.06 (6.7 GHz) & 132.8 & 0.8 & 653 & 3544 & 26.7 \\
G338.93-0.06 (12.2 GHz) & & & 123 & 1431 & 10.7 \\
G339.62-0.12 (6.7 GHz) & 200.3 & 1.1 & 315 & 3513 & 17.5 \\
\hline
\end{tabular}
\end{center}
\end{table}
Figure~\ref{fig:norm_fold} shows a comparative diagram of the normalised folded wave forms for a representative feature from each source. The normalised fold was produced by dividing the time-series into cycles based on its best-fit period, and the flux density measurements in each cycle were divided by the maximum recorded in the cycle. The time-axis was normalised by dividing by the period. Some generalisations can be made regarding the waveforms. G9.62+0.20, G328.24-0.55 and G339.62-0.12 show sharply-peaked asymmetric pulse profiles, while G188.95+0.89 is closer to sinusoidal. The simple colliding wind binary model of \citet{VanderWalt2011} is able to reproduce these waveforms by modification of the orbital parameters of the binary system.
The behaviour of G338.93-0.06, on the other hand, with very sharply-defined minima and no quiescent phase, cannot be explained by the colliding wind binary model. In the model, changes in the free-free emission amplified by the maser is due to a combination of the orbital motion of the secondary star and the partial recombination of the ionized gas at the ionization front. The peak of the maser flare corresponds to periastron passage while the minimum corresponds to apastron. Under the assumption of the adiabatic cooling of the shocked gas, the luminosity of shocked gas varies as $1/r$ where $r$ is the distance between the two stars. The luminosity and therefore the flux of ionizing photons increases rapidly as periastron is approached followed by a slower decay due to the recombination of the partially ionized gas in the ionization front for a rather eccentric orbit. This explains the asymmetric flare profile of some of the maser sources such as seen in, for example, G9.62+0.20E. However, the $1/r$ dependence will always result in a rather slow increase of the flux of ionizing photons just after apastron. It is therefore difficult to explain the sharp increase in the maser flux density as in the case of G338.93-0.06 within the framework of the colliding-wind binary model.
G331.13-0.24 with its changes in pulse profile between cycles and features is also not consistent with a simple colliding wind binary model. We note that G12.89+0.49, while having the shortest period known, also has similar characteristics to G331.13-0.24, with a very stable phase for the minima of the light curve but varying phase for the peak.
G331.13-0.24 is the only source in this sample associated with a known outflow. It is possible that the methanol masers, while not directly associated with the outflow, could be amplifying emission from an episodic or precessing outflow. One of the mechanisms put forward by \citet{MacLeod1996} for strong phase lags in methanol maser flares in G351.78-0.54 was an intermittent thermal jet. A strong bipolar outflow has subsequently been found towards this source \citep{Leurini2009}.
There is also a very close correspondence between 12.2 and 6.7 GHz waveforms, where we are able to achieve sufficient signal to noise, indicating a common mechanism. \citet{VanderWalt2009} have also observed simultaneous flaring at 107 GHz for G9.62+0.19E. The 12.2 GHz masers show higher amplitude variations, in general.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[clip]{folded_normalised_wave_forms.eps}}
\caption{Normalised folded waveforms for each periodic source. The dots are for 6.7 GHz measurements and the crosses are for 12.2 GHz, where available.}
\label{fig:norm_fold}
\end{figure}
The periods appear to be stable for all of the sources. The spectral structure of the sources also appears to be stable in general, although some features may slowly change in intensity over time. The masers tend to return to a similar intensity after a flare, indicating that the maser region itself is not affected by the mechanism causing the periodic modulation. This was confirmed by high resolution observations of G9.62+0.20E during a flare \citep{Goedhart2005}. However, the varying amplitudes of flares and long term trends may be due to local changes in maser path length.
It may be significant that all of these sources have both 6.7 and 12.2 GHz methanol masers, even if they may not be strong enough for monitoring with the 26m telescope. All of the sources, with the exception of G188.95+0.89, appear to have OH maser emission as well. The statistical analysis of \citet{Breen2011a} indicates that these objects are at a more advanced evolutionary stage than those exhibiting only 6.7 GHz maser emission. Most of the periodic sources, with the exception of G188.95+0.89 and possibly G339.62-0.12 have been getting brighter.
Understanding the underlying mechanism of these periodic variations is limited by observational constraints. Monitoring of the flux density of the associated HII regions is challenging due to confusion and the variable uv-coverage of interferometric arrays. However, monitoring of other maser species may enable us to constrain whether the variability lies with the seed or pump photons. Mainline hydroxyl masers are believed to have a common pump mechanism with class II methanol masers and simultaneous monitoring of methanol and hydroxyl masers will be valuable. \cite{Green2012} looked for variations in the OH maser in G12.89+0.89 and found an indication that there may be a drop in intensity coinciding with the same effect at 6.7 GHz. A programme to monitor the OH masers using the HartRAO 26m and the newly commissioned KAT-7 telescope has been started and should give us greater insight into the variability of OH masers associated with periodic methanol sources. \citet{Araya2010} find quasi-periodic variations in the formaldehyde maser in IRAS18566+0408, with correlated variations in some of the 6.7 GHz methanol maser components. The pump mechanism of formaldehyde is not fully understood - \cite{Boland1981} show that the masers can be pumped by the free-free continuum radiation from an HII region, while \citet{Araya2010} argue that the simultaneous flaring of methanol and formaldehyde indicate a common infrared pump and that the variability is caused by periodic accretion of circumbinary disk material.
\section{Conclusions}
We have presented 10 years of monitoring of six periodic class II methanol masers. The periods in this sample range from 132.8 to 509 days. The regularity of the flares indicate a periodic underlying mechanism, while the amplitude of the maser response can vary. Where it is possible to monitor the 12.2 GHz methanol masers, these have been found to flare simultaneously with their 6.7 GHz counterparts.
While the cause of the periodicity is yet to be confirmed, it seems likely that these sources are associated with binary systems. Further work needs to be done in developing time-dependent maser pump models and in finding associated variability in other tracers.
\section*{Acknowledgements}
Part of S Goedhart's work on this project has been supported by the National Research Foundation under grant number 74886.
\bibliographystyle{mn2e}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,373 |
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If BCR wont give me any advice or offer to redo the striping, would an automotive paint shop be able to redo the stripes or would they have to repaint from scratch?
Re: Benjie's Seat and Tank Paint Cracking...Suggestions?
Likely doing the same under the silver. It's on quite a bit thicker, and not breaking through.
Looks like your gel may be breaking.
Looks like lacquer checking. Possibly black lacquer was used for the stripes? Not too many folks using lacquer these days. The phenomenon is not exclusively for lacquer as the same effect can happen with incompatible materials. Regardless, you can't paint over it and not expect to have problems. I don't think there is an issue with the gelcoat, but you'll have to pretty much start back from scratch to ensure the problem is eliminated. You'll be able to tell if the problem is coming up from under the paint. Assuming it is just the black, you can simply sand all the black off and prep for new paint. If it IS in the gelcoat, I'd have to sand off all the gelcoat in order to ensure the problem won't reappear.
That's called crazing. Crazing can occur in one layer or all the way to the base metal. It occurs because of a host of issues- all at the paint shop's fault. It could be too much hardener in the clear coat, or they tried to speed dry the undercoat, or they didn't allow adequate flash time. It could be poor mixing of the paint. I suspect since they sprayed stripes over the base silver, then a clear over top, that they rushed the job. Unfortunately, nothing you can do here except to start over. That's a botched paint job, my friend.
Thanks for the input, everyone. After sending them pictures of the issue I'm at least beginning to understand why they won't return my emails; they'd have to admit fault and agree to repaint. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,101 |
USING_NS_CC;
using namespace CocosDenshion;
AppDelegate::AppDelegate() {
}
AppDelegate::~AppDelegate()
{
}
bool AppDelegate::applicationDidFinishLaunching() {
srand (time(NULL));
// initialize director
auto director = Director::getInstance();
auto glview = director->getOpenGLView();
if(!glview) {
glview = GLView::create("VNTraffic");
director->setOpenGLView(glview);
}
Size screenSize = glview->getFrameSize();
Size designSize = Size(320, 480);
glview->setDesignResolutionSize(designSize.width, designSize.height, ResolutionPolicy::EXACT_FIT);
std::vector<std::string> searchPath;
if (screenSize.width > 768) {
searchPath.push_back("ipadhd");
director->setContentScaleFactor(1280/designSize.width);
} else if (screenSize.width > 320) {
searchPath.push_back("ipad");
director->setContentScaleFactor(640/designSize.width);
} else {
searchPath.push_back("iphone");
director->setContentScaleFactor(320/designSize.width);
}
FileUtils::getInstance()->setSearchPaths(searchPath);
SimpleAudioEngine::getInstance()->preloadBackgroundMusic("background.mp3");
SimpleAudioEngine::getInstance()->preloadEffect("gameover.wav");
// turn on display FPS
director->setDisplayStats(true);
// set FPS. the default value is 1.0/60 if you don't call this
director->setAnimationInterval(1.0 / 60);
// create a scene. it's an autorelease object
auto scene = GameLayer::createScene();
// run
director->runWithScene(scene);
return true;
}
// This function will be called when the app is inactive. When comes a phone call,it's be invoked too
void AppDelegate::applicationDidEnterBackground() {
Director::getInstance()->stopAnimation();
// if you use SimpleAudioEngine, it must be pause
// SimpleAudioEngine::sharedEngine()->pauseBackgroundMusic();
}
// this function will be called when the app is active again
void AppDelegate::applicationWillEnterForeground() {
Director::getInstance()->startAnimation();
// if you use SimpleAudioEngine, it must resume here
// SimpleAudioEngine::sharedEngine()->resumeBackgroundMusic();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,101 |
...to Wave Blasters of Florida. We are a radio controlled model boat club in Fort Pierce, Fl. Our members include both scale and fast electric boats. Our club is affiliated with the North American Model Boat Association (NAMBA) which supports the interests of all our club members.
We meet weekly at Palm Lake Park in Indian River Estates under the pavilion on the east side of the pond. There is lots of shade from the pavilion and trees. The park also has a playground for the kids. Check the Scale and Fast Electric pages for meet dates and times.
1/10th Scale Replica of a 1960's Era Thunder Boat.
Our scale members have some beautiful boats that have taken a lot of time and effort to build. Come out and watch them in all their glory as they cruise around the pond Sunday mornings. You are sure to appreciate the realistic look of these painstakingly crafted boats. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,131 |
Q: UITableView from plist dictionary KEYS, iOS I am attempting to get the name of the custom ringtones in the iTunes directory on a jailbroken iPhone. I can successfully list the custom ringtones, but they re displayed as HWYH1.m4r, which is what iTunes renames the files, but I know theres a way to decipher the actual name of the song.
NSMutableDictionary *custDict = [[NSMutableDictionary alloc] initWithContentsOfFile:@"/iPhoneOS/private/var/mobile/Media/iTunes_Control/iTunes/Ringtones.plist"];
NSMutableDictionary *dictionary = [custDict objectForKey:@"Ringtones"];
NSMutableArray *customRingtone = [[dictionary objectForKey:@"Name"] objectAtIndex:indexPath.row];
NSLog(@"name: %@",[customRingtone objectAtIndex:indexPath.row]);
cell.textLabel.text = [customRingtone objectAtIndex:indexPath.row];
dictionary is returning:
"YBRZ.m4r" =
{
GUID = 17A52A505A42D076;
Name = "Wild West";
"Total Time" = 5037;
};
cell.textLabel.text is returning: name: (null)
A: NSMutableArray *customRingtone = [[dictionary objectForKey:@"Name"] objectAtIndex:indexPath.row];
That line is totally wrong. Your dictionary object it, in fact, an NSDictionary with keys equal to values like 'YBRZ.m4r'. You are requesting a value for a key named 'Name', which doesn't exist. Then, with that returned object, you are sending it a method as if it were an NSArray, which it isn't. And then you expect that to return an NSArray. Again, I don't think it does. It should be more like this:
NSArray *keys = [dictionary allKeys];
id key = [keys objectAtIndex:indexPath.row];
NSDictionary *customRingtone = [dictionary objectForKey:key];
NSString *name = [customRingtone objectForKey:@"Name"];
cell.textLabel.text = name;
Also note, I'm not using NSMutableDictionarys. If you don't need the dictionary to be mutable, you probably should have a mutable dictionary.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,870 |
Delyn Constituency Information
Posted on October 20, 2006, 12:58 am, by admin, under Archived, Welsh Assembly.
Alison served as the Assembly Member for Delyn from the inception of the National Assembly for Wales in 1999 until May 2003.
In the inaugural elections for the Assembly. Delyn presented a unique contest between four women candidates.
Alison Halford was elected to the National Assembly for Delyn and for Labour with the Conservative, Karen Lumley, in second place. On a 44% turnout, 44.7% (10,672) cast their vote for Alison, more than double the votes of her nearest contender. That measure of support for Alison continued throughout her term of office. Not a career politician by choice, Alison had said from the beginning of her term of office that she would not seek re-election and remained true to that position.
The structures and procedures for the Assembly are laid down in the Government of Wales Act 1998 and more detailed processes are set out in the Assembly Standing Orders. For more information, visit the National Assembly Website.
This allows everyone access to a full record of Assembly dealings in plenary and committee sessions including agendas, minutes, associated papers, etc. Members biographies and interests are presented. Questions asked and answers given are recorded.
Alison Halford as Assembly Member for Delyn represented the interests and wishes of the people of Delyn, protecting the interests of her constituents as best she could. Her contributions to Assembly business and debate are fully recorded at the Assembly Website.
Assembly Appointments
Alison was a member of the following committees:
Local Government & Housing Committee
Public Appointments Committee
North Wales Regional Committee
« 22 March 2001
22 March 2001 » | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,758 |
El apeadero Encarnación es una estación ferroviaria ubicada en la ciudad de Encarnación, en Paraguay.
El apeadero reemplaza a la estación de Encarnación, del ferrocarril Carlos Antonio López, que fuera demolida en el marco de las obras de reconstrucción de la costanera de Encarnación. Las tareas fueron realizadas por la Entidad Binacional Yacyretá, para elevar la cota del embalse de la represa de Yacyretá.
El nuevo apeadero es la cabecera paraguaya del servicio ferroviario internacional entre Encarnación y la ciudad argentina de Posadas. Los trenes conectan el apeadero Encarnación con el apeadero Posadas.
Referencias
Encarnación (Paraguay)
Ferrocarril en Paraguay | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,915 |
El bromobenzè és un compost orgànic amb la fórmula C₆H₅Br. És un líquid incolor, encara que amb el temps pod adquirir un color groc. És un reactiu en síntesi orgànica.
Síntesi
El bromobenzè es prepara per l'acció del brom sobre el benzè en presència d'un àcid de Lewis, com el bromur fèrric, que actua com a catalitzador.
El bromobenzè s'utilitza per introduir un grup fenil en altres compostos. Un mètode consisteix en la seva conversió al reactiu de Grignard, bromur de fenilmagnesi. Es pot utilitzar aquest reactiu, per exemple en la reacció amb el diòxid de carboni per preparar àcid benzoic. Altres mètodes inclouen reaccions d'acoblament catalitzades per pal·ladi, com la reacció de Suzuki. El bromobenzè s'utilitza com a precursor en la fabricació de fenciclidina.
Vegeu també
Clorobenzè
Referències
Compostos aromàtics
Compostos de brom | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,655 |
> East-North Central
> Michigan
Related topics: Aerospace & Defense, Automotive, Manufacturing, Technology/R&D
Michigan: Rich History, Bright Future
By: Governor Rick Snyder
Rich History, Bright Future
For a century, Michigan has been the global epicenter of the automotive industry. While Michigan's long-standing auto manufacturing sector has made a triumphant comeback, the state is committed to fostering diverse industries. Multinational companies and small businesses continue to invest in Michigan, proving that a supportive tax climate, talented workforce and rich history make Michigan one of the country's best states for business.
The Future of Transportation
Michigan's famous automotive cluster is going strong, with 63 of the top 100 North American auto suppliers. Automotive companies have announced almost $23 billion in new investments in Michigan since 2010, more than any other state in the U.S.
Today, automakers are choosing Michigan as the testing ground for the future of mobility and transportation. The American Center for Mobility (ACM), set to open later this year, will be the bellwether of autonomous vehicle testing in not only the state, but the nation. ACM will have a 335-acre test facility for connected and automated vehicles, featuring a grid with urban street simulation and a 2.5-mile highway loop, designed to test autonomous cars in every possible situation they will face on the road, regardless of environmental influences.
Recently designated as a federal proving ground for the development and testing of connected and autonomous vehicles, ACM is securing U.S. mobility competitiveness worldwide. ACM will build on research underway at Mcity, the University of Michigan's research site that features 16 acres of roadway and infrastructure. Mcity's major partners include Nissan, Toyota, Ford, Honda and General Motors (GM).
In addition to Mcity and ACM, Michigan has some of the most progressive laws surrounding open road autonomous vehicle testing in the country. Last December, GM announced it will test 300 self-driving Chevrolet Bolt EVs on public roads in Michigan. Ford is already testing driver-monitored autonomous vehicles on Michigan roads and at Mcity.
As automakers increase testing in Michigan, they are also doubling down on manufacturing operations. Ford recently announced it will invest $1.2 billion in Michigan as part of its commitment to develop fully electric and autonomous vehicles in-state. The automaker will create 800 new jobs, adding to 3,600 existing jobs at its assembly plant in Wayne. Plans at Ford's Flat Rock complex include a data center and manufacturing innovation center that will produce electric and autonomous vehicles alongside the iconic Mustang and Lincoln Continental. American automakers are not the only players investing in Michigan — in April, Italian luxury carmaker Maserati decided to move its U.S. headquarters from New Jersey to Auburn Hills.
Detroit is further fostering the future of mobility with Techstars, an accelerator backed by industry heavyweights. Launched in December 2014, classes of 10 startups at a time have been innovating the mobility industry through everything from brakes and batteries to ride sharing and routing.
No. 1 Manufacturing Hub Across Industries
Across industries, Michigan is No. 1 in the nation in new manufacturing jobs, with 173,000 jobs created since 2009. Bolstered by its auto industry, Michigan is also a leader in aerospace and defense manufacturing. The state that produced 14 aircrafts per day during World War II is now one of the top-ranked states for aerospace manufacturing attractiveness. Michigan's defense industry employs more than 105,000 people and had a $9 billion economic impact in 2014.
Nearly 700 aerospace-related companies in Michigan, including General Electric, Honeywell Aerospace, Lockheed Martin and NASA capitalize on the talent, engineering and manufacturing capabilities that support Michigan's automakers. In fact, Michigan has the country's highest concentration of electrical, mechanical and industrial engineers. The University of Michigan's (U-M) aerospace engineering program is ranked No. 3 in the nation (U.S. News). U-M is just one of the state's nine universities with aerospace programs, and 16 universities with nationally ranked undergraduate engineering programs. Just this year, the first AeroAuto Conference united executives from both industries, a testament to the connection between aerospace and automotive in Michigan.
GM has also strengthened the ties between auto and defense manufacturing in Michigan through a partnership with the U.S. Army's Tank Automotive Research Development Center (TARDEC). The initiative tests how GM's fuel cell technology can power the next generation of military vehicles and is part of the company's increasing investment in fuel cell research. In January, GM and Honda announced an $85 million joint venture to mass-produce fuel cell vehicles south of Detroit.
Leading Cybersecurity Initiatives
Today, the state of Michigan is investing in the future of defense by developing cybersecurity technology at both the state and national level. Led by Governor Rick Snyder, Michigan's Cyber Initiative is the world's first comprehensive state-level approach to cybersecurity. Drawn by the state's robust cybersecurity ecosystem, the Department of Defense chose Michigan for key R&D and procurement facilities that are developing technologies with both automotive and military applications.
To ensure all companies throughout the state have access to cybersecurity protection, nonprofit Merit Network established the Michigan Cyber Range (MCR) in 2012. Boasting nearly 4,000 miles of fiber-optic infrastructure and connection to national high-speed network Internet2, MCR offers more than 40 cybersecurity certifications, exercises and workshops for professionals in Michigan and beyond.
Companies offering IT and cybersecurity services have roughly 20,000 sites throughout Michigan, and the state's IT and cybersecurity industry employs nearly 140,000 people. Cybersecurity occupations like computer science, web development and electrical engineering have grown by 17 percent.
An Attractive Place for Business
In the past five years, the state of Michigan has enacted new policies to emerge as one of the country's best states for business. Michigan eliminated the industrial personal property tax, an action that will cut taxes for small businesses by an estimated $372 million by 2020. Combined with a six percent corporate income tax, this action helped Michigan's corporate tax ranking rise to 8th from 49th in the U.S. (Tax Foundation).
Michigan supports small businesses and entrepreneurs with programs like the Michigan Emerging Technologies Fund, which matches federal funds for exceptional research and technical innovation, and Pure Michigan Business Connect, which helps businesses expand their supply chain and access services like legal and accounting assistance at a low cost. The state's 17 SmartZones provide distinct geographical locations where technology firms, entrepreneurs and researchers locate near community assets that assist in their endeavors, promoting public-private collaboration.
Though some of the world's largest companies like Ford, Lockheed Martin, GM and Honda have operations in Michigan, the state's talent pipeline is far from dry. With the country's largest investment into skilled-trades education and top higher education institutions, Michigan is ready to welcome future investments from businesses large and small. T&ID
About Governor Rick Snyder
Content written by Governor Rick Snyder | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,974 |
'use strict';
var chai = require('chai');
chai.expect();
chai.should();
describe('slush-generator module', function() {
describe('#test', function() {
it('should return a hello', function() {
'Hello Slush'.should.equal('Hello Slush');
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 33 |
\section{Introduction}
This supplementary contains derivations of various theoretical results stated in the Main Text,
as well as exploration of a number of model systems. In view of the recent experimental studies on dynamical quorum sensing,
we shall mainly focus on auto-induced collective oscillations of cells mediated by a chemical signal.
However, to the extent that the underlying microscopic processes afford a thermodynamic description,
our approach also applies in other physical contexts, e.g. mechanical or electrical signalling.
We are particularly interested in exploring the nonequilibrium aspects of ``activity'' dynamics in a living cell.
Concepts and tools from the recently developed stochastic thermodynamics~\cite{sekimoto2010stochastic-S}
are used to map out the pattern of energy flow, complementing the descriptive modelling based
on rate equations.
The material is organised as follows. Section~\ref{sect:non-equi-adaptive} contains various mathematical results announced in
the Main Text regarding the nonequilibrium response of an adaptive circuit and the associated energy flow.
In Section~\ref{sect:self-consistency-scheme}, we present a self-consistenct scheme to predict
the onset of auto-induced collective oscillations and its subsequent growth.
The response functions that appear in the discussion can in principle be measured directly in experiments.
In Section~\ref{sect:glycolysis}, we re-analyse glycolytic oscillations in
yeast cell suspensions from the perspective of linear response. The ATP negative feedback loop through the enzyme PFK
is shown to be responsible for both adaptive and oscillatory behaviour of a single cell.
The resulting phase diagram is used to construct a low dimensional model that reproduces the main dynamical features of the
full model when the ATP feedback loop is strongly coupled to the intracellular concentration of acetaldehyde, a small
molecule that diffuses fast through the cell membrane and hence can be used as a metabolic signal.
\section{Nonequilibrium thermodynamics of adaptive response}
\label{sect:non-equi-adaptive}
\subsection{Phase-leading response of an adaptive variable}
Consider the temporal variation $a_t$ of an intracellular observable $a$ induced by a sinusoidal signal $s_t$
from the environment at frequency $\omega$. The amplitude of $s_t$ is assumed to be small, so that it can be
treated as a perturbation. The observable $a$ is either directly or indirectly coupled to the signal $s$.
On the scale of a single cell, both $a_t$ and $s_t$ may contain stochastic components.
In the following, we shall examine the noise averaged response of $a$ to the deterministic part of $s$, i.e., the signal.
As usual, we use $\langle\cdot\rangle$ to denote the noise average.
The following convention on forward and inverse Fourier transforms is adopted,
\begin{equation}
\tilde{f}(\omega)=\int_{-\infty}^{\infty} f(t)\exp(i\omega t)dt,\quad
f(t)=\int_{-\infty}^{\infty} \tilde{f}(\omega)\exp(-i\omega t) \frac{d\omega}{2\pi}.
\end{equation}
In a steady-state, the ratio of the Fourier amplitudes $\langle\tilde{a}(\omega)\rangle$
and $\langle\tilde{s}(\omega)\rangle$ defines the response function
$\tilde{R}_a(\omega)\equiv\langle \tilde{a}(\omega)\rangle/\langle \tilde{s}(\omega)\rangle$,
which can be separated into its real $\tilde{R}_a'$ and imaginary $\tilde{R}_a''$ parts.
(For a cellular variable $a$ that follows stochastic dynamics, the ensemble averaged response is considered.)
The well-known Kramers-Kr{\" o}nig relation from causality requirement on the response
function states~\cite{Sethna2006entropy-S}:
\begin{equation}
\tilde{R}_a'(\omega)=\frac{2}{\pi} \int_0^\infty \tilde{R}_a''(\omega_1)\frac{\omega_1}{\omega_1^2-\omega^2}d\omega_1.
\label{eq:kramers}
\end{equation}
In the case of a perfectly adapting $a$, the response vanishes under a sufficiently slow stimulus, i.e.,
$\lim_{\omega\rightarrow 0}\tilde{R}_a(\omega)=0$. Equation (\ref{eq:kramers}) then requires,
\begin{equation}
\label{eq:adaptation_cond}
\int_0^\infty \tilde{R}_a''(\omega_1)\omega_1^{-1}d\omega_1=0.
\end{equation}
Consequently, $\tilde{R}_a''(\omega)$ must change sign at least once along the frequency axis.
Let $\phi_a=-\arg(\tilde{R}_a)$ be the phase of $-\tilde{R}_a(\omega)$, with the minus sign introduced by convention.
Positive and negative values of $\tilde{R}_a''$ thus translate to phase-lag ($-\pi<\phi_a<0$)
and phase-lead ($0<\phi_a<\pi$) of $a_t$ over $s_t$, respectively.
By virtue of continuity, the sign change of $\tilde{R}_a''(\omega)$ is also expected in the partially adaptive case,
provided the adaptation error $\epsilon\simeq \tilde{R}_a'(0)$ is sufficiently small.
\subsection{Energy outflow from an adaptive channel}
Auto-induced collective oscillations in a dissipative medium require an energy source.
Below, we show that an active cell is able to output energy to a fluctuating $s$ in the presence of an adaptive channel.
The power of the output depends on the strength of the coupling as well as the amplitude and frequency
of the fluctuating signal.
Consider a slightly more general form of Eq.~(1) in the Main Text where the contribution from cell $j$ to the total thermodynamic
force on $s$ is given by $O(a_j)$, which in general is nonlinear in $a_j$.
The work done on $s$ by the cell in a time interval $(0, L)$ can then be written as
\begin{equation}
W_j=\int_0^L O_t\dot{s}_tdt,
\label{eq:Ws_j}
\end{equation}
where $O_t\equiv O(a_j(t))$ and $s_t$ are both fluctuating quantities in general.
We now consider a sinusoidal signal $s_t=s_0+\Delta s \cos(\omega t)$
with a small amplitude $\Delta s$. To the first order in $\Delta s$, we have
\begin{equation}
O_t \simeq O^{(0)}_t + \Delta s|\tilde{R}_O(\omega)|\cos\bigl(\omega t +\phi_O(\omega)\bigr) .
\label{eq:O_t-response}
\end{equation}
Here $O^{(0)}_t$ denotes the stochastic trajectory of $O$ in the absence of the sinusoidal signal.
As usual, the linear response function $R_O$ in the steady-state (ss) to a weak time-varying signal $s_t$ is introduced through
\begin{equation}
\langle O_t\rangle\simeq \langle O\rangle_{ss}+ \int_{-\infty}^t R_O(t-\tau)s_\tau d\tau,
\end{equation}
where $\langle\cdot\rangle$ denotes average over noise. The phase angle $\phi_O(\omega)\equiv -\arg\tilde{R}_O(\omega)$.
Substituting expression (\ref{eq:O_t-response}) into Eq. (\ref{eq:Ws_j}) and taking the limit $L\rightarrow\infty$,
we obtain the time-averaged output power from the cell through this channel (omitting the subscript $j$),
\begin{equation}
\overline{\dot{W}}=\overline{O_t\dot{s}_t} + o\bigl((\Delta s)^2\bigr)
=\frac{1}{2}\omega|\tilde{R}_O(\omega)|\sin\phi_O(\omega)(\Delta s)^2+o\bigl((\Delta s)^2\bigr).
\label{eq:general-W-O}
\end{equation}
Here the overline bar indicates averaging over time, and $o\bigl((\Delta s)^2\bigr)$ denotes terms
higher than second order in $\Delta s$.
Given the relation $O_t=O(a_t)$, adaptation of the cellular variable $a$ to a slow-varying $s$ also
implies the adaption of $O$ to the signal. The causality condition (\ref{eq:kramers}) applied to
$O_t$ then requires $\tilde{R}_{O}''(\omega)\equiv-|\tilde{R}_O(\omega)|\sin\phi_O(\omega)<0$ in a certain frequency range. Consequently, Eq. (\ref{eq:general-W-O}) predicts energy outflow from the channel under a periodic stimulation at
these frequencies.
The discussion leading to Eq. (\ref{eq:general-W-O}) in the previous section can be easily extended to the energy outflow
under an arbitrary signal variation $s_t$ with a power spectrum $\tilde{C}_s(\omega)$,
\begin{equation}
\label{eq:general-W-O1}
\overline{\dot{W}}=-\int d\omega \omega\tilde{R}_O''(\omega)\tilde{C}_s(\omega)+o(\Delta s^2),
\end{equation}
where $\Delta s$ sets the overall amplitude of signal variation.
If the cell were in thermal equilibrium, $a$ would respond passively to a time-varying signal with a phase-lag and dissipate the
energy inflow generated by the stimulation. An adaptive cell, on the other hand, is able to output energy
in the form of work when stimulated in the right frequency range. This form of energy outflow
is different from the heat dissipation arising from keeping the system out of equilibrium as studied in
Refs.~\cite{lan2012energy-S,Pablo2015Adaptation-S,shouwen2015adaptation-S}.
\subsection{The Fluctuation-Dissipation Theorem}
The fluctuation-dissipation theorem (FDT) is generally presented as an identity between the response function
of a chosen variable to an external perturbation and the correlation function of the variable in question with
the one that is conjugate to the perturbation\cite{kubo1966fluctuation-S}.
For Markov systems which are of interest here, FDT holds when the
detailed balance condition on the state-space transition rates is fulfilled. We refer the reader to
Refs.~\cite{diezemann2005fluctuation-S,Wang2016entropy-S} for a detailed discussion, including more rigorous
definitions of various quantities of interest.
Assuming that the signal $s$ affects the cell through coupling to a conjugate variable $O$ which is
proportional to the variable $a$ of interest, i.e., $O=c_0 a$ with $c_0$ a proportionality constant.
In this case, FDT states that
\begin{equation}
\label{eq:FDT}
\tilde{R}_O''(\omega)=\frac{\omega \tilde{C}_O(\omega)}{2T}>0.
\end{equation}
Here, $\tilde{C}_O(\omega)=c_0^2 \langle |\tilde{a}(\omega)|^2\rangle$ is the power spectrum of $O_t$ which is always positive.
Equation (\ref{eq:FDT}) contradicts (\ref{eq:adaptation_cond}), re-affirming that receptor adaptation cannot be realised
without the presence of active processes inside the cell.
In Ref.~\cite{ma2009defining-S}, adaptation through a 3-node incoherent feed-forward motif was considered.
It was later shown that the topology even supports adaptation in an equilibrium setting~\cite{de2013unraveling-S}.
The main difference between these models and the adaptive receptor model in the Main Text (Fig.~3 and Methods) is that,
in the former, $s$ not only couples to $a$ directly, but also to other intracellular variables.
The conjugate variable $O$ is then a combination of $a$ and other intracellular variables.
We leave a detailed investigation of this issue to future work.
\section{A self-consistent scheme for frequency selection and oscillation amplitude determination}
\label{sect:self-consistency-scheme}
The thermodynamic analysis in the preceding section suggests the possibility of a positive feedback loop
formed by a periodic signal and adaptive cells under generic conditions. Collective oscillations emerge when
signal amplification by active cells overtakes signal dissipation in the passive medium. In this section, we examine
this process in further detail and derive equations that can be used to determine the frequency and amplitude of
auto-induced oscillations when the instability takes place. For simplicity, we shall consider a situation where diffusion of
the signalling molecules in the medium is very fast so that spatial variations of $s$ is suppressed. Consequently,
the notion of a well-defined transition to the oscillating state can be introduced.
\subsection{The phase matching condition and threshold cell density}
Given that individual cells couple to each other only through the signal field $s$, a self-consistency procedure similar to the solution
of mean-field models in statistical physics can be employed. In this case, the linear equations governing an eigenmode
with eigenvalue $\lambda$ can be divided into subgroups associated with individual cells.
The internal variables of a given cell appear in one and only one of
the subgroups. Solution of the subset of equations for cell $j$ yields the cell activity $\langle \tilde{a}_j\rangle=\tilde{R}_{a,j}(i\lambda)\langle \tilde{s}\rangle$.
The function $\tilde{R}_{a,j}(i\lambda)$ is the same function introduced in the preceding
section to describe the linear response of $a_j$ to a sinusoidal perturbation at frequency $\omega=i\lambda$.
Likewise, a response function $\tilde{R}_s(\omega)$ from the linearised relaxational dynamics of $s$ can be obtained,
treating contributions from cells as source terms, as in Eq. (1) of the Main Text.
Combining the two steps, we arrive at the following eigenvalue equation
for $\lambda$,
\begin{equation}
\tilde{R}_s(i\lambda)\sum_{j=1}^N\alpha_1\tilde{R}_{a,j}(i\lambda)=1.
\label{eq:eigenvalue_eq}
\end{equation}
When a particular eigenvalue crosses the imaginary axis, its real part vanishes, while its imaginary part $\omega_o$ (the onset frequency) satisfies,
\begin{equation}
\alpha_1N_o\tilde{R}_s(\omega_o)\tilde{R}_{\bar{a}}(\omega_o)=1.
\label{eq:eigenfrequency_onset}
\end{equation}
Here $\tilde{R}_{\bar{a}}(\omega)\equiv N^{-1}\sum_{j=1}^N\tilde{R}_{a,j}(\omega)$ is the averaged single-cell response
function.
Equation (\ref{eq:eigenfrequency_onset}) can be written separately for the phase shift $\phi=-\arg{\tilde{R}}$ and
amplitude $|\tilde{R}|$ of the response functions. For $\alpha_1>0$, we have,
\begin{subequations}\label{eq:oscillation-prediction}
\begin{eqnarray}
\label{eq:phase-matching}
\phi_{\bar{a}}(\omega_o)&=& -\phi_s(\omega_o),\\
\label{eq:amplitude}
N_o&=&\frac{1}{|\alpha_1 \tilde{R}_{s}(\omega_o)||\tilde{R}_{\bar{a}}(\omega_o)|}.
\end{eqnarray}
\end{subequations}
Eq. (\ref{eq:phase-matching}) determines the frequency $\omega_o$ at the onset of collective oscillations,
while Eq. (\ref{eq:amplitude}) gives the threshold cell density $N_o$.
As we mentioned in the Main Text, when the signal is passive, phase lead by the cell is required for Eq. (\ref{eq:phase-matching})
to be fulfilled. The explicit relation presented here complements the energy argument based on Eq. (\ref{eq:general-W-O}),
with the activity-generated thermodynamic force $O_t$ being proportional to $\alpha_1 a_j$.
As it stands, the cell density $N$ does not appear explicitly in the phase-matching condition
(\ref{eq:phase-matching}). Therefore the frequency of collective oscillations can be estimated from separate measurements of
the single-cell response and the medium response. In reality, it is conceivable that properties of the medium
are affected by the presence of cells, e.g., the concentration of the signalling molecules secreted.
Consequently, both $\tilde{R}_{s}(\omega)$ and $\tilde{R}_{\bar{a}}(\omega)$ may have certain weak dependence on $N$.
\subsection{The amplitude equations and frequency shift}
Beyond the initial instability, nonlinear effects need to be treated explicitly to determine the
amplitude and frequency of oscillations. Assuming a periodic state, the signal strength $s(t)$ can be
expressed as a Fourier series that includes the first harmonic as well as higher order harmonics produced by nonlinearities
in the system dynamics. Likewise, the noise-averaged cellular activity $\langle a_j(t)\rangle$ can also be expressed as a
Fourier series in $t$ with the same basic frequency. For weak noise, the trajectory of the system falls on a well-defined
limit cycle whose mean radius $r$ sets the overall amplitude of oscillations, while the amplitude of the $n$th order harmonic
scales as $r^n$. This structure allows for a systematic determination of the amplitudes using perturbation theory.
Below, we illustrate the procedure in the case of cubic nonlinearities in both the dynamics
for $s$ and the dynamics for $a$, and comment on similarities and differences in more general situations.
When the cell's activity is noisy, more sophisticated schemes based on the probability distribution function
of the cellular state need to be introduced (see, e.g. Ref.~\cite{diezemann2012NL_response-S}).
Let us consider a noiseless version of the adaptive dynamics defined in the Main Text (Fig.~3 and Methods), together with
a modified version of Eq. (1) that includes a cubic nonlinearity,
\begin{subequations}
\label{eq:nonlinear-a-s-oscillation}
\begin{eqnarray}
\tau_a\dot{a}_j&=&-(a_j-y_j)-c_3a_j^3+\alpha_2 s\\
\tau_y \dot{y}_j&=&- (a_j+\epsilon y_j)\\
\tau_s \dot{s}&=&- s-d_3 s^3+\alpha_1 \sum_{j=1}^N a_j.
\end{eqnarray}
\end{subequations}
Here $\tau_s=\gamma_s/K_s$ gives the relaxation timescale for the signal. We also set $\alpha_1\to K_s\alpha_1$ for notational simplicity. The two coefficients
$c_3$ and $d_3$ set the strengths of nonlinearities in the cellular and signal dynamics, respectively.
The model has the inversion symmetry $s\rightarrow -s$ and $(a_j,y_j)\rightarrow (-a_j,-y_j)$, all $j$.
Furthermore, if we redefine the sign of $s$ and at the same time change the sign of $\alpha_2$ and $\alpha_1$,
the equations remain the same.
We now seek a periodic solution to Eqs. (\ref{eq:nonlinear-a-s-oscillation}) in Fourier form,
\begin{subequations}
\label{eq:signal-Fourier-series}
\begin{eqnarray}
s(t)&=& B\cos(\omega t) + \sum_{n=2}^\infty B^{(n)}\cos(n\omega t +\phi_s^{(n)}),\\
a_j(t) &=& A_j\cos(\omega t +\phi_{a,j}) + \sum_{n=2}^\infty A_j^{(n)}\cos(n\omega t +\phi_{a,j}^{(n)}),
\qquad j= 1,\ldots, N,\\
y_j(t) &=& C_j\cos(\omega t +\phi_{y,j}) + \sum_{n=2}^\infty C_j^{(n)}\cos(n\omega t +\phi_{y,j}^{(n)}),
\qquad j= 1,\ldots, N.
\end{eqnarray}
\end{subequations}
The amplitudes and phase shifts, all assumed to be real, satisfy a set of equations which can be derived by
substituting Eqs.~(\ref{eq:signal-Fourier-series}) into Eqs.~(\ref{eq:nonlinear-a-s-oscillation}),
and grouping terms according to the order of the harmonic.
Starting from the first harmonic in the expressions (\ref{eq:signal-Fourier-series}), the cubic terms in
Eqs.~(\ref{eq:nonlinear-a-s-oscillation}a) and (\ref{eq:nonlinear-a-s-oscillation}c) generate the first and third order harmonics
according to the identity $(\cos\phi)^3=(3\cos\phi+\cos3 \phi)/4$. Hence terms such as $A_j^3$ and $B^3$ are present in the
equations for the first harmonic. On the other hand, the cubic nonlinearities do not generate even order harmonics if they are not
included in the series initially. Hence, up to the third order in the amplitudes, the equations for the coefficients of
the first harmonic take the form,
\begin{subequations}
\label{eq:cubic-expansion}
\begin{eqnarray}
-i\omega \tau_a \tilde{a}_j&\simeq& - (1+{3\over 4}c_3 |\tilde{a}_j|^2)\tilde{a}_j + \tilde{y}_j +\alpha_2 \tilde{s},\\
-i\omega \tau_y \tilde{y}_j&=& - \tilde{a}_j - \epsilon\tilde{y}_j ,\\
-i\omega \tau_s \tilde{s}&\simeq& - (1+{3\over 4}d_3 |\tilde{s}|^2)\tilde{s}
+\alpha_1 \sum_j \tilde{a}_j.
\end{eqnarray}
\end{subequations}
Here we have introduced the short-hand notations $\tilde{s}=B,\tilde{a}_j=A_j\exp(-i\phi_{a,j})$,
and $\tilde{y}_j=C_j\exp(-i\phi_{y,j})$.
To gain an intuitive understanding of the oscillatory solution as the cell density increases
beyond the threshold $N_o$, we first eliminate the intracellular variable $\tilde{y}_j$ in Eqs. (\ref{eq:cubic-expansion}a)
and (\ref{eq:cubic-expansion}b) to obtain,
\begin{equation}
\label{eq:amplitude-a}
\tilde{a}_j=\tilde{R}_{a,j}^+(\omega)\tilde{s},
\end{equation}
where
\begin{equation}\label{eq:Ra-nonlinear}
\tilde{R}_{a,j}^+(\omega)\equiv {\tilde{a}_j(\omega)\over \tilde{s}(\omega)}\simeq
\frac{\alpha_2}{1+3c_3 |\tilde{a}_j|^2/4-i\omega\tau_a-1/(i \omega \tau_y-\epsilon)}
\end{equation}
is a ``nonlinear response function'' which expresses the ratio of the complex amplitudes of the first harmonic on
the limit cycle.
Similarly, Eq. (\ref{eq:cubic-expansion}c) can be rewritten as
\begin{equation}
\label{eq:amplitude-s}
\tilde{s}=\tilde{R}_{s}^+(\omega)\sum_{j=1}^N\alpha_1\tilde{a}_j,
\end{equation}
where
\begin{equation}
\tilde{R}_s^+(\omega)\simeq \frac{1}{1+3d_3 |\tilde{s}|^2/4-i\omega\tau_s}
\label{eq:Rs*}
\end{equation}
is a ``nonlinear response function'' of $s$ on the limit cycle. It is easy to see that
$\tilde{R}_a^+(\omega)$ and $\tilde{R}_s^+(\omega)$ reduce to their respective linear counterparts $\tilde{R}_{a,j}(\omega)$ and
$\tilde{R}_s(\omega)$ when the oscillation amplitudes vanish.
We now combine Eqs. (\ref{eq:amplitude-a}) and (\ref{eq:amplitude-s}) to obtain the self-consistency condition,
\begin{equation}
\alpha_1N\tilde{R}_s^+(\omega)\tilde{R}_{\bar{a}}^+(\omega)=1,
\label{eq:eigenfrequency_LC}
\end{equation}
which is reminiscent of Eq. (\ref{eq:eigenfrequency_onset}).
Here $\tilde{R}_{\bar{a}}^+(\omega)\equiv N^{-1}\sum_{j=1}^N\tilde{R}_{a,j}^+(\omega)$ is
the averaged single-cell nonlinear response function. When all cells are identical,
$\tilde{R}_{\bar{a}}^+(\omega)=\tilde{R}_a^+(\omega)$.
As before, Eq.~(\ref{eq:eigenfrequency_LC}) can be rewritten
in terms of the phase and amplitude of the nonlinear response functions,
\begin{subequations}\label{eq:oscillation-prediction-LC}
\begin{eqnarray}
\label{eq:phase-matching-LC}
\phi_{\bar{a}}^+(\omega)&=& -\phi_s^+(\omega),\\
\label{eq:amplitude-LC}
\alpha_1 N&=&\frac{1}{|\tilde{R}_{s}^+(\omega)||\tilde{R}_{\bar{a}}^+(\omega)|}.
\end{eqnarray}
\end{subequations}
Formally, Eq. (\ref{eq:phase-matching-LC}) can be used to determine the frequency shift at a finite amplitude of oscillation,
while Eq. (\ref{eq:amplitude-LC}) relates the oscillation amplitude to the cell density $N$. Since the amplitudes enter
quadratically into the nonlinear response functions, they are expected to increase as $(N-N_o)^{1/2}$ just above
the threshold cell density $N_o$, e.g., the transition is of the Hopf bifurcation type.
In the Main Text, we have considered the case $c_3=1$ and $d_3=0$. Numerically, the oscillation frequency is found to decrease
as the coupling strength $\bar{N}\equiv\alpha_2\alpha_1 N$ increases (see also Fig.~\ref{fig:adaptation_linear_signal}{\it A}).
This is consistent with Eq.~(\ref{eq:phase-matching-LC}) whose solution at selected oscillation amplitudes is shown
in Fig.~3{\it F} in the Main Text. As the amplitude of the oscillations increase, $\phi_a^+(\omega)$ decreases on the low frequency side.
Consequently, the intersection point with $\phi_s^+(\omega)=\phi_s(\omega)$ shifts to lower frequencies.
Interestingly, the limit cycle associated with the oscillating state in this model shrinks to a fixed point when $\bar{N}$
exceeds an upper threshold value $\bar{N}_b$. The dependence of $\bar{N}_b$ on the adaptation error $\epsilon$, which
is assumed to be small, can be estimated as follows.
At a fixed point of Eqs.~(\ref{eq:nonlinear-a-s-oscillation}) at $d_3=0$, we have
$s\simeq \alpha_1 N a$ (from $\dot{s}=0$), $y\simeq \alpha_2 s $ (from $\dot{a}=0$),
and $a\simeq \epsilon y$ (from $\dot{y}=0$). Consequently, the upper threshold for oscillations has the scaling
\begin{equation}
\bar{N}_b\sim \frac{1}{\epsilon}.
\label{eq:NH-epsilon}
\end{equation}
Figure~\ref{fig:adaptation_linear_signal}{\it B} shows the numerical values for $\bar{N}_o$ and $\bar{N}_b$
against the adaptation error $\epsilon$ obtained in our simulations, which confirms (\ref{eq:NH-epsilon}).
The oscillating state expands over a larger range of cell densities when individual cells are more adaptive.
\begin{figure}[!h]
\centering
\includegraphics[width=12cm]{figS1_adaptation_linear_signal.PDF}
\caption{\textbf{Collective oscillations in the model Eq.~(\ref{eq:nonlinear-a-s-oscillation})
with nonlinear adaptation ($c_3=1$) and linear signal relaxation ($d_3=0$)}.
({\it A}) Oscillation frequency against the effective cell density $\bar{N}=\alpha_2\alpha_1 N$. ({\it B}) The phase diagram
in the plane spanned by $\bar{N}$ and the adaptation error $\epsilon$.
Other parameters are the same as in Fig.~3 of the Main Text. }
\label{fig:adaptation_linear_signal}
\end{figure}
Next, consider the case of nonlinear signal relaxation ($d_3=1)$ and linear adaptation ($c_3=0$).
The onset of collective oscillations is similar to the previous case (Fig.~\ref{fig:adaptation_nonlinear_signal}{\it A}),
except that oscillations speed up as the cell density increases further. From Eq.~(\ref{eq:Rs*}), we obtain
\begin{equation}
\phi_s^+(\omega)=-\arg \tilde{R}_s^+(\omega)=-\arctan \Big[\frac{\omega}{\omega_s(1+3d_3 |\tilde{s}|^2/4)}\Big],
\end{equation}
which decreases as the oscillation amplitude increases. As shown in Fig.~\ref{fig:adaptation_nonlinear_signal}{\it C},
the intersection point shifts to the right. The predicted signal oscillation amplitude $B=|\tilde{s}|$ and frequency shift agree
quantitatively with our numerical results (Fig.~\ref{fig:adaptation_nonlinear_signal}{\it C}).
\begin{figure}[!h]
\centering
\includegraphics[width=12cm]{figS2_adaptation_nonlinear_signal.PDF}
\caption{\textbf{Collective oscillations in the model Eq.~(\ref{eq:nonlinear-a-s-oscillation}) with linear adaptation ($c_3=0$) and nonlinear signal relaxation ($d_3=1$)}.
({\it A}) Temporal trajectories at various values of $\bar{N}$. ({\it B}) The phase lead $\phi_a$ and lag $-\phi_s^+$ against
$\omega$ at selected oscillation amplitudes. ({\it C}) The predicted oscillation frequency and amplitude as compared with those obtained from numerical simulations. Other parameters are the same as in Fig.~3 of the Main Text. }
\label{fig:adaptation_nonlinear_signal}
\end{figure}
In the more general case when both $c_3$ and $d_3$ are nonzero, we need to first express
$\tilde{s}$ and $\tilde{a}_j$ in terms of a common variable that specifies oscillation amplitude
before applying the phase-matching condition Eq.~(\ref{eq:phase-matching-LC}).
As we see from the discussions above, depending on which of the two cubic nonlinearities is stronger, the
oscillation frequency may shift either to lower or higher values. In general, nonlinearities may also be present
in the dynamics of other intracellular variables which need to be dealt with case by case.
When quadratic nonlinearities are present in the system dynamics, the second harmonic is generated and need to be
considered in the perturbative analysis. Consider for example the equation for $a_j$ with an extra term $c_2a_j^2$.
Following the same procedure that led to Eqs.~(\ref{eq:cubic-expansion}), we find an additional term
$c_2\tilde{a}_j^*\tilde{a}_j^{(2)}$ on the right-hand side of Eq.~(\ref{eq:cubic-expansion}a), where $\tilde{a}_j^{(2)}$
is the amplitude of the second harmonic in $a_j(t)$ (including phase).
The amplitude equation for the second harmonic relates $\tilde{a}_j^{(2)}$ to $c_2\tilde{a}_j^2$ and $\alpha_2\tilde{s}^{(2)}$.
Together with the equation for $\tilde{s}^{(2)}$, amplitudes of the second harmonic can be expressed as a linear
combination of terms $c_2\tilde{a}_j^2$ from different cells. The upshot of this exercise is that coefficient of the
cubic term $|\tilde{a}_j|^2\tilde{a}_j$ in Eq.~(\ref{eq:cubic-expansion}a) should contain additional contributions
proportional to $c_2^2$. The nonlinear response functions (\ref{eq:Ra-nonlinear}) and (\ref{eq:Rs*}) on the limit
cycle can still be defined in the same way, and Eq.~(\ref{eq:eigenfrequency_LC}) still holds formally.
Through $\tilde{s}^{(2)}$, terms $|\tilde{a}_k|^2$ from other cells enter the expression for $\tilde{R}_{a,j}^+(\omega)$.
Two conclusions can be drawn from this fact: i) as in the case of cubic nonlinearities, the transition is still of the Hopf
bifurcation type; ii) $\tilde{R}_{a,j}^+(\omega)$ can no longer be determined by simply measuring the
response of a given cell to a sinusoidal stimulus at finite strength, as it is affected by the oscillation pattern of
other cells in the system due to the quadratic nonlinearity.
\subsection{Requirement on the speed of signal relaxation/clearance for the onset of collective oscillations}
\label{sect:adaptation-error-timescale}
Experiments have indicated that sufficiently fast breakdown of the signalling molecule is needed for DQS
in dicty~\cite{gregor2010onset} and for sustained oscillations in yeast cell suspensions~\cite{richard1994yeast-S}.
Below we derive an upper limit for the signal relaxation time $\tau_s$ to satisfy the phase matching condition
Eq.~(4a) in the Main Text. The result is inversely proportional to the adaptation error $\epsilon$ of the intracellular circuit.
Taking Eq.~(5) for the signal phase shift, $\phi_s=-\tan^{-1}(\omega \tau_s)$, we see that
a longer $\tau_s$ yields a larger signal delay $|\phi_s|$ at a given frequency.
This is illustrated by Fig.~\ref{fig:effect_of_taus}{\it A}, upper panel, where the horizontal frequency axis is shown on logarithmic scale.
For a given $\epsilon$, the intersection of the two phase-shift curves moves to the left, yielding a lower
onset oscillation frequency $\omega_o$ and a larger coupling strength $\bar{N}$ (Fig.~\ref{fig:effect_of_taus}{\it B}).
When $\tau_s$ reaches beyond an upper limit $\tau_s^*(\epsilon)$, the solution disappears (Fig.~\ref{fig:effect_of_taus}{\it B}).
Interestingly, a reduction of $\epsilon$ in Eq.~(9) increases $\phi_a(\omega)$ on the low frequency side, and rescues the solution (Fig.~\ref{fig:effect_of_taus}{\it A}, lower panel).
\begin{figure}
\centering
\includegraphics[width=8.5cm]{figS3_effect_of_taus.PDF}
\caption{Phase-matching at the onset of oscillations for different values of signal timescale $\tau_s$ and
adaptation error $\epsilon$. ({\it A}) Phase shift of the cell activity ($\phi_a$) and signal response ($|\phi_s|$) at selected values of
$\tau_s$ (upper panel) and adaptation error $\epsilon$'s (lower panel).
The onset frequency is given by the intersection of the two curves [Eq.~(\ref{eq:phase-matching})]. ({\it B}) Predicted onset frequency $\omega_o$ and onset coupling strength $\bar{N}_o$ for different values of $\tau_s$.
Oscillations will not be found when the signal relaxation time $\tau_s>\tau_s^*(\epsilon)$.
({\it C}) Numerical results supporting linear scaling between $\tau_s^*(\epsilon)$ and $1/\epsilon$.
The data are obtained from the coupled adaptive circuits under the same parameters (except $\epsilon$ and $\tau_s$) as in Fig.~3 in the Main Text.
}
\label{fig:effect_of_taus}
\end{figure}
The observed inverse dependence of $\tau_s^*$ on $\epsilon$ in Fig.~\ref{fig:effect_of_taus}{\it C} can be justified
from the behaviour of the two phase-shift functions at low frequencies.
Close to $\omega=0$, Eq.~(8) in the Main Text yields $\tilde{R}_a'(\omega)\propto \epsilon$
while $\tilde{R}_a''(\omega)\propto \omega$, as $\tilde{R}_a''$ must be an odd function of $\omega$.
This is confirmed by expanding Eq.~(9) in the Main Text at $\omega=0$.
Consequently, $\phi_a(\omega)\simeq -\tilde{R}_a''/\tilde{R}_a'\propto \omega/\epsilon$.
On the other hand, $\phi_s(\omega)\approx -\omega\tau_s$ in this regime.
The two curves has an intersection at low frequencies provided
\begin{equation}
\tau_s<\tau_s^*\propto 1/\epsilon.
\label{eq:epsilon-tau_s}
\end{equation}
Using the explicit expressions Eqs. (3) and (9) in the Main Text, we obtain $\tau_s^*\simeq\tau_y/\epsilon$ for small $\epsilon$.
The critical cell density and onset oscillation frequency at this maximal $\tau_s$ are given approximately by $N_o\simeq K(\alpha_1\alpha_2\epsilon)^{-1}$ and $\omega_o\simeq \epsilon\tau_y^{-1}$, respectively. This is compared to
$N_o\simeq K(\alpha_1\alpha_2)^{-1}$ and $\omega_o\simeq (\tau_a\tau_y)^{-1/2}$ at $\tau_s<(\tau_a\tau_y)^{1/2}$,
which are insensitive to $\epsilon$ as long as it is sufficiently small.
\\
\section{Glycolytic oscillations in yeast cells}
\label{sect:glycolysis}
Glycolytic oscillations in dense yeast cell suspensions have been known for a long time~\cite{richard2003rhythm-S}.
The phenomenon at cellular level is complex not only because of a large number of enzymes and metabolites involved,
but also due to a multitude of regulatory interactions whose activation pattern and strength are not well understood.
Furthermore, flux diversion into side branches other than the main fermentative pathway can significantly
attenuate or even diminish the oscillations. Yet the oscillations are easy to produce following the standard experimental protocols,
suggesting that certain type of low dimensional mechanism inside a cell is at work.
\begin{figure}[!h]
\centering
\includegraphics[width=14cm]{figS4_glycolysis_kineticMap.PDF}
\caption{\textbf{The network of reactions in a detailed model of glycolysis~\cite{du2012steady-S}}. Letters in blue denote metabolites,
while those in red are the reactions. Directional (bidirectional) arrows indicate irreversible (reversible) reactions.
Abbreviations: Glco, glucose; ACE, acetaldehyde, ADH, alcohol dehydrogenase; AK, adenylate kinase; ALD, fructose-1,6-bisphosphate aldolase; BPG, 1,3-bis-phosphoglycerate; ENO, phosphopyruvate hydratase; F16P, fructose-1,6-bisphosphate; F6P, fructose 6-phosphate; GAPDH, D-glyceraldehyde-3-phosphate dehydrogenase (phosphorylating); G3P, glycerol 3-phosphate; G3PDH, glycerol 3-phos- phate dehydrogenase; G6P, glucose 6-phosphate; GLYCO, glycogen branch; GLK, glucokinase (a hexokinase); P2G, 2-phosphoglycerate; P3G, 3-phosphoglycerate; PEP, phosphoenolpyruvate; PDC, pyruvate decarboxylase; PGI, glucose-6-phosphate isomerase; PFK, 6-phosphofructokinase; PGK, phosphoglycerate kinase; PGM, phosphoglycerate mutase; PYK, pyruvate kinase; PYR, pyruvate; Treha, trehalose branch; SUC, succinate branch; GLYO, glyoxylate shunt.}
\label{fig:kineticMap}
\end{figure}
In the following we explore the possibility of an adaptation route to yeast glycolytic oscillations.
It is known that yeast cells communicate through the intercellular acetaldehyde (ACE) which acts as a redox signal.
The intracellular redox ratio NAD/NADH affects the rate of the key reaction GAPDH separating ATP consuming and ATP
harvesting parts of the glycolytic pathway. Adaptation of the glycolytic flux to a rising (or receding) ACE level may result
from ACE's coupling to ATP homeostatic circuit on the time scale of seconds.
We verify this scenario in a detailed model proposed by du Preez et al.~\cite{du2012steady-S} (referred to as the full model),
and then develop a minimal model that helps us to understand the response phase diagram of the full model.
At intermediate values of the extracellular ACE concentration, both the minimal model and the full model enter an oscillating
state. This part of the phase diagram is flanked by quiescent regions with adaptive response. The width of the adaptive
region can be tuned by altering side branches of glycolysis and downstream pathways, in particular inhibition of the glyoxylate shunt.
We then consider a system of coupled yeast cells, each metabolises according to the minimal model. The extracellular
ACE concentration is set by the cell volume fraction $\phi$. Collective oscillations at low cell densities result from synchronisation
of cells that oscillate on their own. At high cell densities, the elevated ACE level puts individual cells outside their oscillatory
regime when in isolation, yet the population as a whole may still oscillate collectively via the adaptation-driven DQS mechanism.
The width of the oscillatory region can be significantly reduced on the low cell density side by cell-to-cell variations,
and on the high cell density side by side reactions that reduce the adaptation accuracy of pyruvate pool which controls production of ACE. Our model study also highlights the importance of fast turnover of the extracellular ACE for sustained oscillations
as required by the phase-matching condition (Eq.~(4a), Main Text) and noted in previous experimental studies~\cite{richard1994yeast-S}.
\\
\begin{figure}[!h]
\centering
\includegraphics[width=15cm]{figS5_glycolysis_unperturb.PDF}
\caption{\textbf{Spontaneous oscillations in the full model}.
({\it A}) Trajectories of all metabolites at glucose concentration Glco=10.
({\it B}) and ({\it C}) Time-averaged metabolite concentrations and reaction fluxes in descending order. }
\label{fig:unperturb}
\end{figure}
\subsection{Single-cell perturbation study}
The full intracellular reaction network of the kinetic model by du Preez et al.~\cite{du2012steady-S} is shown in
Fig.~\ref{fig:kineticMap}. It contains around 20 reactions and 15 metabolite concentrations as dynamical variables.
The reaction fluxes are highly nonlinear functions of these variables.
Predictions of the model were shown to agree semi-quantitatively with experimental data on yeast glycolytic
oscillations~\cite{du2012steady2-S}. Below we use the same parameter values as adopted
in the original model termed dupree2 in~\cite{du2012steady-S}, unless otherwise stated.
\begin{figure}[!h]
\centering
\includegraphics[width=15cm]{figS6_glycolysis_step_perturbation.PDF}
\caption{\textbf{Response of metabolites to a redox signal at low ACE concentrations}.
Here, $\text{ACE}(t)=\text{ACE}_0[1+0.02 H(t)]$, with $H(t)$ being a Hill function with a large Hill coefficient.
The notation $\delta x$ of a variable $x$ represents its relative change from a pre-stimulus level $\bar{x}$, i.e.,
$\delta x\equiv [x(t)-\bar{x}]/\bar{x}$. Quantities such as PYR and NAD which have too small values are amplified to make
them visible on the plot.
({\it A}) The response of metabolites around G6P in the upper section of the glycolytic pathway;
({\it B}) The response of metabolites around BPG in the middle section of the glycolytic pathway;
({\it C}) The response of metabolites from BPG to PYR in the lower section of the glycolytic pathway;
and ({\it D}) The response of metabolites in the downstream fermentation pathway.
Parameters: $\text{ACE}_0=0.05$ and $\text{Glco}=10$. }
\label{fig:adaptation-glycolysis}
\end{figure}
Fig.~\ref{fig:unperturb} shows an oscillatory solution of the model at the glucose concentration Glco =10 mM.
The oscillation frequency is $\omega_0\approx 21$ min$^{-1}$, corresponding to a period of $\tau_0\simeq 0.3$ min.
The mean concentration of ACE is 0.17 mM (Fig.~\ref{fig:unperturb}{\it B}). Reaction fluxes are concentrated along the linear
pathway from Glco to ETOH, while the side reactions carry much smaller flux (Fig.~\ref{fig:unperturb}{\it C}, blue box).
Below, we present response properties of the model using ACE concentration as the second control variable, in addition
to the extracellular glucose concentration. Time is measured in minutes and concentrations in mM.
To move out of the oscillatory regime, we lower the mean acetaldehyde concentration to $\text{ACE}_0=0.05$.
Experimentally, this can be achieved by adding cyanide (KCN) which reacts with ACE
in the solution~\cite{gustavsson2015entrainment-S}.
Fig.~\ref{fig:adaptation-glycolysis} shows the time course of metabolites under a step-wise increase in the
ACE concentration. The four panels are organised following the order of metabolites along the glycolytic pathway,
with the addition of ATP, ADP and NAD.
Most metabolites adapt at least partially, except F16P and TRIO upstream of the reaction GAPDH
that uses NAD and NADH as cofactors. The redox pair NAD and NADH, being tightly connected to ACE, do not adapt either.
\begin{figure}[!h]
\centering
\includegraphics[width=15cm]{figS7_glycolysis_periodic_perturbation.PDF}
\caption{\textbf{Concentration variations along the glycolytic pathway stimulated by a periodic redox signal}.
Here, $\text{ACE}(t)=\text{ACE}_0[1+0.02 \sin(\omega t)]$.
Organization of metabolites in panels ({\it A})-({\it D}) is the same as in Fig.~\ref{fig:adaptation-glycolysis}.
Parameters: $\text{ACE}_0=0.05$, $\text{Glco}=10$, and $\omega=21$. }
\label{fig:periodic_perturbation}
\end{figure}
We now consider oscillations of the same set of metabolites stimulated by a periodic redox signal at the
frequency $\omega_0$ of spontaneous oscillations mentioned above.
In Fig.~\ref{fig:periodic_perturbation}{\it A}, ATP, G6P and F6P are approximately in phase with each other,
but they are out of phase with Glci at the entry point of the pathway. The non-adaptive F16P has a behaviour of its own.
The phase relations for these metabolites have been measured experimentally, and the results agree well with our numerics~\cite{richard1996sustained-S}.
In Figs.~\ref{fig:periodic_perturbation}{\it B}-{\it C}, metabolites from BPG down to PEP share nearly the same phase
with each other and with ATP. The non-adaptive TRIO lags slightly behind
F16P. In Fig.~\ref{fig:periodic_perturbation}{\it D}, PYR at the end of the glycolytic pathway has an approximately
$90^\circ$ phase lead over ATP, and furthermore a smaller phase lead over ACE and NAD.
Fig.~\ref{fig:phase-relation} shows the phase shifts of metabolites against a sinusoidal signal ACE obtained
from our numerical simulations over a broad frequency range.
Apart from PYR, the phase relationships among metabolites at $\omega_0$ hold also at lower frequencies.
In Fig.~\ref{fig:phase-relation}{\it B}, it is seen that NAD has essentially the same phase as ACE in the frequency interval,
while NADH is completely out of phase. Therefore, on the timescale $\tau_0$, the phase information of ACE is passed
without delay onto the redox ratio NAD/NADH, and fed into the network through the reaction GAPDH.
Around $\omega_0$, the phase lead of NADH over ACE is slightly below 180$^\circ$,
as observed in experiments on glycolytic oscillations~\cite{richard1996acetaldehyde-S}.
Fig.~\ref{fig:phase-relation}{\it C} shows the downstream metabolites from BPG to PEP oscillate in phase with each
other for $\omega\leq\omega_0$, meaning the internal time scales for this part of the pathway are shorter than $\tau_0$.
In contrast, PYR develops a phase lead in the intermediate frequency regime, as indicated by the two black arrows in Fig.~\ref{fig:phase-relation}{\it C}. (Note that in Fig.~5{\it D} of the Main Text, the phase lead extends to
zero frequency indicating that the width of the regime depends on the glycolytic flux.)
The adaptive variable ATP also has a phase lead in the entire low frequency region.
\begin{figure}[!h]
\centering
\includegraphics[width=15cm]{figS8_glycolysis_phase_relations.PDF}
\caption{\textbf{Phase shifts of metabolites against the frequency of a sinusoidal ACE signal}.
({\it A}) Metabolites in the ``preparatory phase'' of the glycolytic pathway, where ATP is consumed to activate the 6-carbon ring
molecule. ({\it B}) Substrate, product and cofactors of the reaction GAPDH
that act as the receptor of the redox signal, together with ATP.
({\it C}) Metabolites in the ``payoff phase'' of the glycolytic pathway, where ATP is harvested.
For the particular values of the extracellular glucose and acetaldehyde chosen,
phase lead of PYR over ACE occurs in the range of frequencies delimited by black arrows.
The blue arrow indicates the intrinsic frequency studied in Fig.~\ref{fig:periodic_perturbation}.
({\it D}) Metabolites that appear in Eq.~(\ref{eq:phase-rule}).
The phase shift of a number of metabolites shows a dip at the low frequency end, indicating a small but finite adaptation
error. Parameters: $\text{ACE}_0=0.05$, $\text{Glco}=10$. }
\label{fig:phase-relation}
\end{figure}
Fig.~\ref{fig:phase-relation}{\it D} shows the following phase relations between ATP and several other metabolites
as summarized by the equations below,
\begin{equation}
\label{eq:phase-rule}
\phi_{ATP}=\pi+ \phi_{ADP}=\pi+ \phi_{AMP} \approx \phi_{BPG}\approx \phi_{PEP} \approx \pi+ \phi_{GLCi}.
\end{equation}
The first two relations among the nucleotides ATP, ADP and AMP simply reflect the conservation of their total number,
and that the fraction of AMP is much lower than the other two.
In-phase relations apply to substrates BPG and PEP of the ATP harvesting reactions PGK and PYK, respectively,
while the out-of-phase relation is observed for GLCi in the ATP consuming reaction GLK. The fact that these relations
hold almost strictly in the entire frequency region suggests that quasi-steady-state conditions apply to these and neighbouring
reactions. It also suggests a prominent role of ATP in synchronising the phase of metabolites distributed along the glycolytic pathway.
In summary, our numerical results suggest the following mechanism of adaptation. Under a stepwise increase of ACE concentration,
the information is passed with negligible delay to the redox ratio NAD/NADH, and then through the delayed reaction GAPDH
to BPG and PEP, transiently boosting ATP production. The transient increase of ATP concentration then reduces the
upstream glycolytic flux by inhibiting the reaction PFK, which in turn decreases the downstream TRIO concentration,
eventually returning the GAPDH flux to its pre-stimulus level.
Although many metabolites adapt, the negative feedback loop of ATP production appears to be the core.
Fig.~5{\it B} in the Main Text shows a more complete phase diagram of the response properties at other values of
$\text{ACE}_0$ and $\text{Glco}$ concentrations, including the region of spontaneous oscillations.
\subsection{A minimal model for glycolytic oscillations}
We constructed a minimal model to test various quantitative aspects of the adaptation mechanism described above.
Reduction in the number of dynamic variables is achieved by lumping consecutive metabolites along the linear pathway
that are phase synchronised into a single variable denoting their total concentration.
This is a reasonable approximation when interconversion among these metabolites is much faster than
the time of interest, e.g. the oscillation period.
Fig.~\ref{fig:MinimalModel}{\it A} illustrates the selected variables and their interactions.
Here, $y$ represents intermediate metabolites that do not adapt (F16P and TRIO), thereby playing the role of a memory node.
The variable $z$ represents metabolites from BPG to PEP along the glycolytic pathway.
The ATP concentration is denoted by $p$, while the concentration of PYR,
substrate for the ACE producing reaction PDC and thus the corresponding cell activity here, is denoted by $a$.
Since NAD (NADH) is always in phase (out of phase) with ACE, we adopt the redox ratio NAD/NADH
as the signal $s$ instead. Motivated by a phenomenological two-component model for glycolytic oscillations
in Ref.~\cite{chandra2011glycolytic-S}, we introduce a minimal model of glycolysis with redox control as follows:
\begin{subequations}\label{eq:MinimalModel}
\begin{eqnarray}
\tau\dot{y}&=&\frac{2p}{1+p^{2h}}-(\alpha_2 s+c_0) y -\epsilon y ,\\
\tau \dot{z}&=& (\alpha_2 s+c_0) y - \frac{2z}{1+p^2},\\
\tau \dot{p} &=&-\frac{2p}{1+p^{2h}}+2 \frac{2z}{1+p^{2}}-\frac{2p^2}{1+p^2}.
\end{eqnarray}
Here, $2 p/(1+p^{2h})$ gives the reaction flux of PFK that consumes ATP and is also inhibited by ATP at high concentrations
(i.e., the negative feedback loop), with the inhibition strength set by the exponent $h (>1/2)$. The entry carbon flux into
the glycolysis pathway is assumed not to be rate limiting, e.g.., one is in a situation of high extracellular glucose concentration.
The term $(\alpha_2 s+c_0) y$ gives the reaction flux of GAPDH, where $c_0$ sets the ``basal'' enzyme velocity at $s=0$. Leakage of TRIO into the side branch is represented by $\epsilon y$.
The term $2z/(1+p^2)$ gives the reaction flux of PYK (and also PGK), which produces ATP but is also inhibited by ATP.
In Eq.~(\ref{eq:MinimalModel}c), the stoichiometric factors 1 and 2 in the first two terms
on the right-hand-side correspond to the ATP consumption and production upstream and downstream of
TRIO, respectively. ATP consumption by the cell outside of glycolysis (e.g., ATPase activity) is modelled by the term
$2p^2/(1+p^2)$, which grows with the ATP concentration until saturation at a maximal value 2.
The output variable $a$ (PYR) is produced by the same flux that produces $p$ (ATP) and degraded at a constant
rate $\alpha_1$,
\begin{equation}
\tau \dot{a}=\frac{2z}{1+p^2}-\alpha_1 a.
\end{equation}
\end{subequations}
The glyoxylate shunt GLYO, which is active at ACE$_0\simeq 0.2$ mM or above in the full model, is turned off.
For simplicity, we have chosen the time constants on the left-hand-side of the equations to be the same. As we show below,
this choice is adequate for recovering the main low frequency properties of the full model.
\begin{figure}[!h]
\centering
\includegraphics[width=13cm]{figS9_glycolysis_schemes.PDF}
\caption{\textbf{A minimal model of glycolysis with redox control}. ({\it A}) The network of metabolites (symbols)
and reactions (boxes). ({\it B}) Response of metabolites upon a stepwise perturbation $s(t)=0.04 (1+0.01 H(t))$.
({\it C}) Response of corresponding metabolites in the full model computed using parameter values given in
FIG.~\ref{fig:adaptation-glycolysis}.
({\it D}) Response properties of the minimal model as the intracellular redox state changes from reductive to oxidative
(left to right). Parameters: $h=3, \alpha_2=\alpha_1=1$, $\tau=0.01$, $c_0=0.02$, and $\epsilon=0.01$. }
\label{fig:MinimalModel}
\end{figure}
Fig.~\ref{fig:MinimalModel}{\it B} shows the response of the dynamical variables to a sinusoidal redox variation
centred around $s_0=0.04$. Except the buffer variable $y$, all other variables show adaptive behaviour, with
$a$ gaining a phase lead of $90^\circ$ over $p$.
For comparison, we show in Fig.~\ref{fig:MinimalModel}{\it C} the response properties of corresponding metabolites
in the full model in the adaptive regime, which are indeed quite similar.
We have examined the response properties of the minimal model at other values of $s_0$
and identified four qualitatively different regimes as shown in Fig.~\ref{fig:MinimalModel}{\it D}.
As in the case of the full model with sufficient glucose (Main Text, Fig. 5{\it B}), spontaneous oscillations (i.e., limit
cycle solution) occur at intermediate values of $s_0$, flanked by adaptive but non-oscillatory regions.
We note in passing that the two-component model of Chandra {\it et al.}~\cite{chandra2011glycolytic-S}
also exhibits spontaneous oscillations when the rate constant $k$ of the pyruvate kinase reaction
(PYK in Fig.~\ref{fig:MinimalModel}{\it A}) takes on intermediate values. As $k$ affects the delay
time of the negative feedback control in ATP production, in this sense it plays a similar role as $s_0$.
However, our model contains an additional buffer node TRIO which is necessary for the adaptive behaviour seen in
Fig.~\ref{fig:MinimalModel}. We have also made the ATP consumption rate dependent on the ATP concentration
to eliminate certain pathological aspects of the Chandra {\it et al}. model at low values of $p$.
Furthermore, our numerical analysis suggests that a sufficiently small but finite adaptation error $\epsilon$ associated
with low flux diversion is needed to reproduce the response diagram Fig.~\ref{fig:MinimalModel}{\it D}.
On the high (oxidative) end of $s_0$, the reaction GAPDH drives down $y$ (TRIO) and hence the flux of the side reaction,
making the system adaptive even when $\epsilon\sim 1$.
Comparing the response diagrams of the minimal model (Fig.~\ref{fig:MinimalModel}{\it D}) and
of the full model at high extracellular glucose concentrations (Main Text, Fig. 5{\it B}), we see that the adaptive regime
on the oxidative side is restricted to a much narrower region in the latter case.
Upon a detailed investigation of the full model we found that, at higher values of
ACE$_0$, the side reaction GLYO is activated. Shutting down the reaction, we obtained a response diagram similar to that of the minimal model (Fig.~\ref{fig:phase_diagram_glycolysis_removing_SUC}).
The reaction GLYO uses NAD as cofactor and consumes ATP (see Fig.~\ref{fig:kineticMap}).
With regard to the change in NAD/NADH ratio upon an upshift of ACE, it has an opposite effect as compared to ADH.
This and ATP consumption by GLYO leads to a sign reversal in the transient response of ATP to ACE upshift
at ACE$_0\simeq 0.2$ mM in the full model (Fig. 5{\it B} in the Main Text).
Inhibition of GLYO eliminates the sign switch and makes PYR adapting to ACE over a much larger region of the phase diagram
(Fig.~\ref{fig:phase_diagram_glycolysis_removing_SUC}).
\begin{figure}[!h]
\centering
\includegraphics[width=14cm]{figS10_phase_diagram_glycolysis_removing_SUC.PDF}
\caption{\textbf{Phase diagram of the modified glycolysis model with the GLYO reaction switched off}.
Now PYR and ATP both adapt over a broad region of the phase diagram, in agreement with
the minimal model. }
\label{fig:phase_diagram_glycolysis_removing_SUC}
\end{figure}
Fig.~\ref{fig:glycolysis_PYK_flux} shows representative time courses of the PYK reaction flux to a stepwise ACE signal,
computed using the original and modified glycolysis model, as well as the minimal model.
Concentration of its product, PYR, is found to be proportional to the PYK reaction flux in all three models, i.e.,
the degradation rate of PYR is a constant.
The original and modified models exhibit nearly identical adaptive response on the low ACE (reductive) side,
but differ on the high ACE (oxidative) side. In the latter case, the PYK flux is significantly higher and also non-adaptive
when the glyoxylate shunt (GLYO) is on. Further numerical investigations of the
full model with blocked GLYO reaction show that it shares the following features of the minimal model
as the oxidation level increases: 1) the frequency inside the oscillatory regime increases;
2) (mean) $p$ (ATP) and $z$ (BPG, P3G, P2G, and PEP concentrations) increase by a moderate amount;
3) $y$ (TRIO and F16P concentrations) decreases; 4) $a$ (PYR concentration) first increases, then decreases.
Experimental time-course measurement with blocked glyoxylate shunt will serve to validate or improve the model assumptions.
\begin{figure}[!h]
\centering
\includegraphics[width=16cm]{figS11_glycolysis_PYK_flux.PDF}
\caption{\textbf{The response of PYK flux to a step perturbation of ACE at $t=10$}.
({\it A}) The response of the original full model at ACE=0.05 and 0.77, respectively.
({\it B}) The response of the modified full model with blocked GLYO reaction at ACE=0.05 and 0.77, respectively.
({\it C}) The response of the minimal model.
Parameters: Glco=10 for ({\it A}) and ({\it B}); the parameters for the minimal model are the same as in Fig.~\ref{fig:MinimalModel}. }
\label{fig:glycolysis_PYK_flux}
\end{figure}
\subsection{Glycolytic oscillation in coupled yeast cells}
To study collective oscillations in a population of cells whose internal dynamics follows Eqs. (\ref{eq:MinimalModel}),
we adopt the following signal dynamics as in Ref. \cite{wolf2000transduction-S}:
\begin{subequations}
\begin{eqnarray}
\label{eq:signal-glycolysis-a}
\tau_s \dot{s}_{in}&=&\alpha_1 a-k_{in} s_{in}- D (s_{in}-s_{ex}), \\
\label{eq:signal-glycolysis-b}
\tau_s \dot{s}_{ex} &=& \phi D(s_{in}-s_{ex})-k_{ex} s_{ex}.
\end{eqnarray}
\label{eq:signal-glycolysis}
\end{subequations}
Here $s_{in}$ and $s_{ex} $ are the intracellular and extracellular signal concentration, respectively; $D$ is the membrane
permeability of the signalling molecule; $k_{in}$ and $k_{ex}$ are the intracellular and extracellular signal degradation rate;
and $\phi$ is the volume fraction of yeast cells, which increases with the cell density, and saturates at 1. The extracellular
signal strength (i.e., acetaldehyde concentration) in the coupled system is a function of $\phi$.
Let us first consider the situation of fast equilibrium between $s_{in}$ and $s_{out}$. Previously, Silvia De Monte \emph{et al.}
proposed a diffusion timescale $\tau_s\approx 0.003$ s by assuming a quasi-stationary concentration profile and that
ACE molecules need to diffuse across a spherical shell with an inner radius $r_1=3\ \mu$m and an outer radius
$r_2=6.5\ \mu$m~\cite{de2007dynamical-S}. This diffusion timescale is much smaller than the oscillation period of $37$ s.
Assuming the time for an ACE molecule to cross the cell membrane is of the order of 1 s or less,
we obtain the following approximate equation for $s=(s_{in}+s_{ex})/2$,
\begin{equation}
\tau_s \dot{s}=\frac{\phi}{1+\phi} \alpha_1 a-\Big(\frac{\phi k_{in}+k_{ex}}{\phi+1 }\Big)s.
\label{eq:reduced-signal-dynamics}
\end{equation}
Up to corrections of order $\epsilon$, the stationary state of the dynamical system defined by Eqs.~(\ref{eq:reduced-signal-dynamics})
and (\ref{eq:MinimalModel}) is given approximately by,
\begin{equation}
p\approx 1, \quad z\approx 1,\quad y\approx \frac{1}{ \alpha_2s + c_0},\quad a\approx \frac{1}{ \alpha_1},\quad s\approx \frac{\phi }{ \phi k_{in}+k_{ex} }.
\end{equation}
The signal strength increases with the volume fraction and saturates at $1/(k_{in}+k_{ex})$.
At small but finite $\epsilon$, corrections to the above expressions become significant at large $y$ or small $s$,
where the side reaction G3PDH in Fig.~\ref{fig:kineticMap} is activated to divert the glycolytic flux.
In the numerical studies presented below, we set the two ACE degradation rates $k_{in}$ and $k_{ex}$ to be small.
The signal strength $s$ varies over a broad range as the cell volume fraction $\phi$ increases.
\begin{figure}[!h]
\centering
\includegraphics[width=14cm]{figS12_glycolysis_coupled_minimum_model_oscillation.PDF}
\caption{\textbf{Collective dynamics of the minimal model coupled via Eq.~(\ref{eq:reduced-signal-dynamics})}.
({\it A})-({\it D}) Temporal trajectories at selected values of the volume fraction $\phi$. The same color scheme of
variables is used. Inset in {\it B} shows the signal trajectory on an enlarged scale.
Parameters: $h=3,\alpha_2=\alpha_1=1, \epsilon=0.01$,
$k_{in}=0.5$, $k_{ex}=0.3$, $\tau=0.01$, $c_0=0.02$, and $\tau_s=0.001$. }
\label{fig:collective_oscillation_glycolysis_illustration}
\end{figure}
Fig.~\ref{fig:collective_oscillation_glycolysis_illustration} shows numerical solutions of the coupled minimal model at
four selected $\phi$ values. Except the case at $\phi=0.01$, oscillations of $s$ and the intracellular variables are seen.
In Fig.~\ref{fig:collective_oscillation_glycolysis_phase_diagram}, we plot the
oscillation amplitudes and time-averaged values of $s$ and $O$ against the cell volume fraction $\phi$.
From the lower panel of Fig.~\ref{fig:collective_oscillation_glycolysis_phase_diagram}{{\it A} we see that,
for the signal dynamics chosen, the lower adaptive regime in Fig.~\ref{fig:MinimalModel}{\it D} is mapped to a
narrow interval of cell volume fraction $0.003<\phi<0.013$.
From Fig.~\ref{fig:collective_oscillation_glycolysis_illustration}, we see that onset of collective oscillations in the
coupled system takes place somewhere between $\phi=0.01$ and $0.014$. More detailed studies indicate that the transition
is not the expected Hopf bifurcation type, but instead emergence of a limit cycle at finite amplitude.
Similar behaviour was seen in the study of the full kinetic model (see Fig. 10 in~\cite{du2012steady2-S}).
On the other hand, experimental work seem to support the Hopf bifurcation scenario~\cite{dano1999sustained-S,de2007dynamical-S}.
We leave this issue to future investigations.
\begin{figure}[!h]
\centering
\includegraphics[width=14cm]{figS13_glycolysis_coupled_minimum_model_phase_diagram.PDF}
\caption{\textbf{Collective oscillations against the yeast cell density}.
({\it A}) Upper panel: oscillation amplitude of the signal $s$ as a function of the cell volume fraction $\phi$.
Lower panel: time-averaged signal concentration against $\phi$. At $\phi_c=0.34$, the signal strength reaches the upper
threshold $s_c=0.72$ for the oscillating state of individual cells (Fig.~\ref{fig:MinimalModel}{\it E}).
The oscillating state at $\phi>\phi_c$ can be considered as DQS driven by adaptation.
({\it B}) Upper panel: oscillation amplitude of the sender node $a$ against $\phi$.
Lower panel: time-averaged value of $a$ against $\phi$. Parameters are the same as in Fig.~\ref{fig:collective_oscillation_glycolysis_illustration}.}
\label{fig:collective_oscillation_glycolysis_phase_diagram}
\end{figure}
Beyond the onset point, oscillation amplitudes vary continuously with the cell density.
For $\phi>\phi_c=0.34$, the time-averaged value of $s$ falls in the upper adaptive regime in Fig.~\ref{fig:MinimalModel}{\it E}.
Since the cell density here already exceeds the threshold value required for collective behaviour of adaptive units,
oscillations continue.
Finally, we present numerical results demonstrating the effect of a slower cross-membrane transport of acetaldehyde
on the collective dynamics. The system dynamics is defined by Eqs.~(\ref{eq:MinimalModel})] for individual cells (with $s=s_{in}$)
together with Eqs.~(\ref{eq:signal-glycolysis}) for the intracellular and extracellular signal concentrations.
Fig.~\ref{fig:glycolysis_glycolysis_effect_D} shows the oscillation amplitude of $s_{in}$ together with the time-averaged
values of $s_{in}$ and $s_{out}$ at selected values of $D$. At $D=100$ and $10$, $s_{in}$ and $s_{out}$ are nearly identical
and the system behaviour is essentially the same as described above under the fast equilibrium assumption.
At $D=1$, the time-averaged value of $s_{ext}$ is noticeably smaller than that of $s_{in}$, indicating a significant
gradient of acetaldehyde concentration across the cell membrane. Nevertheless, collective oscillations via DQS continue over a broad range of the cell density. Interestingly, the oscillation is arrested at very high densities.
Collective oscillations disappear at $D=0.1$. Here, $s_{in}$ remains high due to the slow
intracellular degradation rate $k_{in}$, which places the single-cell dynamics in the upper adaptive regime even when the
cell density is very low. However, the phase delay across the cell membrane changes the response properties of the cell
to external signal variations. In this case, $s_{in}$ should be considered as the sender of the external signal but as one
can see from Eq. (\ref{eq:signal-glycolysis-a}), the adaptation
of $a$ to $s_{in}$ does not translate to adaptation of $s_{in}$ to $s_{ex}$ when $D$ is small. The latter is required for the adaptation route to collective oscillations.
\begin{figure}[!h]
\centering
\includegraphics[width=16cm]{figS14_glycolysis_effect_D.PDF}
\caption{\textbf{Effect of delay in cross-membrane transport of the signalling molecule on collective dynamics}.
Results of numerical integration of Eqs.~(\ref{eq:MinimalModel})] coupled to the two-component signal dynamics
Eq.~(\ref{eq:signal-glycolysis})] at selected values of $D=100,10,1, 0.1$.
In the lower panels, the blue and orange dots correspond to the average of $s_{in}$ and $s_{ex}$, respectively.
Other parameters are the same as in Fig.~\ref{fig:collective_oscillation_glycolysis_illustration}.}
\label{fig:glycolysis_glycolysis_effect_D}
\end{figure}
In summary, under fast equilibration between intracellular and extracellular acetaldehyde concentrations, the coupled system
exhibits collective oscillations over a broad range of cell densities, encompassing the adaptive and oscillatory regimes of a single cell.
Onset of collective oscillations at low cell densities exhibit complex behaviour due to the assumed sensitivity of the
reaction GAPDH to the NAD/NADH ratio. Delay in the cross-membrane transport of acetaldehyde weakens
adaptation of intracellular metabolite concentrations to change in the extracellular acetaldehyde concentration,
and may eliminate collective oscillations altogether when the delay is too long~\cite{richard1994yeast-S}.
At moderate delays, rise in the intracellular
acetaldehyde concentration brings individual cells to the oscillatory even when in isolation. The enhanced oscillation amplitude
at $D=1$ and low cell densities seen in Fig.~\ref{fig:glycolysis_glycolysis_effect_D}, however, is obtained under the assumption
that all cells in the population behave identically. This behaviour is susceptible to cell-to-cell variations as well as temporal noise
in intracellular dynamics. Our model study exposes this and other subtleties that can affect emergence of collective oscillations.
The specific effects we identified in this work could serve to guide the design of future experiments where various model
parameters can be controlled quantitatively, e.g., $k_{ex}$ for extracellular degradation rate of acetaldehyde by adjusting
the flow rate in microfluidic setups~\cite{gustavsson2015entrainment-S}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,522 |
**THE CORROSION OF CONSERVATISM**
_**Why I Left the Right**_
**MAX BOOT**
LIVERIGHT PUBLISHING CORPORATION
A DIVISION OF W. W. NORTON & COMPANY
_Independent Publishers Since 1923_
NEW YORK LONDON
To Robert Weil,
Editor Extraordinaire
All I have is a voice
To undo the folded lie,
The romantic lie in the brain
Of the sensual man-in-the-street
And the lie of Authority
Whose buildings grope the sky
—W. H. Auden,
"September 1, 1939" (1939)
## **CONTENTS**
_Prologue:_ NOVEMBER 8, 2016
1. THE EDUCATION OF A CONSERVATIVE
2. THE CAREER OF A CONSERVATIVE
3. THE SURRENDER
4. THE CHAOS PRESIDENT
5. THE COST OF CAPITULATION
I. RACISM
II. NATIVISM
III. COLLUSION
IV. THE RULE OF LAW
V. "FAKE NEWS"
VI. ETHICS
VII. FISCAL IRRESPONSIBILITY
VIII. THE END OF THE PAX AMERICANA
6. THE TRUMP TOADIES
7. THE ORIGINS OF TRUMPISM
_Epilogue:_ THE VITAL CENTER
_Acknowledgments_
_Notes_
_Index_
## _Prologue_
## **NOVEMBER 8, 2016**
NOVEMBER 8, 2016, WAS ONE OF THE MOST DEMORALIZING days of my life. It was also, in ways that have become impossible to ignore, devastating not just for America in general but for American conservatism in particular.
I had never imagined that Donald Trump could be elected president. If you had suggested to me before 2016 that such a thing was possible I would have replied that it was too far-fetched to contemplate—it sounded like the plot of a dystopian science-fiction movie. Arnold Schwarzenegger would have been a more plausible president—and he wasn't even born in America. I didn't think Trump would win a single Republican primary. Sure, he had been polling strongly in 2015, but I figured that when the actual balloting began my fellow Republicans would sober up and realize that the reality TV star and real-estate mogul was not remotely qualified for the nation's highest office.
Trump had offended my sensibilities from the very first day of the campaign, June 16, 2015, when he had come down the garish escalator at Trump Tower to castigate Mexican immigrants in crudely xenophobic terms. "They're bringing drugs," he said. "They're bringing crime. They're rapists. And some, I assume, are good people." A month later, he launched an odious attack on Senator John McCain, a man whose presidential campaign I had been proud to advise in 2008. This is what Trump, who had gotten five draft deferments, had to say about a war hero who had endured nearly six years of hellish captivity in North Vietnam: "He's not a war hero. He was a war hero because he was captured. I like people who weren't captured." A few months after that, in November 2015, Trump hit another low, mocking a disabled reporter who had the temerity to question his bogus claims to have seen thousands of Muslims in Jersey City, New Jersey, cheering as the World Trade Center came down. Trump then lied about what he had done—even though his cruel japery was recorded on videotape.
There was no possibility, I figured, that the party of Lincoln, Roosevelt, and Reagan would endorse Trump for president. Was there?
When the primaries began and Trump began winning state after state, I thought I had entered _The Twilight Zone_. The torment worsened when he locked up the nomination and Republican after Republican dutifully lined up to endorse his candidacy after having lambasted him in the harshest terms possible. Former governor Rick Perry had called Trump a "cancer on conservatism" before endorsing said cancer—and being rewarded with a cabinet post. Former governor Bobby Jindal had called Trump a "madman who must be stopped" before endorsing said madman. Senator Rand Paul had called him a "delusional narcissist" before endorsing said narcissist. Most painful of all for me, Senator Marco Rubio, whose presidential campaign I had served as a foreign policy adviser, went from denouncing Trump as a "con artist" to endorsing said con artist. House Speaker Paul Ryan got my hopes up by hesitating to endorse Trump, but in the end, he too bent the knee. This was not the Republican Party I knew. Or thought I knew. How could so many Republicans for whom I had such respect have betrayed everything that they—and I—believed in? What was going on? How could all of these conservatives turn into Trump toadies? I was angry and bewildered. My faith in the Republican Party was shaken and has never recovered.
But at least I comforted myself that in the general election there was no way the American people could possibly elect someone like Trump. I had come to America as a six-year-old from the Soviet Union in 1976 and had grown to revere the country that had offered asylum to my family. I was convinced that America was the greatest and most selfless country in the world. Now I had faith that the voters would in their wisdom choose Hillary Clinton, who was a deeply flawed and seriously uncharismatic candidate, to be sure, but also extremely knowledgeable, resolutely centrist, and amply qualified. I had never voted for a Democrat in my life, but for me it was an easy call. Here I was, a conservative Republican, voting for Clinton; I figured that there would be plenty of others who would do the same. If Trump couldn't even count on the undivided support of the GOP, there was no way he could win.
Like countless other commentators, I was sure Trump was finished on October 7, 2016, when a videotape emerged in which he could be heard bragging that because he was a "star" he could do anything he wanted to women—even "grab them by the pussy." Numerous Republicans withdrew their endorsements and urged Trump to drop out. Yet when he refused to withdraw, many of the same Republicans came crawling back to re-endorse him. The race tightened as Election Day approached. Yet I was still certain—foolishly, naïvely, pathetically certain—that Trump could not win. My Pollyannaish faith in America had blinded me to what was to come, and that faith has not survived the debacle to come.
I agreed to spend election night at the Comedy Cellar nightclub in downtown New York, offering commentary on the results along with other pundits and comedians at a forum organized by _Foreign Policy_ magazine. I was nervous in the afternoon but was reassured by rumors that the exit polls showed a Clinton victory. I was on stage, chatting with the other panelists, when around 8 p.m. I saw my partner, Sue, growing increasingly agitated across the room. She kept looking at her phone and getting more upset. I sneaked out my own phone and saw what was disturbing her—the _New York Times_ had just moved Florida into Trump's column. It now looked as if he had a path to victory. As the night wore on, swing state after swing state went for Trump. Clinton went from the odds-on favorite to an increasing long shot.
By the time Sue and I got home to our apartment on the Upper West Side at 10 p.m. or so, it was obvious that the unthinkable was about to become the inevitable: Donald Trump was going to be elected the 45th president of the United States. A friend came over from the Clinton election-night party at the Javits Center; she was crying and in shock. I swilled a Scotch and took some sleeping pills—something I don't normally do—and tried to sleep. And, yes, I know you're not supposed to combine sedatives with alcohol, but you're also not supposed to elect a bigoted bully as president of the United States. This was a day for disregarding the rules. Even with chemical inducements, however, my sleep was fretful and disturbed because I knew that I would awaken to a nightmare. My America had become Trump's America. My Republican Party had become Trump's party. My conservative movement had become Trump's movement.
The first thing I did the next morning—the dawn of what I felt was a new annus horribilis—was to go online and change my voter registration. I had been a Republican since turning eighteen just before the 1988 presidential election. Now, at the age of forty-seven, I became an independent. Politics is a team sport. Suddenly I was without a team. I was politically homeless. In an instant I felt alienated from some of my oldest friends and fellow travelers—conservatives with whom I had been in one fight after another over the past quarter-century. How was it possible that 90 percent of Republicans had supported a charlatan who had only recently been a Democrat and who had few fixed convictions outside of narcissism and nativism, racism and sexism? My sense of alienation has only deepened as I have watched the Trump presidency in action. No other president has been more hostile to the values of conservatism as I conceived it.
Conservatism, American-style, means different things to different people. There is, after all, an inherent tension in advocating a conservative vision in a liberal society in which social, economic, and technological change is constant. American conservatism is very different from the kind of "blood and soil" conservatism that has long been characteristic of Europe. Continental conservatism is chauvinistic and pessimistic; American conservatism is optimistic and inclusive. For me, conservatism means prudent and incremental policymaking based on empirical study; support for American global leadership and American allies; a strong defense and a willingness to oppose the enemies of freedom; respect for character, community, personal virtue, and family; limited government and fiscal prudence; freedom of opportunity rather than equality of outcome; a social safety net big enough to help the neediest but small enough to avoid stifling individual initiative, enterprise, and social mobility; individual liberty to the greatest extent possible consistent with public safety; freedom of speech and of the press; immigration and assimilation; and colorblindness and racial integration. Looming above them all are two documents that I revere, as should every American. The Declaration of Independence defines the United States as a nation bound together not by shared heritage or blood but rather by a shared belief in the "self-evident" truths "that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness." The "pursuit of Happiness" is a critical concept, putting personal freedom at the center of our political enterprise. While the Declaration lays out the goals of self-government, the Constitution defines how we can achieve them. It protects our liberties, limits the government's power, and ensures that the rule of law prevails. We honor, defend, and respect the Constitution, and the offices, laws, and norms that derive from it. All Americans, of all political persuasions, are expected to defer to the Constitution, but it should be of particular concern to conservatives who proclaim their desire to conserve what makes America great.
That, to me, is American conservatism. That is what I believe. Those are the ideas I have tried to advance as a writer and commentator. To judge by his words and actions, Trump does not understand or believe in a single one of these principles. Yet he remains wildly popular among Republicans and conservatives. When 2016 began I could hardly find a Republican who had anything positive to say about Trump. By the beginning of 2018 it was hard to find a Republican who had anything negative to say about him—at least in public.
How can this be? Did I not understand all along what American conservatism was all about? Did I miss essential features that Trump had discerned and used to his benefit? Or had conservatism morphed under the magnetic pull of Trump's outsized personality to become something very different from the movement I had grown up in?
The modern conservative movement was inspired by Barry Goldwater's canonical text from 1960, _The Conscience of a Conservative_. I believed in that movement, and served it my whole life, but under the pressure of Trumpism, conservatism as I understood it has been corroding—and so has my faith in the movement. Hence this book's title. I am perceiving ugly truths about America and about conservatism that other people had long seen but I had turned a blind eye to. I no longer like to call myself a conservative, a label that has become virtually synonymous with Trump toady. I now prefer to think of myself as a classical liberal.
I would like to be able to quote Ronald Reagan's quip when he became a Republican—"I didn't leave the Democratic Party, the Democratic Party left me"—but in truth my beliefs are shifting because of the rise of Trumpism and other contemporary developments such as the failure of the Iraq invasion, the Great Recession of 2008–2009, the #MeToo movement, and the spread of police videotapes revealing violent racism. My ideology has come into conflict with reality—and reality is winning. I have undertaken a painful and difficult intellectual journey, leaving behind many of the simple verities that I clung to for decades as a "movement" conservative. I am now forced to think for myself, and that is not an easy thing to do. But given the epochal events that have shaken America, this self-reflection is necessary, indeed overdue. I only wish more conservatives were willing to engage in similar self-examination instead of resorting to glib insults of "libtards" and "snowflakes" or reflexive defenses of the man who has usurped their party.
I am no longer a Republican, but I am not a Democrat either. I am a man without a party. This is a record of my ideological journey so far—and of my attempts to come to grips, honestly and unflinchingly, with the phenomenon known as Trumpism. The question that haunts me is: Did I somehow contribute to the rise of this dark force in American life with my advocacy for conservatism?
Whatever the case, I am now convinced that the Republican Party must suffer repeated and devastating defeats. It must pay a heavy price for its embrace of white nationalism and know-nothingism. Only if the GOP as currently constituted is burned to the ground will there be any chance to build a reasonable center-right political party out of the ashes.
How did we get to the point where I—a lifelong Republican—now wish ill fortune upon my erstwhile party? To find the answer, I invite you to turn the page with me, literally as well as figuratively.
What follows is not a full-blown memoir or autobiography. But to make you understand why I—and other #NeverTrump conservatives, all too few in number—feel such a strong sense of betrayal at the hands of Trump and his Republican Party, it is important for you to understand how I became a conservative in the first place and what it felt like to be a conservative in the heyday of the movement. My history, I feel, can help the reader to make sense of late-twentieth-century conservatism—now, in the early twenty-first century, practically unrecognizable. I take the story up to the present day, explaining why I left the Republican Party because of my profound opposition to Trump, how Trump continues to traduce conservative principles, and what the future holds for me and other conservatives who cannot imagine being members of a Trumpista party. Put another way, this is a tale of first love, marriage, growing disenchantment, and, eventually, a heartbreaking divorce. Today we are locked in a bitter custody battle over the future of the Republican Party: Will it return to its previous principles or will it remain forever a populist, white-nationalist movement in the image of Donald Trump?
This book, I strongly suspect, will infuriate many of my old comrades on the right who will conclude that I have gone soft in the head or sold out my beliefs to gain popular acceptance in liberal circles. I, in turn, am convinced that they are the ones who have gone off the rails by embracing a demagogue who seems to equate bigotry with conservatism. There is a gulf between us that cannot be bridged, at least not while Trump is still in office. Likewise, what follows is unlikely to satisfy the hard left. No matter how strongly I come out against Trump and his hateful works, I find it is never enough for the most doctrinaire leftists who seem to think that no step short, perhaps, of ritual suicide will atone for my "war crimes," which upon closer examination seem to consist of supporting an invasion of Iraq that was backed by bipartisan majorities in both houses.
This book is not addressed to the far left or the far right. It is written with the center-left and the center-right in mind. My hope is that my ideological odyssey will inspire others—that I can be part of a larger, bipartisan movement in America toward greater moderation and civility in our politics. Or, if that doesn't happen, and if the present trend toward extremism continues, I will at least register my dissent in the strongest terms I know.
I love America. I am devoted to conservative principles. I want to defend what I hold dear when I see it under unprecedented attack from within—with the greatest threat posed by a man at the very pinnacle of power. This is how I became a conservative and why I no longer feel part of a movement whose betrayal of its principles is abhorrent to me.
**THE CORROSION OF CONSERVATISM**
## _1_.
## **THE EDUCATION OF A CONSERVATIVE**
I WAS BORN IN WHAT DONALD TRUMP WOULD CALL A "shithole" country. My experience of becoming a conservative is unique—but so is everyone else's. I cannot tell all of their stories, so I will tell mine in the hope that others will recognize at least a glimmer of their own experiences. My tale may not be typical, but my journey from embrace of the conservative movement to exit is nevertheless representative of a certain number of #NeverTrumpers. The major difference is that I had to come a much longer way than most of them to join the right. Before I could become a conservative, I first had to become an American.
The Soviet Union in 1969, the year of my birth, was a grim place ruled by a geriatric oligarchy that repressed its own people and ruined its own economy. Store shelves were bare and lines long. Moscow, where my family lived, was better off than the provinces, but it was still an impoverished and illiberal place. I am especially sensitive to growing signs of anti-Semitism in Donald Trump's America because that prejudice was so pervasive in Leonid Brezhnev's Soviet Union. Jews—their "nationality" stamped in their internal passports, just like in Nazi Germany—had to work harder to get into a university or find a job because they were routinely discriminated against. Most synagogues had been shuttered. There were no Hebrew schools. Despite the official passports, most Jews, including my parents, were thoroughly assimilated and thought of themselves as Russians, but that's not the way ethnic Russians viewed them. My father got his share of beatings as a boy from anti-Semitic bullies who called him a _zhid_ (kike).
Little wonder that my parents—young, educated, urban Jews—were anxious to emigrate. But they couldn't find a way out of this countrywide prison. My father became a refusenik, one of the dissidents who was refused permission to leave. He agitated for freedom by passing around samizdat publications and talking to Western reporters. That my parents and hundreds of thousands of other Soviet Jews were finally able to leave was due largely to neoconservative foreign policy.
American Jews began agitating in the early 1970s for their Soviet brethren to be allowed to emigrate—a fundamental right enshrined in Article 13 of the 1948 Universal Declaration of Human Rights. As Gal Beckerman recounts in his 2010 book, _When They Come for Us We'll Be Gone: The Epic Struggle to Save Soviet Jewry_ , this became one of the first major human-rights campaigns in US foreign policy, presaging later agitation over apartheid, Bosnia, and Darfur. Senator Henry "Scoop" Jackson, a New Deal Democrat from Washington, became the most prominent advocate for the Soviet Jews. Jackson wasn't Jewish; his family, having come from Norway, were the kind of immigrants Trump would have liked. But, unlike Trump, he believed that the United States needed to stand up for human rights, and he worried that the Nixon administration was caving into Communism with its policy of détente. Jackson's nemesis was Secretary of State Henry Kissinger, who was a Jewish immigrant himself but also a Realpolitiker who did not believe that human rights should intrude into cold-blooded calculations of national interest.
Jackson, working with his aides Dorothy Fosdick and Richard Perle, won the battle by mobilizing the support of liberal Democrats and conservative Republicans—an alliance unthinkable today—to link US diplomatic and trade ties to the Soviet Union with its willingness to allow Jewish emigration. His triumph was symbolized by the passage in 1974 of the Jackson-Vanik Amendment, which forbade the United States from granting most-favored-nation trade status to any nonmarket (i.e., Communist) economy that did not respect the freedom to emigrate. Under pressure from the United States, Soviet leader Leonid Brezhnev opened the door a crack and allowed some Jews out.
In later life I would support giving moral concerns a prominent place in US foreign policy, a stance that has been associated with neoconservatism. It's not hard to see why I came to this position: I wouldn't have become an American, at least not when I did, if the Realpolitik approach had prevailed.
MY FATHER DID, IN FACT, get out of the Soviet Union in 1973 a few steps ahead of the KGB; the KGB had just issued a warrant for his arrest, but the bureaucracy was so ineffective that the word did not reach the passport control office at the airport in time to prevent him from flying away. My father settled in Houston, where he went to work for NASA as a Russian-English translator on the Apollo-Soyuz space program, a centerpiece of the Nixon-Kissinger policy of détente. My parents had divorced when I was only two and when we—my mother, grandmother, and I—followed him to America in 1976, we did not wind up in the same city. We went first to Italy, where Soviet immigrants were expected to wait for an entry visa to the United States. I have a memory of New Year's Day 1976 in Rome—the holiday that Russians typically celebrated rather than Christmas or Hanukah. The only present I received was a little toy boot filled with candy. It was all my mother could afford. After a few months in Italy, we were allowed to move to Los Angeles. Why L.A.? Because my aunt and uncle—a sculptor and a physician—had just moved there with their two children. This is what nativists would call "chain migration": the practice of immigrants bringing relatives from the old country to join them in America.
My widowed and retired grandmother, a babushka who had once worked as an engineer in Russia, settled into an apartment in Hollywood, supported by my mother and by the Social Security Administration's Supplemental Security Income payments. (Thank you, America.) Before long, my mom—a thin woman with a regal bearing, a Bohemian taste in clothes, and an indescribable but vaguely European accent—and I moved some sixty miles to the east. She had gotten a job teaching at the University of California, Riverside, so we relocated to a small town in the desert as far removed as imaginable from the snowy metropolis where I had been born. At first, we lived in an apartment complex, and later, in a small home at the edge of a dun-colored mountain range full of rattlesnakes and mesquite. In Russia, mom had taught English to Russians. In America, she would teach Russian to Americans.
I am immensely, indescribably grateful to my mother for bringing me to America, a country that I would come to love as my own—because it indeed became my own. If I had remained in Russia, I can't imagine how I would have survived under the Putin regime, which is re-creating many of the worst elements of Soviet despotism that my father agitated against in the late 1960s and early 1970s. Putin may be even more ruthless than Brezhnev in killing inconvenient dissidents, although usually in ways that are difficult to trace back to the Kremlin. Little could I imagine then that an admirer of the Russian despot would eventually lead the Land of the Free.
In hindsight I marvel at the way my parents, in their twenties, were able to pack up, abandon everything and everyone they had known, and start new lives in a country they had never visited. I do not know if I could be so brave. There was no going back in those days: on our way out of Russia we were stripped of our Soviet nationality. From the Soviet standpoint, we were becoming nonpersons. We would be lucky to reach our Russian relatives, including my father's parents, on the telephone; visiting them was out of the question until the Soviet Union collapsed in 1991, and of course no one knew in the 1970s that the USSR wouldn't last forever. We had no money and no prospects. If some generous country did not take us in, we would become stateless persons. The only two countries that were willing to succor us were Israel and America. Because my mother spoke English but not Hebrew, we decided to come to America. I am as thankful to America for taking us in as I am to my mother for getting us there, and I am greatly saddened that the United States has extended refugee status to so few others in the Trump era. I think of how bleak our lives would have been if we had not been allowed to come to this distant shore, and I feel the pain of those suffering and in want who are today denied entry.
Ours was a modern version of the immigrant saga shared by most Americans, only instead of arriving by ship at Ellis Island, we reached Kennedy Airport by jetliner. But the agonies of assimilation had not changed much since the turn of the twentieth century, even if my parents, unlike earlier generations of Jewish immigrants from Eastern Europe, did not have to toil as garment workers or junk peddlers.
Everything about this new country was strange and wondrous. We were mystified when we visited a temple of haute cuisine called McDonald's that served an exotic dish known as french fries. There were no McDonald's in Russia back then. No fast food of any kind. My mom's first car was a used Buick Electra with giant tail fins. I remember sitting in the back while she learned to drive; there were few private cars in Russia back then and no driver's ed. She couldn't simply leave me with a babysitter because she couldn't afford one. I was a "latchkey kid" who came home from school by myself while still in elementary school—an experience that induced a combination of befuddlement and dread in my young heart. Where, I wondered, was my mother? I cried tears of consternation and frustration, but she was doing the best she could. My parents had little money in the Soviet Union and were allowed to take next to nothing out. We were only able to finance the transition to a new life with loans from the Hebrew Immigrant Aid Society—loans that my scrupulously honest mother later repaid. And working as a junior lecturer at a state university wasn't exactly a lucrative job. My mother was lucky to have found employment at all. She would later say that it was either teaching Russian or washing dishes.
I was only seven years old when we arrived in the bicentennial year—and desperate to fit in. Everything and everybody was new, strange, and intimidating. It was such a traumatic experience that I have blotted a lot of it out of my memory. I do remember being bewildered by meeting African American, Asian, and Latino kids: while the Soviet Union was a polyglot empire, the only black people to be found in Moscow were a few African students who encountered terrible prejudice while being indoctrinated at Patrice Lumumba University. The idea of a racially and ethnically diverse society was, at first, strange to me, but I would quickly come to see it as one of the glories of America—a hard-won achievement that is now under siege.
My mother enrolled me in swim classes, piano lessons, and Hebrew school to give me the athletic, musical, and religious education that she had lacked. I became a slow if confident swimmer, but I can't speak a word of Hebrew or play a note on the piano. Still, I have to give her credit for trying. And even if most of my extracurricular education did not stick, all of those years of Hebrew school did ensure that I grew up identifying with the Jewish people and the state of Israel in a way that my parents had not been free to do back in the USSR.
There were no English as a second language classes in those days, at least none that I was aware of. I was thrown straight into regular elementary school classes—first in Hollywood, then in Riverside. It was like learning to swim by being thrown into the deep end of a pool. Eventually, after a period of floundering around, I dog-paddled to safety. Before long I became so proficient in English that, to my mother's initial dismay, I refused to speak Russian anymore. I even lost any trace of a Russian accent—an achievement that, I like to joke, may have hindered my future career as a foreign policy pundit because I lack the gravitas that comes from having a thick foreign accent like Henry Kissinger or Zbigniew Brzezinski. I later learned that my parents—my mother in Riverside, my father in Houston—discussed how to handle my refusal to speak Russian and decided to leave me alone. They didn't want me to become a linguistic half-breed who did not speak either language perfectly, and they were sufficiently alienated from their homeland that they didn't want me to have much to do with it either. So we developed an odd routine at home where my mother would speak to me in Russian and I would answer in English. Ironically, in later years, my mother would become a pioneer of heritage language studies, teaching the children of immigrants how to speak their parents' tongue. I like to think that I inspired her professional success with the C that I received in the only Russian-language class I ever took in college.
By now—more than forty years after coming to America—I have all but forgotten Russian and have few if any memories of Russia beyond what I can see in grainy black-and-white photos showing me as a Soviet toddler bundled up like a polar explorer to brave the harsh Moscow winter. I have never visited Russia since leaving. I feel entirely American. Or rather I felt that way before the rise of Trump and his demonization of immigrants.
THE FIRST BIG DISLOCATION of my childhood that remains in my adult consciousness was moving to America. The second and third were acquiring a stepfather and moving from Riverside to Los Angeles. The former event occurred in 1978, the latter in 1981. My mother proved such a proficient Russian professor that she was hired away by UCLA, a larger and more prestigious University of California campus. Three years before moving, she married a dark-haired and handsome, if socially awkward, geophysicist who was a fellow Russian émigré. In his heavy accent, he would recount tales of his adventures researching earthquakes during his travels in Soviet Central Asia—akin to the American West. He made a positive impression on me initially by whirling me around and playing with me. I remember nine-year-old me sitting on the floor of a room in our Riverside house, watching Saturday morning cartoons, when Mom came in to tell me that she was getting married. I was so excited I couldn't sit still. I had a stepfather! I even wanted to adopt his last name—an impulse that passed. He would not prove to be a hands-on parent. A typically unworldly scientist, he was always intensely focused on his work—predicting earthquakes using complicated and, to me, incomprehensible mathematical models. Everything else, including his new family, receded into the background.
I would see my biological father for only a few months each year when I was shipped off to Houston to spend the stifling southern summer with him and his younger second wife, a hairdresser also newly arrived from Russia. From a child's perspective, my father seemed to be living the quintessential American life of the swingin' seventies, complete with driving a big car made in Detroit, hosting pool parties at his new townhouse, playing tennis in short shorts, partying in shiny silk shirts, and, of course, consuming lots of cigarettes and booze. A character out of Philip Roth or Saul Bellow—a young Jewish striver on the make—he was full of strike-it-rich schemes that never quite materialized. Never lacking in self-confidence, he thought of himself as a major intellectual and resented the world for not sharing his self-assessment. But it was hard to hold his overactive imagination against him. He was a skilled raconteur who would hold listeners in thrall for hours talking about his favorite subject—himself. An irresponsible ne'er-do-well who did not contribute financially to my upbringing or attend my high school or college graduations, bar mitzvah, or wedding, my sybaritic father could nevertheless charm anyone he met—except possibly his own son. My ascetic mother, who died of cancer while I was writing this book, was considerably more responsible, sober, and hard-working but also more of a scold and not nearly as entertaining. My father was an exhibitionist; my mother, intensely private. He was portly; she, thin. It was no surprise they got divorced; the wonder was that they were ever married to begin with.
In the late 1970s and early 1980s, my father always seemed to be busy during the day, hustling for work as a freelance translator, soon to become an advertising copywriter, or . . .doing something else. I learned when I became a little older that he was a prodigious womanizer despite his beer belly, baldness, and generally unprepossessing appearance. While he was busy elsewhere—I never quite knew where—he would park me at an arcade with a bag of quarters to entertain myself playing Pac-Man, Space Invaders, and Frogger. But it wasn't as if he neglected my intellectual development.
WHEN I WAS THIRTEEN my father got me a subscription to _National Review_. As an émigré from a Communist country and a self-styled intellectual, he naturally embraced the most anti-Communist strand of American political life, and he wanted to introduce me to the ideas associated with the "conservative movement." Its Bible was _National Review_ , a magazine launched in 1955 by William F. Buckley Jr., a wealthy young Yale graduate, to stand "athwart history, yelling Stop." I grew up reading _NR_ , as it was known to its devotees, and thereby became immersed in the worldview of its regular writers—not only Buckley but also Richard Brookhiser, Joseph Sobran, Erik von Kuehnelt-Leddihn, Jeffrey Hart, Brian Crozier, John Simon, Ernest van den Haag, Ralph de Toledano, D. Keith Mano, and other stalwarts. They were hardly household names even in those days, but I can list their names more easily today than I can the star players on the Los Angeles Rams during their thrilling and unlikely run to the Super Bowl in 1979. The _NR_ writers were a learned, worldly, elitist, and eccentric lot far removed from the simpleminded, cracker-barrel populists who have taken control of the conservative movement today. The Austrian-born Kuehnelt-Leddihn, for example, was a self-described monarchist who could speak eight languages and read seventeen others. Their brand of conservatism was known as fusionism, a term coined by the philosopher Frank Meyer for an inclusive approach combining free-market economics with traditional social views and a hawkish, anti-Communist foreign policy. This became my brand of conservatism too, at least when I was young—minus the Catholicism of Buckley and many of the magazine's other mavens.
In addition to faithfully reading _NR_ , I read many of the books that had influenced its contributors or been written by them, including Whittaker Chambers's _Witness_ , a searing chronicle of the author's break with Communism; F. A. Hayek's _The Road to Serfdom_ , the influential free-market manifesto; Russell Kirk's _The Conservative Mind_ , a compendium of the conservative canon; and Edmund Burke's _Reflections on the Revolution in France_ , the ur-text of modern conservatism, describing the English political philosopher's rejection of the radicalism of the French Revolution in favor of gradual, organic change. I doubt that Donald Trump has heard of any of these books, much less read them. I doubt, in fact, that his recreational reading has ever extended beyond _Golf_ magazine or possibly _Playboy_ when it still flourished. But these tomes were part of the common curriculum for conservatives of my generation.
My growing interest in politics was evident at my bar mitzvah in 1982 at the Wilshire Boulevard Temple in Los Angeles, a magnificent old pile of a synagogue that looked like a grand cathedral from Europe and that was located in a part of the city where few Jews lived anymore. It was presided over by another impressive relic, Rabbi Edgar Magnin. By then in his nineties, he was part of a department store dynasty and had been the "rabbi to the stars" since the birth of Hollywood in the 1910s. In my speech at the end of the ceremony I did not speak on the usual Torah theme. Even then I was more interested in worldly rather than religious affairs—of which I have remained lamentably and invincibly ignorant. I even had to read my Torah portions from an English transliteration. My speech, delivered in a newly purchased beige suit that I loved, was a defense of Israel's recent invasion of Lebanon. It displayed my precocity, my attachment to Israel, a country I had not yet visited—and my questionable judgment, since the invasion would turn out to be a fiasco that would embroil Israel in a Vietnam-like quagmire.
In case you haven't figured it out yet, I was a nerd. Massively so. I wasn't a math or science geek—I was never good at those subjects, which is perhaps one reason why I did not grow up to become a Silicon Valley billionaire. I was a humanities nerd, history division. That's why I used my bar mitzvah to share my views of geopolitics. Many people are drawn to conservatism by the study of economics or political philosophy. My route was via history. In my bedroom I did not have pictures of typical teen idols. I had a giant portrait of Winston Churchill—the famous picture snapped by the peerless photographer Yousuf Karsh in 1941 showing the great man glowering. (Karsh induced that facial expression by plucking Churchill's cigar out of his mouth while he was posing.)
Like many American conservatives, I idolized Churchill and read everything by and about him that I could get my hands on. He impressed me not only because of his resolution and foresight in rejecting appeasement in the 1930s—he was a hawk who was amply vindicated by subsequent events—but also because of his graceful literary style and his appetite for adventure and the good life. In eighth grade at my public junior high school in the San Fernando Valley, I even made a deal with my English teacher to let me read Churchill's _History of the English-Speaking Peoples_ instead of _The Adventures of Huckleberry Finn_. (I did eventually read Mark Twain's classic.) But the Churchill book I liked best was _My Early Life_ , his rollicking account of growing up in an aristocratic British family and then serving as a young officer and war correspondent on the frontiers of the British Empire. From Churchill my interest expanded to World War II and then to military history generally.
My mother and stepfather, as typical members of the Russian intelligentsia, had highbrow tastes—they loved classical music, ballet, opera, and museums. Even living in Riverside, they would regularly shop at the one German deli in town where they could get sausages, beer, mustard, pickles, and other delicacies that reminded them of what they ate back in Russia. I lived in a halfway house between Russian and American culture but became increasingly Americanized as I grew older. While I never developed a taste for Wonder Bread, Coke, or cereal—those quintessential products of the American industrial-food complex—I did become an avid consumer of peanut butter and that American delicacy, french fries; a devotee of classic rock—the Beatles and the Clash, Tom Petty and Billy Joel—rather than of Bach, Beethoven, and Mozart; and an avid watcher of television in an era where there were only a handful of channels. _Mission Impossible_ , _Rockford Files_ , _Starsky & Hutch_, _Dukes of Hazzard_ , _CHiPs_ , _Greatest American Hero_ —I loved all the action shows.
I read a lot but can't claim that all my reading was highbrow. Well, I could, but I wouldn't be telling the truth. I devoured a lot of comic books, which I carefully collected in plastic packets in the confident expectation that someday I would make a mint by reselling them. (I recently threw away all my comics in the process of cleaning out my parents' old home, thereby parting with another youthful illusion.) I didn't like the alienated superheroes, Spider-Man and the X-Men, that are so popular with angst-ridden teenagers. I liked the more mainstream, all-American heroes Batman and Superman; I particularly empathized with the Man of Steel because he was a visitor from another planet. As my TV-viewing and comic-book-reading habits make clear, I was becoming a typical suburban kid of the 1980s—achieving my great ambition of fitting into this new society to which I had been transplanted.
My mother and stepfather pretty well left me alone. They neither encouraged nor discouraged my headlong rush toward assimilation. They were busy at work, and this was a more easygoing era when no one thought anything of smoking around a kid or driving around without seatbelts. There were of course no cell phones or internet in the 1980s, so even though I watched too much TV and read too many comic books, there was plenty of time left over for thinking and dreaming. An only child, and not a particularly outgoing one, I spent a lot of my childhood by myself, lost in my own head. I recall that one of my persistent daydreams was to travel the world on a great big yacht with my father—a subconscious indication of how much I missed the old man even though I grew increasingly exasperated in my interactions with him as the years went by.
My dad's self-absorption and bumptiousness grated on me as I grew older; his stories became less amusing when I heard them for the fifth or sixth time. I saw less of him as I entered high school because I refused to spend my summers with him anymore. Perhaps for this reason our brands of conservatism diverged—a reminder of how many different views can lodge under that catchall title. He became a European-style reactionary who pined for the days of monarchy and admired the eighteenth-century anti-Enlightenment thinker Joseph de Maistre. After divorcing the Russian hairdresser, marrying an English woman, and moving to London in 1988, he often expressed his disdain for the United States as a proletarian place devoid of true culture. He liked the European Union even less: he compared it, absurdly enough, to Nazi Germany or the Soviet Union because of its supposedly undemocratic character. I, on the other hand, liked the European Union because it subsumed nationalist competition in Europe, and I loved the United States, my adopted homeland. His brand of conservatism seemed to me, a kid who had grown up in America, alien and strange. My brand of conservatism, which drew on the nineteenth-century English classical-liberal tradition, was firmly in the mainstream of American thought. At least in those days.
I WAS ALWAYS INTERESTED IN current events—hardly surprising given how closely the fate of my family had been tied to the twists and turns of the Cold War. My parents were faithful subscribers to the _Los Angeles Times_ and _Newsweek_ , and I recall unruly mounds of old issues of _Newsweek_ piled up in our attic like fish on a deep-sea trawler. I read _Newsweek_ along with _National Review_ to get the facts from the former and opinions from the latter—a quaint distinction that still existed in those pre-internet days.
I was also a faithful devotee of the Sunday morning show on ABC, _This Week with David Brinkley_. Like millions of other Americans, I couldn't wait for the avuncular Brinkley to fill me in on "the news since the Sunday papers." My favorite part of the show was the roundtable at the end, when Brinkley would chew over the news of the day with Cokie Roberts, a genial Every Mom, and George F. Will, a bow-tied Tory delivering High Church judgments in crisp, complete sentences. It was a very civilized conversation, although I sensed that Will sometimes became exasperated with Roberts's liberal pieties.
Other kids dreamed of becoming athletes or astronauts. I dreamed of becoming a syndicated columnist like George F. Will or William F. Buckley Jr. I regularly watched Buckley's PBS show, _Firing Line_ , in which he would engage in erudite conversation with a single guest, often a fellow intellectual, for thirty to sixty minutes. It was as far removed from the sound-bites and snark that dominate cable news today as quill pens and foolscap are from laptop computers and iPads. I can still picture in my mind's eye Buckley clutching his clipboard, his tie askew, slumping, in that peculiarly aristocratic way, ever lower in his chair until he was almost horizontal, arching his eyebrow, and delivering a devastating critique or cutting witticism with a slight smile flickering at the edge of his mouth. _Firing Line_ elevated conservatism; the shows that conservatives watch today, primarily on the Fox News Channel, debase and dumb down the movement.
In 1984 Buckley came to speak at UCLA. I was in high school and drove with my mom across the hills from the San Fernando Valley to hear him deliver a typically dazzling oratorical performance punctuated by his Brobdingnagian vocabulary and coruscating wit. He radiated sophistication and joie de vivre. His books served up not only his conservative opinions but also his jet-set lifestyle—yachting across the Atlantic, skiing at St. Moritz, dining with John Kenneth Galbraith and David Niven. _This_ was who I wanted to be. It goes without saying that Buckley, with his mid-Atlantic accent and aristocratic languor, would be derided by today's populists as a "globalist," "cuck," and "RINO" (Republican in name only) who doesn't sound or act like a "real" American.
One of the most important contributions that Buckley made from his perch at _National Review_ was to drive assorted cranks and extremists out of the ranks of the right. He had initially sympathized with some of their views, particularly in his uncompromising anti-Communism and opposition to desegregation, but in the 1960s he exiled the John Birchers—conspiracy theorists so unhinged that they claimed Dwight D. Eisenhower was a Communist agent. Later he would excommunicate the writers Joseph Sobran, Samuel Francis, and Pat Buchanan for their blatant anti-Semitism. All the more painful, then, to see the intellectual heirs of these disreputable elements crawl out of the gutter to seize control of Buckley's movement.
Buckley clearly saw through Donald Trump, whom he identified as a narcissist and demagogue all the way back in 2000, when the real-estate developer was first flirting with a presidential campaign. "When he looks at a glass, he is mesmerized by its reflection," Buckley wrote. "If Donald Trump were shaped a little differently, he would compete for Miss America. But whatever the depths of self-enchantment, the demagogue has to say _something_. So what does Trump say? That he is a successful businessman and that that is what America needs in the Oval Office. There is some plausibility in this, though not much. The greatest deeds of American Presidents—midwifing the new republic; freeing the slaves; harnessing the energies and vision needed to win the Cold War—had little to do with a bottom line."
But by the time Trump began running for president in 2015, Buckley had been in his grave for seven years. And even if he were still alive, he would not have exercised the same influence he once had. Buckley was a gatekeeper of the conservative movement. But there are no gatekeepers anymore. The democratization of politics via the internet has empowered the cranks and conspiracy-mongers, while making it impossible for more erudite eminences, to the extent that they still exist, to shape the conversation as Buckley once did.
IF BUCKLEY AND WILL were my intellectual heroes, my political hero was Ronald Reagan. How I loved that man. As a Soviet refugee, I thrilled to his anti-Communist rhetoric and his unapologetic defense of freedom. Discarding détente, Reagan said on May 17, 1981, "The West won't contain communism, it will transcend communism. . . . It will dismiss it as some bizarre chapter in human history whose last pages are even now being written." But it wasn't just his hard line against the Soviet Union and his championing of human rights and democracy that drew me to him. It was also his perceived goodness, his sincerity, his gentleness, his sheer niceness. Living in Riverside, a college town, when he was elected in 1980, I remembered predictions from liberals that if you voted for Reagan today, World War III would break out tomorrow. I never believed it. I believed in the Gipper. He was a movement conservative, just as I was becoming. He had read the books of the conservative canon—and subscribed to _National Review_ —long before I was born.
I vividly recall the shock when I heard on March 30, 1981, that the president had been shot. The news arrived when I was sitting in the library of my elementary school in Riverside. A teacher wheeled out a large, clunky TV on a metal cart, antenna protruding from the back, and turned it on so we could watch one of the network newscasts. I was desperately worried until I heard that he would live.
I wished the best for Reagan, but always fretted when he sparred with the press. I was concerned that he would make one of his famous gaffes revealing that he wasn't in command of the facts, thereby providing ammunition for his numerous critics. Usually it was a needless worry, but he stumbled often enough to leave supporters like me on tenterhooks every time he spoke extemporaneously. When he was speaking from a teleprompter, however, there was no one better—as he proved on January 28, 1986. That was the day when the space shuttle _Challenger_ exploded on takeoff, killing the entire crew and one passenger—Christa McAuliffe, who was to be the first teacher in space. Like countless other school kids across the nation, I was watching the launch. In an instant wonder turned to horror and we were left with nothing to see but the smoky contrail in the sky. Reagan was due to deliver the State of the Union that day. He turned it into a beautiful tribute to the dead astronauts. Peggy Noonan wrote the words, but no one other than Reagan could have said them so movingly: "We will never forget them, nor the last time we saw them, this morning, as they prepared for their journey and waved goodbye and 'slipped the surly bonds of earth' to 'touch the face of God.'" Along with the entire nation, I was touched and uplifted. That moment cemented my deep bond with the president. He became for me what JFK had been for an earlier generation: an icon and an inspiration. He made conservatism optimistic and inclusive—a sharp contrast to how dark and divisive the movement would become in more recent years.
The one time I saw Reagan in person was a thrilling occasion for me—and possibly even for him, although not because I was present. The last rally of his last political campaign was held on November 5, 1984. He was cruising toward a big reelection victory and decided to end his political career where it had begun—in Los Angeles. He spoke that day at Pierce Community College in Canoga Park, California, an unremarkable suburb full of strip malls and tract homes not far from the equally dreary suburb where I lived. I recall getting there hours ahead of time with a friend and roasting in the hot sun waiting for the president to make his appearance. The erstwhile matinee idol was natty as usual in his dark suit, his tie in a Windsor knot, a white handkerchief displayed in his breast pocket. He only spoke for about thirty minutes but did not disappoint. "I've come to the people of the San Fernando Valley to ask for support many times before," Reagan said in that soft-spoken, appealing, aw-shucks way of his, "and I'd like to ask you this last time to be with us tomorrow." The crowd of twenty thousand roared back, "Four more years! Four more years!" Normally I hate crowds and groupthink, but I happily joined in.
Nobody would ever confuse this lovefest with the kind of hate-mongering that would become characteristic of Trump rallies. Reagan did not attack his opponent or the news media or minority groups or anyone else. He simply offered an optimistic vision epitomized by his slogan "Morning in America." Today, by contrast, the conservative movement seems to think that it's two minutes to midnight and that an extremist agenda is justified in order to save the country from impending doom at the hands of liberals. I am incredulous that anyone could possibly compare Reagan to Trump. The former had dignity, grace, humor, and class that the latter can only dream of. Reagan may not have known as much about public policy as intellectuals expected, but compared to Trump, he might as well have been a political science PhD, having spent decades reading, writing, and speaking about the issues. Reagan offered hope; Trump, fear.
Reagan inspired me to work on my first political campaign as a high school student, volunteering for the Ed Zschau for US Senate campaign in 1986, the year that the Chernobyl nuclear reactor blew up in the Soviet Union and the Iran-Contra scandal blew up in Washington. I spent countless hours laboring without remuneration for Zschau, a young, moderate Republican running against the aged liberal lion, Senator Alan Cranston. Zschau fell just short, but the Republican governor, George Deukmejian, was easily reelected to a second term. That was back in the days when there was a viable Republican Party in California—before Senator Pete Wilson was elected governor in 1990 and embarked on an anti-immigrant agenda that alienated the state's growing Latino population. Wilson, a far more moderate and reasonable figure, did to the California GOP what Donald Trump may well be doing to the national GOP.
The Zschau campaign didn't win, but I loved the experience—even the drudge work, which is most of what I did. I stuffed envelopes, made copies, answered mail—whatever was necessary. The spirit of camaraderie among the campaign staff in the Sunset Boulevard office left a deep impression on me. I relished the sense that we are all conservatives working together to defeat a liberal Democrat notorious for his dovish approach to the Soviet Union and his hawkish approach toward Ronald Reagan.
THERE WERE OTHER INFLUENCES on my embryonic conservatism, not just politics and history but writing and criticism. The curmudgeonly and conservative Baltimore newspaperman H. L. Mencken, who died in 1956, was a particular favorite. I savored his hilarious, stylish, and lacerating essays even if I did not share his contempt for democracy or the ordinary Americans that he lampooned as the "booboisie." I was even less in sympathy with the racist and anti-Semitic sentiments he revealed in letters published long after his death. In more recent years, however, I have concluded that Mencken's skepticism about "the swinish multitude" was well justified. The ignorance and extremism he denounced in early-twentieth-century America—exemplified, in his view, by the populist rabble-rouser William Jennings Bryan, who argued against teaching the theory of evolution—is more evident than ever in the early twenty-first century. I often think of Mencken's description of democracy—"Democracy is the theory that the common people know what they want, and deserve to get it good and hard"—when I think of the triumph of Trump.
A master of the bon mot, Mencken had a description that could not be bettered of how he became a writer. He spoke for me when he wrote:
It happens that I was born with an intense and insatiable interest in ideas, and thus like to play with them. It happens also that I was born with rather more than the average facility for putting them into words. In consequence, I am a writer and editor, which is to say, a dealer in them and concoctor of them.
There is very little conscious volition in all this. What I do was ordained by the inscrutable fates, not chosen by me. In my boyhood, yielding to a powerful but still subordinate interest in exact facts, I wanted to be a chemist, and at the same time my poor father tried to make me a business man. At other times, like any other relatively poor man, I have longed to make a lot of money by some easy swindle. But I became a writer all the same, and shall remain one until the end of the chapter, just as a cow goes on giving milk all her life, even though what appears to be her self-interest urges her to give gin.
I only differed from Mencken in never entertaining any hopes of becoming a chemist or businessman. I did think of becoming a lawyer—an ambition encouraged by my father—but only because I doubted my ability to make a living as a writer.
I felt the urge to write as early as junior high school, when a friend and I came up with a prototype for an unofficial school newspaper. My ambition would be realized in high school when some friends and I would start an "underground" newspaper called _The Forum_. We loved making trouble for the administrators at Grover Cleveland High School in Reseda, California, a public school that, with its low-slung buildings surrounded by fences and its windows covered in bars, looked from a distance like a minimum-security prison. _The Forum_ 's publication was suspended by the principal after we published an article accusing the mathematics department of "rampant ineptitude" because of students' low scores on standardized state tests. We promptly called up the _Los Angeles Times_ and were featured in an article as First Amendment heroes. It was, of course, a lot easier to champion free speech in an American public school than in the Soviet Union, but that experience impressed on me as a teenager the importance of freedom of the press—a realization that has not dawned on Trump even in his seventies. By my senior year, I was editor not only of the underground newspaper but also of the official student newspaper, _Le Sabre_. I was also the Los Angeles County champion in mock trial. You might say that my careers as a lawyer and a newspaper mogul both peaked before the age of eighteen.
I was respected by my peers but hardly beloved; I was voted "most likely to succeed," not "most popular." I was a good student but not a quiet or obedient one. I loved to mouth off in class, challenging the leftist views of my teachers at the Cleveland Humanities Magnet—a school within a school for kids especially interested in the liberal arts—with the conservative orthodoxy I was imbibing from _National Review_. I'm sure I was insufferable, but I reveled in dissenting. Did I mention that I was also captain of the debate team? I loved arguing with anyone about everything. My best friend in high school, who later became a lawyer, was an equally opinionated liberal, and we would spend hours happily debating with each other. When we got to college—we both went to the University of California at Berkeley, which cost next to nothing for in-state residents in those days—our girlfriends quickly tired of this forensic ritual. But we loved it.
In retrospect I think that arguing, in print and person, gave me a way to interact with the world that I found comfortable, whereas less antagonistic interactions were harder for me, because I was naturally shy. I loved getting a rise out of people, which helps to explain why I wrote contrarian if not particularly wise articles for _The Forum_ , such as one arguing that money can buy happiness.
MY EXPERIENCE AT a liberal high school was perfect preparation for my undergraduate career as a conservative troublemaker in Berkeley, a town that never seemed to have left the sixties behind. The counterculture was the dominant culture in Berkeley even in the conservative 1980s. The Spartacist League bookstore still sold Marxist tracts; hippies still congregated in People's Park; and peace signs and tie-dyes could still be seen all along the main drag, Telegraph Avenue. My favorite all-night hot-dog stand, Top Dog, even served its wieners with a side of libertarian politics. There would be a riot during my senior year in 1991 to protest the university's attempts to place volleyball courts in People's Park, thus re-creating the riot that broke out in 1969 when the university tried to build a sports field atop this same weed-infested block that had become a magnet for drug dealers and addicts.
I lived initially in Slavic House, an elegant, university-owned house on fraternity row reserved for students of Slavic languages; my mother pulled a few strings to get me in, even though I had no intention of studying any Slavic language, because I had lost out in the housing lottery for space in the normal high-rise dormitories that looked like Soviet workers' housing. It was at Slavic House, as a sophomore, that I met my future wife during a party fueled by vodka-spiked punch. She first heard that I existed when another resident told her that there was a "fascist" living down the hall with a "Bush-Quayle '88" sticker on his door. It didn't take much to be branded a fascist in the ultraliberal atmosphere of Berkeley! But while I did not hide my views, I was eager to make my mark in campus journalism, not politics.
One of the first things I did upon arriving in the fall of 1987 was to head to _The Daily Californian_ , the official student newspaper, which was headquartered in a pink stucco building a few blocks from campus. I spent much of my first two years covering the Berkeley City Council. It was a fascinating experience because Berkeley was one of the few towns in America with its own foreign policy. The city council was regularly passing resolutions condemning Reagan's hawkish foreign policy and Israel's occupation of Palestine. I rolled my eyes every time the council members launched into their predictable diatribes cribbed from the pages of _The Nation_. I also covered numerous rallies and sit-ins on campus. Once, demonstrators occupied Sproul Hall, the campus administration building, and, while I watched, they had a lively debate to figure out why—were they asking for a nuclear freeze, divestment from South Africa, greater diversity on campus, an end to animal testing, or some other demand? In other words, protest first, cause second. This juvenile posturing increased my scorn for the left—not that I was above juvenile posturing myself. I was just posturing from the right rather than the left.
Before long I transitioned from reporting to editorializing. In the fall of 1989 I did a semester abroad at the London School of Economics. When I came back, I ran in the _Daily Cal_ 's election to become a columnist. In true communal fashion, all the editors and columnists were elected by the staff. A lot of people were inclined to vote against me because of my conservative views, but, as I recall, Jim Herron Zamora, a future editor in chief and a true newsman, urged a yes vote because he thought it would be fun to "wind up Max and point him at City Hall." I won and immediately penned a column in January 1990 about the cognitive dissonance I felt returning from Europe after the fall of the Berlin Wall, which I saw as vindication of Ronald Reagan's Cold War policies and proof of the enduring appeal of American-style liberal democracy. The US triumph in the Cold War made me proud to be an American—and a conservative. "While Eastern Europe is being transformed by a new longing for economic prosperity and political freedom," I wrote, "Berkeley remains mired in a rigid, New Left mindset created 30 years ago. Nostalgia for the politics and protests of the 1960s dominates this town's collective consciousness. The signs of it are everywhere. Note the prevalence of tie-dyes, lovebeads, and Birkenstocks—all symbols of a world that has not existed for 20 years."
That set the tone for the weekly columns that I would write for the next year and a half until my graduation. I loved making a bonfire of Berkeley's liberal pieties, including its rent-control regulations that made housing hard to find, its tolerance of homeless people that made the streets dangerous to walk, and its affection for the Communist dictator of Nicaragua that made a mockery of human rights. I expressed standard conservative views—for example, attacking affirmative action quotas and supporting the Gulf War. Like many conservatives of the day, I often protested political correctness that stifled free speech, writing that anyone who disagrees with the left "is labeled (pick one) 'fascist,' 'homophobe,' 'racist,' or 'elitist.'" "The biggest threat to the First Amendment," I wrote, "comes from the left, not the right." (That is a view I no longer hold in the age of Trump.)
My articles caused considerable consternation on campus, where conservative opinions were rarely heard. One reader tossed a brick through the window of the _Daily Cal_ building in protest. Another sent me a bullet in the mail with my name taped on it. I wasn't the slightest bit intimidated. I loved the attention and notoriety. And the violent reaction only confirmed in my mind the rightness of my views, in both senses of the word.
IN EARLY 2018, I went back to Berkeley as part of a book tour, and, with an hour to kill before my lecture, visited the university library to read my old columns on microfilm. Some of them I still endorse, including my criticisms of rent control, an anticapitalist conceit that reduced the stock of rental housing not only in Berkeley but also in New York, where I would move a few years later. But some things I wrote now make me cringe—for example, my reaction to a 1990 hostage siege at an off-campus hangout called Henry's. A deranged, heavily armed gunman held thirty-three people hostage, killing one and wounding seven, before he was gunned down by the Berkeley police SWAT team. In my column, I for some reason endorsed the death penalty, even though the assailant was already dead. Presumably this was because support for capital punishment, which had been suspended by the Supreme Court in 1972 before being reinstated four years later, was such a big conservative cause in those days. Just as predictably for someone indoctrinated in conservatism, I resisted calls for stricter gun control. "We don't need even tougher restrictions, because the status of gun law has little relation with the ability of determined maniacs to acquire weapons," I wrote. "Gun control only deters law-abiding citizens from defending themselves. (Imagine how the outcome of the Henry's incident may have been different if one of the patrons had been armed.)"
Was I really suggesting it would have been a good idea for some boozy customer to whip out a Glock and open up? Apparently so. Then, as now, this was the conventional wisdom on the right, which exaggerates the defensive use of firearms by law-abiding citizens and minimizes their use in homicides and suicides. I joined in fetishizing the Second Amendment as a license to sell just about any gun to just about anyone—even though the historical record and plain text clearly indicate that this language was intended to prevent the federal government from shutting down state militias. It has taken me decades to break free of the pro-gun orthodoxy. I now favor stricter gun controls, which are permissible even under the Second Amendment, because the only explanation for the prevalence of gun violence in America—we have six times as many firearms homicides as Canada and nearly sixteen times as many as Germany—is the ready availability of guns. But in those days I faithfully echoed the conservative party line.
While mine was an often lonely voice in "Berzerkeley," there were plenty of other conservatives on other college campuses in the 1980s and 1990s. Reagan had made conservatism cool, and right-wing donors were funding conservative newspapers to battle what they regarded as a stifling leftist orthodoxy on campus. The most notorious publication was the _Dartmouth Review_ , which produced the future far-right superstars Dinesh D'Souza and Laura Ingraham. As _Mother Jones_ magazine later noted: "While [D'Souza] helmed the _Review_ , it published a 'lighthearted interview with a former Klan leader'—accompanied by a staged photo of a black person hanging from a tree—and an assault on affirmative action titled, 'Dis Sho Ain't No Jive, Bro,' which was written in Ebonics. ('Now we be comin' to Dartmut and be up over our 'fros in studies, but we still be not graduatin' Phi Beta Kappa.') The 'Jive' article caused Jack Kemp, a conservative icon mindful of the right's problems with minority outreach, to resign from the _Review_ 's advisory board." The _Review_ also outed members of the Gay Students Association at a time when "coming out" was still a risky thing to do; one of the outed students reportedly "became severely depressed and talked repeatedly of suicide."
While there was undoubtedly some overlap between my writings and those in publications like the _Review_ , my college journalism was less sensationalistic and less offensive. I was not a racist; as a Jewish immigrant, I identified with other minorities. To my shame, I must admit that I shared some of the homophobic attitudes of that less-enlightened time—prejudices that I have long since outgrown—but I would never have dreamed of singling out anyone for public humiliation. There were, from the start, sharp limits on how far I would go in pushing my political views. Provoking people, for me, was not an end in itself. I liked Ronald Reagan's approach—or what I took to be his approach—of offering an outstretched hand rather than a clenched fist. That was not the case with many of my conservative contemporaries, who made a career of becoming ever more transgressive to get attention. Little did I suspect that this all-out assault on liberalism would set the stage for the political triumph of a boorish real-estate developer who in the 1980s was regularly mocked as a "short-fingered vulgarian" in the now-defunct _Spy_ magazine, one of my favorite periodicals besides _National Review_.
In the 1990s, I felt solidarity with other young conservatives who were determined to complete the conservative revolution that Reagan had begun. My faith in the movement was strong; it would be decades before my certitudes would begin to corrode as I watched conservatism morph into something I could not recognize. When I graduated from college in the spring of 1991, shortly after the American victory in the Gulf War and shortly before the collapse of the Soviet Union, I knew I wanted to contribute to the great cause.
I just had to figure out how.
## _2._
## **THE CAREER OF A CONSERVATIVE**
MY PERSPECTIVE ON CONSERVATISM WOULD SHIFT as I went from being an outsider—a fanboy on the left coast—to an insider: a professional journalist, historian, and foreign policy pundit in the Acela Corridor. I would become personally acquainted with many of the conservative sages whose writings I had grown up reading. In some cases, familiarity would breed contempt. But in many other instances, my respect would only grow as I got to know my boyhood heroes personally. At one point, I would declare that the smartest people I knew were conservatives: a statement I now regard as the height of folly. That I would say such a thing is indicative of the mindset that I developed as a "movement" conservative. I was not just drinking the Kool-Aid; I was bathing in it. I understand what Fox News viewers experience because I experienced a version of the same brainwashing myself. This was a process of indoctrination—largely self-indoctrination, I should add—that took decades and that I am only now escaping.
That's not to say that I am leaving behind all, or even most of, what I believed then. Rather, as I approach my fifties, I am sorting out for myself what makes sense and what doesn't in the conservative Weltanschauung. That is not something I was capable of doing in my twenties, when I was just being initiated into the world of the right. Back then, my enthusiasm for conservatism was excessive and indiscriminate, as is so often the case with a proselyte.
I COULD HAVE GONE to work as an entry-level reporter for some newspaper after graduation, most likely the _Los Angeles Times_ ; I had had college internships at the _Times_ bureau in Sacramento and the _Baltimore Sun_ bureau in Washington, and I had worked as the _Times_ 's stringer in Berkeley. But I didn't relish covering the "night cops" beat or working in some dull suburban bureau—the normal initiation rituals for cub reporters. So instead I went to graduate school, studying history at Yale University. Between my college graduation and the start of grad school I got married to my college girlfriend. While I was taking classes at Yale, she applied to law school—and got into Harvard but not Yale. In 1992, having received an MA from Yale after just a year amid the urban decay of New Haven, I moved with her to Cambridge so she could go to Harvard Law School.
Living in a small duplex in north Cambridge—this was Tip O'Neill's old district—was an education for me. In California, where I had grown up, people of pallor were an unvariegated mass known collectively as Anglos. In the Boston area, by contrast, they were not simply white people—they were Italians, Irish, Poles, Portuguese, etc. This was my introduction to the white ethnic, working-class politics that Donald Trump would exploit so skillfully across the Rust Belt. Our landlord was an Italian American barber who had lived in this area his whole life and regarded a trip to Harvard Square—a mile away—as a journey to a strange and bewildering land. Above his mantel he had a picture of his hero, John F. Kennedy. He was socially conservative but strongly identified with the Democratic Party. If he is still alive, I wonder if today he might not be displaying a Trump photo.
I left grad school—temporarily, I thought—to work as a junior editor at the _Christian Science Monitor_ in Boston. The newspaper had a distinguished pedigree; at one time it had been one of only two national newspapers in America, the other being the _Wall Street Journal_ , and it was known for its outstanding international coverage and a sober, serious approach to the news. But the national rollout of the _New York Times_ usurped the _Monitor_ 's position, and its costly and ill-conceived foray into television sapped it of precious operating capital. By the time I arrived, many of its most distinguished editors and correspondents had left. That was perfect from my perspective because it gave me—a twenty-two-year-old kid—the chance to assume responsibilities far beyond my modest level of experience. I not only worked as an assistant editor on the national desk but also wrote numerous articles, reveling in the thrill of having my byline appear in print.
I had actually applied to, and gotten into, the Harvard history department, but by the time it was time to go back to grad school in the fall of 2003, I decided to stay in journalism. Editing and writing was more fun than going to class and then heading to a dive bar to commiserate with fellow grad students about the awful job market for history PhDs. I knew that it would be especially hard for me to find a job in the academy because of my conservative views and interest in military history—both intellectual deviations frowned upon by the liberal professoriate. Journalism seemed like a more promising and interesting alternative.
Because the _Monitor_ was distributed by the US Postal Service, it had absurdly early deadlines—around 1 p.m. in those days. I recall trudging through fields of snow up to my knees at the crack of dawn to reach the Alewife T stop in north Cambridge to take the train to Back Bay. It was a culture shock for a Californian, but a return to my heritage as a Muscovite. The early deadlines made it hard to stay on top of the news, so we focused more on analysis, but in a rigorously even-handed way. For the time being I put aside my conservative views and concentrated on learning the newspaper trade as a "straight" journalist—not an editorialist. The _Monitor_ was strict about maintaining its neutrality and integrity, so I was trained quite differently from young writers today who are hired by click-hungry websites with the expectation that they will inject "attitude" and opinion into their writing from the start. Indeed, online publications such as _Infowars_ and _Breitbart_ and broadcasters such as Fox News Channel and RT have entirely destroyed the demarcation line between opinion and news—they peddle their self-serving fantasies as if they were reality. My time at the _Monitor_ taught me to be more devoted to getting the facts right. Hence I am especially offended that America now has a president who repeats the same lies over and over, long after they have been called out by the fact-checkers.
One of the oddities of the _Monitor_ was that almost everyone who worked in the home office—a massive modernist building located next to the Romanesque Mother Church in Boston's fashionable Back Bay neighborhood—was a Christian Scientist. I was practically the only non-Scientist in headquarters, although many of the correspondents in far-flung bureaus were also nonbelievers. Luckily, as a Russian Jewish immigrant in America and a conservative Republican at Berkeley, I had had plenty of experience being a minority. I discovered that Scientists were nice people, but it took a while to adjust to the peculiarities of their religion. Because they believe that the entire physical world is only a figment of "the immortal Mind" of God, they don't celebrate birthdays, acknowledge deaths (people are said to "pass"), or use medical metaphors (no "budget headaches" for them). They are best known for relying on faith healers known as "practitioners" rather than medical doctors. When someone was absent from the office for a few days, you never knew whether she had a cold or terminal cancer—either way, she wasn't going to admit that anything was physically wrong. I got tired of having people ask, when I told them I worked for the _Monitor_ , whether I had health insurance. The answer was yes—heretics like me could use our health benefits to see a real doctor. In truth, some of the Scientists would sneak aspirin themselves.
My exposure to the Christian Science faith deepened my appreciation for the diversity of America and made me realize I could like people very different from myself even if there were far more of "them" than there were of people like me. I wish more Trump supporters, anxious about the changing demographics of America, would have a similar epiphany.
WHILE I ENJOYED WORKING AT the _Monitor_ , my dream was to enter the world of conservative punditry. I set my sights on the editorial page of the _Wall Street Journal_ , which I had taken to reading in college. The _Journal_ was known for its advocacy of supply-side economics, the notion that cuts in taxes and regulations can spur economic growth. It was a philosophy that Ronald Reagan adopted as his own. _Journal_ editorials were reported and influential and did not hesitate to disagree with the more liberal views of the _Journal_ 's news pages. I wasn't particularly interested in the financial markets, but the _Journal_ had expanded to become a more general interest newspaper that set the tone for much of the Republican Party. It reached a far bigger audience than a niche publication like _National Review_ —it was, in fact, the largest circulation daily newspaper in America, with 1.7 million subscribers.
I would regularly mail clippings of my _Daily Californian_ and _Christian Science Monitor_ articles to the _Journal_ editorial page (this was in the days before the internet and email), receiving in reply nothing more than form letters. I did not have much hope that my aspirations would be realized until one day in early 1994, when I was notified that Robert L. Bartley, the editorial page editor, would be in Boston and wanted to meet me. He was a legendary journalist who had taken over the editorial page in 1972 at age thirty-four and had won a Pulitzer Prize in 1980. He did not just run an editorial page; he ran his own news-gathering operation in competition not only with other newspapers but also with the _Journal_ 's news side. Bartley was one of the most influential opinion shapers among conservatives. I was intimidated and excited to meet him.
I trudged through the snow to the stately old Parker House hotel (this was where Parker rolls had been invented) for one of the more unexpected and important meetings of my life. I had imagined that Bartley would quiz me about my political views. In preparation I had prepared elaborate disquisitions on my Burkean conservative philosophy. Instead he preferred to talk about my work in journalism. He wanted skilled journalists, not untrained ideologues, working for him. He would later say, "Journalistically, my proudest boast is that I've run the only editorial page in the country that actually sells newspapers." Our conversation was halting and stilted; Bartley, slight and bespectacled, was shy and retiring in person, not an easy person to talk to in spite of his perpetual Mona Lisa smile. I thought I had blown the interview until he casually mentioned that he had a couple of job openings—one for an editorial writer on economics, the other for an assistant op-ed editor. Which one would I prefer? Somehow I had been hired.
I was horrified at the prospect of becoming the economics editorialist for the nation's premier business newspaper. I had never taken a class in the subject and had no interest in it. Bartley was serenely unperturbed. I later learned that he liked to take writers who did not know much about the subject and train them in his way of thinking. He did not want to hire an economist because most professional economists disdained supply-side economics—the philosophy that he had embraced at the urging of economists Arthur Laffer and Robert Mundell and former editorial-page writer Jude Wanniski. In spite of Bartley's confidence in my abilities to write economics editorials, I opted for the op-ed editing job and prepared to move to New York.
I was thrilled to have gotten this opportunity. When I joined the _Wall Street Journal_ in 1994 at the age of just twenty-four, I felt as if I had already fulfilled my life's ambitions. My mother was predictably proud; my father was just as predictably unimpressed. I considered myself supremely lucky to be at the epicenter of conservative politics and journalism. I had joined what Michael Kinsley had called "a central cog in the vast right-wing conspiracy." It was a description meant to be disparaging but one that we embraced. I was dazzled by new colleagues such as John Fund, a walking encyclopedia of American politics; Paul Gigot, the Pulitzer Prize–winning "Potomac Watch" columnist (and Bartley's eventual successor), who seemed supremely plugged into Washington; and Dorothy Rabinowitz, an unconventional woman of indeterminate age who dressed in all black and was always making amusing and acerbic observations on the passing scene.
Now that I was working in the heart of the conservative moment, I got to meet its other luminaries. I was invited on a cruise around Manhattan to celebrate _National Review_ 's fortieth anniversary, and I went to Bill Buckley's townhouse for dinner with him and his editors. I found the experience every bit as enchanting as I had imagined, if also a bit intimidating. Buckley was the height of geniality, but his social X-ray wife, Pat, was more ferocious and standoffish. The evening ended, as I recall, with Buckley playing the harpsichord.
This was part of my introduction to conservative society in New York—a small but intense circle of right-wingers who tended to stick together in this liberal city. There was a conservative social circuit that included annual fundraising dinners hosted by the Manhattan Institute and _Commentary_ magazine, where big-bucks donors mingled with down-at-the-heels writers like me; modest Christmas parties for the highbrow crowd at the offices of the _New Criterion_ ; monthly gatherings of the Monday Meeting, a networking opportunity for a motley collection of conservative activists promoting every cause from the gold standard to the horse-drawn carriages of Central Park; regular fundraisers hosted by wealthy GOP donors such as Paul Singer and Mallory Factor to fete and cross-examine candidates for high office; book talks hosted by the influential philanthropist and deeply learned patron of history Roger Hertog; and more informal occasions such as an annual holiday gala in the airy SoHo loft of James and Heather Higgins, prominent conservative activists and philanthropists who were close to Newt Gingrich and other leading Republicans in Washington. A special mark of favor was to be invited to spend the weekend at the Hamptons estate of one of these wealthy luminaries. Needless to say, such invitations would be extended only to those who faithfully toed the party line. (I haven't been receiving a lot of them lately from Trump supporters.)
I'm sure that similar pressures exist on the left that can be just as straitening. This is how all social, ideological, or religious movements police their members—by making clear that agreement will be rewarded with greater social standing and support, and disagreement punished with ostracism. This pressure is so amorphous and pervasive that, like oxygen, you are only aware of it when it is gone.
My arrival as a "made guy" of the conservative moment was ratified a few years later, in 2007, when I won the Eric Breindel Award for Excellence in Opinion Journalism, given annually to a writer who exhibits "love of country and its democratic institutions" and "bears witness to the evils of totalitarianism." The award had been established by Rupert Murdoch in honor of the late editorial page editor of the _New York Post_ , one of many newspapers that he owned. I received the honor, along with a munificent $20,000 check, at a dinner cohosted by Murdoch and Roger Ailes, the impresario behind Fox News. I strained to understand Murdoch's Australian-accented mumbling and roared at Ailes's coarse humor. I could not possibly imagine then that a few years later I would be bemoaning their creation, Fox News Channel, as a pernicious influence on American life—as, in fact, a threat to this country's democratic institutions.
In joining this conservative counterculture, I got to meet Norman Podhoretz and Irving Kristol, two of the most prominent "neocons," meaning literally "new conservatives." They had been Democrats who had drifted to the right in the 1960s and 1970s and eventually became enthusiastic supporters of Ronald Reagan. Kristol was the editor of the _Public Interest_ , Podhoretz of _Commentary_ —two little magazines whose intellectual impact was out of all proportion to their meager circulations. I became friendly, too, with their sons, John Podhoretz and William Kristol, who were neocon royalty of a sort. Together with the well-known pundit Fred Barnes, they launched a new magazine called the _Weekly Standard_ in 1995 to serve as a standard-bearer for a new generation of conservatives. I became a contributing editor to the _Weekly Standard_ and spent a decade (2007–2017) as a blogger for _Commentary_. Bill Kristol became a mentor and good friend, whose genial, wisecracking company I regularly sought out when I was in Washington or he was in New York.
Perhaps because the Kristols and Podhoretzes were Jewish and could trace their ancestry back to Eastern Europe, just like me, I found myself drawing closer to these neoconservatives than to the heavily Catholic and more socially conservative _National Review_ crowd. Before long I would find myself described as a "neocon," a term that was often used as a slur to mean "Jewish conservative" or simply "ultrahawk." I bristled at the description because I wasn't a "new" conservative, never having been a Trotskyite or even a Democrat. Neither had Bill Kristol or John Podhoretz. We were all "right from the beginning" as much as the arch-nativist Pat Buchanan, who used that phrase as the title of his autobiography. Buchanan's attacks, however, were good preparation for the routine charges of disloyalty and demands that we move "back" to Israel or Russia or some other country that Jewish conservatives have faced in the age of Trump.
I FOUND IT HARD AT FIRST to figure out how things worked at the _Wall Street Journal_ editorial page. The newspaper not only preached laissez-faire doctrine but also practiced it. There was a daily meeting in the doorless office of Dan Henninger, the deputy editorial page editor, but the most striking characteristic of these gatherings was the long periods of silence. This took some getting used to for a hypervocal Jewish kid from Los Angeles. A product of Minnesota and Iowa, Bob Bartley was so shy and introspective that he was comfortable sitting for long minutes without saying anything, just staring at the ceiling. The meetings became almost a staring contest to see who could go the longest without speaking. I had to resist the urge to fill the dead air. People would wander in and out and nothing ever seemed to get decided, yet somehow at the end of the day, every day, a complete editorial page emerged as if by magic. Bartley managed by indirection, making vague, cryptic comments and leaving his subordinates to intuit what he would want them to do. Remarkably enough, this hands-off approach worked.
Bartley did not write editorials himself every day, but when he did they were inevitably razor sharp. Somehow this diffident bookworm would turn into a ninja warrior on the printed page, severing his adversaries' rhetorical aortas with a few flicks of his metaphorical sword. I am similarly paradoxical: mild in person, fierce in print. My partner Sue would later tell me that she was surprised when she met me, having known me only through my writing, to discover that I wasn't an "asshole."
I advanced rapidly at the editorial page, in part because I was deemed good at my job, but also because of the high turnover. My first boss at the op-ed page was Amity Shlaes. Then she left for another division of Dow Jones, the publisher of the _Journal_ , and was replaced by David Brooks. Then he left for the _Weekly Standard_ and was replaced by David Asman. Then he left for a new TV network called Fox News. I was the last man standing. In 1998, at the age of twenty-eight, I became the op-ed editor of the _Wall Street Journal_. I was in charge of three signed articles, called editorial features, that appeared every day next to the editorials, known in-house as "randos" because they appeared under the banner of "Review & Outlook."
I fully subscribed to the views of the editorial page. Its motto was "Free Men and Free Markets" (later changed to the gender neutral "Free People"), and it advocated tax cuts, free trade, immigration, a strong defense, and an internationalist foreign policy. We didn't talk much about social issues. Bob was hardly a fundamentalist, but he also didn't want to offend the religious right, whose support Republican candidates needed to win, so he generally steered clear of saying anything at all on the subject. I was becoming more liberal on social issues, but I didn't disagree with his pragmatic judgment, in part because my own views had been molded over the years by reading the _Journal_ 's editorial page.
Bartley's obsession in the 1990s was "Whitewater," a catchall phrase for alleged financial wrongdoing by the Clintons. We ran rando after rando on this subject; one of them ("Who Was Vince Foster?") was even blamed for contributing to the suicide of a White House lawyer who had been a Hillary Clinton law partner in Little Rock. Phrases such as "Rose law firm" and "Mena airport" constantly cropped up; I had a flash of recognition years later when I saw the Tom Cruise movie _American Made_ , which featured a real-life drug smuggler and sometime CIA operative operating out of the Mena, Arkansas, airport. The evidence of Clinton misconduct was hardly conclusive, and the editorials were so carefully lawyered that it was hard for me to figure out just what they were supposed to have done wrong. I still can't say, for example, what the connection was supposed to be between Governor Clinton and Mena Airport. But Bartley was so proud of his work that he published six volumes of Whitewater editorials as stand-alone books.
These editorials could be seen as contributing to the growing tendency in American public life not just to disagree with one's political opponents but also to try to annihilate them—a trend that ultimately culminated in the election of Donald Trump, a Republican who vituperates Democrats as traitors and charges that "they certainly don't seem to love our country very much." The left was far from innocent in this regard—"to Bork" had become a political term for character assassination after the mauling that Judge Robert Bork had received in his Supreme Court confirmation hearing in 1987—but the right certainly bore its share of guilt. Both sides used the past outrages perpetrated against their partisans to justify their own outrages against the other side. It was an eye for an eye, a tooth for a tooth, until the body politic would be in danger of becoming sightless and toothless.
In the end President Clinton would be brought low, at least temporarily, not by his greed but his libido. His ill-conceived dalliance with his White House intern Monica Lewinsky opened the way for the Republican-controlled House to impeach him for perjury because he lied under oath about their assignations. I shared the outrage of other conservatives over Clinton's tawdry behavior and the way he had demeaned the presidency, and I was dismayed that liberals excused his conduct. Unlike many other conservatives, however, I did not think that Clinton was the devil incarnate. He was a center-of-the-road president who cooperated with a Republican Congress to eliminate the budget deficit and reform the welfare system. I thought he was a deeply flawed man, but I appreciated the achievements of his presidency.
Ever a contrarian, I decided, as one of my first initiatives as the _Journal_ 's op-ed editor, to commission an article critical of supply-side economics from a Princeton professor named Paul Krugman—not yet a celebrated _New York Times_ columnist. That didn't work out so well. Bartley saw the article on the lineup and came into my office to let me know in his soft-spoken way that he didn't want to run it. I had to call Krugman to apologize and pay a kill fee; he graciously passed up an opportunity to publicly embarrass the newspaper. There was an official liberal slot opposite the editorial page—a rotating column written by authors such as Christopher Hitchens and Alexander Cockburn—and Bartley wasn't interested in seeing a lot of contrary viewpoints beyond that. His reasoning was that other newspapers such as the _New York Times_ provided plenty of space for the liberal orthodoxy; he wanted the _Journal_ 's editorial pages to be a forum for the right. I barely managed to avoid losing my job over the Krugman debacle; Bartley apparently thought about removing me but decided to give the tyro another chance. This was an early indication to me that groupthink could be just as tenacious on the right as on the left.
While I could not run too many head-on attacks against the _Journal_ 's editorial line, I did manage to sneak all sorts of nonpolitical articles onto the page by authors who weren't conservative ideologues. For instance, I convinced a former hedge fund manager named Andy Kessler to write on business news, and a former boxer turned philosophy professor named Gordon Marino to write on boxing news. I have always had an intolerance of orthodoxy, even when I agreed with it, and I was interested in producing a lively page full of unexpected insights rather than simply rote repetitions of the party line. I was greatly aided in this task by an outstanding cast of assistant editors, including a young Bret Stephens and a young Kimberly Strassel. The civil war that would eventually rip apart the conservative movement would lead Bret to become a prominent #NeverTrumper and Kim a prominent pro-Trumper.
ALTHOUGH WORKING PRIMARILY AS an editor, I also got a chance to write unsigned editorials. It was a heady experience for a twenty-something to write in the royal "we" with all the eminence and majesty of America's largest newspaper behind his every word. I'm sure that the people reading my words imagined that their author was some sixty-year-old graybeard rather than a twenty-seven-year-old know-it-all punk. Because our legal columnist had just left the editorial page, there was nobody to write about legal issues so, without anyone assigning me to do it, I started to cover the legal beat. That meant writing editorials castigating ambulance-chasing "tort lawyers," who were blamed by the editorial page for winning unfair judgments from corporations.
I even broke a story in 1996 when I went down to Houston to interview several employees of John O'Quinn, a well-known Texas trial lawyer, who accused him of paying "accident runners" to sign up clients after airplane crashes—an ethical no-no that might have gotten him disbarred. O'Quinn had gotten rich with a good-ole-boy swagger that he used to charm juries in backwoods jurisdictions to award his clients vast sums from large, out-of-state corporations. He had won billions from makers of breast implants, cigarettes, and pharmaceuticals. This was populism for profit—O'Quinn had figured out how to monetize the resentment of small-town Americans against the coastal elites.
O'Quinn's rage after my article appeared was something to behold. He sent letters to the _Journal_ calling me "goofier than a road lizard" and threatening to sue the newspaper for every penny it possessed. Given O'Quinn's track record as one of the most successful litigators in America—he was said to have earned 40 million dollars the previous year—this was no idle boast. I was genuinely worried, even if I did have the _Journal_ 's legal team behind me. But O'Quinn turned out to be, as they say in Texas, all hat and no cattle. No lawsuit was ever forthcoming. He was just trying to intimidate us into backing down. Donald Trump employs exactly the same kind of bluster, right down to threats of lawsuits that are never filed. He might as well be O'Quinn's long-lost New York cousin. Actually, Trump has gone him one better, because he's figured out how to take advantage of ordinary people not only to grow richer but also vastly more powerful than O'Quinn could have ever imagined. My run-in with O'Quinn, who died in 2009, was an early education in the kind of scoundrels who hide behind the banner of populism.
My focus on the legal profession led me to write my first book, _Out of Order: Arrogance, Corruption and Incompetence on the Bench_ , which came out in 1998. It was a jeremiad against judges who offended conservative sensibilities by either allowing plaintiffs' lawyers to win unreasonably large judgments or who engaged in "judicial activism," legislating their own views from the bench. These were standard conservative critiques that I was parroting without truly understanding the underlying issues. I did not deal convincingly with the inherent tensions in the case against judicial activism: I condemned the 1954 decision in _Brown v. Board of Education_ , for example, by claiming that school desegregation was not mandated by the Constitution, yet I applauded the blow that it had struck against racism. Despite (or perhaps because of) my shallow reasoning, I convinced Robert Bork, a hero on the right, to write the foreword. If he had written the whole thing, it would have been a much better book. I am not proud of _Out of Order_ ; I feel about it much as a well-established actress might feel about a porno film that she did when she was just starting out. The best thing about the book is that it provided an advance that enabled my wife and me, with our first baby on the way, to make a down payment on a house in the New York suburbs.
I WAS DETERMINED THAT my next book would be better. Over the next four years, I spent nights, early mornings, and weekends, while working full time at the _Journal_ , writing a history of America's small wars. I got interested in the subject because the 1990s was the decade of US interventions in Somalia, Haiti, Bosnia, and Kosovo. I wanted to find the historical antecedents of these operations and wound up writing a history that stretched from the Barbary Wars of the early nineteenth century up to the Vietnam War and beyond. I felt much better about this book than about _Out of Order_ because history, and specifically military history, was my first love. This time I was not simply parroting the conclusions of other scholars; I was striking out into virgin intellectual terrain.
I had a manuscript of the book finished by September 11, 2001. The day started off normally enough—I took a Metro-North train from Westchester County to Grand Central Station. En route, I heard rumors that an airplane had crashed into the World Trade Center. I imagined a Cessna flying into one of the buildings by accident and proceeded downtown as usual by subway. It turned out to be the last train running. When I got off at City Hall, a scene straight out of hell confronted me. Sirens were wailing, people were running, and everything and everybody was covered in a ghostly film of white dust. I stumbled forward and watched the second tower fall. And then a giant ball of soot and smoke came roaring down the street like the massive boulder at the beginning of _Raiders of the Lost Ark_. I ran away along with everyone else. After having walked all the way to midtown, I caught a commuter train back to sylvan Larchmont. Arriving in my comfortable suburb from what was now a war zone was surreal: I had gone in the space of a few hours from watching people plummet to their deaths to taking the recycling container out to the curb. I worked remotely that afternoon, helping the paper come out as usual even though our headquarters was a shambles—the _Journal_ was located in the World Financial Center across the street from the World Trade Center, and the collapse of the Twin Towers punctured its windows, covering everything inside in soot and ash. None of my colleagues were killed in the attack but some had close calls. The editorial page staffers would spend the next few weeks working out of a temporary office in Princeton, New Jersey. It was an intense period full of work and purpose—we felt as if we were helping America to make sense of the worst terrorist attack in history.
I made my own small contribution to the national conversation with a cover story in the _Weekly Standard_ that appeared a month after 9/11: "The Case for American Empire." For a number of years this was the most influential article I had written; it helped to spark a broader debate about "American empire" that was taken up by Niall Ferguson, Andrew Bacevich, and others. I would lose track of how often that article's signature line was quoted—"Afghanistan and other troubled lands today cry out for the sort of enlightened foreign administration once provided by self-confident Englishmen in jodhpurs and pith helmets." I now recognize that in my youthful zeal to shock, my use of the word "empire" may have backfired by generating more heat than light. I wasn't calling for the United States to acquire colonies but rather to engage in Kosovo-style nation-building in countries such as Afghanistan to prevent them from becoming once again a breeding ground for extremism. I still believe this is a good idea, and one that doesn't require a massive American military presence. Without a political solution, no amount of military action can achieve decisive results. But it doesn't help to generate political support for this idea in the postcolonial age by invoking the example of nineteenth-century empires. It's been years since I've made the case for the United States to become more unapologetically imperialist. I still argue, however, for paying more attention to nation-building, albeit of the "small footprint" variety that doesn't involve sending lots of American soldiers.
In the spring of 2002 my second book— _The Savage Wars of Peace: Small Wars and the Rise of American Power_ —came out. Its thesis was that "small wars" had played an important, if unrecognized, role in shaping American power and would remain as commonplace in the future as they had been in the past. The US military, I argued, did not have the luxury of concentrating only on the kind of big, conventional wars that it preferred to fight. _The Savage Wars of Peace_ was much better received and much more successful than my first book. I would be gratified in subsequent years to hear from many soldiers and marines who read it either before deploying to Iraq or Afghanistan or while on deployment; they would tell they had benefited from the historical perspective that it provided. I had, of course, written the book before 9/11, but it was more timely than ever when it came out after the attacks and while the United States was engaged in a new "small war" in Afghanistan.
Not long after the book's release, I got a call from Leslie Gelb, president of the Council on Foreign Relations, a think tank and membership organization in New York that was viewed with suspicion by conservatives who thought it was a bastion of liberal globalists. An idiosyncratic and accomplished figure who had been both a high-ranking government official (he had overseen the preparation of the _Pentagon Papers_ for the Johnson administration) and a _New York Times_ columnist, Les told me that he was impressed by my book and wanted to hire me as a senior fellow because he wanted greater ideological diversity on his staff. You might say I was an affirmative action hire—for my views, not my skin color. I was not immediately won over; I liked my job at the _Journal_. But Les was hard to say no to when he was enthusiastic about something—and he was enthusiastic about bringing me to the Council. In hindsight I am very glad he won me over. I have found working at the Council to be the best job that I have ever had, and quite possibly the best job that anyone has ever had—anyone, that is, who isn't concerned about making a fortune. Neither Les nor his successor as Council president, the distinguished policymaker and scholar Richard Haass, has ever told me what to write or say. They fully supported me as I plunged deeper into military affairs in the years ahead, writing a series of books that focused on military history and too many articles to count on the controversies of the day. It was like being at a university without student papers to grade—and without the stifling political correctness of the classroom.
One of the great benefits of being at the Council was getting to know our visiting military fellows—colonels and captains from the military services who are in residence with us for a year. Spending time with them introduced me to American military culture. I was able to expand my understanding by making regular trips "down range" to observe military operations in Afghanistan and Iraq. Sometimes I would travel by myself or with other think tankers; at other times I would lead tour groups of prominent Council members interested in educating themselves about the post-9/11 wars. This is the kind of education that Donald Trump never bothered to get before becoming president. Although he lived and worked for decades only a few blocks from the Council's headquarters in an elegant Upper East Side townhouse, he never showed any interest in its work or in US foreign policy more broadly. Even as president he has never once visited a war zone. He loves military symbols—hence his desire for a military parade in Washington—but shows no understanding of how the armed forces actually operate.
JUST LIKE DONALD TRUMP and most other people, I was a supporter of the war in Iraq. The difference is I'm willing to admit it, not pretending that I was opposed to the war all along. I backed the overthrow of Saddam Hussein, one of the worst tyrants on the planet, and I had faith—obviously unwarranted—that the United States could build a democratic Iraq after his downfall. But I would not have advocated military intervention if I were not convinced that containment wasn't working and that Saddam posed a threat to the United States and its allies with his weapons of mass destruction program. None of this set me apart from most Americans in 2003. The war had the support of 72 percent of the public initially, and it was authorized by both houses of Congress. Among those who voted for the use of force were Senators Hillary Clinton, Joe Biden, Chuck Schumer, and John Kerry. What made me different from many of these fair-weather hawks was that, as the situation spun out of control from the summer of 2003 on, I did not change my ornithological coloration to become a dove. I criticized the mistakes made by the Bush administration—for instance, after the 2004 Abu Ghraib prisoner-abuse scandal, I scandalized fellow conservatives by calling for the ouster of Secretary of Defense Donald Rumsfeld, who, I felt, was mismanaging the war. (This made for an awkward meeting with Midge Decter, wife of Norman Podhoretz and an influential neoconservative thinker in her own right, who had just published a hagiography of Rumsfeld.) I was anguished by the deteriorating course of the conflict and felt some measure of responsibility for this quagmire. But I was convinced that an American pullout would make the situation even worse—a judgment amply vindicated by subsequent experience. President Obama's troop withdrawal in 2011 made possible the rise of the Islamic State of Syria and Iraq.
Instead of leaving Iraq, I became convinced that US troops needed to implement a classic counterinsurgency strategy of the kind I had described in _The Savage Wars of Peace_. This required sending more troops and pushing them out to provide security to the populace, rather than hunkering down in sprawling Forward Operating Bases isolated by Hesco barriers and blast walls from the Iraqi people. General David Petraeus implemented precisely this strategy when he took command in early 2007, and the result was a fall in violence by 90 percent over the next two years. I was not sure that the surge would work, but I became an advocate of giving this strategy a chance. I had previously met Petraeus, a rare general with an Ivy League PhD, while he was a division commander in Iraq in 2003, and we became friends. He invited me to visit Iraq—and later Afghanistan—to offer him my recommendations. These trips were occasionally dangerous (on one occasion in Mosul, a Humvee directly in front of mine hit a roadside bomb and we took machine gun fire from a nearby building), but always fascinating and inspirational. I came away deeply impressed by the morale, skills, and dedication of our troops. There has been no finer fighting force in history, and, though ill-prepared to fight a counterinsurgency, they rapidly improvised and figured out what to do.
My advocacy of the Iraq War and my refusal to support a pullout led the far left and far right to call me a "neocon warmonger" and "chickenhawk" who was part of a "cabal" that had "lied" America into the war. The aspersion that this was a "neocon war" seemed designed to play into ancient prejudices, on both the far left and far right, about conniving and disloyal Jews, since the "neocons" blamed for the conflict—Paul Wolfowitz, Douglas Feith, Richard Perle—were Jewish officials of limited influence. This is exactly the kind of calumny that Trump now spreads in inveighing against "globalist" elites who are supposedly betraying America. In truth the decision to go to war had been made by President George W. Bush, in consultation with colleagues such as Dick Cheney, Condoleezza Rice, Colin Powell, and Don Rumsfeld, none of whom was remotely a "neocon." Those of us who supported the invasion were, as one of my friends said, like hapless passengers who got into a vehicle with a drunk driver and could not escape as the car careened across the center divider.
For years I felt defensive about my support for the war and refused to repent. Stubborn and self-righteous, I did not want to cede any ground to my critics. Now, looking back with greater introspection and humility after the passage of more than fifteen years, I can finally acknowledge the obvious: it was all a big mistake. Saddam Hussein was heinous, but Iraq was better off under his tyrannical rule than the chaos that followed. I regret advocating the invasion and feel guilty about all the lives lost. It was a chastening lesson in the limits of American power. It is not nearly as easy to remake a foreign land by force as I had naïvely imagined in 2003, and even the conservative "best and brightest"—Cheney, Powell, Rumsfeld, Rice, and all the rest—can make mistakes that are every bit as dumb as those that their more liberal counterparts made in Vietnam.
One of the perverse consequences of this catastrophe was that—along with Hurricane Katrina in 2005 and the Great Recession in 2008–2009—it disillusioned many Republicans with the traditional leadership of their party and made them receptive to an outsider like Donald Trump who was unabashed in his hatred of the war and its architects. So, much to my chagrin, I now realize that the failed policies I advocated in 2003 helped, thirteen years later, to elect a president who stands in opposition to nearly everything that I believe in.
It was a lesson in the unintended effects of a militaristic foreign policy that I should have learned earlier. But some conservatives still have not learned it, as witness the agitation in some quarters in 2017 for a preventative war against North Korea before President Trump launched talks with Kim Jong Un. For my part, Iraq cured me of any enthusiasm for what my boss Richard Haass has labeled "wars of choice." Listening in early 2018 to hard-liners like future National Security Adviser John Bolton advocate a first strike against North Korea—an act that could easily trigger a nuclear war—I recognized an echo of my callow, earlier self. Bolton, a conservative firebrand since his days as a student at Yale University in the early 1970s, is whom I used to be.
MY POLICY ADVOCACY WAS not limited to the printed page. I was also becoming involved in politics at a higher level than the drudge work I had done as a teenager on the Zschau campaign. In 2007 I became a foreign policy adviser to the presidential campaign of Senator John McCain, whom I had gotten to know after he had read and liked _The Savage Wars of Peace_ , which, unbeknownst to me, featured the exploits of one of his ancestors—an army officer who had fought Pancho Villa in 1916. McCain became one of those rare politicians, like Ronald Reagan, that I revered. He had exhibited superhuman courage in enduring more than five years of torture in North Vietnam. He could have won early release because his father was an admiral who commanded the US Pacific Fleet. But the POWs had a strict rule—first in, first out—and McCain was not going to betray his honor even if the cost of staying was nearly unendurable physical suffering. I still cannot believe that Trump, who sat out the Vietnam War with five draft deferments and claimed that avoiding sexually transmitted diseases was "my personal Vietnam," has the temerity to criticize one of America's greatest war heroes for being captured. I find it much easier to believe that Trump's graceless example encouraged one of his White House aides to "joke" that McCain's opposition to Gina Haspel's nomination to run the CIA didn't matter because the senator had brain cancer and "he's dying anyway."
McCain showed his character not just as a POW but also as a politician. In 2008, he corrected a woman at a rally who told him, "I can't trust Obama. I have read about him and . . . he's an Arab." "No ma'am," McCain replied. "He's a decent family man, a citizen that I just happen to have disagreements with on fundamental issues, and that's what this campaign is all about." How easy would it have been for McCain to traffic in conspiracy theories and demagoguery. But he refused to do it—and his refusal, along with his own missteps (such as choosing Sarah Palin as his running mate), helped cost him the presidency.
McCain spent his career in Congress as a leading champion of America's role as a global leader. Every year he led a delegation of lawmakers to the Munich Security Conference, a gathering of transatlantic movers and shakers. I was lucky enough to accompany the congressional delegation on their air force airplane on a few occasions. It was a heady experience, trading small talk and wisecracks with famous lawmakers and former policymakers. But what most impressed me was when I walked up to the front of the aircraft once and saw McCain reading a fat volume of history. I thought often of that scene in later years as I witnessed the ascendancy of a Republican president who was incapable of reading long briefing papers, much less books. Advising McCain on foreign policy was the easiest job I've ever had because he knew more about the subject than any of his advisers. His standing on economic and domestic policy was shakier, however, and it cost him badly when the financial markets began to melt down in the fall of 2008. Barack Obama took advantage of the turmoil and his opposition to the Iraq War to win a longshot bid for the presidency.
Four years later, in 2012, I advised another honorable man who won the Republican presidential nomination even though I had some doubts about his conservative bona fides. Many other conservatives were also lukewarm about Mitt Romney even though his personal character, thoughtfulness, and work ethic were unimpeachable. Romney needed more help in the foreign policy field because he had been a governor, not a senator, and I tried to provide it even though I was hardly a member of the inner circle. Obama was a formidable adversary, not only because of his natural charisma and oratorical skill but also because he could boast of having killed Osama bin Laden and kept General Motors alive. I remained highly critical of his presidency, however, because I feared that his "lead from behind" foreign policy was ceding American global leadership. Obama had won the presidency in no small measure because of his opposition to the Iraq War, which showed the high cost of American interventionism. But he swung too far in the other direction, toward noninterventionism, by pulling US troops out of Iraq in 2011 and refusing to intervene in the Syrian civil war. The result would be the rise of ISIS and the creation, in Syria, of what Petraeus would call a "geopolitical Chernobyl" spewing its toxins across the region and the world.
One of the turning points of the 2012 campaign was Romney's assertion that Russia was "without question our No. 1 geopolitical foe"—a claim that Obama ridiculed with the devastating line, "The 1980s are now calling to ask for their foreign policy back because the Cold War's been over for twenty years." Looking back, it's obvious that Obama was wrong and Romney right—even if not even Romney could anticipate how the Kremlin would subvert American democracy. All the more incredible, then, that Republicans have gone from backing a candidate who saw Russia as the enemy to one who refuses to say a negative word about its leader, even though Russia is a far greater menace today than it was in 2012.
By 2015, with another presidential campaign once again looming, I was convinced that we needed a president who, in the manner of Ronald Reagan, would unapologetically assert American power, stand with our allies, and defend freedom from the onslaught not only of terrorist groups but also of rogue states such as Iran and North Korea and rising near-peer competitors such as China and Russia. Jeb Bush came into the Republican race as the early frontrunner, but he wasn't conservative enough for me. That's how uncompromisingly conservative (and naïve) I was—Jeb Bush wasn't good enough for me! Today I would give anything in the world to make him president.
Rather than support Bush, I signed up as a foreign policy adviser with the presidential campaign of Senator Marco Rubio, an eloquent young senator from Florida who appealed to me because he was also a son of immigrants (in his case from Cuba) and spoke movingly of the need to defend and expand the sphere of freedom. I knew there was a good chance he would lose—but I figured that if he did, it would be to an eminently qualified if somewhat boring candidate like Bush.
I had fatally overestimated my fellow Republicans. My disillusionment was to be painful and prolonged; in fact, existential. In the process of being disabused of my illusions about the GOP I would also lose my faith in the conservative movement in whose bosom I had been nursed for decades.
I now wonder: What did I miss? How could all these eminences that I had worked with, and respected, sell out their professed principles to support a president who could not tell Edmund Burke from Arleigh Burke? Even as late as the winter of 2016 I would not have thought it possible.
## _3._
## **THE SURRENDER**
AT FIRST, LIKE MOST PEOPLE IN THE POLITICAL AND policy worlds, I did not take the Trump campaign all that seriously; I assumed that Donald Duck would have as much chance of becoming president as Donald Trump. He appeared to be a second-rate entertainer and first-rate self-promoter running for the presidency to expand his name recognition and boost the ratings of his TV show, _The Apprentice_. He had no government experience or policy knowledge. His past was full of affairs and divorces, of corporate bankruptcies and of stiffed vendors, of shady ventures like Trump University that led to lawsuits and recriminations. He was an opportunist who had regularly flip-flopped between the Democratic and Republican Parties, and he had given copious campaign donations to liberal Democrats. He hardly looked like the natural candidate of a Republican Party that, ever since the ascendancy of Newt Gingrich in the 1990s and then of the Tea Party in the 2000s, had been veering to the right. My assumption, like most others, was that Jeb Bush and Chris Christie would fight it out in the "Establishment lane," while Marco Rubio and Ted Cruz would compete for the hearts of conservatives. Rand Paul, an arch-isolationist, looked to be the most dangerous candidate in the race. Trump just seemed like a freak show—a less-qualified and less-knowledgeable version of Ben Carson, another candidate who was not remotely prepared to be president.
"I don't expect Trump's inability to articulate comprehensible policies to end his ascendancy in the polls anytime soon," I wrote in mid-August 2015, a couple of months after Trump had launched his campaign, "but I do have enough faith in the American political process to hope and even expect that his outright buffoonery will stop him from winning the Republican nomination, much less the presidency." I added a caveat, however, that I should have taken more seriously. Citing the examples of inept officeholders from James Buchanan, the president who had brought America to the brink of the Civil War, to Jesse "The Body" Ventura, a former wrestling star turned governor of Minnesota, I wrote: "But given the history that so many democracies have of making such foolish mistakes in choosing leaders, and given Trump's own history of surviving an advanced case of foot-in-the-mouth disease, there is a small part of me that wonders whether I am being Pollyannaish in expecting that the White House will not eventually have a neon 'Trump' sign on top of it."
My warning would turn out to be prophetic. And yet in the months ahead I would forget my own insight, so convinced was I that no candidate could possibly win by violating so many of the supposed rules of the American presidential selection system. This is one of the traps that a historian can all too easily fall into: assuming that because something hasn't been done before, it can never be done in the future. Trump, supremely ignorant of the past, would prove to have a surer grasp on how to shape the future.
In the Rubio campaign, like most others, we were stockpiling policy advisers, pumping out position papers, and briefing the candidate to respond in a cogent and yet calculated manner to the news of the day. Trump didn't bother with any of that. He didn't have a normal campaign apparatus, and he didn't read briefing papers. He just flew around the country in his own airplane, speaking seemingly off the cuff, making promises to "build the wall," "take their oil," "drain the swamp," and "win so much" that resonated well with his audiences but that struck sophisticates like me as nonsensical. Pressed on the specifics of his plans, he would reply with vapid generalities, promising "we're going to have great plans," "I want to get you something great," and "we will make it stronger and smarter than ever, ever, ever before."
In December 2015, at a Republican debate in answer to a question from radio host Hugh Hewitt, Trump revealed ignorance of the "nuclear triad"; Rubio had to patiently explain to him that the term referred to the US armed forces' ability to launch nuclear weapons from land, sea, and air. Sitting at home, watching the debate, I chortled. _Gotcha!_ Advantage, Rubio. These were the kind of gaffes that would embarrass a normal candidate, but Trump and his acolytes couldn't care less. "The real problem with The Donald is not that he is ignorant but that he is aggressively ignorant," I wrote at the time. "He thinks that his lack of knowledge is a virtue, demonstrating his regular guy quality." The _real_ real problem would turn out to be that Trump was right and I was wrong—the kind of policy cramming that most candidates did prior to running or while running did indeed turn out to be superfluous. Trump was revealing that ignorance was no bar for a presidential candidate. It would prove to be a greater obstacle when he was elected; not having done the normal preparation work for a presidential run, he would prove to be spectacularly unqualified to assume the most difficult and powerful position in the world.
IT WASN'T JUST THAT Trump was ignorant. The larger problem was that much of what he said just wasn't so. He had a proclivity for conspiracy theories, having spent years pushing the "birtherism" hoax that Barack Obama had forged his birth certificate. He lied incessantly: PolitiFact found in mid-2016 that 78 percent of all of Trump's fact-checked statements were either mostly or entirely false, compared to only 16 percent for Hillary Clinton.
Trump also had a long history of creepy, sexist comments that long predated the release of the _Access Hollywood_ tape in the fall of 2016. In August 2015, after Megyn Kelly asked him tough questions during a presidential debate, he implied that it was because she was menstruating. The next month he suggested that no one would vote for his Republican rival Carly Fiorina because her face was so unattractive.
Trump's history of racism was just as long as his record of sexism, stretching all the way back to 1973, when Trump and his father settled charges brought by the Justice Department that their company had refused to rent to African Americans. When Ronald Reagan was endorsed by the Ku Klux Klan, he eloquently rejected the "politics of racial hatred and religious bigotry." When Trump was given three chances by CNN on February 28, 2016, to reject an endorsement from Klan leader David Duke, he refused to do so—and used as excuse the astonishing claim that he didn't know enough about the Klan. Trump subsequently blamed his failure on a malfunctioning earpiece, his version of "the dog ate my homework." This was part of a pattern with Trump. One study in early 2016 found that an incredible 62 percent of his retweets were of white supremacists.
Trump's defenders pointed out that he palled around with black celebrities such as Jay Z, Don King, and Mike Tyson, and that his daughter converted to Judaism when she married Jared Kushner. But at most this demonstrated that Trump is not a doctrinaire neo-Nazi; he is too scattershot to be doctrinaire about _anything_. The evidence suggested he was a more casual bigot along the lines of Archie Bunker, the fictional _All in the Family_ TV character from the New York borough of Queens, where Trump was born. That Trump may like particular individuals did not prevent him from stereotyping and stigmatizing entire minority groups—not just African Americans but also Mexicans and Muslims.
Trump began his campaign by castigating all Mexican immigrants, not just those who had arrived illegally, as "rapists and murderers," promising to build a wall along the Mexican border that Mexico would pay for, and to deport all eleven million undocumented immigrants in the United States. Trump was unwittingly imitating the animus of earlier generations of nativists against earlier waves of newcomers, including the Irish, Italians, Jews, Chinese, and Japanese. Each one of those ethnic groups was once accused of being alien to Anglo-Saxon America—ethnically, culturally, or religiously—and yet each one assimilated, just as Latinos are now doing.
Trump seemed to forget—if he had ever known—that even German Americans like him had encountered pervasive hatred and discrimination during World War I. That did not stop him from slinging accusations of disloyalty against Muslims—the latest group caught on the wrong side of one of America's wars, just as I had been as a Russian American in the last decade of the Cold War. He falsely claimed that "thousands" of Arab Americans in Jersey City had cheered on 9/11, and when a reporter showed that this simply did not happen, Trump made fun of his disability. Trump upped the ante in early December 2015. After a terrorist attack in San Bernardino, he called for a "total and complete shutdown" of all Muslims entering the country—an abhorrent proposal that was contrary to the freedom of religion guaranteed by the First Amendment.
WHILE SCAPEGOATING MEXICANS, African Americans, and Muslims, Trump had nothing but praise for an actual enemy of America. In an interview on _Morning Joe_ on December 18, 2015, Trump praised Russia's dictator Vladimir Putin as a better "leader" than President Obama. When Joe Scarborough asked him about allegations about Putin killing journalists and political opponents, Trump replied, "I think our country does plenty of killing also, Joe, so you know."
Conservatives like me had spent decades inveighing against the tendency on the left to engage in moral relativism—to suggest that America, because of its sins, was no better than any other country. We had labeled this "anti-Americanism." Reagan's United Nations ambassador, Jeane Kirkpatrick, whose name graces the position I occupy at the Council on Foreign Relations, said: "There is no more misleading concept abroad today than this concept of . . . superpower equivalence." In 2011, Representative Paul Ryan said, "If you ask me what the biggest problem in America is, I'm not going to tell you debt, deficits, statistics, economics—I'll tell you it's moral relativism." Now here was Trump doing the very thing that conservatives had been criticizing, and in the process he was siding with Russia, a country whose aggressive designs Republicans had been opposing since the Bolshevik Revolution in 1917.
Trump did not just admire Putin. He gave every indication of wanting to emulate his authoritarian example. "Be careful," Trump often told anyone speaking out against him, as if he were a dictator or at least a mob boss. He expressed a desire to change the libel laws so that when journalists wrote negative articles about him, "we can sue them and win lots of money." Trump would routinely instigate violence at his rallies against protesters. At a February 2016 rally in Las Vegas, Trump said, from behind a phalanx of Secret Service bodyguards, that he'd like "to punch a demonstrator in the face" and lamented that the man "wasn't being carried out a stretcher." The parallels with fascist rallies in the 1930s were inescapable and alarming—even if "see no evil" Republicans purported not to notice them.
And those rallies did not _just_ occur in Europe: In 1939, as recounted by historian Gordon F. Sander, the German American Bund filled Madison Square Garden with twenty thousand cheering Nazis—America's own version of the Nuremberg rallies. The führer of "Swastika Nation" was a German American auto worker named Fritz Julius Kuhn. After being introduced as "the man we love for the enemies he has made" (sound familiar?), he proceeded to denounce those enemies with anti-Semitic twists on their name. Like Trump, Kuhn had a gift for derogatory nicknames—at least derogatory to his anti-Semitic followers. He dubbed Franklin D. Roosevelt, who had called Nazism "a cancer," "Frank D. Rosenfeld." District Attorney—and future governor and Republican presidential candidate—Thomas Dewey was "Thomas Jewey." Mayor Fiorello LaGuardia, who was part Jewish, was "Fiorello Lumpen LaGuardia." His insults were greeted with cries of "Free America!"—a precursor of "Make America Great Again." A Jewish demonstrator became so agitated that he tried to rush the podium to make Kuhn stop spewing his filth, but he was tackled and beaten by the Bund's own brownshirts. Americans were horrified by newsreel footage of Nazis beating a Jew in New York City. Yes, it can happen here—and did. District Attorney Dewey got his revenge later that year by prosecuting Kuhn for tax evasion; the Nazi leader was sentenced to two-and-a-half years in prison. There would be poetic justice if Trump were to face a similar fate: brought down not for his hate-mongering but for his shady business practices.
To the extent that Trump had coherent policy proposals, they were a throwback to the isolationism and protectionism of the 1930s—and a repudiation of the free trade and internationalism that the Republican Party, and the conservative movement, had championed since World War II. Trump threatened to pull US troops out of Germany, South Korea, and Japan, and to impose massive tariffs on our trade partners. He even adopted as his slogan "America First," the very same phrase that had been used by Charles Lindbergh and a cohort of other isolationists in the prewar period—de facto allies of the German American Bund who were sympathetic to Hitler and hostile to Jews. This should have been a tocsin ringing loudly—but it was ignored by Republican voters, ignorant of history, who seemed to have no conception of how toxic this phrase was.
MY OPPOSITION TO TRUMP went far beyond disagreement with his policy impulses; I would not dignify them by calling them "ideas" or "proposals." I was morally offended by him, and apoplectic that anyone could think that he was remotely qualified to fill the office of Washington, Lincoln, and Roosevelt. In November 2015 I tweeted: "Trump is a fascist. And that's not a term I use loosely or often. But he's earned it." This was not an epithet I casually tossed around, having been branded a fascist myself at Berkeley for holding moderate conservative views. The word has no exact, commonly agreed upon definition, but many of the attributes of fascism cited by Georgetown University historian John McNeill clearly applied to Trump—including "hyper-nationalism," "militarism" (Trump had just bragged at a GOP debate: "I'm the most militaristic person on that stage"), "glorification of violence," "fetishization of masculinity," "leader cult," "lost-golden-age syndrome," "self-definition by opposition," "mass mobilization," "tendency to purge the disloyal," and "theatricality."
Trump's lack of restraint caused me to loosen my own restraints. I had grown up in a culture in which, even after the convulsions of the 1960s and 1970s, the President—a title that was always capitalized—was treated with deference and respect. I was influenced by watching old black-and-white movies from the fifties in which the chief executive was such a mighty personage that he could only be glimpsed in silhouette or from the back—seeing his face would have seemed as sacrilegious as glimpsing the face of God. That attitude had carried over to my work as a political pundit. Even when I disagreed with presidents such as Barack Obama or George W. Bush, I was careful to treat them with the respect that their accomplishments and office demanded. A similar habit of deference carried over to presidential candidates, any one of whom (almost) could be the future commander in chief. But I could not treat Trump as a normal candidate when he was transgressing every norm not just of presidential politics but also of civilized society. I had some trepidation about calling out a presidential candidate—and a Republican to boot—in terms normally reserved for foreign tyrants, but my indignation propelled me forward. I could not stay silent.
In early 2016, I wrote that Trump was "a liar, an ignoramus, and a moral abomination." I added: "I have never previously described any presidential candidate in such harsh terms—not even close—but there is no other way to accurately describe him. There simply isn't." In March I wrote that Trump was "emerging as the number one threat to American security. Yes, that's right—a bigger threat than ISIS, North Korea, Russia, China, Iran, or all the rest." I have not revised my view in the intervening time: I still believe that Trump poses the greatest threat to US security. Writing in the _Weekly Standard_ , my conservative Council on Foreign Relations colleague Benn Steil and I warned that Trump's "policies would not make America 'great.' Just the opposite. A Trump presidency would represent the death knell of America as a great power." That, too, is a view that I believe has been vindicated by events.
I may have gone a bit further rhetorically than some other conservatives, but I was hardly alone in my alarm at the rise of Trump. One of the few conservatives who endorsed Trump early on felt compelled to do so anonymously. Michael Anton, a dandy in bespoke suits who had also graduated from Berkeley and who would subsequently become the spokesman for Trump's National Security Council, used a Roman pseudonym, Publius Decius Mus, to pen a pro-Trump essay in the _Claremont Review of Books_. I knew Anton and assumed that, having worked for President George W. Bush and Mayor Rudolph Giuliani, he was a normal, middle-of-the-road conservative who just happened to have an unusual degree of interest in the details of men's wear ("Prints are safe. Everyone wears prints") and French cooking. So I was unprepared for the role he now assumed as the leading Trumpian intellectual—admittedly an honor for which there was slim competition, given how anti-intellectual the candidate and most of his followers were. Anton argued: "2016 is the Flight 93 election: charge the cockpit or you die. You may die anyway. You—or the leader of your party—may make it into the cockpit and not know how to fly or land the plane. There are no guarantees. Except one: if you don't try, death is certain. To compound the metaphor: a Hillary Clinton presidency is Russian Roulette with a semi-auto. With Trump, at least you can spin the cylinder and take your chances."
This was an argument, as even Anton had to concede, that sounded "histrionic" to "ordinary conservative ears." It didn't just sound histrionic; it _was_ histrionic. It was, in fact, the kind of argument that apologists for dictators—dandies, aesthetes, and eccentrics among them—had made in the past, claiming that the alternative to their revered leader was so dire that it justified the imposition of despotism. Such beliefs typically had gained popularity at a time of existential crisis, usually following an economic meltdown or military defeat, when the threat came from Marxists or those who could credibly be depicted as such. That had been the pattern of fascism in countries such as Italy, Argentina, Chile, Hungary, Japan, Spain, and Germany. Anton's innovation was to sound this apocalyptic alarm at a time of peace and prosperity about _Hillary Clinton_ —a boring, middle-of-the-road policy wonk who was considered far too conservative by Bernie Sanders–style progressives.
I remember serving on a Defense Department advisory board with Clinton when she was in the US Senate. The alphabetical order of our names led to us being seated next to each other. I found her to be congenial and smart. Not only was she utterly free of political cant, at least in this private setting, but she was interested in the details of defense planning to a degree that was unusual for a politician. It never occurred to me that this amiable lady whom I chatted with and liked would be depicted as the harbinger of doom by my fanatical compatriots on the right.
Anton's screed was widely mocked in the conservative circles in which I traveled. There was not a single mainstream conservative early on who was willing to embrace Trump, which helped to explain why he could not sign up any credible policy advisers and had to hire dubious characters such as Carter Page and George Papadopoulos, who were subsequently revealed to have links to the Kremlin. Along with more than a hundred other Republican national security experts, I signed an open letter that appeared on the War on the Rocks website pledging eternal opposition to Trump. The letter, organized by my friends Eliot Cohen and Bryan McGrath, concluded that "as committed and loyal Republicans, we are unable to support a Party ticket with Mr. Trump at its head. We commit ourselves to working energetically to prevent the election of someone so utterly unfitted to the office." A few of those who signed the letter would subsequently try to disown it to win positions in the Trump administration, but they would not succeed in erasing this mark of Cain. Trump put stock in personal loyalty above all, and the national security experts who signed the War on the Rocks letter had disqualified themselves from service in his administration by showing that they were more loyal to the country than to its president.
Similar opposition to Trump was expressed by a bevy of prominent conservatives who wrote essays that _National Review_ published shortly before the first primaries in January 2016 under the headline: "Against Trump." _NR_ 's editors warned: "Donald Trump is a menace to American conservatism who would take the work of generations and trample it underfoot in behalf of a populism as heedless and crude as the Donald himself." Contributing to this issue was a who's who of the conservative movement, including Ronald Reagan's attorney general Edwin Meese III, economist Thomas Sowell, syndicated columnist Cal Thomas, radio hosts Glenn Beck and Michael Medved, _Weekly Standard_ editor Bill Kristol, _Commentary_ editor John Podhoretz, evangelist Russell Moore, Club for Growth president David McIntosh, and former attorney general and judge Michael Mukasey.
With the stars of the right aligned against Trump, I figured, what chance did he have to win over ordinary conservative voters?
THE CONSERVATIVE MOVEMENT had always been an uneasy alliance, not only between different strains of ideology (cultural conservatives, libertarians, supply-siders, deficit hawks, internationalists, isolationists, etc.) but also between the leaders in Washington and New York and the masses in the rest of the country. Conservative elites like to accuse liberal elites of being "out of touch." It was an appeal to the supposed wisdom of the crowd that I had used on occasion myself—imagining that, even though I was part of a small, embattled minority among the liberal cognoscenti of Manhattan, I was in sync with the good people of Manhattan, Kansas (a place that, of course, I had never visited). "Middle America" was, along with "entrepreneurs," one of those imagined archetypes that conservative intellectuals employed much as Marxists appealed to the authority of "the workers" and "the proletariat." But it turned out that conservative eggheads were just as far removed from the heartland of America as the "limousine liberals" and "smoked-salmon socialists" that we hypocritically loved to ridicule. (Personally, I enjoy smoked salmon, a.k.a. lox, so much that I have it on a bagel nearly every weekend.)
Republican voters could not have cared less what elite conservatives like us were saying. Trump went over our heads by speaking directly to the country at his televised rallies. His lowbrow slogans resonated much better with ordinary Republican voters than our highfalutin arguments. The news networks covered Trump obsessively, providing him with what was later estimated to be $2 billion of free television advertising. He was getting more TV coverage than all of his rivals combined because there was a seductive frisson of excitement about his rallies that was lacking in the more scripted and sedate rallies of his rivals. Everyone wondered: What crazy thing would he say next?
Trump may not know much about policy, but he is a genius at self-promotion—a Jay Gatsby for our time. He has much in common with the land promoters who bamboozled English immigrants into coming to the New World in the seventeenth century with fanciful tales of riches—what Trump would describe as "truthful hyperbole." Or with the kind of charming con men who peddled patent medicines in the nineteenth century and then, in the twentieth century, penny stocks and time-shares. But his bunkum would take on a more sinister aspect when he deployed it not to just to dupe people into handing him their money but also supreme power.
DESPITE HIS LEAD IN THE POLLS, Trump narrowly lost the Iowa caucuses, finishing just behind Ted Cruz and just ahead of Marco Rubio. This stronger-than-expected showing for Rubio prompted a brief flurry of hope in our camp that he could win the New Hampshire primary and then run the table. But Rubio sabotaged himself in a February 6, 2016, debate when he robotically repeated the same canned lines four times in response to attacks from Chris Christie. Watching the debate on TV, I felt as if I were witnessing an automobile accident; all that was missing was the horrible screech of crumpling metal. But when I contacted the campaign, staffers feigned insouciance, claiming the situation wasn't as bad as it seemed. It was. Although we didn't realize it at the time, the Rubio campaign was basically finished that night. Trump won a commanding victory in New Hampshire, followed far behind by Cruz and Rubio.
Yet the other candidates still refused to engage in all-out attacks on Trump on the theory that his ramshackle campaign would inevitably fall apart, leaving one of the "serious" candidates to inherit his supporters. The professional politicians were being too clever by half, in the manner of aristocratic German conservatives such as President Paul von Hindenburg and Chancellor Franz von Papen, who welcomed Adolf Hitler's ascension as German chancellor in 1933 on the assumption that the jumped-up, loudmouthed little corporal would be easily manipulated by his social betters. The Republican candidates would have been better advised to be forthright from the beginning about Trump's unsuitability for high office and let the voters make up their minds. With the momentum Trump was gaining, he was becoming unstoppable.
Conservatives hoped that the South Carolina primary, on February 20, would slow Trump down. South Carolina voters had a reputation for being conservative and religious, and we told ourselves that they would punish Trump's blatant immorality and his trashing of the last Republican president, George W. Bush. No such luck. Again Trump won. Cruz was lucky to hold his home state of Texas on March 1. Rubio was not so lucky: on March 15 he lost Florida in spite, or because, of his desperate attempts to slow down Trump by imitating his personal insults. With his genius for needling his opponents, Trump had called Rubio "Little Marco"; Rubio retaliated by mocking Trump for having "small hands" and a "spray tan." It turned out, however, that while personal insults worked _for_ Trump, they did not work _against_ him. Voters seemed to expect nothing better from the reality TV star, while they wanted more from a senator like Rubio, who would subsequently apologize for his descent into the gutter.
With Rubio gone, John Kasich and Ted Cruz were the only Trump alternatives left. I voted for Kasich in the New York primary on April 19, but by then it was obvious that Trump was going to claim the GOP nomination.
AS TRUMP BEGAN TO EMERGE as the inevitable Republican nominee, something ominous occurred: Republicans genuflected before their new master. This could be explained by the Republicans' demonization of Democrats; by their knee-jerk loyalty to the GOP brand, regardless of whether its nominee shared any of their professed principles or not; by their fear of the Republican masses, whose passions Trump had shown a disturbing skill in whipping up; and by the sheer lust for power that is unfortunately characteristic of most officeholders and seekers. As countless toadies had done with demagogues of the past, so now most Republican leaders showed that that they were willing to discard their principles as mindlessly as a Styrofoam fast-food container if by doing so they could enhance their own positions and avoid the wrath of a powerful and vindictive leader.
Marco Rubio went from proclaiming "Never Trump" to endorsing Trump within a matter of months. How could he possibly support someone he had just described as a "con man" who was "too erratic" to be entrusted with the nuclear codes? Distraught, I called one of his aides, who lamely explained to me that "Never Trump" had only applied to the primaries. I was incredulous. Clearly I had misread Rubio. I had thought he was a man of principle, overlooking his track record of backtracking if those principles risked making him unpopular with the base. He had, for example, disowned attempts to reach a compromise on immigration in the Senate in 2013 after a backlash from nativists. Now he was showing a similar lack of courage and consistency, in fact a decided opportunism. This same phenomenon was evident across the conservative movement: all too many people who had only recently shared my abhorrence of Trump were boarding the Trump train as it picked up speed.
I remember having a conversation over a backyard cookout in Washington, D.C., with one of my closest friends, someone with whom I had worked on the Rubio campaign. He was more conservative than I was on social issues, but if anything that should have caused him to recoil even more strongly against the thrice-married libertine. So I was shocked when he told me that Trump would not only win the general election against Hillary Clinton but that, despite all his faults and flaws, he _deserved_ to win. It did not take long for his seemingly reluctant endorsement to turn to full-blown enthusiasm. Today my old friend propagates conspiracy theories about how Democrats and the Deep State are supposedly plotting against Trump.
I had already said publicly and repeatedly, to the horror of many of my fellow conservatives, that I would vote for Clinton; even many fellow #NeverTrumpers could not bring themselves to support the Democratic nominee. I did not find it hard to cross this psychological Rubicon. In one of my less-considered quotes, I had told the _New York Times_ that "I would sooner vote for Josef Stalin than I would vote for Donald Trump." That was not, as Trump supporters disingenuously claimed, because I was sympathetic to Stalin, who had exiled my own grandfather to Siberia after World War II for having been taken prisoner by the Germans while serving in the Red Army. (Like Trump, Stalin did not like people who were captured.) It was just my clumsy if emphatic way of saying I was "Never Trump" all the way. As I told the _Times_ : "There is no way in hell I would ever vote for him. I would far more readily support Hillary Clinton, or Bloomberg if he ran." My friend, who had served in a previous Republican administration, had a different view. He told me that politics is inherently tribal, and he had signed up with the Republican tribe, no matter who its chief was.
The difference between my friend and me is that I had always seen my primary allegiance as being to conservative principles rather than to the Republican Party. I had become a Republican because I viewed the GOP as the party most sympathetic to my ideals, but if it was now anointing a standard-bearer hostile to my views, I was not going to support the GOP. I had not realized how tribal politics was and how divorced it could be from principles or conviction. I was about to get an education that would dispel my naïve faith in the conservative movement and the American political system.
The rate of Republican surrender to Trump varied, as did the amount of anguish expressed along the way. But in the end almost everyone submitted in an astonishing domino effect. Senator Mitch McConnell, the Senate majority leader, went quietly, issuing a tepid press release in early May 2016 announcing that "I have committed to supporting the nominee chosen by Republican voters, and Donald Trump, the presumptive nominee, is now on the verge of clinching that nomination." Representative Paul Ryan, the House Speaker, said he was "not there yet" in terms of endorsing Trump, raising my hopes that he would prove to be a principled tribune of Reaganesque conservatism, but on June 2 he too caved.
With classic bad timing, Ryan delivered his endorsement just before Trump attacked a federal judge overseeing a civil case against him because of his Mexican ancestry. Ryan rightly described this as "the textbook definition of a racist comment," but he had little to say when the textbook expanded into multiple volumes. Ryan, a conservative of my own age who had worked for such movement icons as Bill Bennett and Jack Kemp, had been one of my heroes. I had viewed him as smart, principled, and brave because of his willingness to touch the third rail of American politics—he advocated reforming expensive entitlement programs such as Social Security and Medicare that were bankrupting the country. But now he showed that he was simply another politician like all the rest. Either he lacked real principles or he convinced himself that backing Trump could somehow advance those principles—a calculation that would predictably backfire.
Ted Cruz pleasantly surprised me by holding out longer against a nominee who had insulted both his wife and father. Like most people in Washington, I had seen Cruz as an insufferable opportunist who cynically exploited the populist whims of the Republican electorate when he knew better. During his Senate campaign in 2012, for example, he had denounced my employer, the Council on Foreign Relations, as a "pit of vipers" that was "working to undermine our sovereignty," even though his accomplished wife, Heidi, a Goldman Sachs banker, was herself a Council term member. I was in an unaccustomed position, therefore, in rooting for Cruz when he braved boos from the delegates by refusing to endorse Trump at the Republican convention. But eventually he reverted to form by running up the white flag, reportedly under pressure from his donors. He went from calling Trump a "pathological liar," "utterly amoral," "a narcissist at a level I don't think this country's ever seen," and "a serial philanderer" . . . to endorsing Trump. By 2018, he would be celebrating Trump as "a flash-bang grenade thrown into Washington by the forgotten men and women of America."
Cruz proved he was every bit as opportunistic and unprincipled as Trump—but not nearly as successful at faking candor. His oleaginous manner oozed insincerity. Ironically such flip-flops only confirmed Trump's attacks on the Washington "swamp"—not that he had any intention of draining the swamp. He would actually make Washington swampier than ever.
THE PRESSURE ON MOST Republican officeholders to support "the nominee," as reluctant endorsers preferred to call him, was simply too strong to resist, coming as it did not only from their base, which thrilled to Trump's politically incorrect rhetoric, but also from their donors, who were mobilizing behind Trump in the expectation that he would push through a tax cut that would benefit them. Similar pressure was being felt by radio and television hosts, who felt the need to appease their pro-Trump audience. Even Hugh Hewitt, who had shown up Trump's staggering ignorance and had earned the candidate's wrath in return, became an enthusiastic supporter—apparently after having been instructed by his employer, Salem Media Group, to get aboard the Trump train. I had often been a guest on Hewitt's radio show, and had found him to be a smart, well-informed interviewer. I had thought he was a cut above the Fox rabble-rousers; I was wrong.
Fox News Channel, of course, went all-in, with personalities such as the cast of _Fox & Friends_, Jeanine Pirro, Sean Hannity, and Bill O'Reilly—and later Laura Ingraham and Tucker Carlson—competing to outdo one another in their sycophancy to Trump. Was this out of principle or expedience? Were they telling their audience what it wanted to hear or did they actually believe what they were saying? Probably a bit of both, although I suspect Carlson of greater cynicism than Hannity, simply because I believe Carlson is smarter and knows better—or should. Indeed, Carlson often resorted to the shabby rhetorical device of bashing Trump critics rather than simply praising Trump—anti-anti-Trumpism being the last resort of the conservative scoundrel. Radio hosts such as Rush Limbaugh and Mark Levin likewise swallowed their doubts; they too were all-in.
The few conservative radio or TV personalities who resisted—for example, Charlie Sykes in Wisconsin and Erick Erickson in Georgia—soon found themselves losing their audience. Those who worked for think tanks and advocacy organizations were just as vulnerable to pressure from their funders. The president of one small, conservative think tank told me that he agreed with me about Trump but couldn't say so in public for fear of offending his board of directors.
Even most of those who had contributed to _National Review_ 's "Against Trump" issue backtracked. A year later, David Frum noted that of the twenty-one signatories, "only six continue to speak publicly against his actions. Almost as many have become passionate defenders of the Trump presidency, most visibly the Media Research Center's Brent Bozell and the National Rifle Association's Dana Loesch." The few who held out from Trump's blandishments tended to be either religious minorities—Jews or Mormons—who had ancestral memories of persecution or national security experts who knew the high stakes involved in the presidency. As it happens, I checked off both boxes.
In March 2016, I had written that Trump was a "character test" for the GOP: "Do you believe in the open and inclusive party of Ronald Reagan? Or do you want a bigoted and extremist party in the image of Donald Trump?" To my growing horror, most Republicans were failing the test. Conservatives with whom I had been working on anti-Trump briefing papers for the Rubio campaign emerged, in the blink of an eye, as enthusiastic Trumpkins. I still have their anti-Trump emails to me, and while writing this book I reread them with amusement and disbelief. Like so many other conservatives, they evidently viewed the GOP as a cult from which there is no escape even if the cult leader changes.
ONLY A PRECIOUS FEW prominent conservatives refused to endorse Trump. The roll call of honor included former presidential candidates Mitt Romney and Jeb Bush; former RNC chairmen Michael Steele and Ken Mehlman; Senators Susan Collins, Lindsay Graham, Ben Sasse, Lisa Murkowski, and Jeff Flake (John McCain, Rob Portman, Mark Kirk, Cory Gardner, Dan Sullivan, and Kelly Ayotte initially endorsed and then repudiated Trump); a few House members, including Adam Kinzinger, Charlie Dent, Mia Love, and Barbara Comstock; Governors John Kasich, Charlie Baker, and Larry Hogan; and a smattering of #NeverTrump pundits, political consultants, intellectuals, and writers like me. A partial list of the latter includes Stuart Stevens, Michael Murphy, Steve Schmidt, Mark Salter, Matthew Dowd, Nicolle Wallace, Jennifer Rubin, William Kristol, Robert Kagan, David Brooks, Ross Douthat, Michael Gerson, John Podhoretz, Jonah Goldberg, Tom Nichols, David Frum, David French, Linda Chavez, Anne Applebaum, Bret Stephens, Stephen Hayes, Ana Navarro, Rick Wilson, Evan McMullin, Mindy Finn, Joe Scarborough, Charlie Sykes, and George F. Will.
I was proud to stand with all of them, but we were too few in number. There were enough of us for a dinner party, not a political party. I expected many more to join our ranks and was shocked by how many others—men and women whom I had known and admired for years—sold out their professed principles. It was as if they had descended into the "sunken place" in _Get Out_ to be brainwashed like the African American characters in that classic horror film. The historian Richard Brookhiser, a longtime stalwart at _National Review_ , summed up the Trump effect: "Now the religious Right adores a thrice-married cad and casual liar. But it is not alone. Historians and psychologists of the martial virtues salute the bone-spurred draft-dodger whose Khe Sanh was not catching the clap. Cultural critics who deplored academic fads and slipshod aesthetics explicate a man who has never read a book, not even the ones he has signed. . . . Straussians, after leaving the cave, find themselves in Mar-a-Lago. Econocons put their money on a serial bankrupt."
The Republican Party as I had known it was "dead," I wrote in May 2016, adding that, "as far as I'm concerned," the anti-Trump holdouts "are the real Republican Party, in exile. I only hope that they—and I—can return from the wilderness after November." That was not an easy article to write; it represented my first public break with my tribe, and of course I hoped that the rupture would eventually be repaired. I was being overoptimistic. Little did I suspect that the Trump wing would become the dominant one, and that the rest of us would be consigned to an exile that gives every appearance of becoming permanent.
AS THE TRUMPKINS GAINED ascendancy within the GOP, attacks on conscientious objectors like me increased in vehemence. We were told that we were suffering from "Trump derangement syndrome" and were guilty of "virtue signaling" so that we could win social acceptance at "Georgetown cocktail parties." In reality I had been happy for years to be an outspoken conservative in liberal cities like Los Angeles, Berkeley, New Haven, Cambridge, and New York, and had never felt any desire or need to kowtow to liberal sensibilities. I reveled in being a minority and a dissenter; remember, I love to debate. I had never attended a single "Georgetown cocktail party" before opposing Trump and still have not been to one. I had, however, developed a network of conservative friends and supporters, and now I felt increasingly alienated from them. I might have called this book _Ex-Friends_ were that title not already taken by Norman Podhoretz, who lost friends when he moved from left to right.
Some of my ex-friends told me that I sounded "angry" when talking about Trump, implying that emotion clouded my judgment. They suggested I needed to be "helpful" to Trump, rather than critical. In my view, they were the deluded ones whose judgment has been clouded by power worship. I do, however, plead guilty to being angry. I was and remain furious at what Trump is doing to our democracy and how he is demonizing the most vulnerable among us. And I'm angry with all those people who are _not_ angry—who are, in fact, complacent in the face of his attack on our institutions or even serve as his willing accomplices.
Often these conservatives would preface their remarks with "I'm no fan of Trump _but_ , . . ."—the "but" being key—and then proceed to make clear that they were indeed fans but were ashamed to admit it because they knew how vile Trump was considered to be by polite society. This was almost identical to the way certain people would say "I'm no racist but, . . ." before proceeding to prove that they were indeed racists by opining that people of color were "stupid," "criminal," or "lazy." But a large section of the Republican base was unashamed by Trump's transgressions and indeed celebrated him for breaking "politically correct" taboos.
I had lived in the United States for more than forty years without experiencing overt anti-Semitism; admittedly I had resided in liberal enclaves such as Los Angeles and New York, where a lot of Jews live. In 2015–2016, however, my Twitter account and sometimes my email inbox filled up with anti-Semitic, pro-Trump vitriol. I was called "a traitor to America" and told that "Jews want Whites to think . . . ethnic identity's a vice." Some charming Twitter troll posted a picture of me being executed in a gas chamber by a smiling Trump dressed in a Nazi uniform. Others suggested that I should leave America and move to either Israel or Russia.
My experience was hardly unusual; other Jewish commentators critical of Trump received similar treatment—in some cases much worse than I did because they had more obviously Jewish-sounding names. The Anti-Defamation League found that from August 2015 to July 2016 there were 2.6 million anti-Semitic tweets, reaching 10 billion total impressions. Admittedly about 70 percent of them came from just 1,600 accounts, suggesting a small but fanatical movement. The ADL also determined that the number of anti-Semitic acts in the United States increased from 942 in 2015 to 1,267 in 2016 and 1,986 in 2017.
Most Jews, even conservative Jews, were alarmed by Trump despite his pro-Israel statements and his Jewish son-in-law because of his covert appeals to anti-Semitism. Witness all of his retweets of white nationalists and all of his attacks on "international banks" that were supposedly plotting "the destruction of U.S. sovereignty in order to enrich these global financial powers." To illustrate the danger posed by "global special interests," Trump's closing campaign commercial flashed photographs of such readily identifiable Jews as financier George Soros, Federal Reserve chair Janet Yellen, and Goldman Sachs CEO Lloyd Blankfein. It sounded all too similar to what Nazi propaganda minister Joseph Goebbels had said at the Nuremberg rally in 1935 when he warned of "the absolute destruction of all economic, social, state, cultural, and civilizing advances made by Western civilization for the benefit of a rootless and nomadic international clique of conspirators."
The racist Alt Right—a new name for age-old bigotry—was energized by Trump's candidacy, and why not? They liked what they heard from the most unapologetically racist major-party nominee in many decades—and quite possibly ever. One of their own—Stephen Bannon—even went from publishing the openly racist Breitbart website to chairing Trump's campaign and later working in the White House.
THE RUSSIAN GOVERNMENT also turned out to be a big fan of Trump—hardly surprising given how effusively he praised Vladimir Putin. By the time of the Democratic National Convention in late July 2016, it was obvious, at least to me, that the Russians were meddling in America's democracy to help elect Trump. WikiLeaks, a left-wing, anti-American website run by the fugitive Julian Assange, released twenty thousand emails stolen from the DNC. These leaks forced the resignation of DNC chairwoman Debbie Wasserman Schultz because they seemed to show that the DNC had been biased in favor of Hillary Clinton over challenger Bernie Sanders. This served Trump's purposes by encouraging Sanders supporters to sit out the general election or to vote for a third-party candidate, Jill Stein, who was suspiciously friendly toward Putin.
CrowdStrike, a cybersecurity firm hired by the DNC, traced the source of the leaks to two groups of hackers—"Cozy Bear" and "Fancy Bear"—associated with two Russian intelligence agencies. As I noted at the time, "Moscow's virtual fingerprints are all over this operation, including hyperlinks in Cyrillic and Internet protocol addresses linked to previous Russian hacks. In short, this appears to be a Russian intelligence operation designed to damage Clinton." Yet Trump denied that Russia was behind the hacks—"It could be Russia, but it could also be China. It could also be lots of other people," he said during the first presidential debate in the fall. "It also could be somebody sitting on their bed that weighs 400 pounds." That Trump would not admit reality—that the Russians were behind the hacking—revealed a guilty conscience on his part because he was making active use of the stolen emails. Former CIA director Michael Hayden described Trump as a "useful idiot" of Vladimir Putin; former acting CIA director Michael Morell called him an "unwitting agent."
In September 2016, President Obama dispatched national security officials to Capitol Hill to plead for bipartisan unity in confronting the Kremlin. Senate Majority Leader Mitch McConnell shamefully put party above patriotism by refusing to cooperate. This was another low point for the Republican Party—which in those days was still my party. The GOP was so determined to win the presidential election that it was willing to overlook Russian meddling in the electoral process. Obama should have done more to stop the Russians, but if he had done so without Republican support, he would have been vulnerable to Trump's cynical charges that the election was "rigged." By refusing to confront this foreign threat, the GOP had made itself complicit in something close to treason.
THROUGHOUT THE FALL, Trump traduced the most basic norms of American democracy: he called for his opponent to be locked up and refused to say that he would accept the election results if he lost. Republicans pretended not to notice. All considerations of morality or ethics had to be suspended in the name of defeating Hillary Clinton. This was the reductio ad absurdum of the "win at all costs" mindset that the Republican Party had cultivated for decades. The GOP wanted to prevail even if its nominee was a faux Republican who was likely to do lasting damage to American democracy.
The final blow to Republican self-respect occurred in October with the release of the _Access Hollywood_ tape. Trump is clearly heard bragging about groping women: "When you're a star, they let you do it. You can do anything. . . . Grab them by the pussy. You can do anything." Before long nineteen women would come forward to testify that this was no mere "locker room talk"—that Trump had in fact assaulted them. The future of Trump's candidacy was briefly in doubt. Prominent Republicans withdrew their endorsements and called on him to pull out. His poll numbers plummeted. But he refused to budge, and as his support began climbing again, some of the very same Republicans who had un-endorsed him, re-endorsed him. In lieu of principles, these politicians had poll numbers.
Evangelical Christian leaders didn't even bother going through the charade of temporarily abandoning Trump; their devotion to him was so total and unshakable that Trump might as well have been the Messiah. Another notorious rascal, Governor Edwin Edwards of Louisiana, had once said in what would become a familiar trope: "The only way I can lose this election is if I'm caught in bed with either a dead girl or a live boy." In Trump's case, it's doubtful if even that would have been enough to alienate his evangelical acolytes. He could literally have killed someone in the middle of Fifth Avenue, as he bragged, and he still would not have lost their backing.
"My view is that people of faith are voting on issues like who will protect unborn life, defend religious freedom, create jobs, and oppose the Iran nuclear deal," said Ralph Reed, the chairman of the Faith and Freedom Coalition. "I think a 10-year-old tape of a private conversation with a TV talk show host ranks pretty low on their hierarchy of concerns." If only Reed and other evangelicals had extended similar sympathy and understanding to Bill Clinton after his own sex scandal. In the past they had always maintained that political leaders needed to be unimpeachable in their personal conduct. But they were theological silly putty: they had no problem twisting their supposed convictions to support whatever political outcome they favored.
Of all the GOP's toadies and hypocrites, the fundamentalists were the most egregious: these supposed champions of morality were willing to support a candidate who regarded the sins proscribed in the Ten Commandants as his personal to-do list. The more commandments he violated, the better they liked him. It called to mind H. L. Mencken's quip: "Religion is a conceited effort to deny the most obvious realities."
TRUMP WAS BUOYED not just by his fervent partisans but also, ironically, by a law enforcement professional whom he would subsequently fire and vilify. FBI director James Comey announced just eleven days before the election, in contravention of Department of Justice guidelines, that he was reopening an investigation of the Hillary Clinton email scandal based on new emails found on a laptop belonging to Anthony Weiner, the estranged husband of Clinton aide Huma Abedin. _Uh-oh_ , I muttered when I heard the news in a Manhattan studio where I was preparing to tape a podcast. _Here we go again_. Just a few days later Comey announced that this new investigation had not found any criminal wrongdoing, any more than a previous investigation had. I rejoiced when I heard the news. But by then it was too late: Comey had thrust the Clinton emails, the source of Trump's endless calls to "Lock her up," back into the center of the campaign. I don't believe Comey did this to help Trump—he was trying to protect the FBI's reputation from right-wing critics who would have claimed a cover-up if Clinton had won—but the effect was the same.
Along with the stolen emails from the Russians, the FBI's unwise updates on the Clinton investigation—while keeping quiet about the ongoing investigation into Trump-Russia ties—would help seal the fate of Clinton's campaign. But that was obvious only in hindsight. In the days before the election, sealed off in my coastal enclave, I was serenely oblivious of the fate about to befall the country. I made the fatal mistake for an analyst of conflating my own preferences with those of other people.
I ended the campaign with an explanation in _Foreign Policy_ magazine of why "This Lifetime GOP Voter Is with Her." I knew that my scribblings would not make any difference to the outcome, but I wanted to do everything I possibly could to stop Trump so that, no matter what happened, my conscience would be clear. "In the final analysis, the strongest case for Clinton is what she is not," I wrote. "She is not racist, sexist, or xenophobic. She is not cruel, erratic, or volatile. She is not a bully or an authoritarian personality. She is not ignorant or unhinged. Those may be insufficient recommendations against a more formidable opponent. But when she's running against Donald Trump it's more than enough."
Few other Republicans agreed with me. Trump won 90 percent of GOP votes, roughly the same percentage as previous nominees. He lost the popular vote by nearly three million votes but eked out a narrow Electoral College victory by a margin of fewer than 80,000 votes in three Rust Belt states.
THE NEW YORK REAL-ESTATE SCION had figured out a way to appeal to what used to be known as the Reagan Democrats—the white working-class voters in the Midwest who had been part of the New Deal coalition but by the 1960s had become disenchanted by the party's liberal positions on issues such as national security, crime, abortion, and civil rights. Many of them had supported George Wallace in 1968, Richard Nixon in 1972, and Ronald Reagan in 1980 and 1984. Bill Clinton and Barack Obama, two of the most gifted orators in modern American politics, had won at least some of them back; Hillary Clinton—a woman whose intimidating intelligence was not leavened by a common touch—had not.
Trump had beguiled these unsophisticated voters with the same kind of appeal to nostalgia made by so many demagogues in the past. He promised to return America to an imagined paradise of the 1950s, a time when blue-collar workers had high-paying jobs and people of color were powerless and invisible to mainstream white society. This message resonated among people struggling with a devastating opioid epidemic and years of economic stagnation. Between 1980 and 2014, the top 1 percent of the country experienced 205 percent growth in personal income; the bottom 50 percent saw only 1 percent growth. And in those same years, nearly 40 percent of US factory jobs disappeared. Those statistics, as much as anything else, helped to explain why so many people were so desperate for salvation that they were willing to turn to a reality TV host as their savior. They had lost confidence in Washington because of such epic blunders as the Iraq War—yes, the war I supported—and the financial meltdown in 2008—yes, the economic crisis that my laissez-faire ideology helped to bring about. Trump gave them hope that he would blow up what they viewed as a dysfunctional political system—and that somehow something good would grow from the ruins.
I knew that shuttered steel mills and abandoned coal mines would not reopen even if Trump were elected, as subsequent events have confirmed, but I failed to grasp the extent of despair in the heartland or the ways in which my own free-market ideology and faith in globalization had contributed to this economic devastation. Those of us who support capitalism—even with welfare-state protections—tend to take for granted that it will result in the greatest prosperity for the greatest number of people. That is an easy ideology to hold when, like me, you are one of the well-educated beneficiaries of a rapidly changing economy living in a booming coastal enclave surrounded by other upwardly mobile strivers. And it may even be true in the long run but, in the meantime, the costs of "creative destruction" can be prohibitively high for those who lack the skills to benefit from the transformation wrought by the Information Revolution.
I was insensitive to those costs even though I was not nearly as insulated from ordinary life as Trump is. I do not, after all, shuttle between my many properties on a private airplane; I have only one property, an apartment in Manhattan, and I fly economy class like everyone else. Yet somehow, despite his "champagne wishes and caviar dreams" lifestyle, Trump managed to position himself as the tribune of the "forgotten men and women of our country." It was a stroke of brilliant marketing rooted in his undoubted ability to read and exploit the mood of customers, viewers, and now voters.
Thanks to his expert demagoguery, Donald Trump was president-elect, and Republicans controlled all three branches of the federal government for only the second time in the last 84 years, along with a majority of governorships and state legislatures. It was an epochal achievement—and a cause for celebration—if you were a Republican. But I no longer was. The day after the election, I reregistered as an independent after a lifetime of supporting the GOP. As I explained in the _Los Angeles Times_ : "I can no longer support a party that doesn't know what it stands for—and that in fact may stand for positions I find repugnant."
## _4._
## **THE CHAOS PRESIDENT**
A CONGENITAL OPTIMIST, I TRIED TO FIND CAUSE FOR good cheer immediately after the election—kind of like the Jewish troublemakers in Monty Python's _Life of Brian_ who sing "Always look on the bright side of life" while being crucified. On the day after the most unexpected and dispiriting election result in American history, I cited Adam Smith's words upon being told that British troops had been defeated by the American rebels at Saratoga in 1777: "There is a great deal of ruin in a nation." "That proposition is about to be put to the test by President-elect Donald Trump," I wrote in _Foreign Policy_. "Yes, I can barely believe that I am actually writing those words: 'President Trump.' I never thought he was remotely qualified for the highest office, and I never thought he would win. I was obviously wrong about the latter. Now I have to pray that I was wrong about the former. . . . I'm hoping against hope that he will grow in the White House—that the office will make the man. Because if that doesn't occur, the consequences are too ghastly to contemplate."
Okay, I wasn't _that_ optimistic. But I was so desperate for the country to avoid catastrophe that I was even willing to hold out a small olive branch to the newly triumphant Trumpites. In _USA Today_ four days later, I noted: "The temptation now for me and my fellow #NeverTrumpers is to want nothing to do with a candidate we considered unfit for office. The temptation for Trump is to want nothing to do with people who considered him unfit. For the good of the country, I hope the two sides can come together." I had no personal ambitions—I would not have worked for Trump even in the unlikely eventuality that I had been asked to do so—but I encouraged friends to take administration positions. A few of my friends would indeed assume administration jobs—some quite senior. I supported their willingness to serve, even if they felt compelled to cut off all ties with a dissenter like me lest they be accused of disloyalty by the thought police. One official—a close friend of many years—pointedly crossed my name off a guest list at a party thrown in her honor by a Washington powerbroker. I was dismayed but not surprised. As the saying, attributed to Harry Truman, has it: If you want a friend in Washington, get a dog.
Like many other policy experts, I hoped that Trump would allow himself to be guided by wise advisers. Those hopes were briefly buoyed when he appointed what I wrongly judged to be a strong cabinet, with the exception of Rex Tillerson, who seemed completely unqualified to be secretary of state. The events of January 20, 2017, and the weeks that followed, would reveal how hollow those hopes that Trump would grow in office were—they were as naïve as my expectation that Trump would lose the election.
TRUMP SIGNALED THE DIRECTION of his administration with the most dystopian, dispiriting, and divisive inaugural address ever. Not for him the inspirational tone of a John F. Kennedy, Ronald Reagan, or Barack Obama. He spoke of an America full of "rusted-out factories scattered like tombstones across the landscape of our nation," a land rife with "crime and gangs and drugs that have stolen too many lives and robbed our country of so much unrealized potential." This was a very jarring vision of America—paranoid, angry, xenophobic—for someone like me who came here in 1976 from the Soviet Union as a six-year-old boy. To me, and to countless other immigrants (including, ahem, Trump's own grandparents), America appeared to be not the hellhole he describes but a land of unimaginable wealth and opportunity. Trump, by contrast, was echoing the darkest depiction of America promulgated by a long line of Russian dictators, culminating in Vladimir Putin.
In his first days in office, Trump claimed that the "fake news media" had lied about the size of his inauguration crowds and that he would have won the popular vote were it not for "millions" of ballots cast by illegal immigrants. White House aides, called upon by the president to defend his falsehoods, had to resort to the disturbing explanation that his "alternative facts" were just as good as the actual facts. Trump then pulled out of the Trans-Pacific Partnership, a proposed free-trade zone incorporating twelve Pacific Rim nations, and signed an ill-conceived executive order barring visitors from seven Muslim countries. No one knew the details: Were existing visas canceled? Did the ban apply to US green card holders? The result: mass confusion at airports. Within hours, federal judges began intervening to block this Draconian decree, forcing the administration back to the drawing board.
Other inexperienced presidents had gotten off to a bad start—the Kennedy administration had been shaken by the failure of the Bay of Pigs invasion on April 17–20, 1961—but none had begun his term with so many self-inflicted debacles in the very first days. The impression of chaos was reinforced when Trump's national security adviser, retired lieutenant general Michael Flynn, was fired after just twenty-four days on the job—the shortest tenure on record. Flynn would subsequently plead guilty to a felony for lying to FBI agents about his dealings with Russia's ambassador. That Flynn had been appointed in the first place, after having taken money from the governments of Turkey and Russia, was an indication of the new administration's astonishing lack of ethics—and sheer incompetence. The ineptitude would be on display daily, with White House aide Kellyanne Conway referring to a nonexistent "Bowling Green massacre" and the president himself talking as if the abolitionist orator and ex-slave Frederick Douglass, who died in 1895, were still alive.
THIS TRAGEDY OF ERRORS was chalked up by Trump apologists to his inexperience as a businessman serving for the first time in government. Eventually, on July 31, 2017, White House chief of staff Reince Priebus was fired and replaced by John Kelly, who was supposed to bring greater discipline to a White House in crisis. And yet a year later Kelly was losing influence and the chaos was rising. By early 2018, Trump had gotten rid of anyone likely to stand as an impediment to his impetuosity. Secretary of State Rex Tillerson and National Security Adviser H. R. McMaster—who had counseled Trump to stay in the Iran nuclear deal—were fired. Economic adviser Gary Cohn, who had counseled against trade wars, quit when Trump imposed steel and aluminum tariffs. John Bolton, an advocate of regime change in Iran and North Korea, became Trump's third national security adviser in fifteen months, and the hardline CIA director Mike Pompeo took over as secretary of state. Trump was already on his sixth communications director, and hundreds of critical positions across the government remained empty. John Dowd stepped down as the president's lead outside lawyer, to be replaced by former New York mayor Rudolph Giuliani. Even Trump's emotional support dogsbodies, bodyguard Keith Schiller and Communications Director Hope Hicks, departed.
Trump's White House saw an astonishing 43 percent turnover among senior staff in its first year, compared to 9 percent for Barack Obama and 6 percent for George W. Bush. Trump averaged one major firing or resignation every nine days. Amid this turmoil, Trump was forced to admit that he had paid off a porn star who had alleged that they had engaged in an affair. Before long FBI agents were raiding the offices of Trump's personal lawyer, Michael Cohen, who had facilitated the payoff. He was revealed to have taken millions of dollars from companies, including one linked to a prominent Russian oligarch, that were eager to influence the administration.
In explaining why he would not join Trump's legal team, the prominent conservative attorney Ted Olson, a former solicitor general in the George W. Bush administration, explained: "I think everybody would agree: This is turmoil, it's chaos, it's confusion, it's not good for anything. We always believe that there should be an orderly process, and, of course, government is not clean or orderly ever. But this seems to be beyond normal."
Actually, this was the new normal. If Trump's administration has proven anything, it is that Jeb Bush was right when he said that Trump was "a chaos candidate," and that he would be "a chaos president." Trump's tenure, in fact, felt more like the reign of a Roman emperor than a normal American president. There were echoes of Rome's most capricious rulers: emperors like Tiberius, who moved his whole court to the isle of Capri, where he could frolic with young boys and girls ("Pans and nymphs") and pronounce death sentences by the score while he left the commander of his Praetorian Guard, Lucius Sejanus—a forerunner of generals such as John Kelly, H. R. McMaster, and James Mattis—to run the government in his absence. Or Tiberius's infamous successor, Caligula, who may have been clinically insane: he proclaimed himself a god, was famously said to have made his horse a consul, and, when he ran out of gladiators to be slaughtered at the Coliseum, had his guards throw an entire section of spectators into the arena to be eaten by wild beasts. (This was the ancient version of the kind of professional wrestling match in which Trump participated before his election.) Or Nero, who did not actually fiddle while Rome burned in AD 64 but did impale and burn to death scores of Christians to propitiate the people's anger over the deadly conflagration. This was a precursor of the kind of scapegoating of minorities that Trump specializes in.
Trump's recreational pursuits were tamer than the Romans'—he cavorted with strippers and playmates, not underage "Pans and nymphs," and instead of attending gladiator matches he played golf, _lots and lots_ of golf. Indeed, if he keeps playing golf at the current pace, he will spend nearly one-fourth of a four-year term at the golf course. And, much to his regret no doubt, he could not simply impale his enemies as the emperors had done. But his sloth and ineptitude were nearly as great. Trump was unable to function effectively because he was functionally illiterate: He could read in theory but chose not to, getting his information from television instead, Fox News to be exact. A former aide said that "everything that needs to be conveyed to the President must be boiled down to . . . 'two or three points, with the syntactical complexity of 'See Jane run.'" Trump routinely made decisions based on his "gut" rather than the kind of study and staff work that had characterized previous presidents. As Susan Glasser noted in the _New Yorker_ : "Many of this President's major decisions—from appointing Cabinet secretaries to pulling out of the Iran nuclear deal—are completely opaque and, in many cases, shockingly process-free." The result was frequent fiascos such as the aborted nomination of the White House physician, Rear Admiral Ronny Jackson, to be secretary of veterans' affairs, his chief qualification having been his effusive praise of Trump's health. Or, far worse, the mishandling of Hurricane Maria in Puerto Rico, which resulted in at least 1,400 deaths—and possibly as many as 4,000.
Most Americans recoiled from the "carnage," just as most Romans recoiled at the excesses of their emperors. (Hence the fact that many more emperors were killed or overthrown—voting an emperor out of office not being an option—than died on the throne of natural causes.) Despite robust economic growth, the public gave Trump record-low approval ratings that averaged around 40 percent—higher than warranted but lower than just about any previous president at a similar point in his tenure. By early 2018, after a slight uptick in his ratings, Trump would be in Jimmy Carter territory. And yet between 80 percent and 90 percent of Republicans stuck with him, applauding enthusiastically when the rest of the country was lustily booing or simply looking away in disgust.
WHAT WAS IT THAT these Republicans saw that the rest of us did not? The case for Trump was based on:
* A strong economy—even though Trump inherited, and did not create, the robust economic conditions, including low unemployment, low interest rates, and a roaring bull market. In fact, Trump's policies jeopardized the economy. His tariffs and attacks on companies such as Amazon spooked the markets, making stocks as volatile as the president himself. The Dow average had risen 37 percent between Obama's inauguration and April 2010; between Trump's inauguration and April 2018, only 19 percent. Likewise average monthly job creation since Trump's inauguration—186,200 new jobs a month—was lower than under Obama, who over the previous four years had averaged 215,875 new jobs a month. Trump deserved some credit for presiding over the expansion but not nearly as much as Obama for presiding over the recovery from the Great Recession of 2007–2009. Indeed, a researcher at the Brookings Institution found that, measured against the five other presidents who inherited a growing economy since 1960, Trump's record, far from being exemplary, is tied for last place, lagging behind even Jimmy Carter's.
* Trump's "defeat" of ISIS—even though that organization is far from eradicated and even though the game plan for combating it was largely devised and implemented by Obama. By early 2018, moreover, Trump was threatening to pull the remaining 2,000 US troops out of Syria, thereby jeopardizing all of the gains against ISIS and opening eastern Syria to Iranian expansion.
* Trump's pullout from treaties abominated by the right—the Paris Climate Accord and the Iran nuclear deal. His supporters refused to admit that global warming is real and that the Paris accord did not mandate job-killing regulations; compliance was entirely voluntary. As for the Iran nuclear deal, I had opposed it myself, but I did not think it made sense to pull out when Iran was complying with its terms, as even Trump's own secretary of state admitted. Trump had no Plan B for containing Iranian power, beyond imposing unilateral sanctions and hoping for the best.
* Trump's move of the US embassy in Israel from Tel Aviv to Jerusalem, fulfilling a pledge that other presidents had made but failed to implement. But while the location of the US embassy had obvious symbolic importance, it was of scant strategic significance. Far more important to Israel's security was the growing Iranian domination of Syria, which Trump showed little interest in combating, forcing Israel to launch an increasing number of air strikes on Iranian positions next door.
* Trump's summit with Kim Jong Un. Trump did not succeed in winning any significant concessions beyond empty promises to "work towards the complete denuclearization of the Korean peninsula"—the same kind of assurances North Korean leaders had been making, and breaking, since 1992. In return, Trump legitimized North Korea on the world stage, effectively undermining the international sanctions regime. Trump showered sickening praise on one of the world's worst human-rights violators—in his telling, Kim has "got a great personality," "he's a funny guy, he's very smart, he's a great negotiator," and "he loves his people." Trump even agreed to cancel US–South Korea joint military exercises, which he called, in an echo of Pyongyang's propaganda, "provocative . . . war games." Conservatives had criticized President Obama for far less after he shook hands with Cuban dictator Raul Castro and signed a nuclear deal with Iran that Trump said was the "worst deal ever." Trump's deal with North Korea was far worse than the Iran nuclear deal. And yet conservatives showered Trump with praise after he returned from Singapore and proclaimed, "There is no longer a Nuclear Threat from North Korea"—his version of Neville Chamberlain's 1938 boast that the Munich Conference would result in "peace for our time."
* The passage, near the end of Trump's first year in office, of a massive tax cut bill, even though, as we shall see, the cost of that legislation could prove to be prohibitive and its impact on the economy negligible. This was Trump's major and virtually sole legislative achievement—and it was the product of congressional negotiations into which Trump had little input. While Trump's tax-and-spend policies helped further stimulate the economy, his former economic adviser Gary Cohn worried that their impact could be wiped out by the trade wars Trump started.
* A host of regulations that Trump ordered repealed—even though the extent and impact of that deregulatory push was wildly exaggerated by the White House. Trump claims that "in the history of our country, no president, during their entire term, has cut more regulations than we've cut," but Ronald Reagan achieved far more deregulation. _Bloomberg BusinessWeek_ found that Trump was using typical sleight of hand to obscure the fact that "hundreds of the pending regulations had been effectively shelved before Trump took office. Others listed as withdrawn are actually still being developed by federal agencies. Still more were moot because the actions sought in a pending rule were already in effect." Goldman Sachs researchers determined that the actual scale of the Trump deregulation was so limited that it had no appreciable impact on the economy.
* Trump's selection of conservative judges Neil Gorsuch and Brett Kavanaugh to the Supreme Court and a host of lesser-known conservatives to the lower courts—even though in all other ways Trump would attack the rule of law that judges are supposed to uphold. For much of Trump's first year in office, any criticism of the president would be greeted by a predictable refrain: "But, Gorsuch . . ."
* Trump wasn't Hillary Clinton. From Clinton's attempts as First Lady to reform health care to the murder of the US ambassador in Libya while she was secretary of state, conservatives had done an effective job of turning this centrist and capable, if uncharismatic, politician into a caricature of a far-left, anti-American "feminazi." The coup de grace was Trump's nickname—"Crooked Hillary"—bestowed because she had violated State Department regulations, although apparently not the law, by using a private server for some of her official emails. (As president, Trump would use a private cell phone for sensitive conversations that represented a much more significant potential security breach.) If you are convinced that there is no greater evil that can be visited upon America than a Clinton presidency, then you are prepared to see a Trump presidency as your salvation. I cannot count how many emails and tweets I have gotten from Trump fans who respond to my criticisms of the president with: "But, Hillary . . ."
* Last, and perhaps most important, Trump "triggered" liberals. The very fact that he sparked so much opprobrium was taken by conservatives as a compelling argument—probably, in fact, the _most_ compelling argument—in his favor. Sure, conservatives would concede, sometimes Trump goes too far or says something he should not, but "at least he fights." For much of the right, there is no higher end in politics than to annoy liberal "elites"—or now also conservative #NeverTrump elites.
That's it: the conservative case for Trump.
For progressives, of course, few of these arguments, save Trump's outreach to Pyongyang, are remotely persuasive. They are not in favor of deregulation or tax cuts because they are acutely conscious of the costs to society, and of course they were not petrified (if also not enthusiastic) about the prospect of another Clinton presidency. Trump's achievements, such as they are, are harder to dismiss for conservatives. Even those who are critical of Trump have to acknowledge that his presidency hasn't been all bad—there have been a few good things achieved. For example, I favored the move of the US embassy to Jerusalem, the intensification of efforts to defeat ISIS, the imposition of harsh sanctions on North Korea, and the cut in the corporate tax rate to bring the US tax code into conformity with other major industrialized countries (although I thought it should have been done in a revenue-neutral way). If Trump can convince North Korea to carry out its pledges of denuclearization, that will obviously be a great thing—if also nearly impossible to verify.
But the presidency is not an á la carte buffet. It is a prix-fixe set menu. You cannot pick and choose what you want. You have to digest everything, good and bad. And in the course of achieving these minuscule policy victories, conservatives would have to swallow enough rancid garbage to give them a severe case of indigestion—even if most of them would never admit it.
## _5._
## **THE COST OF CAPITULATION**
AS A HISTORIAN, I AM PARTIAL TO PARALLELS FROM the past, but it is nearly impossible to find equivalent events, at least in US history, to the rise of Donald Trump. He resembles demagogues such as Huey Long, Joseph McCarthy, and George Wallace, who made skillful use of the mass media, first radio and then television, to prey on the fears of their constituents and vilify minorities—whether "the rich," Communists, or African Americans—in order to accrue power. The major difference, of course, is that none of those men became president. Trump did. No historical comparison is ever exact, but for anyone wondering what the Kingfish, Tailgunner Joe, or the Fighting Judge would have done in office, Trump provides some hints.
The parallels between Wallace—the segregationist governor of Alabama who repeatedly sought the presidency, achieving his greatest success as a third-party candidate in 1968—and Trump are particularly striking. Wallace might as well have been speaking for Trump when he said, "Hell, we got too much dignity in government now, what we need is some _meanness_." So, too, other descriptions of Wallace from his biographer Marshall Frady have an uncanny resonance.
Frady wrote: "It has become Wallace's conviction—more than conviction, visceral sensation—that he exists as the very incarnation of the 'folks,' the embodiment of the will and sensibilities and discontents of the people in the roadside diners and all-night chili cafes, the cabdrivers and waitresses and plant workers, as well as a certain harried Prufrock population of dingy-collared department-store clerks and insurance salesmen and neighborhood grocers: the great silent American Folk which have never been politically numbered as the Wallace candidacy has now numbered them."
Also: "In fact, he seems to regard formal political organization with a vague contempt, as a sign of political effeteness, an absence of vitality—as if he is already naturally blessed with what political organization exists to create. His simple directness is, at once, part of his absurdity and part of his genius."
And: "He seems empty of any private philosophy or persuasions reached in solitude and stillness. He is made up, in mind and sensibilities, of the clatter and chatter and gusting impulses of the marketplace, the town square, the barbershop. His morality is the morality of the majority. 'The majority of the folks aren't gonna want to do anything that ain't right,' he insists. He is the ultimate product of the democratic system."
Finally: "Not only are abstract ethics alien to him, but he entertains a particular antipathy to people who live and act from them. It's something like the Dionysian principle applied to politics. 'Hell, intellectuals, when they've gotten into power, have made some of the bloodiest tyrants man has ever seen,' he maintains. 'These here liberals and intellectual morons, they don't believe in nothing but themselves and their theories. They don't have any faith in people. Lot of 'em don't really _like_ people, when you get right down to it.'"
The major difference is that Wallace reflected the style of white, rural Alabama, Trump of white, urban Forest Hills, Queens—but the similarities are striking, especially their anti-elite and anti-intellectual appeals, their contempt for the normal way of running campaigns, their unwillingness to abide by ethical norms, their championing of working-class whites, and their demonization of minorities. Both men were even alike in their disdain for alcohol.
Trump has not imposed fascism, as many, including me, had feared, but then neither did Wallace, even if he did uphold American apartheid as long as he could. It's hard to wreck a democracy in a short period, even in a political system much less robust than America's. Even for Vladimir Putin the task of turning Russia from a rickety democracy to a full-blown dictatorship took years after he was first appointed president in 1999. The courts, press, oligarchs, regional governors, opposition leaders, and other checks on his authority had to be painstakingly removed, and sometimes killed, one by one. A cult of personality had to be constructed, featuring the Russian dictator engaged in manly pursuits such as bare-chested horseback riding. Recep Tayyip Erdogan in Turkey and Abdel Fattah el-Sisi in Egypt have followed Putin's example in snuffing out the remnants of democracy in their countries. A similar process, not yet as advanced, has taken place in countries such as Hungary, the Philippines, and Poland, where far-right populists have won power in recent years.
If Trump were operating in a republic with a less-lengthy constitutional tradition or weaker institutional safeguards, he might well be another Vladimir Putin or a Benito Mussolini, Juan Peron, or Hugo Chavez. He would not be another Hitler: The Nazi leader was uniquely evil, and it does a disservice to the victims of Nazism to suggest that Trump is the second coming of the Führer, even if there are some disturbing parallels. (Likewise, it unfairly glorifies his critics to suggest that we are "the Resistance," as if we were the French Maquis risking torture and death to fight the Nazi-backed Vichy regime.) Trump is more of a garden-variety strongman, and if he were ruling in Italy in the 1920s, Argentina in the 1950s, or Russia or Venezuela in the 2000s, he would undoubtedly be a dictator by now.
America's robust checks and balances—a free press, an independent judiciary, and an apolitical civil service, in particular—keep Trump from fully acting on his authoritarian impulses. He can't even go as far as his fellow populists Viktor Orban in Hungary or Rodrigo Duterte in the Philippines, or the Law and Justice Party in Poland. But Trump has cozied up to dictators and sought to redefine America's role in the world, picked fights with US allies, started trade wars, demonized the free press (and sought to financially punish the _Washington Post_ 's owner), spewed nonstop lies, and whipped up hatred of minorities such as rich African American athletes and poor Mexican immigrants—all the while waging unrelenting war on the rule of law in order to save himself from criminal investigation. And, just like those earlier demagogues Huey Long, Joe McCarthy, and George Wallace, Trump has won plaudits from a substantial portion of the American electorate while assaulting the highest ideals of America because he has claimed to be protecting the country from insidious threats—disloyal elites, criminals of color, immigrants, terrorists, perfidious trade partners and allies—that threaten America's "greatness."
The Founding Fathers anticipated the rise of demagogues. _Federalist 10_ warns: "Men of factious tempers, of local prejudices, or of sinister designs may, by intrigue, by corruption, or by other means, first obtain the suffrages, and then betray the interests, of the people." But the Founders could not have foreseen the ability of this particular demagogue to directly reach millions of people via Twitter, Facebook, or Fox News Channel. Nor could they have anticipated the spread of "fake news" that allows Trump's followers to live in a world of "alternative facts" at odds with reality. We are in uncharted territory—the kind of unexplored land that, on at least one sixteenth-century globe, was marked "Here be dragons."
It's impossible to know where we are headed: How can you predict what a president who is capable of saying or doing just about anything will do next? Trump is so unpredictable that sometimes he even does sensible things, much as his critics may hate to admit it. His very volatility is the source of his power; if he were ever to become boring and predictable like other politicians, he would be finished. But even if we cannot know where he is going next, we can at least chart the surrealistic journey up to this point.
What follows is a brief and incomplete list of Trump's transgressions against common decency and good sense—and quite possibly the law itself. Some—perhaps much—of what follows may be familiar to you, but there has been so much craziness emanating from the administration that it's hard for even the most dedicated consumer of news to keep track of it all. It is, therefore, imperative to briefly summarize just what Trump has done that is so wrong. This is the grim reality that should offend any person with a shred of decency. This is what the president's acolytes either ignore or excuse. This is why I am so disgusted with Trump—and his toadies.
**I. R ACISM**
Any recounting of the toll of the Trump presidency must begin with the damage that the president is doing to race relations in America. This is the most vexing, emotional, and important issue in American history, and it is one on which conservatives, as we shall see in the next chapter, have had a checkered record. But the Republican Party is, or perhaps more accurately _was_ , the party of Lincoln—the party that freed the slaves. That commitment to civil rights lasted at least into the 1960s. A higher percentage of House and Senate Republicans supported the 1964 Civil Rights Act and the 1965 Voting Rights Act than did Democrats. In more recent years Republicans had justified their opposition to racial "quotas" and "set-asides" by claiming that the right response to the racial discrimination of the past was not a new form of discrimination but, rather, strict adherence to the creed of color blindness. We liked to quote the immortal words of the Rev. Dr. Martin Luther King Jr.: "I have a dream that my four little children will one day live in a nation where they will not be judged by the color of their skin but by the content of their character."
There is no reason to assume that Donald Trump shares this dream—and much reason to suspect that he does not. His entire career has been full of racist slurs and acts. His presidency has been no different. After neo-Nazis clashed with counterdemonstrators in Charlottesville, Virginia, on the weekend of August 11–12, 2017, Trump claimed there were "very fine people on both sides." He went on to side with the neo-Nazis in opposing the removal of Confederate monuments. "Many of those people were there to protest the taking down of the statue of Robert E. Lee," the president said. "So this week, it is Robert E. Lee. I noticed that Stonewall Jackson is coming down. I wonder, is it George Washington next week? And is it Thomas Jefferson the week after? You know, you really do have to ask yourself, where does it stop?" This is moral sophistry of a high order. Washington and Jefferson were indeed slave owners. But they also created a system of government that, while stained by the original sin of slavery, nevertheless established certain "unalienable rights" that would eventually be vindicated after the struggles of the Civil War, Reconstruction, and the civil rights movement of the 1950s–1960s. By contrast, what is it that we are supposed to be grateful to the Confederates for? For triggering the bloodiest conflict in American history? For fighting to keep their fellow citizens in bondage? The fact that Trump cannot make these basic distinctions is indicative of his deeply prejudiced worldview.
Yet more evidence of Trump's racism came just a few weeks after Charlottesville, when he pardoned Joe Arpaio, the sheriff of Maricopa County, Arizona, who had been found guilty of criminal contempt of court for ignoring a federal judge's order not to arrest Latinos solely because he suspected them of being in the country illegally.
The trend continued in the fall of 2017, when Trump launched a crusade against African American NFL players who kneeled during the playing of the national anthem to protest police brutality. At a rally on September 22, Trump said, "Wouldn't you love to see one of our NFL owners when someone disrespects our flag to say, 'Get that son of a bitch off the field right now . . . he's fired!'" Trump kept going in a similar vein as the NFL season unfolded, and even after—for example, saying that any player who didn't kneel during the anthem "shouldn't be in this country." This was exactly the kind of impingement on free speech that conservatives routinely complained about when it was committed by college leftists—but they applauded Trump's attempts to stifle peaceful protests. Finally, in the spring of 2018, the NFL owners caved in and mandated fines for any players who kneel during the anthem. "The issue of kneeling has nothing to do with race," the president insisted. "It is about respect for our Country, Flag and National Anthem. NFL must respect this!" But there was no doubt that the racist undertones of his attacks on rich, privileged African American athletes resonated with his white, working-class base.
A _Politico_ reporter who journeyed to Johnstown, Pennsylvania—a onetime coal-mining town that had long since been hollowed out—found that Trump supporters were ecstatic about his vilification of the NFL, and for all the wrong reasons. One Trump voter, the owner of a catering company, griped: "Shame on them. These clowns are out there, making millions of dollars a year, and they're using some stupid excuse that they want equality—so I'll kneel against the flag and the national anthem?" "You're not a fan of equality?" reporter Michael Kruse asked. "For people who deserve it and earn it," he replied. "All my ancestors, Italian, 100 percent Italian, the Irish, Germans, Polish, whatever—they all came over here, settled in places like this, they worked hard and they earned the respect. They earned the success that they got. Some people don't want to do that. They just want it handed to them." So only European immigrants work hard? Professional athletes, who work with superhuman stamina to hone their skills, just want everything "handed to them"? Another Trump supporter—a retired meat packer whose son had died of a heroin overdose—was even more explicit. Do you know what NFL stands for? she asked. "Niggers for life."
No, not all Trump supporters are racist. But virtually all racists, it seems, are Trump supporters. And all Trump supporters implicitly condone his blatant prejudice. At the very least they don't consider racism to be a reason to turn against the president. For a disturbingly large number of Trump voters, it is the primary reason to support him. A 2018 study published by the National Academy of Sciences concluded that support for Trump was not primarily an economic phenomenon—the lowest-income voters actually supported Clinton—but, rather, motivated mainly by the "status threat felt by the dwindling proportion of traditionally high-status Americans (i.e., whites, Christians, and men)." In other words, the study found, it was white anxiety about the looming demographic reality that white people will soon be a minority in America that, more than anything else, drove voters to back a candidate who "emphasized reestablishing status hierarchies of the past."
**II. N ATIVISM**
Donald Trump wasn't the first president to promise to "make America great again." Ronald Reagan used that very phrase in a speech on Labor Day, 1980, delivered at Liberty State Park in Jersey City with the Statue of Liberty as a backdrop. The difference between the two Republican presidents is that Reagan used much of this speech not to bash immigrants but to hail their contributions to America. "Through this 'Golden Door,' under the gaze of that 'Mother of Exiles,' have come millions of men and women, who first stepped foot on American soil right there, on Ellis Island, so close to the Statue of Liberty . . . ," Reagan said. "They came to make America work. They didn't ask what this country could do for them but what they could do to make this refuge the greatest home of freedom in history." Not only did Reagan celebrate immigrants rhetorically, he also signed legislation in 1986 that legalized three million undocumented immigrants, and he set in motion the negotiations that produced the North American Free Trade Agreement. His dream—never realized—was to allow free travel between the United States, Canada, and Mexico. He wanted to open opportunities, not build walls.
Donald Trump has a very different vision. Far from praising the contributions of immigrants, he regularly cites a poem called "The Snake" about a talking reptile that fatally bites a woman who has taken it in and nurtured it with "milk and honey"; in Trump's telling, the real "snakes" are immigrants. Trump says of the illegal immigrants he is deporting: "These aren't people, these are animals." He claims that he is only speaking of MS-13 gang members, but he is using his denunciations of these criminals to signal his repugnance of all illegal immigrants, indeed of _all_ immigrants aside, presumably, from European supermodels. This is exactly the kind of dehumanizing language—labeling minority groups as scum, vermin, subhumans, rats, cockroaches, lice, etc.—employed by dictators and ethnic-cleansers in countries from Nazi Germany to Rwanda. Trump even complains about immigrants in sanctuary cities "breeding" as if they were, yes, animals.
The Department of Homeland Security created, at Trump's instigation, a Victim of Immigration Crimes Enforcement office to focus on the supposed menace of immigrants, although immigrants, even illegal immigrants, are much less likely to commit crimes than are the native born. But Trump doesn't care about the facts. In early 2018, he told congressional leaders that he prefers immigrants from white countries such as Norway to those from "shithole countries" in Africa and the Caribbean. In point of fact, a study of African immigrants in America found that they "attain higher levels of education than the overall U.S. population as a whole." It is hard to escape the suspicion that what Trump doesn't like about African immigrants is the color of their skin, not the content of their character.
Trump has displayed similar animus against Muslims. He has not only tried to bar newcomers from Muslim nations but has also been quick to label all attacks by Muslims as "terrorism" committed by "animals," while staying quiet about hate crimes against Muslims and labeling mass shootings by non-Muslims as evidence of a "mental health" problem. He even retweeted hateful and false anti-Muslim videos posted by a far-right British leader.
In her landmark book _The Second Sex_ , the French philosopher Simone de Beauvoir noted how essential it was for all social groups to differentiate themselves from "the Other": "In small-town eyes all persons not belonging to the village are 'strangers' and suspect; to the native of a country all who inhabit other countries are 'foreigners'; Jews are 'different' for the anti-Semite, Negroes are 'inferior' for American racists, aborigines are 'natives' for colonists, proletarians are the 'lower class' for the privileged." Stigmatizing "the Other" is especially important for authoritarian rulers. For Trump, Mexicans and Muslims have become "the Other" that rally his base behind his leadership despite all of the scandals during his presidency.
The overall number of arrests by the Immigrations and Customs Service increased by 25 percent in Trump's first year in office, and arrests of immigrants without criminal records soared by 164 percent. There are all too many people like Jorge Garcia, a landscaper with a wife and two kids who lived in Detroit for thirty years before being deported to Mexico. Trump even mandated that parents be separated from their children at the border—an unspeakably cruel policy designed to discourage illegal immigration. In the first six weeks alone, some two thousand children were taken away from their parents—sometimes literally ripped out of their arms. Still more deportations may be in the offing because in the fall of 2017 Trump withdrew legal protection from "Dreamers," more than seven hundred thousand immigrants who had been brought illegally to America as young children. Trump claimed to support a deal in Congress to legalize the Dreamers, yet set impossible conditions for Democrats and sabotaged the prospect of compromise at every turn. On Easter Sunday 2018, he tweeted, "NO MORE DACA DEAL!" (DACA stands for Deferred Action for Childhood Arrivals.) Court orders have maintained legal protections for Dreamers, but for how much longer?
The end of DACA hit me particularly hard because almost half of those affected arrived in the United States before their sixth birthday. In other words, they were about the same age I was when I came here. It made me wonder: What would I do now, at nearly fifty years of age, if I were deported to a country that I have not seen in more than forty years and whose language I no longer speak? How would I survive? In my case it would be a particularly pressing problem, given how critical I have been of Russia's current president. (Putin's propaganda outlet, RT, has attacked me by name.) The risk of political persecution would be all too real for me—as it is for Dreamers who might be deported to repressive countries. And what would happen to my family—to my partner, to my children, to my stepchildren? None of them is Russian. A move would be even more jarring for them than for me.
Trump's war on immigrants is making me feel like I am no longer a "real" American. Increasingly I feel like a Jew, an immigrant, a Russian—anything but a normal, mainstream American. That may be precisely what Trump and his most fervent supporters intend. They are redefining what it means to be an American. The old idea that anyone who embraces America's ideals can become an American is out. White House aide Stephen Miller even repudiated the words on the Statue of Liberty that Ronald Reagan celebrated in 1980: "Give me your tired, your poor,/Your huddled masses yearning to breathe free." Instead, American-ness is being redefined in blood-and-soil terms. I find myself increasingly forced to think of my ethnic identity instead of the national identity I adopted as a boy in 1976. That is discomfiting for me and a tragedy for America. Yet most Republicans either excuse or—more frightening—applaud Trump's blatant xenophobia.
**III. C OLLUSION**
We know two things for a fact about the 2016 election. First, Donald Trump won a narrow victory in the Electoral College while losing the popular vote. Second, we know, in the words of the US intelligence community, that "Russian President Vladimir Putin ordered an influence campaign in 2016 aimed at the US presidential election," and that "Putin and the Russian Government aspired to help President-elect Trump's election chances." This "high confidence" intelligence community estimate was subsequently endorsed by a federal grand jury that, at the request of Special Counsel Robert S. Mueller III, indicted twenty-five Russians involved in this influence operation. Trump's former national security adviser, Lieutenant General H. R. McMaster, said that the evidence of Russian tampering was "inconvertible," and his successor, John Bolton, described it as an "act of war." Even the Senate Intelligence Committee's Republican majority admitted that the Russians tried to help Trump.
It strains credulity to claim, as Trump supporters do, that there was no relation between the Kremlin's intervention and Trump's victory. If the Russian operation had no impact on the results, why did Trump mention WikiLeaks—used as a conduit for Democratic Party documents stolen by Russian hackers—141 times during the last month of the campaign? It's true that the Russian spending was only a pittance compared to the $2.4 billion spent on the 2016 presidential campaign, but the Kremlin propaganda blitz reached 126 million Americans via Facebook alone. By all indications, the Russian operation was the most successful foreign attack on America since 9/11. It did not kill anyone, but it did undermine faith in American democracy. What we still do not know is whether there was conscious collusion between the Trump campaign and the Kremlin, but the weight of circumstantial evidence points in that direction.
According to the Moscow Project of the Center for American Progress: "In total, we have learned of at least 80 contacts between Trump's team and Russia linked operatives, including at least 23 meetings. . . . None of these contacts were ever reported to the proper authorities. Instead, _the Trump team tried to cover up every single one of them_." The evidence of collusion grows stronger as more information emerges about these contacts.
Former Trump foreign policy adviser George Papadopoulos learned well in advance of their public release that the Russians had "thousands of emails" with "dirt" on Hillary Clinton; he relayed to the campaign offers of Russian "cooperation." The entire high command of the Trump campaign, including Paul Manafort (himself linked financially to the Russian oligarchy), Donald Trump Jr., and Jared Kushner, met at Trump Tower on June 9, 2016, with a lawyer from Moscow, closely connected to the Russian government, who promised to "incriminate" Clinton. Trump Jr. communicated with WikiLeaks, and Trump adviser Roger Stone with both WikiLeaks and Guccifer 2.0, a Russian intelligence officer, while Manafort and his deputy, Rick Gates, were in active contact with a former—and possibly current—Russian intelligence officer. After Trump won, his national security adviser, Michael Flynn, held secret conversations with Russian ambassador Sergey Kislyak. An even closer Trump adviser—his son-in-law Jared Kushner—also met after the election with Kislyak and expressed interest in setting up a private back channel to Moscow via the Russian embassy in Washington to bypass the US government.
It is hard to imagine an innocent explanation for all of the Trump-Kremlin contacts—or for all the lying that Trump officials have done about them. It is harder still to imagine that Trump was unaware of what his team was up to. He was, after all, personally involved in putting out a false statement that the June 9 meeting at Trump Tower was about "adoptions." It will not, however, be easy to prove the president's personal complicity.
Trump defenders try to defend him against charges of collusion by citing all of the actions that he has supposedly taken against Russia. "Probably no one has been tougher to Russia than Donald Trump," Donald Trump says, referring to himself in the third-person style favored by monarchs and dictators. In truth his record on Russia has been hard and soft—in a word, incoherent.
Trump can point to the fact that he has authorized the sale of lethal weaponry to Ukraine, expanded sanctions on Russia, launched two pinprick strikes on the Syrian regime allied with Russia, and expelled sixty Russian diplomats in retaliation for the attempted Russian murder of a former Russian agent and his daughter in Britain. But much of this was simply for show: even though the Russians could replace the sixty expelled diplomats, Trump was said to have been furious when he found out that his aides had maneuvered him into so many expulsions. When Putin ordered the elimination of 755 US diplomatic positions in Russia, Trump praised him. When the Russian autocrat won a rigged reelection victory, Trump called to congratulate him, disregarding a briefing paper from his own staff warning him "DO NOT CONGRATULATE." Trump even announced in 2018 that he wanted to invite Russia back to the Group of Seven meetings from which it had been expelled four years earlier after its invasion of Ukraine. The following month, the president was shockingly supine when confronted with Putin's lies at a Helsinki summit, choosing to take the word of the Russian despot over the findings of the US intelligence community.
Just before retiring in April 2018, Trump's national security adviser H. R. McMaster admitted: "We have failed to impose sufficient costs" on Russia. Although Trump hasn't made US policy as pro-Russia as Putin might have hoped, largely because of the Russiagate scandal, his chaotic governance style, hostility to US allies, and aversion to American global leadership have allowed Russia to keep expanding its power from Ukraine to Syria and beyond. Putin could barely conceal his glee in 2018 when Trump launched trade wars with America's NATO allies and called the European Union a "foe."
Imagine what Republican lawmakers would have said if the president accused of colluding with the Kremlin was Hillary Clinton. Charges of "treason" would fill the air. Indeed, such charges are common today, with Republicans peddling specious allegations that Clinton sold American uranium to Russia or somehow conspired with the Kremlin to concoct charges that Trump colluded with the Russians. But rather than thinking worse of Trump as evidence of collusion accumulates, Republicans are starting to think better of Putin because of the coziness between the two men. Gallup found that favorable views of Putin among Republicans jumped twenty points between 2015 and 2017. Admittedly, even in 2017, only 32 percent of Republicans had a positive view of the Russian despot, but that's more than three times higher than among Democrats. And a Pew poll found that only 38 percent of Republicans viewed "Russia's power and influence" as a major threat to the United States compared to 63 percent of Democrats. This is a disturbing indication of how GOP voters are willing to follow Trump wherever he leads—even into the arms of the Kremlin.
**IV. T HE RULE OF LAW**
It is ironic that so many conservatives premise their support for Trump on his willingness to appoint conservative judges because Trump has mounted such a wide-ranging and unprecedented assault on the rule of law. It's not just that he has savaged judges that have ruled against him in cases such as his attempts to limit Muslim immigration, leading his own Supreme Court appointee Neil Gorsuch to denounce his "disheartening" and "demoralizing" attacks on the judiciary. And it's not just that he has abused his pardon power to bypass Justice Department procedures and grant clemency to unrepentant felons such as Dinesh D'Souza and Joe Arpaio because they support him politically—thereby signaling to his own aides who are under criminal investigation that they should not cooperate with the FBI. Even worse, Trump has sought to politicize law enforcement in a way that no president has done in half a century. After the Watergate scandal, rules and regulations were established by the Justice Department to prevent presidential tampering with law enforcement of the kind that Richard Nixon and other presidents had carried out for partisan advantage. Trump shows no awareness that any such restrictions even exist.
The president tried to pressure then-FBI director James Comey into pledging him personal loyalty and going easy in his investigation of Mike Flynn. When Comey refused, he was fired on May 9, 2017, ostensibly for mishandling the Hillary Clinton email investigation. But that cover story swiftly crumbled. On May 11, Trump admitted to Lester Holt of NBC News that he was determined to fire Comey "regardless" of the recommendation from Deputy Attorney General Rod Rosenstein in order to stop the investigation of the "Russia thing." Trump isn't the first president to attempt to obstruct justice. But he is the first to admit what he was doing on national TV, and his admissions led to the appointment of former FBI director Robert S. Mueller as a special counsel.
Trump was only narrowly dissuaded from firing Mueller. Instead he first tried to get Attorney General Jefferson Sessions III to reverse his recusal, which had cleared the way for Mueller's appointment, and then, when Sessions quite properly refused, publicly castigated him for being "very weak," "disgraceful," and an "idiot." This was a spectacle without precedent in US history: a president harshly condemning his own attorney general for refusing to politicize law enforcement. When Sessions would not either quit or accede to Trump's unethical demands, the president vented his "fire and fury" on the FBI, forcing out Comey's deputies Stewart Baker and Andrew McCabe. Trump tweeted: "After years of Comey, with the phony and dishonest Clinton investigation (and more), running the FBI, its reputation is in Tatters—worst in History! But fear not, we will bring it back to greatness." Later he referred to the FBI as a "den of thieves." Trump showed no awareness that the FBI answered to him or that his assault on its reputation could hinder its job of protecting the American public from criminals, spies, and terrorists. When Comey hit back at Trump in a best-selling memoir, Trump called the former FBI director an "untruthful slime ball," "a proven LEAKER & LIAR," and "the WORST FBI Director in history, by far!" and called for him to be prosecuted.
Trump then moved on to attacking the special counsel by name—something that his surrogates at Fox and in Congress had been doing all along. "Why does the Mueller team have 13 hardened Democrats, some big Crooked Hillary supporters, and Zero Republicans?" Trump tweeted. "Another Dem recently added . . . does anyone think this is fair? And yet, there is NO COLLUSION!" In point of fact, Mueller was a Republican just like Trump. Both men also were born into wealth. But there the resemblance ended. Mueller had devoted his life to public service, from volunteering to fight in Vietnam in the 1960s, where he was wounded and earned numerous commendations, to serving as a line prosecutor handling homicide cases in the District of Columbia in the 1990s when he could have been collecting a fat paycheck from a big law firm. Trump, by contrast, had never served any cause greater than his own id.
Mueller is the best of America; Trump the worst. All you need to know about the diseased state of today's Republican Party is that it reviles Mueller and reveres Trump.
SOME REPUBLICANS WARNED Trump not to fire Mueller, but the leaders of the House and Senate refused to move legislation that could have protected the special counsel by adding judicial oversight to any decision to remove him. Republicans not only showed themselves to be unwilling to defend Mueller—some of them also actively joined in Trump's attempts to obstruct his investigation. The foremost culprit was Representative Devin Nunes of California, chairman of the House Intelligence Committee. I had met Nunes before the rise of Trump and had thought him to be thoughtful and reasonable. How misleading first impressions can be! In thrall to Trump, Nunes revealed himself to be an unscrupulous partisan who did not hesitate to misuse his authority to protect the president at all costs.
First Nunes claimed that Obama national security adviser Susan E. Rice had illegally "unmasked" Trump aides in surveillance transcripts. Trump's own national security adviser, H. R. McMaster, concluded that Rice "did nothing wrong." Nunes next directed his staff to prepare a memorandum alleging that the FBI had obtained a surveillance warrant for Trump campaign adviser Carter Page based on former British spy Christopher Steele's work, while hiding Steele's partisan funding. This was false. The Steele dossier was only one piece of evidence among many, and the Justice Department did reveal that Steele was paid by an anti-Trump political entity.
Nunes's failure to make the case did not, of course, lead either him or the White House to retract their scurrilous allegations of a Deep State plot against Trump. Republicans tried embarrassing the FBI with outtakes from the private texts between FBI agent Peter Strzok and his girlfriend, FBI attorney Lisa Page, both of whom had already been removed from the Mueller probe. Trump said their texts were "BOMBSHELLS!" and evidence of "treason"—the kind of accusation of disloyalty that dictators routinely make against their critics. While the Strzok and Page texts critical of Trump and other political figures were embarrassing, the Justice Department's inspector general found that there was no evidence that their personal views influenced any investigation.
When the texts didn't pan out, Nunes opened a new front by effectively outing an informant that the FBI had used to investigate whether Russia was infiltrating the Trump campaign. Trump then hyperbolically claimed that this perfectly proper use of a human source in a counterintelligence investigation, which he dubbed "Spygate," constituted "one of the biggest political scandals in history!" "The day that we can't protect human sources is the day the American people start becoming less safe," said FBI director Christopher Wray. It's safe to say that lickspittle Republicans such as Nunes care more about protecting Trump than they do the American people. Even when their fellow Republican, Representative Trey Gowdy, admitted that the FBI had acted properly, Trump and Nunes kept pushing their conspiracy theory.
Nothing that Nunes unearthed remotely supported the hyperbolic demands of Trump supporters who wanted the leaders of the Justice Department and FBI to be "taken out in cuffs." If anyone was breaking the law, it was Trump with his attempted obstruction of justice. He practically admitted as much in a tweet, writing: "No Collusion or Obstruction (other than I fight back)." That's a pretty big exception!
Yet the dishonest Trump-Nunes campaign of demonization against the FBI was succeeding with Republican voters: an April 2018 survey found that "more than half of Republicans now think that the FBI is actively biased against Trump." Republican candidates for Congress even sought votes by calling for the prosecution of Clinton, Comey, and other enemies of the state. Republicans were casting themselves ever further into dishonor, disgrace, and disrepute by helping the president to undermine the rule of law—the very foundation of the American republic.
**V. "F AKE NEWS"**
_The Washington Post_ reports that Donald Trump began his presidency by making an average of 4.9 false or misleading statements a day. By his second year in office, like a true Stakhanovite, he had ramped up production to an average of six falsehoods a day. By May 2018, he had uttered the 3,000th falsehood of his presidency and was now serving up nine whoppers a day. Trump made Richard Nixon seem like an honest man by comparison.
Trump did not just lie. He showed reckless disregard for the truth. He boasted that he had told Prime Minister Justin Trudeau that Canada had a trade surplus with the United States even though he didn't know if it did. (It doesn't.) Trump's own US trade representative reports that Canada runs an $8.4 billion trade deficit with the United States. But Trump didn't back down. He tweeted: "We do have a Trade Deficit with Canada, as we do with almost all countries (some of them massive)." And he kept on repeating this fake fact as he imposed steep tariffs on aluminum, steel, and other products manufactured in Canada. Like Trump's claims that General John J. Pershing slaughtered Muslims in the Philippines, or that his inauguration drew record crowds, or that he would have won the popular vote if millions of illegal immigrants had not voted, this is another example of a would-be dictator's desire not just to sneak lies by us but to shove them down our throats. Trump is signaling that he doesn't care what the truth is. From now on the truth will be whatever he says, and he expects every loyal follower to faithfully parrot the official party line, no matter how nonsensical. Trump even had the audacity to claim "I never fired James Comey because of Russia!" a year after having admitted on videotape that, yes, this was precisely why he had fired Comey.
The frightening thing is that Trump's insistence on redefining reality is working, at least with his base. The video news site NowThis posted a hilarious and horrifying clip showing Fox News talking heads hyperventilating over President Barack Obama's promise to meet with the leaders of hostile states such as North Korea (Mike Huckabee: "President Obama likes talking to dictators!"), before going on to effusively praise President Trump for doing just that by meeting with Kim Jong Un. If Trump were to bomb North Korea tomorrow, his cultish followers would praise that decision as avidly as they praised his appeasement of Kim. Trump is sucking a substantial portion of America into his Orwellian universe. The rest of us have to struggle simply to remember that war _isn't_ peace, freedom _isn't_ slavery, ignorance _isn't_ strength. Every lie that is accepted as the truth chips away at the foundations of our democracy. As Rex Tillerson said after his dismissal, in a pointed jab at his ex-boss: "If our leaders seek to conceal the truth or we as people become accepting of alternative realities that are no longer grounded in facts, then we as American citizens are on a pathway to relinquishing our freedom."
HAND IN HAND with Trump's war on the truth has been his campaign against the truth-tellers—the press. All presidents have chafed at critical media coverage and many have lashed out at what they regard as inaccurate reporting, at least in private. But Trump has taken media-bashing to a whole new level.
During the 2016 campaign he called journalists "sick people" and said, "I really don't think they like our country." Reporters covering his rallies said they felt menaced by his incited supporters. As president, Trump tweeted in February 2017: "The FAKE NEWS media (failing @nytimes, @NBCNews, @ABC, @CBS, @CNN) is not my enemy, it is the enemy of the American People!" This was an extraordinary escalation of his assault on the First Amendment. Here was an American president adopting the language used by Adolf Hitler (the Nazis referred to the Lügenpresse, or "lying press") and Josef Stalin, who called the press "vrag naroda" (enemy of the people).
Trump regularly threatened to revoke licenses from broadcast networks that angered him and to loosen the libel laws to make media outlets easier to sue. He is obsessed with Amazon because its CEO, Jeff Bezos, owns the _Washington Post_ , where I am a columnist. Trump repeatedly inveighs against the "Amazon Washington Post" and complains—wrongly—that Amazon does not pay taxes and costs the US Postal Service money. Trump even asked the postmaster general, Megan Brennan, to double the rate Amazon is charged—a demand that Brennan refused because shipping rates are set by contract. In another sign that Trump may be punishing media companies he doesn't like, his attorney, Rudolph Giuliani, said that the president personally intervened to block AT&T's merger with Time Warner, the owner of CNN. (Giuliani later walked back his statement, and a federal judge overruled the administration's objections to the merger.) Trump is following the playbook of strongmen such as Viktor Orban, Vladimir Putin, and Recep Tayyip Erdogan, who silenced the press not by imposing censorship but by imposing financial pressure on independent news organizations to either force them out of business or into the hands of friendly owners.
To be sure, Trump has not actually made good on most of his bloodcurdling threats against the media because the First Amendment provides them such strong protection. If anything, he has been good for the media business, driving newspapers such as the _New York Times_ and the _Washington Post_ to new circulation highs with their exposés of administration scandals. But he is poisoning the civic culture of the United States, raising distrust of the press, and making it impossible for people on opposing sides of the political divide to agree on a commonly accepted set of facts. In one poll taken in late 2017, more than 60 percent of Trump supporters said that the media are the enemies of the people, while only 19 percent had confidence in the media. Trump's attacks on the media are not just undermining the First Amendment. They are also emboldening authoritarian rulers around the world: the ruling parties in, among other countries, Myanmar, Poland, Egypt, Kuwait, Turkey, Syria, Libya, and the Philippines have adopted the "fake news" mantra to attack press freedom.
Trump does not understand what John McCain, a previous standard-bearer of the GOP, wrote—namely that "journalists play a major role in the promotion and protection of democracy and our unalienable rights, and they must be able to do their jobs freely. Only truth and transparency can guarantee freedom." Trump is intent in discrediting the truth, destroying transparency, and undermining democracy—and he is doing so with nary a protest from most Republicans.
**VI. E THICS**
One of the great non-mysteries of the Trump administration is why cabinet members think they can behave like aristocrats at the court of the Sun King. The Department of Housing and Urban Development spent $31,000 for a dining set for Secretary Ben Carson's office while programs for the poor were being slashed. The Environmental Protection Agency paid for Administrator Scott Pruitt to fly first class and be protected by a squadron of bodyguards so he didn't have to mix with the great unwashed in economy class. Two other cabinet secretaries—Veterans Affairs secretary David Shulkin and Health and Human Services secretary Tom Price—were fired over excessive travel expenses. Pruitt eventually got the boot too.
Why would cabinet members act any differently when they are serving in the least ethical administration in our history? The "our" is important because there have been more crooked regimes—but only in banana republics. The corruption and malfeasance of the Trump administration is unprecedented in US history. There are only a few points of comparison: Crédit Mobilier and the Whiskey Ring during the Grant administration, both scandals involving groups of shady businessmen who defrauded the government with the help of federal officials, including reportedly President Grant's private secretary. Teapot Dome during the Harding administration, when the secretary of the Department of the Interior was convicted of taking bribes in return for providing oil companies with leases on federal land, including the Teapot Dome reservation in Wyoming. And, during the Nixon administration, Watergate—a catchall designation for Nixon's attempts to obstruct an investigation of the burglary of the Democratic National Committee carried out by his campaign and his misuse of the FBI and IRS to spy on his political opponents—along with the bribe taking, extortion, and tax fraud committed by Vice President Spiro Agnew while he was governor of Maryland. There have been other notable scandals, such as Iran-Contra and the Monica Lewinsky affair, but neither rose to the same level—the Iran-Contra affair was not done for the personal profit of the Reagan administration officials who traded arms for hostages to Iran and then funneled some of the proceeds to support Nicaraguan rebels, and, while Bill Clinton was caught lying under oath, it was to conceal a tawdry sexual liaison, i.e., essentially a private matter.
By any historical standard, the Trump administration is in an unethical league of its own—and the president has set the tone. Trump's former national security adviser Mike Flynn and deputy campaign manager Rick Gates have pleaded guilty to felonies; his onetime campaign manager, Paul Manafort, faced thirty-two criminal charges, including conspiracy against the United States; and his personal lawyer was under federal investigation. Manafort even went to prison while out on bail for attempting to tamper with witnesses against him.
Trump skirted nepotism rules by hiring his daughter Ivanka and son-in-law Jared Kushner to work in the White House. Ivanka, who had not divested her ownership stake in her clothing company, has used her high visibility to market her products. In the meantime, according to the _New York Times_ , Kushner's family company received hundreds of millions of dollars in loans from companies whose executives met with him in his capacity as a senior White House aide. The _Washington Post_ reported that officials in the United Arab Emirates, China, Israel, and Mexico had discussed how they could manipulate Kushner "by taking advantage of his complex business arrangements, financial difficulties and lack of foreign policy experience." It's hard to imagine that anyone who wasn't married to the president's daughter would have received a top-level security clearance under those circumstances—especially after having amended his disclosure forms on numerous occasions to note meetings and income he had "forgotten" to record. But Kushner's father-in-law got away with far worse.
President Trump broke with decades of precedent by refusing to reveal his tax returns, suggesting he has something to hide—whether it's shady money from foreign sources or simply that his net worth is not as high as he claims. Ronald and Nancy Reagan went so far as to donate to charity all of their income from television, radio, and movie residuals so as to avoid any appearance of a conflict of interest. Trump wouldn't even divest himself of his business holdings. He simply turned over the management of the Trump Organization to his two adult sons, Eric and Donald Jr. They continued to promote real-estate developments across the world, many of them involving local businessmen closely connected to governments eager to ingratiate themselves with the Trump administration. When the Trump Organization got into a dispute over a hotel in Panama, its lawyers demanded that the president of Panama intervene to help and warned of "repercussions" for the country if he didn't. The president of Panama had to upbraid Trump's company for acting improperly.
And while Trump was conducting trade negotiations with China, a Chinese state-owned bank provided $500 million in financing for a project in Indonesia that includes "Trump-branded residences, hotels and golf course." China also provided seven new trademarks for products sold by Ivanka Trump. Within days, Trump shocked national security professionals by announcing that he would lift sanctions on the Chinese telecom giant ZTE. At the very least the president was in violation of the Constitution's Emoluments Clause; at worst, this sequence of events gives the impression that China may have been bribing him.
Meanwhile, back in Washington, the Trump International Hotel has become a favored venue for foreign governments, political organizations, and lobbyists to stay while attempting to influence the Trump administration. Citizens for Ethics and Responsibility in Washington calculated that "political groups spent more than $1.2 million at Trump properties during the president's first year in office. Prior to President Trump's 2016 campaign, annual spending by political committees at Trump properties had never exceeded $100,000 in any given year going back to at least 2002." Trump's for-profit "Winter White House," Mar-a-Lago, doubled membership fees to $200,000 as soon as Trump won the presidency; for that money, well-heeled members can have access to the president of the United States.
Trump's sins extend, of course, well beyond the financial sphere. They include sexual affairs followed by payoffs that recall the conduct of Democratic senator John Edwards and allegations of sexual misconduct similar to those that have brought down other public figures from Harvey Weinstein to Charlie Rose. During his first year in office, Trump had on his White House staff a senior aide who was accused of beating his wives. And in the 2017 Alabama US Senate race, he endorsed a candidate who was credibly accused of molesting underage girls. Trump has a propensity to engage in misconduct himself and to excuse it in others—as long as, like him, they are powerful white men.
Republican voters, in turn, have a dismaying propensity to forgive Trump any sin, including his vulgar, boastful, insulting, and illiterate way of expressing himself—more fitting for an elementary school playground than the Oval Office. Remarkably, according to one poll, 61 percent of Republicans consider Trump a good role model for their children—a view held by only 2 percent of Democrats. What kind of children are these Republicans raising anyway? If Trump is their role model, kids will grow up to be name-calling, lying, narcissistic, ignorant, greedy, prejudiced bullies.
Even those who resist Trump can be contaminated by his conduct. Decades of social science research has revealed that "aggression, bullying and incivility mutate into social super-viruses": "people lie and cheat more after they've seen someone get away with it," "when political leaders are uncivil on social media, it catalyzes aggression in supporters and opponents alike," and "after experiencing incivility at work, 94 percent of us respond with incivility of our own—most commonly with anger and a desire to retaliate." Evidence of the Trump effect is evident in a Pew survey that found 51 percent of Republicans under the age of thirty-four—the ones most influenced by Trump's example—believe that personal insults are sometimes fair game in politics, compared to only 29 percent of young Democrats and fewer than 40 percent of older Republicans. Trump's misbehavior is a stain that will not easily wash away. By legitimating hitherto taboo behavior, he will leave an unsightly mark on American society for decades to come.
Yet conservatives, even (or especially) religious conservatives, couldn't care less: 75 percent of white evangelicals supported Trump in one 2018 poll. The backing of these supposed moralists is Trump's "get out of jail free" card, at least when it comes to matters of morality. It is one big reason why so far, at least, he has avoided the fate of Gary Hart, driven out of the 1988 presidential campaign by evidence of infidelity; Bill Clinton, impeached as a result of his affair with an intern; Bob Livingston, denied the House speakership in 1998 because of his own affairs; or other politicians felled by promiscuity. Liberals, with their live-and-let-live attitude, cannot credibly attack the president for consensual sexual conduct, and conservatives choose not to. Hence, he escapes the consequences of his actions unless he can be shown to have violated the law—no easy matter to establish.
**VII. F ISCAL IRRESPONSIBILITY**
Republicans have long cultivated a reputation—not necessarily deserved—as the party of fiscal austerity. But with Trump's accession, any pretensions to frugality have been tossed into the bonfire, along with the GOP's reputation as the party of law and order and family values. We are a long way removed from 1953, when President Dwight D. Eisenhower said, "There must be balanced budgets before we are again on a safe and sound system in our economy." Or even from 2012, when Representative Paul Ryan said: "We have a debt crisis right in front of us, and what brings down Empires—past and future—is debt."
In 2012, when Ryan spoke those words, federal debt stood at $16 trillion. By 2018 the debt was more than $21 trillion—and climbing, largely because of the spending increases and tax cuts passed by Republicans like Ryan. In 2017 Republicans in Congress approved, on a party-line vote, a tax bill that is projected to add $1.9 trillion to the debt. This was a far cry from the 1986 tax reform act, passed under Ronald Reagan, which was revenue neutral. Then in 2018, a bipartisan coalition in Congress blew through spending caps by approving $300 billion in additional spending over the next two years.
The Committee for a Responsible Federal Budget estimates that, as a result of Republican profligacy, the federal government will be running trillion-dollar deficits "indefinitely." That's roughly double the deficit in Obama's last full year in office—$585 billion. So much for Trump's election-year promise to eliminate the entire federal debt within eight years. The United States is the only major industrialized country whose debt-to-GDP ratio is projected to get worse in the years ahead. Perennial fiscal basket cases such as Greece and Italy are reducing their debt load, while the United States is rapidly expanding it. Republicans are turning economic logic on its head. Periods of economic expansion should be used to balance the budget. Then, when a downturn hits, that's the time for stimulatory spending increases and tax cuts. Running stratospheric deficits while the economy is booming leaves us defenseless to fight a future recession.
It's hard to argue with Senator Rand Paul, who, during a lonely protest on the Senate floor, said, "If you were against President Obama's deficits, and now you're for the Republican deficits, isn't that the very definition of hypocrisy?" But, of course, he's a hypocrite too, having voted for the massive tax cut.
The only way to restore fiscal sanity is to either increase revenue or restrain entitlement spending. Paul Ryan made entitlement reform a centerpiece of his career, but there is no chance of Congress taking badly needed action because Trump, the self-styled "king of debt," couldn't care less. Ryan all but admitted defeat when he announced his retirement in 2018. The president's profligacy is leading the country to fiscal ruin and the Republican Party to intellectual ruin. As former Republican senator Judd Gregg, a onetime chairman of the Senate Budget Committee, writes, the GOP has "no claim any longer to being the party of fiscally responsible government, or to being good stewards of the government and its fiscal health. The Republican Congress now represents a party with very few significant defining principles other than the promotion of the president's impulses at that moment."
**VIII. T HE END OF THE PAX AMERICANA**
Ever since 1945, American foreign policy has been premised on defending and extending political and economic freedom. President after president, both Democrat and Republican, going back to the days of Harry S. Truman, has promoted collective security, international law, free trade, and human rights. Donald Trump has decisively broken from this tradition. He is hostile to both democracy and free trade.
Trump has loudly and repeatedly signaled his intention to abandon more than seventy years of America's commitment to reducing trade barriers and tariffs. With the mindset of a New York real-estate developer used to competing over scarce land, he imagines that every deal has a winner and a loser. He cannot fathom that free trade can be a win for both sides—the country that manufactures a product and the country that buys it. It is doubtful he has ever heard of David Ricardo's theory of "comparative advantage," which holds that countries should specialize and trade in what they are best at producing—cloth for nineteenth-century Britain, wine for Portugal—rather than trying to make everything themselves. Trump imagines that if any country is running a trade surplus with the United States, it is taking advantage of America rather than performing an invaluable service by providing products that Americans want to buy. By the same logic my dentist is ripping me off when he charges $300 for a tooth cleaning unless he buys $300 worth of my books at the same time.
Trump's pullout from the Trans-Pacific Partnership (TPP) was an economic and geopolitical gift to China, which can now pursue its own Regional Comprehensive Economic Partnership, an alternative to the TPP that is designed to facilitate Chinese hegemony over East Asia. Already, China has signed free-trade agreements with twenty-one countries, compared with only twenty for the United States, and it is negotiating more than a dozen additional pacts. Far from lowering trade barriers, as his predecessors did, Trump has demanded changes in existing accords, such as the North American Free Trade Agreement and the Korea-U.S. Free Trade Agreement, while imposing 25 percent tariffs on steel and 10 percent tariffs on aluminum, followed by $37 billion of tariffs on Chinese goods.
Trump welcomes a trade war—he says that "trade wars are good, and easy to win"—but no serious economist would agree. Moody's economists estimate that a trade war with China could cost 190,000 American jobs. Another study finds that steel and aluminum tariffs alone could cost as many as 400,000 American jobs. But Trump seems as oblivious to these grim predictions as he is to the history of trade wars.
The Smoot-Hawley Tariff Act of 1930 triggered a trade war that spread the Great Depression from the United States to the rest of the world. The resulting economic meltdown contributed to the rise of totalitarian regimes in Germany and Japan and led to the outbreak of World War II. After 1945, US policymakers pursued a free-trade policy. They midwifed the creation of the General Agreement on Tariffs and Trade in 1947 and the World Trade Organization in 1995. Free trade became one of the pillars of the Pax Americana, along with support for democracy, international law, and collective security. This altruistic approach paid off: today, the United States has 4.4 percent of the world's population and 24.3 percent of its gross domestic product.
Granted, the impact of free trade has not been uniformly positive; for many Americans it has contributed to the despair they feel. Even if economic change is primarily driven by technology (one study found that 88 percent of the loss of US manufacturing jobs between 2006 and 2013 was due to automation and related factors), it is easy to blame trade with other countries for hollowing out industrial towns and throwing workers onto the unemployment line. To some extent this is even true—trade does contribute to economic dislocation even if it is not the primary cause. An influx of immigrants also can contribute to the impression among white, working-class Americans that "their" country is being lost. But the right answer is to ameliorate the suffering of those left behind by providing retraining and social welfare benefits—not to shut down free trade and immigration and thereby impose heavy costs on the entire country. There can be no return to some kind of imagined autarkic paradise of the past during which the United States did not depend on the free movement of goods and people. I still believe in the old _Wall Street Journal_ slogan so familiar to me from my years at that proudly capitalist publication: "Free People and Free Markets." Too bad so few conservatives are willing to fight for those ideals anymore. Indeed, congressional leaders blocked attempts by some Republican lawmakers, such as Senator Bob Corker, to roll back Trump's steel and aluminum tariffs.
I fear that America's farsighted postwar trade policy—and all of the economic and security benefits it delivered—may not survive Trump's mindless acts of vandalism. Trump has launched a war not just on "unfair" trade practices but also on the very idea of an open, rules-based international system of trade. Trump seems aware of the negative consequences of his tariffs, which is why his commerce department was considering exemptions from steel and aluminum tariffs for companies that claimed they could not find the metals they need domestically. But his case-by-case approach turns trade deals into sweetheart arrangements that undermine the hope of American policymakers going back to the 1940s to create a rules-based trading system under which disputes could be adjudicated impartially and on the merits.
Republicans aren't troubled in the least. A _Washington Post_ –ABC News poll in early 2018 showed that 66 percent of Republicans thought that a trade war with China would be good for US jobs, compared to only 36 percent of all adults. Once the party of free trade, the GOP has embraced protectionism because of Trump.
TRUMP'S AVERSION TO free trade is matched by his hostility to America's traditional, democratic allies. He has long believed that the United States was getting ripped off and taken advantage of by its closest friends. In 1990, for example, when the United States was at the height of its power, he said, "Our 'allies' are making billions screwing us." Once in office, Trump allowed his more internationally minded advisers to talk him out of pulling US forces out of South Korea, Japan, or Germany, as he had once threatened to do. He even reluctantly affirmed NATO's Article V mutual-defense provision but then questioned why America's sons should fight for Montenegro. He abandoned not only the TPP but also the Paris climate accord and the Iran nuclear deal. He imposed tariffs on America's NATO allies and abruptly pulled out of a joint communiqué with other world leaders at a Group of Seven summit in Quebec in June 2018. Trump impulsively accepted a summit meeting with North Korean dictator Kim Jong Un and then showered Kim with praise despite not getting any real concessions in return. These are indications of how the administration "grown-ups" could not contain his unilateralist (and erratic) instincts for long.
Trump's words spoke volumes about his contemptuous attitude toward allies. During his first year in office, he had testy exchanges with, among others, the prime minister of Australia and the president of Mexico. Not even America's closest ally was safe from Trump's animadversions. In June 2017, after a terrorist attack in London, Trump blasted that city's first Muslim mayor, Sadiq Khan, tweeting: "At least 7 dead and 48 wounded in terror attack and Mayor of London says there is 'no reason to be alarmed!'" Trump blatantly misrepresented the mayor's remarks—Khan had said there was no need to be alarmed about a heightened police presence on the streets, not about terrorism. A few months later, in November 2017, Prime Minister Theresa May rebuked Trump for posting anti-Muslim videos produced by a far-right British leader. Trump instantly hit back, lecturing the prime minister, "Don't focus on me, focus on the destructive Radical Islamic Terrorism that is taking place within the United Kingdom. We are doing just fine!"
Trump's relationship with the chancellor of Germany was just as testy as with the prime minister of the United Kingdom. It is perhaps no coincidence that the leaders of both countries are intelligent, strong-willed women; with the notable exception of his own daughter Ivanka, Trump does not appear comfortable dealing with independent women. During the 2016 campaign, Trump bashed Angela Merkel for admitting Muslim refugees into Germany. "I think what she did in Germany is a disgrace," he said, adding that he was no longer "a fan." When the two leaders met at the White House in early 2017, Trump pointedly refused to shake Merkel's hand. Then in May, following a G7 summit in Italy during which Trump clashed with the European leaders over the Paris climate accord, Merkel emerged to say that Europe could no longer count on the United States. "We Europeans truly have to take our fate into our own hands," she declared.
Trump has been even more vitriolic in attacking Canadian prime minister Justin Trudeau, the leader of a country whose troops have fought and bled alongside Americans for more than a century. After Trudeau vowed to retaliate for US tariffs imposed under the pretense that his country posed a "national security threat" to the United States, Trump blasted him as "very dishonest & weak." Trump's aides piled on, with economic adviser Larry Kudlow accusing Trudeau of a "betrayal" and trade adviser Peter Navarro saying there's a "special place in hell" for the Canadian prime minister. Navarro subsequently apologized, but the contrast between the administration's praise for Kim Jong Un and its attacks on Justin Trudeau was striking. In Trump's telling, he had established a "special bond" with Kim but there is a "special place in hell" for Trudeau.
Just about the only democratic leaders with whom Trump developed cordial relationships were Shinzo Abe of Japan and Emmanuel Macron of France—the former because he played golf with Trump (and probably let the American win), the latter because he took Trump to a military parade in Paris. But not even the budding Abe-Trump bromance could spare Japan from being included on the list of nations subject to Trump's ill-advised steel and aluminum sanctions, and Macron could not prevent Trump from pulling out of the Iran nuclear deal or imposing tariffs on the European Union. By June 2018, Macron and Trump were exchanging hostile tweets—and it was entirely Trump's fault.
While clashing with democratic leaders—especially those who happen to be women or minorities—Trump has gotten along disturbingly well with dictators. He positively purred after the Saudis in May 2017 projected a five-story-sized photo of him onto the side of his hotel in Riyadh. In July 2017, Trump had a tête-à-tête with Vladimir Putin at the G-20 summit in Hamburg that began with the American president telling the Russian autocrat what an "honor" it was to see him. In November 2017, Trump was even more laudatory after a visit with Xi Jinping in Beijing. He said that his feeling toward Xi is "an incredibly warm one," and described Xi as a "highly respected and powerful representative of his people." Correction: Xi is not the "representative" of his people. He is their dictator, and it's impossible to know how respected he is because anyone who is disrespectful to him is likely to be locked up. Xi is currently orchestrating the most intense cult of personality that China has seen since the days of Mao Zedong—and Trump is doing his level best to help. He declared that it's "great" that Xi is making himself president for life and added, supposedly in jest, "maybe we'll have to give that a shot someday."
Trump has a kind word for every strongman he chats with. He congratulated Rodrigo Duterte of the Philippines because "of the unbelievable job" he was doing "on the drug problem"—a problem that Duterte is addressing by unleashing death squads that have killed thousands of Filipinos. He said that Recep Tayyip Erdogan is "getting very high marks" as he was crushing civil society and congratulated him on a rigged referendum that spelled the death knell for Turkish democracy. And, of course, he claimed that North Korea's Kim Jong Un, an odious human-rights violator, loves his people and in turn is loved by them. He even expressed admiration for how "tough" Kim was in accumulating absolute power ("I mean, that's one in 10,000 that could do that")—a feat that Kim pulled off by, among other steps, killing his own uncle and half brother. His relationship with Vladimir Putin is so obsequious that former CIA director John Brennan and former director of National Intelligence James Clapper suggested that Trump might have been compromised by the Kremlin.
All American presidents have been forced to deal with dictators. Recall Franklin Roosevelt's apocryphal aphorism about Anastasio Somoza of Nicaragua: "He's a son of a bitch, but he's our son of a bitch." But no president has been so greasy and extravagant in his blandishments. For Trump, these dictators' attacks on democratic norms are not a problem but rather policies to be praised—and possibly even emulated. "He speaks and his people sit up in attention," Trump said of Kim Jong Un. "I want my people to do the same." In 2018 Freedom House downgraded America in its annual "Freedom in the World" report, which noted that "core institutions were attacked by an administration that rejects established norms of ethical conduct across many fields of activity." Similarly, the watchdog group Reporters Without Borders downgraded America's level of press freedom in 2018, down to forty-fifth in the world, behind Surinam, Burkina Faso, and Rumania.
Trump seemed to have no idea of the damage he was doing to America's standing in the world. But it was profound and long-lasting.
FOR YEARS, AS A SELLER of real estate and star of reality TV, Trump made a living wooing, if not bamboozling, customers and viewers. His selling skills were honed enough that he convinced voters to elect him as president in spite of his near-total lack of qualifications. He is a real-life incarnation of "Professor" Harold Hill, the protagonist of _The Music Man_ , who convinces the denizens of a midwestern town that he is a bandleader even though he is bereft of musical skills. Yet once in office Trump has proved to be the worst salesman that America has ever had. Far from winning over other countries, he is actively repelling and repulsing them.
According to Gallup, "approval of U.S. leadership across 134 countries and areas stands at a new low of 30%." That's lower than the 34 percent approval during the last year of George W. Bush's administration in the wake of fiascos such as the Iraq War. Even more ominously, the number of Arab youth who see the United States as an enemy shot up from 32 percent in 2016 to 57 percent in 2018. Meanwhile, a poll found that only 14 percent of Germans consider the United States a reliable partner, compared to 36 percent for Russia and 43 percent for China. That the citizens of one of America's staunchest and most important allies now look more favorably upon our illiberal foes is a testament to Trump's unrivaled wrecking abilities. After Trump pulled out of the Iran nuclear accord over the objections of the Europeans, Donald Tusk, the president of the European Commission, tweeted: "Looking at latest decisions of @realdonaldtrump someone could even think: with friends like that who needs enemies."
Trump is entirely focused on American hard power—military and economic might. An administration official described his worldview as follows: "His dream would be to have a strong military that protects our homeland. We'd wall ourselves off and strike at our discretion and then retreat to defending our homeland." Or, as another administration official put it, the Trump Doctrine is: "We're America, bitch." This type of chest-thumping unilateralism was not a viable policy even in the early twentieth century; it certainly won't work in the twenty-first century in a world that has been brought more closely together by communications and transportation technologies.
What Trump doesn't realize is that much of America's success as a superpower has rested on its "soft power." America is a superpower by invitation: we have troops in more than 170 countries and alliances with at least 60 countries because most other nations do not feel threatened by American power. Anti-Americanism is a fact of life, but the United States simply has not engendered the same kind of fear and loathing that less altruistic, more militaristic would-be hegemons have done—notably, Habsburg Spain, monarchist and Napoleonic France, Wilhelmine and Nazi Germany, and the Soviet Union. Each of those superpowers provoked other nations to ally against its expansionist designs, eventually leading to its downfall. There is no similar international coalition against the United States because it has been viewed as a more benign actor. In the past, America's adversaries—China and Russia—were the isolated ones. These illiberal powers have a few satraps but almost no real friends. They are regarded with suspicion and hostility by their neighbors. But now Trump's unilateralism is leaving an opening for these illiberal states to usurp American power.
Britain was said by the great Victorian historian J. R. Seeley "to have conquered and peopled half the world in a fit of absence of mind." America is losing its global power in the same way. Through ignorance and malice, Trump is destroying the foundations of American influence that previous leaders spent three-quarters of a century erecting. When it comes to "soft power," he is engaging in unilateral disarmament—and that in turn will have dire consequences for American security and prosperity.
American power had been eroding even before Trump came to office, thanks to the growth of competitors such as China and the foreign policy mistakes of George W. Bush, who was too interventionist, and Barack Obama, who was too noninterventionist. Trump has accelerated the decline. He might even have made it irreversible. What ally will trust America ever again? Even if a future president reverts to a more internationalist and free-trade policy, the rest of the world will be acutely conscious of the risk that the American electorate might elect another isolationist and protectionist president in the future. Trump may well be ending the Pax Americana and helping to usher in either a Chinese Century or a new global disorder in which there is no international law and life is "nasty, brutish and short."
## _6._
## **THE TRUMP TOADIES**
_T_ _HIS_ IS THE COST THAT AMERICA HAS PAID FOR THE Trump presidency—so far. His supporters, meaning almost all Republicans have rationalized the president's racism and nativism, the evidence that his campaign colluded with the Kremlin to undermine American democracy, his attempted obstruction of justice and general assault on the rule of law, his efforts to spread lies and undermine the press, his violations of the most basic ethical norms, his staggering fiscal irresponsibility, and his assault on seventy-plus years of American international leadership in promoting free trade and free societies. Trump is undermining the foundations of American democracy. And in return they have gotten . . . what? A few conservative judges, a big but unnecessary tax cut, the repeal of some regulations, some gains against ISIS, a substance-free summit with North Korea. Trump claims to be the world's best deal maker, but he is offering his followers the world's worst deal. Republicans are expected to loyally support him, and in return he traduces most of what they claimed to believe in.
Yet almost no Republicans are willing to speak out against him. It is hard to know who is worse: Trump or his enablers. I am inclined to think it is the latter. Trump does not know any better; he has no idea of how a president, or even an ordinary, decent human being, is supposed to behave. But many of his supporters do know better, and they are debasing themselves to curry favor with him because he controls the levers of power.
Before Trump won the presidency, here is some of what Republicans had to say about him, as compiled by the _New York Times_. They called him a "malignant clown," "national disgrace," "complete idiot," "a sociopath, without a conscience or feelings of guilt, shame or remorse," "graceless and divisive," "predatory and reprehensible," flawed "beyond mere moral shortcomings," "unsound, uninformed, unhinged and unfit," "a character and temperament unfit for the leader of the free world," and "A bigot. A misogynist. A fraud. A bully." Since the election? For the most part the sound of silence—even though all of these characterizations of the president have been validated many times over.
Having become enablers of Trump's transgressions against decency, common sense, and quite possibly the law itself, Republicans find themselves drawn ever deeper into a web of complicity and rationalization. They are defending Trump because they are trying to convince themselves that they have not made a terrible mistake. Like many dupes—for example, the students of Trump University who paid thousands of dollars to be taught the secrets of Trump's success—they do not want to admit that they were conned. Many Republicans overcompensate and wind up praising Trump in unctuous terms more fit for "Dear Leader," the late dictator of North Korea, than for the president of a constitutional republic.
TRUMP HAS DEVELOPED an authoritarian-style cult of personality with the shameful connivance of those around him. At a televised cabinet meeting in June 2017, Trump's appointees took turns lavishing praise on their insecure and needy boss. Vice President Mike Pence told him: "It is the greatest privilege of my life to serve as vice president. The president is keeping his word to the American people." Agriculture secretary Sonny Perdue said: "I just got back from Mississippi and they love you there." Chief of Staff Reince Priebus: "On behalf of the entire senior staff around you, Mr. President, we thank you for the honor and the blessing that you've given us to serve your agenda and the American people." Labor secretary Alexander Acosta: "I am privileged to be here—deeply honored—and I want to thank you for your commitment to the American workers." Treasury secretary Steven Mnuchin, a Yale-educated, Olympic-class sycophant who had previously opined that the overweight president "has got perfect genes" and is "unbelievably healthy," chipped in: "It was a great honor traveling with you around the country for the last year, and an even greater honor to be here serving on your cabinet."
This nauseating display of toadyism, like something out of an imperial court, was capped by Trump's self-praise. At a time when he had no significant legislative achievements, the president had the gall to boast that he had rivaled the achievements of Franklin Roosevelt's first hundred days—a period of unparalleled legislative activity in which Congress created, among other agencies, the Tennessee Valley Authority, the Agricultural Adjustment Act, the Civilian Conservation Corps, and the Federal Emergency Relief Administration. "I will say that never has there been a president, with few exceptions—in the case of FDR he had a major Depression to handle—who's passed more legislation, who's done more things than what we've done," Trump claimed. "We've been about as active as you can possibly be, and at a just about record-setting pace."
All of that immoderate flattery is almost tame compared to what Trump's outside admirers say about him. After Trump's State of the Union address in 2018—a decent speech by his standards but hardly an oratorical masterpiece— _Washington Times_ writer Charles Hurt enthused: "President Trump has officially transformed himself from merely a great American president into a historic world leader keeping lit the torch of freedom for all people around the world. . . . Mr. Trump joins Reagan, Margaret Thatcher, Pope John Paul II and Martin Luther King Jr. in the pantheon of great champions of freedom from the past half-century." Thus did Hurt do even one better than White House aide Stephen Miller, who said: "President Trump is the most gifted politician of our time, and he's the best orator to hold that office in generations." Such over-the-top praise for Trump's nearly incomprehensible rhetorical effusions has become standard among his followers.
After Trump's trip to the Middle East in May 2017, Robert Charles, a veteran of the Reagan and both Bush administrations, gushed on Fox's website: "The world from which President Trump returns on his historic trip to Muslim Saudi Arabia, Jewish Israel, Christian Vatican, and agnostic Brussels, is different from the world prior to his trip. . . . We have not seen this kind of leadership in a very long time, not in the Middle East—not anywhere. Hope exists where it did not before this trip, because of his personal outreach, resolve and authenticity." It almost seemed like a letdown after the comparisons to Reagan, Thatcher, the Pope, and Martin Luther King Jr., but _Washington Examiner_ writer Steve Cortes likened Trump to John Wooden, the UCLA basketball coach who won a record-setting ten national championships and became known as the "Wizard of Westwood": "If President Donald Trump has more win streaks like his present one, he might well become known as the 'Wizard of Washington.' From the economy to important victories across the vast Asian continent, America is winning under Trump's leadership."
It is no exaggeration to say that Trump's most perfervid followers literally worship him. Candace Owens, communications director of a pro-Trump group called Turning Point USA, tweeted: "I truly believe that @realDonaldTrump isn't just the leader of the free world, but the savior of it as well." (Trump predictably extolled her as a "very smart" thinker.) The evangelical leader Franklin Graham actually said: "I just appreciate that we have a man in office that understands the power of prayer and the need for prayer." This, of a president who notoriously revealed that he had never heard of the New Testament book II Corinthians and who reportedly wasn't sure whether Presbyterians like himself are Christians. Representative Jim Jordan, leader of the far-right Freedom Caucus in the House, denied that he had ever heard Trump tell a lie. Indeed, Trump supporters routinely praise him for his "honesty," by which they mean, apparently, not actually telling the truth but making "politically incorrect" remarks—their term of art for racist, xenophobic, or otherwise offensive statements.
Avowals of Trump's infallibility are widespread—just as they have been for other authoritarians. In Fascist Italy, for example, a popular slogan had it: "Il Duce is always right." Trump's economic adviser Peter Navarro updates this refrain when he says: "My function, really, as an economist is to try to provide the underlying analytics that confirm his intuition. And his intuition is always right in these matters." The classicist Victor Davis Hanson, whose historical work I admire, compared Trump to such "tragic heroes" as George S. Patton, Shane, Marshall Will Kaine (Gary Cooper) in _High Noon_ , Ethan Edwards (John Wayne) in _The Searchers_ , and the protagonists of _The Magnificent Seven_. Some praise for a plutocrat who avoided the draft and has never shown any willingness to sacrifice anything for anyone. Naturally, Trump's followers nominated him for a Nobel Peace Prize even before he met with Kim Jong Un.
All of this extravagant flattery recalls nothing so much as the cult of personality that developed around Marshal Philippe Pétain, the First World War hero who during the Second World War became ruler of France's Vichy regime. As recounted by historian Julian Jackson: "Images of Petain were produced on an industrial scale. One could buy Petain posters, postcards, calendars, plates, cups, chairs, handkerchiefs, stamps, coloring books, matchboxes, tapestries, paperweights, medals, vases, board games, ashtrays, penknives, barometers. One could have him in Aubusson tapestry, Baccarat glass, Sèvres porcelain, or plastic." Other despots have had similar merchandising operations: I still have a lighter that I bought years ago in Lebanon's Bekaa Valley, the stronghold of Hezbollah, which projects an image of Hassan Nasrallah, leader of this terrorist organization. In a similar vein, Trump's official website sells "Make America Great Again" hats for a mere twenty-five dollars, water bottles, cups, T-shirts along with a presidential medal, a Trump-Pence flag, "patriotic coolies," a "collectible ornament," lapel pin, and mini-megaphone. The website of the Trump Organization offers Trump-branded knit caps, polo shirts, cocktail glasses, wine glasses, coffee mugs, slippers, robes, chargers, umbrellas, luggage tags, playing cards, "trinket dishes," deodorant, cologne, teddy bears, key chains, clocks, and even pet bandanas, dog throw toys, and dog collars—although sadly not Trump steaks, deodorant, and urine tests, which are no longer on the market.
Just as Trump's followers buy souvenirs of their hero, so too they echo the effusions of Pétain's admirers, who celebrated him as "an envoy from God" and literally sang his praises—one popular ditty labeled him the "savior of France," just as Trump is celebrated as the savior of America. One half-expects some Trump toady to match the rapturous enthusiasm of René Benjamin, author of three hagiographies of Pétain, who grew dizzy with delight after spotting the marshal's overcoat: "After several moving and happy meetings [with Pétain] I had one which I believe was more extraordinary than all the others. I found myself one day alone with his overcoat. Yes, his overcoat, which was lying just like that on the armchair in his study. It was a magnificent moment. I was overcome. Then all of a sudden I become as motionless as the coat when I noticed that the seven stars were gleaming like the seven stars of wisdom of which the ancients tell us."
Whoever pens the first paean to Trump's "really beautiful" and "very classy" overcoat can expect at the very least a positive mention from Fox News if not an actual appointment to the White House staff. Perhaps it will be the anonymous troll who attacked the historian and journalist Jon Meacham, who criticized the president in his typically thoughtful fashion, by tweeting: "Trump isn't an aberration, he is America's savior. He will go down as the greatest president since Washington. Elitist globalist leftists like yourself are a cancer on America's culture & future. You must be eradicated like all disease." To state the obvious: This kind of invective against the president's critics and this kind of worship of the president is not normal or healthy. It is a sign of the corrosion not just of the conservative movement but of our democracy.
IT IS NOT QUITE FAIR to say that all Republicans have become Trump toadies. Only most of them. The consequences for not faithfully toeing the party line can be severe: in April 2018, the conservative website Red State fired, in the words of one source, "everyone who was insufficiently supportive of Trump." In June, South Carolina representative Mark Sanford, who had dared to criticize Trump in the past (while still voting with him), lost his primary election after the president endorsed his opponent. To be an anti-Trump Republican in this climate requires moral courage that few politicians or media personalities display. All too few Republicans are resisting Trump's incipient authoritarianism just as, sadly, too few politicians of any party resisted previous assaults on civil liberties in US history. Assaults such as the Alien and Sedition Act of 1798, which was used by President John Adams's Federalists to prosecute twenty-five prominent Republicans. The Espionage Act of 1917 and the Sedition Act of 1918, which were used by President Woodrow Wilson to exclude from the mails "disloyal" publications such as _The Nation_ and _The Masses_ , to deport thousands of radicals such as the anarchist Emma Goldman, and to jail others, including Socialist Party leader Eugene V. Debs, who received nearly a million votes in the 1920 presidential election while sitting in prison. Particularly egregious was Executive Order No. 9066 in 1942, which was used by Franklin Roosevelt to consign 120,000 Japanese Americans to detention camps. Or the McCarthyism in the 1950s, which ruined the lives of numerous civil servants and movie industry figures who were accused of being Communist subversives, often with scant evidence.
Political courage is esteemed so highly precisely because it is so rare. Only one Republican senator publicly upbraided Joe McCarthy early on—Margaret Chase Smith of Maine, the only female member of the Senate. She rose on the Senate floor in 1950 to denounce Republicans for their "selfish political exploitation of fear, bigotry, ignorance and intolerance." She did not name McCarthy, but everyone knew who she had in mind when she spoke about those "who shout loudest about Americanism" while they "by their own words and acts, ignore some of the basic principles of Americanism—the right to criticize; the right to hold unpopular beliefs; the right to protest; the right of independent thought." Her words fell in a vacuum because almost all of her Republican colleagues had calculated that, however boorish McCarthy was, they had to cooperate with him because he had tapped into genuine anti-Communist sentiments in the country shortly after China's fall to Communism, the Soviet Union's acquisition of the atomic bomb, and the outbreak of the Korean War. In such a climate, to defy the Red Scare was judged too great of a risk even for a war hero. On the campaign trail in 1952, Republican nominee Dwight D. Eisenhower planned to criticize McCarthy for casting outrageous aspersions on the loyalty of his mentor, General George C. Marshall, a man who had done as much as anyone to make possible US victory in World War II. Eisenhower's advisers begged him to take out his defense of Marshall—and, at the last minute, he did.
Similarly, today, most Republicans practice situational ethics to convince themselves that there is some advantage to be gained—for their party or their country or themselves—by catering to Trump. One of the few outspoken critics has been Senator Jeff Flake, who in the fall of 2017 eloquently attacked Trump and his Republican enablers. "Mr. President, I rise today to say: enough. We must dedicate ourselves to making sure that the anomalous never becomes the normal," Flake said, while announcing that he would not run for reelection because his views made him toxic for Arizona Republicans. Senator Bob Corker of Tennessee was even more direct, calling Trump "utterly untruthful," warning that his staff has "to figure out ways of controlling him," and noting that "he lowers himself to such a low, low standard and debases our country." But then Corker, too, decided not to seek reelection—before briefly changing his mind and groveling before Trump to secure his support. Even after giving up hopes of staying in office, Corker said he "probably" would still have voted for Trump over Clinton.
Another Republican who did not stay silent was John McCain. He warned in an eloquent speech about the dangers of Trumpism, denouncing the "half-baked, spurious nationalism cooked up by people who would rather find scapegoats than solve problems." And he voted, along with Republican senators Lisa Murkowski and Susan Collins, against a bill strongly supported by Trump to repeal Obamacare. But McCain was a storied Republican leader of yesteryear who had been left behind by his party's embrace of Trumpism.
A few other Republicans, such as Senators Ben Sasse and Lindsay Graham, have occasionally criticized Trump, but most ordinary Republicans seem to adore him, while most of their leaders seethe against him quietly, behind closed doors, not daring to agree in public with his critics. Typical is the Republican congressman who told Erick Erickson: "It's like Forrest Gump won the presidency, but an evil, really fucking stupid Forrest Gump. He can't help himself. He's just a fucking idiot who thinks he's winning when people are bitching about him." Naturally this congressman doesn't want to be quoted by name, because he represents a district that Trump won, and he has regularly defended Trump on Fox News and other outlets. Were this congressman to share his true opinion of Trump publicly, he would have difficulty winning reelection. In the 2018 primaries, Republican candidates routinely accused their opponents of being insufficiently devoted to the Maximum Leader, and those who once criticized Trump tried to explain that they didn't really mean it.
Republican donors have been just as pusillanimous: many of them are troubled by Trump's behavior, but they continue to contribute to Republican candidates as if this were still the party of Paul Ryan rather than of Donald Trump—or as if Paul Ryan were still the high-minded fighter for conservative principles he once appeared to be, rather than another pathetic appeaser of the demagogue in the White House who chose to retire rather than fight for what he supposedly believes in. Donors prefer to ignore what the GOP has actually become—a vehicle for waging a Trump-style culture war, feeding Trump's egomania, and protecting Trump from accountability for his actions. Obstruction of justice has practically become a plank of the new Republican Party. "There is no Republican Party. There's a Trump party," says former House Speaker John Boehner. "The Republican Party is kind of taking a nap somewhere."
One of the few GOP donors who has switched to funding Democrats is the Boston hedge fund billionaire Seth Klarman. Once New England's top GOP donor, he gave more than $222,000 to seventy-eight Democrats running for Congress since 2016. "The Republicans in Congress have failed to hold the president accountable and have abandoned their historic beliefs and values," Klarman told the _Boston Globe_. "For the good of the country, the Democrats must take back one or both houses of Congress." Klarman is right, and it's a disgrace that more Republicans aren't willing to put aside their partisanship for the good of the country.
Republicans in Washington cover their cowardice by claiming they need to appease Trump to pass their policy agenda, but even after the passage of a tax cut, Republicans remained as invertebrate as ever. In a sense, the arch-populist Stephen Bannon is right to exult that "the establishment Republicans are in full collapse." Trumpism is triumphant—at the moment. But Jeff Flake is also right that "this spell will eventually break." At least I hope he's right. And if he is, the judgment of history will not be kind to so many of my old friends and fellow travelers who propitiated a man so unfit for the highest office in the land.
## _7._
## **THE ORIGINS OF TRUMPISM**
YOU KNOW HOW, AFTER YOU WATCH A MOVIE WITH A surprise ending, you sometimes replay the plot in your head on your way out of the theater to find the clues you missed the first time around? That's what I've been doing lately with the history of conservatism. It would be nice to think that Donald Trump is an anomaly who came out of nowhere to take over an otherwise sane and sober movement. But it just isn't so. Trump is a unique force in American politics, but, in many ways, he is merely the culmination of the right's ruin rather than its cause. Or, put another way, he is a symptom of a deeper, underlying disease.
I have been reading about the origins of the modern conservative movement in books written by liberal scholars such as Geoffrey Kabaservice, E. J. Dionne Jr., Corey Robin, and Rick Perlstein rather than conservative hagiographers, and realizing how much I missed when I was growing up. Upon closer examination, it's obvious that the whole history of modern conservativism is permeated with racism, extremism, conspiracy-mongering, ignorance, isolationism, and know-nothingism. Even those who were not guilty of these sins too often ignored them in the name of unity on the right. I disagree with liberals who argue that these disfigurations define the totality of conservatism; conservatives have also espoused high-minded principles that I still believe in, and the bigotry on the right appeared to be ameliorating in recent decades. But there is no doubt that there has always been a dark underside to conservatism, and one that I chose for most of my life to ignore. It's amazing how little you can see when your eyes are closed!
THE UR-CONSERVATIVES of the 1950s were revolting not against a liberal administration but against the moderate conservatism of Dwight D. Eisenhower—a doctrine that one of his advisers labeled "Modern Republicanism." Ike ended the war in Korea and refused to send US ground troops to Indochina. He preferred to counter Communist advances with covert actions in countries such as the Philippines, Guatemala, and Iran. He balanced the budget in three of his eight years in office, cut federal civilian employment, reduced the debt, and presided over nearly a decade of uninterrupted peace and prosperity. Yet ideological conservatives viewed Eisenhower as a sellout; John Birchers thought he was a Communist agent.
Why the animus against this war hero? Conservatives were furious that Eisenhower made no attempt to liberate the "captive nations" of Eastern Europe, "roll back" Communism, and undo FDR's Yalta "betrayal" because he knew that to do so could have resulted in World War III. They were further enraged that he did not try to repeal the New Deal because he knew that a minimal social safety net was needed for capitalism to maintain popular support. Rather than acting as a conservative revolutionary, Eisenhower marginally expanded government by creating a new Department of Health, Education and Welfare and by building the interstate highway system. Eisenhower further offended the right by working behind the scenes, after his initial appeasement of McCarthyism, to defeat Joseph McCarthy and end his irresponsible Red Scare. Worst of all, from the viewpoint of contemporary conservatives, Eisenhower was a moderate on racial issues. He appointed Chief Justice Earl Warren, a former Republican governor of California who, rejecting Eisenhower's "gradualist" vision of integration, presided over the Supreme Court's unanimous school desegregation decision, _Brown v. Board of Education_. Ike then sent the 101st Airborne Division to Little Rock, Arkansas, to desegregate Central High School in the face of white resistance.
This was the "liberal"—really moderate Republican—status quo against which William F. Buckley Jr., Barry Goldwater, Ronald Reagan, and other conservatives were rebelling in the 1950s. Buckley—yes, my boyhood hero—coauthored a book in defense of Joe McCarthy, _McCarthy and His Enemies_ (1954), followed by a defense of the House Un-American Affairs Committee, _The Committee and Its Critics_ (1962). Buckley also editorialized in _National Review_ against desegregation. One notorious 1957 editorial, "Why the South Must Prevail," claimed that "the white community in the South is entitled to take such measures as necessary to prevail, politically and culturally . . . because, for the time being, it is the advanced race." The editors went on, shockingly enough, to assert: "The great majority of the Negroes of the South who do not vote do not care to vote, and would not know for what to vote if they could." That is as blatant and ugly a statement of racism as one might have the misfortune to read. To his credit, Buckley recanted those views in later years; while running for mayor of New York in 1965, he even endorsed affirmative action. But many other conservatives refused to disown prejudice and bigotry.
Most Republicans in Congress voted in 1964 and 1965 for landmark civil rights legislation, but the conservative hero Barry Goldwater did not. In his 1960 bestseller _Conscience of a Conservative_ —ghostwritten by Buckley's brother-in-law, Brent Bozell—Goldwater wrote that "the federal Constitution does not require the states to maintain racially mixed schools. Despite the recent holding of the Supreme Court [ _Brown v. Board of Education_ ], I am firmly convinced—not only that integrated schools are not required—but that the Constitution does not permit any interference whatsoever by the federal government in the field of education." Goldwater was not personally a racist—he had integrated the Arizona National Guard—but he was happy to make common cause with racists in order to wrest the South from the Democrats.
Goldwater was just as extreme when it came to foreign affairs. Abjuring Eisenhower's efforts to maintain the peace, he suggested that Americans needed to overcome their "craven fear of death." If another major uprising occurred in Eastern Europe, like the one in Hungary in 1956, he counseled, "We ought to present the Kremlin with an ultimatum forbidding Soviet intervention, and be prepared, if the ultimatum is rejected, to move a highly mobile task force equipped with appropriate nuclear weapons to the scene of the revolt." In other words, Goldwater was willing to risk nuclear war to free a single "captive nation" from Communist control. I used to think that Goldwater's reputation as an extremist was a liberal libel. Reading his actual words—something I had not done before—reveals that he really was an extremist.
The delegates to the 1964 Republican convention who chose Goldwater as their presidential nominee fully endorsed his far-right views. They voted down planks committing the party to enforce civil rights laws and to repudiate extremist groups such as the Ku Klux Klan and the John Birch Society. The Republicans gathered at the Cow Palace in San Francisco lustily applauded Goldwater's assertion that "extremism in the defense of liberty is no vice" and that "moderation in the pursuit of justice is no virtue," while booing and jeering Governor Nelson Rockefeller of New York when he tried to deliver a more moderate message. Liberal Republicans in attendance were scared by the vehemence of the crowd, comparing it to Nazi rallies. This was the same sort of reaction that Trump would later elicit, and with a message similar in tone. Only in 1964 it didn't work: Goldwater won only six states (all except his home state in the South) and went down to a landslide defeat. But his example continued to inspire conservatives for decades, making clear that extremism is embedded in the DNA of the modern conservative movement.
The Goldwater precedent would prove especially important when it came to civil rights. In 1964, the GOP ceased to be the party of Lincoln and became the party of southern whites. All of the Republican presidential nominees in the future would harvest racist votes, whether consciously or not, because from then on the GOP would be the party of white privilege, and the Democrats, of minority rights. "States' rights"—a euphemism for segregation—became the new Republican rallying cry. As I now look back with the clarity of hindsight, I realize that, whatever Republican candidates claimed to stand for, what a lot of their voters heard was: this is someone who will put minorities in their place. Or who, at the very least, will not grant them any more rights, as the Democrats would do. I am now convinced that coded racial appeals—those dog whistles—had at least as much, if not more, to do with the electoral success of the modern Republican Party than all of the domestic and foreign policy proposals crafted by well-intentioned analysts like me. This is what liberals have been saying for decades in accusing the Republican Party of racism. I never believed them. Now I do, because Trump won by making the racist appeal, hitherto relatively subtle, obvious even to someone like me, who used to be in denial. The polite term for the voters that Republicans appeal to is "Jacksonians" after the populist president, general, and slave-owner Andrew Jackson. The more accurate description is white nationalists.
REPUBLICAN PRESIDENTS, in fairness, have proven a lot more moderate in office than the red-hot rhetoric of the campaign trail would suggest. Richard Nixon pursued a "Southern Strategy" to woo whites in the South and invaded Cambodia, but he also created the Occupational Safety and Health Administration and the Environmental Protection Agency, instituted wage and price controls, launched openings to Moscow and Beijing, and pulled US troops out of Vietnam. Gerald Ford continued Nixon's détente, much to the fury of the right. Ronald Reagan opposed the great civil rights legislation of 1964–1965, vilified "welfare queens" (who were presumably African American), and began his general election campaign in 1980 by proclaiming his adherence to "states' rights" just outside Philadelphia, Mississippi, the very town where in 1964 three civil rights activists had been murdered by the Ku Klux Klan. As president, he delivered on conservative promises by cutting taxes and rebuilding the military, but he also raised taxes, failed to cut spending, legalized undocumented immigrants, and launched negotiations with Soviet leader Mikhail Gorbachev that were viewed as naïve by critics on the right. After leaving office, he even publicly supported a ban on the sale of assault rifles—a position that no prominent Republican dares to espouse today.
His successor, George H. W. Bush, shamefully catered to racist sentiment in 1988 by allowing his campaign, led by the unscrupulous Lee Atwater, to demonize Michael Dukakis by association with Willie Horton—a black convict in Massachusetts who, on furlough, raped a white woman and assaulted her fiancée. (The furlough program had actually been started by Dukakis's Republican predecessor.) But in office, Bush the elder proved to be the consummate moderate. He offended conservative sensibilities by urging the Soviet Union to go slow in its dissolution, refused to march on Baghdad after victory in the Gulf War, signed the Clean Air Act and the Americans with Disabilities Act, and, worst of all from the conservative perspective, agreed to raise taxes in return for spending cuts. A substantial portion of the GOP rebelled against Bush after he broke his "no new taxes" pledge, even though his willingness to raise taxes and cut spending set the stage for balanced budgets and a robust economic recovery.
His son, George W. Bush, the last "normal" Republican president, called himself a "compassionate conservative" and worked with Democrats to pass the No Child Left Behind Act, an expansion of Medicare, and a bank bailout during the financial meltdown of 2008. Bush even tried to pass immigration reform and went out of his way to visit a mosque after 9/11—acts that probably did as much to alienate the hardcore right as his mishandling of Iraq and Hurricane Katrina did to repel the rest of the country.
The very moderation of Republican presidents has stoked the fury of the right. The pattern was set early on, in 1964, with Phyllis Schlafly's best-selling tract _A Choice Not an Echo_ , which would help launch the conservative movement. Her eccentric list of grievances ranged from resurrecting Joe McCarthy's discredited claims that the State Department and CIA were permeated with "Communist agents" to a new complaint: she thought that it was scandalous for President Lyndon Johnson to award the Medal of Freedom to the distinguished literary critic Edmund Wilson because "he has had four wives."
Schlafly was baffled why Republican candidates had lost presidential elections in 1936, 1940, 1944, 1948, and 1960. It could not be that Democrats fielded more attractive candidates or that most Americans did not share her far-right ideology. "It wasn't any accident," she wrote, ominously, of GOP setbacks. "It was planned that way. In each of their losing presidential years, a small group of secret kingmakers, using hidden persuaders and psychological warfare techniques, manipulated the Republican National Convention to nominate candidates who would sidestep or suppress the key issues." These nefarious "kingmakers" were New York financiers who only pretended to be Republicans but in fact favored "a continuation of the Roosevelt-Harry Dexter White-Averell Harriman-Dean Acheson-Dean Rusk policy of aiding and abetting Red Russia and her satellites." Harry Dexter White was a Soviet agent, whereas Harriman, Acheson, and Rusk were Democratic Cold Warriors determined to contain the Soviet threat, but to Schlafly there was no difference between them. And how did these "kingmakers" manipulate the GOP to ensure the defeat of its candidates? By promulgating "false slogans" such as "Politics should stop at the water's edge," "We must unite behind our President who has sole power in the field of foreign affairs," and "Foreign policy should be bipartisan." In other words, for Schlafly the very idea of bipartisanship was evidence of incipient treason.
This was not the ranting of some marginal oddball. Schlafly, the recipient of undergraduate and law degrees from Washington University and a master's degree from Harvard/Radcliffe, was one of the leading lights of the right. In the 1970s she would lead the successful campaign against the Equal Rights Amendment. _A Choice Not an Echo_ sold millions of copies. Her work drew on a long tradition of conspiratorial literature that claimed to see the manipulations of the Jesuits, Freemasons, Illuminati, and other bogeymen in the unfolding of American history. Trump's claim that he is going to "Make America Great Again" after it has been betrayed by disloyal elites is only the latest manifestation of this populist derangement.
THE HISTORY OF THE Republican Party over the past several decades is the story of moderates being driven out and conservatives taking over—and then of those conservatives in turn being ousted by those even further to the right. In the 1960s and 1970s there were many liberal or centrist Republicans such as Governors Nelson Rockefeller of New York and George Romney of Michigan (the father of Mitt Romney), New York mayor John Lindsay, and Senators Edward Brooke of Massachusetts, Mark Hatfield of Oregon, William Scranton of Pennsylvania, John Chaffee of Rhode Island, Jacob Javits of New York, and Clifford Case of New Jersey. They supported civil rights and environmental protection while also favoring an internationalist foreign policy, a tough-on-crime policy, and fiscal rectitude, at least in theory. (In practice, Rockefeller and Lindsay proved fiscally profligate, but so did many Republicans who were far more conservative.) Movement conservatives often displayed more venom against these apostates than against liberal Democrats, much as Mensheviks and Bolsheviks hated each other more than they hated their capitalist enemies. One by one, the "Rockefeller Republicans" were driven out, leaving the Democrats in control of most of the northeastern and West Coast states.
The GOP became the party of midwestern isolationists and southern segregationists—a marriage of Robert Taft Jr. and Strom Thurmond—even if the isolationist strain was muted as long as the Communist threat existed. Where once the GOP had been a "big tent" party, it now became an ideological conservative organization, and each generation of the right is more extreme than the one that came before it. A telling moment came in 1996, when the Republican presidential nominee, Bob Dole, visited an aged Barry Goldwater in Arizona. Once upon a time, Dole and Goldwater had defined the Republican right, but by 1996, Dole joked, "Barry and I—we've sort of become the liberals." "We're the new liberals of the Republican Party," Goldwater agreed. "Can you imagine that?"
The rightward lurch of the GOP was symbolized by Newt Gingrich, a conservative firebrand from the South. He waged unrelenting war on Democrats such as House Speaker Jim Wright and masterminded the campaign that allowed Republicans to claim control of the House in 1994 for the first time in forty years. I remember celebrating that glorious occasion at an election-night cocktail party on the Upper West Side with fellow staffers from the _Wall Street Journal_ editorial page. For us, the moment felt as glorious as the Bolshevik Revolution in 1917 had been for leftists. Like the radical journalist Lincoln Steffens, we had seen the future and we were convinced that it would work. Only it didn't. Like so many ideologues, Gingrich proved incapable of governing. He forced a government shutdown in 1995 to make President Bill Clinton agree to Republican budget priorities but had to back down after his move proved to be a public relations debacle. Gingrich was then toppled by his own caucus.
Before long, the Republican Party was shaken by a new insurgency, the Tea Party, which arose during the Obama administration. Its zealots made Gingrich seem like a squish by comparison. The new icon of the right, Senator Ted Cruz, forced a government shutdown of his own in 2013 to repeal Obamacare and pave the way for his presidential bid. This was another gambit that failed miserably but did not shake the far right's domination of the Republican Party. As E. J. Dionne noted in _Why the Right Went Wrong_ , "between January 1995 and January 2015, the proportion of Republicans who called themselves 'very conservative' nearly doubled, from 19 percent to 33 percent."
The ascendance of these extreme views increasingly made the House Republican caucus ungovernable. The three dozen or so far-right members of the House Freedom Caucus drove House Speaker John Boehner, himself a conservative, into retirement in 2015, after his deputy, House Majority Leader Eric Cantor, lost his reelection bid to a Tea Party insurgent in Virginia the previous year. Boehner's successor as speaker, Paul Ryan, lasted only three years and also found himself accused of being insufficiently conservative in spite of having a perfect score from the National Right to Life Committee and the National Federation of Independent Business and an 89 percent rating from the American Conservative Union. Ryan's downfall signaled the final repudiation of an optimistic brand of Reaganesque conservatism focused on enhancing economic opportunities at home and promoting democracy and free trade abroad. The Republican Party would now be defined by the tenebrous vision of Donald Trump, with his depiction of Democrats as America-hating, criminal-coddling traitors and his invective against immigrants, Mexicans, and Muslims. Trump had beaten the Republican rabble-rousers at their own game. Completely unrestrained by logic or morality, he could go where even the most extreme conservatives had previously feared to tread.
THE MODERN HISTORY OF the GOP is a warning to be careful of who you pretend to be because sooner or later you will become that person. Republicans have long flirted with populism, conspiracy-mongering, and know-nothingism. This is why they became known as the "stupid party." Stupidity is not an accusation that could be hurled against such early Republicans as Abraham Lincoln, Theodore Roosevelt, Elihu Root, and Charles Evans Hughes. But by the 1950s, it had become an established shibboleth that the "eggheads" were for Adlai Stevenson and the "boobs" for Dwight D. Eisenhower—a view endorsed by Richard Hofstadter's 1963 book _Anti-Intellectualism in American Life_ , which contrasted Stevenson, "a politician of uncommon mind and style, whose appeal to intellectuals overshadowed anything in recent history," with Eisenhower—"conventional in mind, relatively inarticulate." The Kennedy presidency, with its glittering court of Camelot, cemented the impression that it was the Democrats who represented the thinking men and women of America.
Rather than run away from the anti-intellectual label, Republicans embraced it for their own political purposes. In his "time for choosing" speech, Ronald Reagan said that the issue in the 1964 election was "whether we believe in our capacity for self-government or whether we abandon the American Revolution and confess that a little intellectual elite in a far-distant Capitol can plan our lives for us better than we can plan them ourselves." Richard M. Nixon appealed to the "silent majority" and the "hard hats," while his vice president, Spiro T. Agnew, issued slashing attacks on an "effete corps of impudent snobs who characterize themselves as intellectuals" and the "nattering nabobs of negativism." (The latter phrase, ironically, was written by speechwriter William Safire, who would go on to establish a reputation as a libertarian columnist and grammarian for the conservatives' bête noire, the _New York Times_.) William F. Buckley Jr. famously said, "I should sooner live in a society governed by the first 2,000 names in the Boston telephone directory than in a society governed by the 2,000 faculty members of Harvard University." More recently, George W. Bush joked at a Yale commencement: "To those of you who received honors, awards and distinctions, I say, well done. And to the C students I say, you, too, can be president of the United States."
Many Democrats took all this at face value and congratulated themselves for being smarter than the benighted Republicans. Here's the thing, though: the Republican embrace of anti-intellectualism was, to a large extent, a put-on—just like their espousal of far-right rhetoric on the campaign trail. In office they proved far more moderate and intelligent. Eisenhower may have played the part of an amiable duffer, but he may have been the best prepared president we have ever had—a five-star general with an unparalleled knowledge of national security affairs. When he resorted to gobbledygook in public, it was in order to preserve his political room to maneuver. Reagan may have come across as a dumb thespian, but he spent decades honing his views on public policy and writing his own speeches. Nixon may have burned with resentment of "Harvard men," but he turned over foreign policy and domestic policy to two Harvard professors, Henry A. Kissinger and Daniel Patrick Moynihan, while his own knowledge of foreign affairs rivaled Ike's.
There is no evidence that Republican leaders have been demonstrably dumber than their Democratic counterparts. During the Reagan years, the GOP briefly became known as the "party of ideas" because it harvested so effectively the intellectual labor of conservative think tanks like the American Enterprise Institute and the Heritage Foundation and publications like the _Wall Street Journal_ editorial page, _National Review_ , and _Commentary_. Scholarly policymakers such as George P. Shultz, Jeane J. Kirkpatrick, and Bill Bennett held prominent posts in the Reagan administration, a tradition that continued into the George W. Bush administration—amply stocked with the likes of Paul D. Wolfowitz, John J. Dilulio Jr., and Condoleezza Rice. This was the Republican Party that attracted me as a teenager in the 1980s and maintained my loyalty for decades to come.
In recent years, however, the Republicans' relationship to the realm of ideas has become more and more attenuated as talk-radio hosts and television personalities have taken over the role of defining the conservative movement that once belonged to thinkers like William F. Buckley Jr., Irving Kristol, Norman Podhoretz, and George F. Will. The Republicans' populist pose has become all too real. A sign of the times is that Bill Bennett, possessor of a PhD from the University of Texas at Austin and a JD from Harvard Law School, stoops to attack George Will, a Princeton PhD, for his criticism of Vice President Mike Pence by mocking his "penchant for writing columns filled with big words that most Americans never use and can't even define." Presumably a dictionary counts as elitist foppery.
The turning point in the Republican transformation was the rise of Sarah Palin after John McCain made the mistake of selecting her as his running mate in 2008—a move that he later regretted, wishing he had selected his friend, Democratic senator Joe Lieberman of Connecticut, instead. Palin showed that she was a dim bulb when she was asked during the campaign which sources she relied on for the news. Caught off-guard, she could not answer and had to deflect with unconvincing generalities: "I have a vast variety of sources." This was akin to an admission that she did not read newspapers or magazines beyond, possibly, _Field & Stream_ or _Guns & Ammo_. I can't say I was terribly surprised. As a McCain foreign policy adviser, I had briefed her and found her to be nonresponsive and uninterested in foreign policy issues. The most memorable takeaway from our meeting at a midtown Manhattan hotel was that she wore earrings in the shape of the state of Alaska.
Palin's lack of preparation for high office could perhaps be excused as the provincialism of a small-state governor who had not asked for the national spotlight (Alaska's population is smaller than San Francisco's). But rather than return to her duties after the election or try to educate herself on the issues, Palin resigned as governor in 2009 and sought to cash in on her celebrity by becoming a full-time media personality. She then proceeded to litter the land with inanities that have few parallels in our history. There was no malapropism that she did not employ. She even invented a new word—"refudiate"—by conflating "repudiate" and "refute" and tried to suggest that she was a Shakespearean sage who was enlarging our vocabulary. A sample of Palin's other bizarre statements: "Well, if I were in charge, they would know that waterboarding is how we'd baptize terrorists." "But obviously, we've got to stand with our North Korean allies." "We can send a message and say, 'You want to be in America, A, you'd better be here legally or you're out of here. B, when you're here, let's speak American.'" "I think on a national level, your Department of Law there in the White House would look at some of the things that we've been charged with and automatically throw them out." (There is no Department of Law in the White House or anywhere else.)
Conservatives applauded this inanity, making Palin one of the biggest stars on the right-wing rubber-chicken circuit until she was eclipsed by the rise of the even more vulgar and vacuous Donald Trump. The rise of Palin and now Trump indicates that the GOP really truly has become the stupid party. Its primary vibe has become one of indiscriminate, unthinking, all-consuming anger.
THAT ANGER IS STOKED by the "alternative media" of the right. Its origins can be traced back to the founding of the newspaper _Human Events_ in 1944, Regnery Publishing in 1947, and _National Review_ in 1955. In later years, two publishing houses—Basic Books and the Free Press—played an important role in producing works of conservative scholarship, including many tomes that I read while growing up, such as Alan Bloom's _The Closing of the American Mind_ , Charles Murray's _Losing Ground: American Social Policy, 1950–1980_ , George Gilder's _Wealth and Poverty_ , and James Q. Wilson's _Bureaucracy_.
The alternative media did not become a mass phenomenon until Ronald Reagan's Federal Communications Commission decided in 1987 to stop enforcing the "fairness doctrine," a 1949 regulation which had mandated that all television and radio broadcast outlets had to present both sides of controversial public issues. This deregulatory decision made possible the debut in 1988 of Rush Limbaugh's national radio show, which did not pretend to offer anything but a conservative perspective on the news. Revealingly, Limbaugh called his fans "dittoheads" because they mindlessly echoed his prejudices—or he theirs; the pandering went both ways. Many other right-wing "talkers" followed.
I worried about the impact of the talk-show populists as far back as 1994, when I wrote a _Wall Street Journal_ op-ed headlined "Down with Populism!" shortly after Republicans had taken control of the House for the first time in forty years. I argued that the GOP should not "'Rush' to embrace talk show democracy" because of the dangers of mob rule. I quoted my boyhood favorite, H. L. Mencken: "Least of all do I admire the puerile, paltry shysters who constitute the majority of Congress. But I confess frankly that these shysters, whatever their defects, are at least appreciably superior to the mob." The expression of mob rule I was most worried about was a new conservative TV network called National Empowerment Television that has long since faded away. I had no idea that Fox News Channel would be founded in two years' time and that it would make my worst fears of populism run amok come true. Coincidentally, 1996 was also the year that the Drudge Report, an online bulletin board for right-wing fever dreams, was launched.
Limbaugh, Fox, and Drudge still remain three of the most popular outlets on the right, but they have been joined by radio hosts such as Mark Levin and Michael Savage, celebrity authors and talking heads such as Ann Coulter, Milo Yiannopoulos, and Dinesh D'Souza, and websites such as Breitbart News, TheBlaze, Infowars, and Newsmax. The original impetus for these outlets was to offer a different viewpoint that people could not get from the more liberal TV networks, newspapers, and magazines. But soon the alternative media moved from propounding their own analyses to concocting their own "facts," turning into an incubator of outlandish conspiracy theories such as "Hillary Clinton murdered Vince Foster," "Barack Obama Is a Muslim," or even "Michelle Obama Is a Man."
The career of Dinesh D'Souza, one of the right's biggest media stars in spite of being a convicted felon (who has now been pardoned by President Trump), is indicative of the downward trajectory of conservatism. After a checkered career in conservative journalism at Dartmouth, he made his name with _Illiberal Education_ , a well-regarded 1991 book published by the Free Press, which denounced political correctness and championed liberal education. Then he wrote a widely panned 1995 book, also from the Free Press, claiming that racism was no more. It was all downhill from there. In 2014 he pleaded guilty to breaking campaign finance laws. More recently, as the _Daily Beast_ notes, he has become a conspiratorial crank who has suggested that the white supremacist rally in Charlottesville was staged by liberals and that Adolf Hitler, who sent fifty thousand homosexuals to prison, "was NOT anti-gay." D'Souza managed to sink even lower in February 2018 by mocking stunned Parkland school-shooting survivors after the Florida legislature defeated a bill to ban assault weapons: "Worst news since their parents told them to get summer jobs." He was joined in this repugnant japery by Laura Ingraham, who made fun of school-shooting survivor David Hogg for not getting into the college of his choice (she later apologized), and by Jamie Allman, a Sinclair broadcasting commentator who was fired for saying that he would like to sexually assault Hogg with a "hot poker."
It is hard to imagine anything more cruel and heartless, but for these opportunists it's all in a day's work. As D'Souza wrote in his 2002 book _Letters to a Young Conservative_ , "One way to be effective as a conservative is to figure out what annoys and disturbs liberals the most, and then keep doing it." That, in a nutshell, is the credo of today's high-profile conservatives: say anything to "trigger" the "libtards" and "snowflakes." The dumber and more offensive, the better. Whatever it takes to get on (and stay on) Fox News and land the next book contract! Hence inane outbursts such as this tweet from Fox News's twenty-five-year-old blonde commentator, Tomi Lahren: "Let's play a game! Go to Whole Foods, pick a liberal (not hard to identify), cut them in line along with 10–15 of your family members, then take their food. When they throw a tantrum, remind them of their special affinity for illegal immigration. Have fun!" (I only mention Lahren's appearance because the employment of female hosts in short skirts, and preferably with blonde hair, is such an integral part of Fox's strategy to attract the elderly white men who form its core audience.)
Such rhetorical sallies are as lucrative as they are illogical. D'Souza has grossed tens of millions of dollars with documentaries attacking Hillary Clinton and Barack Obama as anti-American subversives. Sean Hannity makes roughly $30 million a year and flies on his own private jet even while railing against "overpaid" media elites.
Naturally, just as drug addicts need bigger doses over time, these outrage artists must be ever more transgressive to get the attention they crave. Ann Coulter's book titles have gone from accusing Bill Clinton of _High Crimes and Misdemeanors_ to accusing all liberals of _Treason_ , of being _Godless_ and even _Demonic_. Her latest assault on the public's intelligence was called _In Trump We Trust: E Pluribus Awesome!_ If this is what mainstream conservatism has become—and it is—count me out.
WHILE DINESH D'SOUZA, Ann Coulter, Laura Ingraham, Alex Jones, and many others have played a role in dumbing down conservatism, no institution has been more harmful in this regard than the Fox News Channel, whose debut must count as one of the most baleful milestones in modern American political history. Fox has turned itself into the American version of RT. Not only does Fox usually go to great lengths to avoid criticizing President Trump but it also regularly peddles insidious conspiracy theories on his behalf. To try to undermine the "incontrovertible" evidence that the Russians hacked into the Democratic National Committee, for example, some Fox commentators pinned the blame on a DNC staffer named Seth Rich, even going so far as to claim that his murder—ascribed by District of Columbia police to a botched robbery—was the work of the Democrats. Rich's parents sued Fox for the "pain and anguish" inflicted on them. (Fox retracted the story and moved on to propagating other crazy claims.)
The Seth Rich hoax is only the tip of the conspiratorial iceberg at Fox, which has also pushed claims that Obama wasn't born in America, that Obamacare would create "death panels," that Hillary Clinton sold America's uranium to Russia, and that a Deep State is plotting against Trump. (Little wonder that, according to one poll, 74 percent of Americans believe in the existence of a Deep State—a concept that Trump borrowed from Egypt and Turkey to suggest that his own government is plotting against him.) Fox, naturally, has taken the lead in smearing Special Counsel Robert Mueller and calling for his investigation—which they call, echoing Trump, "a witch hunt"—to be terminated before its conclusion.
What makes Fox's ravings so scary is that they are not just influencing the public but also the president. Matthew Gertz of Media Matters for America found an insidious feedback loop between Trump and the TV personalities he watches so faithfully. Many of the president's deranged tweets—e.g., his claim that his "nuclear button" is "much bigger & more powerful" than Kim Jong Un's or that Hillary Clinton aide Huma Abedin should be imprisoned—are lifted straight from Fox. When Fox reported that a "caravan" of Central American migrants was about to invade America, Trump echoed its hysteria. On reflection, then, it's not quite right to say that Fox is Trump's RT. Putin is smart enough not to believe his own propaganda. Trump isn't. Fox may be said to have created Trump, but now the president and the network have such a symbiotic relationship that it's not clear who is in charge. Trump even relies on Fox talking heads as trusted advisers. Sean Hannity, the _Washington Post_ reports, "is so close to Trump that some White House aides have dubbed him the unofficial chief of staff." Hannity even shared a lawyer with Trump—a fact that he did not feel compelled to disclose to his viewers. Such disclosure would be de rigueur for any journalistic organization. But that's not what Fox is—it's a disseminator of disinformation.
I GOT MY OWN SMALL TASTE of Fox's modus operandi in July 2017 when I appeared on Tucker Carlson's show from a studio in Los Angeles. I was there—I thought—to discuss President Trump and US policy toward Russia. Instead, as I later recounted in _Commentary_ , I got the equivalent of a barrel of raw sewage dumped on my head. I should have known what I was getting into when the Fox producers refused to give me any makeup; they wanted to make sure that I did not look as good as the heavily powdered and carefully coiffed host in Washington.
Carlson was still smarting from a confrontation the previous night with Ralph Peters, a retired army officer who had accused him of sounding like "Charles Lindbergh in 1938" for his advocacy of an alliance with Russia. Because I had retweeted Peters's comment, Carlson appeared determined to take out his fury on me. His very first question was: "To dismiss anyone who doesn't share your view as a Nazi sympathizer seems cheap and a shortcut and not really befitting a self-described genius like yourself. Why would you say something like that?"
"Well, rest assured, Tucker, I'm not actually saying that you are a Nazi sympathizer," I replied, while wondering when I had described myself as a genius. I went on to state my view that Russia is a major threat to the United States, not a potential ally. I was "very disturbed," I added, to hear Carlson "yukking it up" at the top of his show with guest Mark Steyn—a clever writer I had once invited to write for the _Wall Street Journal_ editorial page—about Donald Trump Jr.'s eagerness to accept Russian help in the 2016 election.
During the course of the next ten or so minutes, Carlson and I traded barbs about Russia, Syria, Iran—and my foreign policy perspicacity or lack thereof. I tried to stick to the issues, but he kept interrupting me with smirky sarcasm, obnoxious laughter, and ad hominem insults. My brain raced to formulate retorts as I heard Carlson's rapid-fire abuse in my earpiece, the image on the monitor in front of me disorientatingly delayed by five or six seconds. It was hard to get in a word edgewise. "You have been consistently wrong in the most flagrant and flamboyant way for over a decade," he charged. This turned out to be principally a reference to the invasion of Iraq, which he had also supported before turning against it. "You're humiliating yourself," he said, and "this is why nobody takes you seriously." (If no one takes me seriously, why was I invited on his show?) I was so discredited, Carlson said, that maybe I "should choose another profession. Selling insurance, house painting, something you're good at." Clearly, he had an inflated impression of my house-painting and insurance-selling skills.
Ironically, while dishing out a nonstop stream of invective, Carlson kept accusing _me_ of not engaging in substantive debate: "This is precisely the style of debate that prevents people from taking you seriously. . . . It's almost impossible to have a conversation with you because your responses are so childish. . . . You are incapable of giving a factual answer."
Feeling the same adrenaline charge as when I got into fights in elementary school and junior high school, I tried to give as good as I got. For example, when Carlson said that "your judgment has been clouded by ideology," I shot back that _his_ "judgment has been clouded by ratings because you feel compelled to be a spokesman for Donald Trump in order to win ratings on the Fox News Channel." I thought the exchange was a draw, despite Carlson's home-court advantage (it's not easy debating with someone you cannot look in the eye), but I left aghast at his rudeness and unprofessionalism. If this is how he treats guests on his TV show, I'm afraid that if I were ever invited to his house—admittedly unlikely—I would be served a hemlock cocktail.
Maybe there's a good case to be made for an "America First" foreign policy, but Carlson wasn't making it. His shtick is sarcasm, condescension, and mock-incredulous double takes. And he doesn't hesitate to lie when it suits his purposes. After our interview, his team posted a clip on Facebook under this headline: "Col. Ralph Peters suggested that anybody who disagrees with him on Russia would also be a Hitler sympathizer. Historian Max Boot agrees." Neither Peters nor I ever said any such thing; in fact, I'd said the opposite on his show.
Carlson's circus act gets ratings and so do the similar performances of Sean Hannity, Laura Ingraham, Lou Dobbs, Jeanine Pirro, and other Fox "personalities." But the cost is high. They are debasing political discourse and degrading our democracy in return for seven- and eight-figure paydays. No one did a better job of revealing what Fox has become than Ralph Peters, a conservative uber-hawk and former army intelligence officer who in early 2018 resigned in disgust as a Fox commentator. In a scorching letter of resignation leaked to BuzzFeed, he wrote, "Fox has degenerated from providing a legitimate and much-needed outlet for conservative voices to a mere propaganda machine for a destructive and ethically ruinous administration." He went on: "Four decades ago, I took an oath as a newly commissioned officer. I swore to 'support and defend the Constitution,' and that oath did not expire when I took off my uniform. Today, I feel that Fox News is assaulting our constitutional order and the rule of law, while fostering corrosive and unjustified paranoia among viewers. Over my decade with Fox, I long was proud of the association. Now I am ashamed."
I TOO AM ASHAMED—not because I've ever been a Fox employee but because I used to view it as a benign force that was merely selling the conservative agenda, developed by policy wonks like myself, to a larger audience. Now I know better. This is part of my general awakening to disturbing trends not just in the conservative movement but also in the larger American society that conservatism grows out of and reflects.
In college, as I first recounted in _Foreign Policy_ , I used to be one of those smart-alecky young conservatives who would scoff at the notion of "white male privilege" and claim that anyone propagating such concepts was guilty of "political correctness." As a Jewish refugee from the Soviet Union, I felt it was ridiculous to expect me to atone for the sins of slavery and segregation, to say nothing of the household drudgery and workplace discrimination suffered by women. I wasn't racist or sexist. (Or so I thought.) I hadn't discriminated against anyone. (Or so I thought.) My ancestors were not slave owners or lynchers; they were more likely to have been victims of the pogroms.
I saw America as a land of opportunity, not a bastion of racism or sexism. I didn't even think that I was a "white" person—the catchall category that has been extended to include everyone from a _Mayflower_ descendant to a recently arrived illegal immigrant from Ireland. I was a newcomer to America who was eager to assimilate into this wondrous new society, and I saw its many merits while blinding myself to its dark side.
Well, live and learn. A quarter-century is enough time to examine deeply held shibboleths and to see if they comport with reality. In my case, I have concluded that my beliefs were based more on faith than on a critical examination of the evidence. In the last few years, in particular, it has become impossible for me to deny the reality of discrimination, harassment, even violence that people of color and women continue to experience in modern-day America from a power structure that remains for the most part in the hands of straight white males. People like me, in other words. Whether I realize it or not, I have benefited from my skin color and my gender—and those of a different gender or sexuality or skin color have suffered because of it.
This sounds obvious, but it wasn't clear to me until recently. I have had my consciousness raised. Seriously.
This doesn't mean that I agree with America's harshest critics—successors to the New Left of the 1960s who saw this country as an irredeemably fascist state that they called "Amerikkka." Judging by historical standards or those of the rest of the world, America remains admirably free and enlightened. Minorities are not being subject to ethnic cleansing like the Rohingya in Burma. Women are not forced to wear all-enveloping garments like in Saudi Arabia. No one is jailed for criticizing our supreme leader as in Russia.
We are becoming more aware of oppression and injustice, which has long permeated our society, precisely because of growing agitation to do something about it. Those are painful but necessary steps toward creating a more equal and just society. But we are not there yet, and it is wrong to pretend otherwise. It is even more pernicious to cling to the conceit, so popular among Donald Trump's supporters, that straight white men are the "true" victims, because their unquestioned position of privilege is now being challenged by "uppity" women, gays, and people of color.
Similarly, I reject the premise, so popular in conservative circles, that the white identity politics promoted by Trump is simply an understandable response to the identity politics of African Americans, Latinos, Asians, and other minorities. That is only a compelling argument if your historical consciousness begins in the sixties—the decade that conservatives like to blame for everything they dislike about modern America. In fact, white identity politics has been pervasive ever since the European settlement of North America in the sixteenth century. Southern states had apartheid policies that lasted legally until the 1960s, backed up by racist police and terrorist groups such as the Ku Klux Klan that stood ready to imprison, lynch, or otherwise eliminate anyone who ran afoul of the white power structure. The civil rights revolution was a long-overdue attempt to right the scales and deliver justice for Americans of color, but it was only partially successful—and it left a long legacy of smoldering white resentment that was exacerbated by measures designed to promote integration such as school busing and affirmative action. Seen in the context of the long sweep of history, the identity politics of African American, Latino, or Asian American activists in recent years is a reaction to, rather than the cause of, white-nationalist sentiment. Again, this isn't news to many people. But it is a new realization for me.
I used to take a reflexively pro-police view of arguments over alleged police misconduct, thinking that cops were getting a bum rap for doing a tough, dangerous job. I still have admiration for the huge majority of police officers, but there is no denying that some are guilty of mistreating the people they are supposed to serve. Not all the victims of police misconduct are minorities—witness a blonde Australian woman shot to death by a Minneapolis police officer after she called 911, or an unarmed white man shot to death by a Mesa, Arizona, officer while crawling down a hotel hallway—but a disproportionate share are. The videos do not lie. One after another, we have seen the horrifying video evidence of cops arresting, beating, even shooting black people who were doing absolutely nothing wrong or who were stopped for trivial misconduct. For African Americans, and in particular African American men, infractions like jaywalking or speeding or selling cigarettes without tax stamps can incite corporal, or even capital, punishment without benefit of judge or jury. African Americans have long talked about being stopped for "driving while black." I am ashamed to admit I did not realize what a serious and common problem this was until the videotaped evidence emerged. The iPhone may well have done more to expose racism in modern-day America than the NAACP.
Of course, the problem is not limited to the police; they merely reflect the racism of our society, which is not as severe as it used to be but remains real enough. I realized how entrenched this problem remains when an African American friend—a well-educated, well-paid, well-dressed woman—confessed that she did not want to walk into a department store carrying in her purse a pair of jeans that she planned to give to a friend later in the day. Why not? Because she was afraid that she would be accused of shoplifting! This is not something that would occur to me, simply because the same suspicion would not attach to a middle-aged, middle-class white man. Likewise, I would never be asked to leave a Starbucks, much less arrested for loitering, even if I didn't buy anything. But that was the fate of two African American men in Philadelphia in 2018. Trump's victory has revealed that racism and xenophobia are more widespread than I had previously realized. Previous Republicans may have dog-whistled to the racist right; with Trump it's more of a wolf whistle.
As for sexism, its scope has been made plain by the horrifying revelations of widespread harassment, assault, and even rape perpetrated by powerful men from Hollywood to Washington. As with the revelations of police brutality, so too with sexual harassment: I am embarrassed and chagrined that I did not understand how bad the problem is. I had certainly gotten some hints from my female friends of the kind of harassment they have endured, but I never had any idea it was this bad or this common—or this tolerated. I now realize, thanks to the #MeToo movement, something I should have learned long ago: that feminist activists had a fair point when they denounced the corrosive impact of a "patriarchal society," even if choosing to become a mother and homemaker is hardly evidence of oppression, as some radicals might suggest. While the abuse of, and discrimination against, women is less severe than it used to be, it remains a major problem in spite of the impressive strides the United States has taken toward greater gender equality.
This doesn't mean that I am about to join the academic brigade in protesting "microaggressions" and "cultural appropriations" and agitating against free speech. I remain a classical liberal, and I am disturbed by attempts to infringe on freedom of speech in the name of fighting racism, sexism, or other ills. But I no longer think, as I once did, that "political correctness" is a bigger threat than the underlying racism and sexism that continues to disfigure our society decades after the civil rights and women's rights movements. If the Trump era teaches us anything, it is how far we still have to go to realize the "unalienable Rights" of all Americans to enjoy "Life, Liberty and the pursuit of Happiness," regardless of gender, sexuality, religion, or skin color.
WHEN I FIRST WROTE about my "awakening" to "white privilege" in December 2017, I got a lot of positive responses from moderate liberals and honest conservatives—but also quite a bit of blowback. The far right naturally mocked me for "virtue signaling" and diagnosed me as suffering from "Trump derangement syndrome"—ironic charges given that if anyone has been deranged by Trump, it is all of the conservatives who jettisoned their supposed beliefs to support him. Tucker Carlson sneered: "Max Boot will say anything if they just let him invade Iran." (I never advocated invading Iran, but of course if Trump did invade, Tucker would support him.) But the far left was also unimpressed, basically taking the attitude of "What took you so long?" A self-described "anarcho-psychonaut" (whatever that is) wrote in a blog post that I was "spectacularly evil" underneath the headline: "Iraq-Raping Neocons Are Suddenly Posing as Woke Progressives to Gain Support." Former CNN anchor Soledad O'Brien tweeted a more civilized version of the leftist lament: "Sorry, but why is this 'gutsy'? Admitting you've walked around your whole life oblivious to even the basic experiences of women and people of color is just . . . pathetic."
O'Brien may be surprised to hear me say it, but she has a good point (even if I wasn't the one who claimed that my writing was "gutsy"). It is pathetic, I suppose, that I didn't focus on the underbelly of American society—and of the conservative movement in particular—until relatively recently. But then, as George Orwell wrote: "To see what is in front of one's nose needs a constant struggle." I don't expect any commendations for seeing what should have been obvious all along. All I can say in my own defense is that many never see it at all, even when what is in front of their noses is a president who evinces sympathy for neo-Nazis, dictators, alleged child molesters, and accused wife-beaters.
Most conservatives appear willing, even eager, to shed earlier attempts to moderate their movement and to make it more acceptable to polite society. They are embracing the kind of extremism personified in the past, in their different ways, by the likes of Joe McCarthy, Barry Goldwater, Strom Thurmond, and Phyllis Schlafly. Trump benefits from, and accelerates, this move to the fringe, and he channels it in a white-nationalist, rather than a libertarian, direction. But he did not start the trend—and it won't end when he is gone. Indeed, if the modern history of conservatism is any guide, Trump's successors might actually be worse than he is. If Trump has a saving grace, it is that he is so ignorant and impetuous. He is incapable of effectively implementing his worst impulses in the face of entrenched resistance from government professionals, the judiciary, and the press corps. A future Trump might be smarter and more disciplined, and thus more dangerous. That's a frightening thought, given how much damage even the scattershot Trump has done.
## _Epilogue_
## **THE VITAL CENTER**
I AM NO LONGER A REPUBLICAN. AM I STILL A CONSERVATIVE? Honestly, I'm not sure, because I don't know what conservatism stands for anymore beyond reflexive support of Donald Trump and all his malign works. I certainly don't like to call myself a conservative anymore because the movement has been so thoroughly taken over by Trump and his amen chorus. Even before the rise of Trump, however, the tide of extremism was rising on the right, although I chose to ignore the warning signs.
You tell me where I belong on the political spectrum:
* I am socially liberal. I am pro-LGBTQ rights and pro-choice. I am not religious but am respectful of those who are—as long as their beliefs do not impinge on anyone's individual rights.
* I am fiscally conservative. I think we need to reduce the deficit and get entitlement spending under control without, however, shredding the social safety net. The bipartisan Simpson-Bowles Commission came up with a plan in 2010 to accomplish this objective.
* I am in favor of free markets _and_ the welfare state. Far from being incompatible, the latter ensures the success of the former. Markets aren't perfect, and government has a responsibility to ameliorate their failings. The welfare state is an inherently conservative institution, as its creator, Otto von Bismarck, realized: the German chancellor pioneered in the 1870s governmental programs to take care of the sick, aged, and disabled in order to forestall the advance of socialism.
* I am pro–free trade. I think we should be concluding new trade treaties rather than pulling out of old ones, because free trade has not only contributed to America's prosperity but also enriched the entire world. But I have been reminded by Trump's victory that globalization has left many of our fellow citizens behind. They deserve government aid not only because it is the right thing to do but also, on Bismarckian grounds, because if they do not receive it they are more likely to fall prey to populist nostrums that will damage the entire country—and the world.
* I am pro-environment. I think that climate change is a major threat that we need to address, and I don't want to indiscriminately open federal lands to strip mining and oil digging.
* I am pro–gun control. I don't see any legitimate reason why civilians should be able to buy military-grade assault weapons or to buy any gun at all without passing an extensive background check and a test in firearms safety—the kind of requirements routinely imposed in other democracies that have much lower rates of gun violence.
* I am pro-immigration. As an immigrant myself, I think that immigrants are the source of America's greatness and that we would benefit from more of them. Rather than trying to kick out eleven million undocumented immigrants, we should offer them a path to citizenship so that they can become full-fledged members of our society. And rather than reduce legal immigration, we should increase it to address a shortage of native-born workers. But I am newly cognizant that the policies I favor could exacerbate a populist backlash—as has happened not only in the United States but also in a number of European countries. It is imperative to screen newcomers carefully and to make the case to white Americans that changing demography is no threat to their wellbeing.
* I am in favor of free speech and oppose the identity politics of both minority groups and the white majority. I believe in the melting pot, integration, and color blindness, while recognizing that America must struggle to erase the stain of racism. I oppose both overly restrictive campus speech codes _and_ the NFL's ban on kneeling to protest while the national anthem is played.
* I am strong on defense: I think we need to maintain a capable military to cope with multiple enemies—from near-peer threats such as China and Russia, to rogue states like Iran and North Korea, to non-state actors like Al Qaeda and ISIS. I believe it is imperative to maintain America's military deployments in the three centers of global power and wealth—Europe, the Middle East, and East Asia—to keep the peace and deter aggressors. But, as a chastened hawk, I also recognize that we should be cautious about the use of force and shy away from preventive wars.
* I am an internationalist: I believe it is in America's self-interest to promote and defend freedom and a rules-based international order, as we have been doing in one form or another since at least 1942. Above all, I believe that America needs to stand with our allies, especially democratic allies. Isolationism was not a viable option even in the 1930s; it is certainly not possible in today's interconnected world.
You would think these political views would make me unexceptional. Taken individually, each of these opinions scores overwhelming support from the American public. But there is no party that reflects this compendium of convictions. Because I hold these views, I am a political pariah—a man without a party. Neither Democrats nor Republicans are appealing to someone of my center-right outlook. I am politically homeless.
After years of thinking of myself as a "movement conservative," I now realize that I am actually a "Rockefeller Republican," although "Eisenhower Republican" is a better description, because Ike was a more competent and impressive figure. Whatever the name, this is a breed of centrist Republican that is as extinct as the dodo, at least at the national level. (There are still some Rockefeller Republicans, such as Maryland governor Larry Hogan and Massachusetts governor Charlie Baker, in blue states.) Which is why I'm not registered as a Republican anymore. Ironically, given how far to the right the GOP has moved, even earlier icons of conservatism such as Barry Goldwater and Ronald Reagan, if they were still alive, would now be derided as RINOs—Republicans in name only.
One sign of my isolation is that I am regularly pilloried by both the far left and far right, often in terms that are virtually indistinguishable. I'm used to being described by the far left as a bloodthirsty neocon warmonger for the original sin of having supported the invasion of Iraq, along with 72 percent of the American public. This is the same mindset exhibited by Randa Jarrar, the California State University, Fresno, professor who celebrated the death of First Lady Barbara Bush, writing "I'm happy the witch is dead," because she "raised a war criminal." This kind of intolerance and incivility from the left only encourages more of the same from the Trumpist right. If Jarrar did not exist, Fox News Channel would have to invent her, so perfectly does she fit its narrative of the loony left.
While still being excoriated by the far left as a war criminal, I am simultaneously vilified by the far right as a dangerous left-winger. David Horowitz's _FrontPage_ magazine accused me of going "full leftist" for acknowledging that racism and sexism remain pervasive problems. (Horowitz went from being a leftist radical in the 1960s to a rightist radical in more recent years; his ideology changes, but he remains consistent in his embrace of immoderation.) _Breitbart_ called me, with ironic quotation marks, the "Washington Post's ostensibly new 'conservative' columnist," because, among other apostasies, I support gun control and immigration. _American Greatness_ wrote that I am a "soulless, craven opportunist" whose "brain is broken," because I compared President Trump's indifference to the 2016 Russian election assault to a president ignoring 9/11. For the same offense, Jack Posobiec—an Internet troll notorious for pushing the theory that Hillary Clinton was running a child-sex ring out of a Washington pizza parlor—said I was "sick" and a "Russian propagandist." In the Orwellian language of the far right, someone who wants to combat Russian aggression is a "Russian propagandist," whereas someone who echoes Russian propaganda is putting "America first."
I respect #NeverTrumpers such as Bill Kristol and Tom Nichols who have remained Republicans because they hope to wrest that party back from the extremists. But I have concluded that the battle is lost, at least for the time being. Sadly, David Horowitz and Laura Ingraham, Dinesh D'Souza and Jack Posobiec, Donald Trump and Devin Nunes are far more representative than I am of the right today—and quite probably in the past too, even if it's taken me a long time to realize it.
Every day that goes by, I am ever more thankful that, after spending my entire adult life as a Republican, I left the GOP the day after Trump's election. I now ardently wish harm upon my former party because it has become an enabler of Trump's assault on the rule of law and the norms of civilized society. In the republic's hour of peril, most Republicans are either cheering the president's attack on liberal democracy or pretending it doesn't exist. I agree with my friends the centrist writers Benjamin Wittes and Jonathan Rauch when they call for voters to support Democrats "mindlessly and mechanically" because "the Republican Party, as an institution, has become a danger to the rule of law and the integrity of our democracy." I would not, however, call this option "mindless": supporting the opposition party is a mindful and considered response to a grotesque situation in which the majority party in Congress refuses to resist presidential abuses.
My fondest hope is that the Republican Party is soundly defeated in elections to come. And, yes, I know there are some "decent" Republicans who are inwardly cringing at what their party has become. But the key word is "inwardly": few Republican officeholders or media personalities are willing to oppose Trumpism publicly because to do so would likely be ruinous to them personally. So these invertebrates become accomplices to misdoing on a scale that would have revolted the Founding Fathers. For the time being, I echo the thirteenth-century French abbot who, when asked by Crusaders how to tell devout Catholics from apostates, reportedly advised them to kill them all and let God sort them out. Voters should simply vote against _all_ Republicans as long as Trumpism remains such a dire threat to our republic. And they should keep on doing so however long it takes to purge the taint of Trumpism. Republican candidates must realize how wrong it is—morally wrong if not politically wrong—to appeal for votes by demonizing minorities or undermining the rule of law.
IF I AM NOT A REPUBLICAN, why am I not a Democrat? I became an independent instead because, although Democrats are sounding a lot more sensible than Republicans these days, I cannot fully embrace their party either.
To Democrats' credit, they have been rightly outraged by Trump's conduct, and they have tried to hold him to account despite their minority status in Congress. Trump's embrace of Vladimir Putin has even led many Democrats to adopt the kind of tough-on-Russia foreign policy that many once criticized. Democrats certainly understand the imperative of American global leadership on issues such as climate change and gay rights. But many still don't seem to get the importance of a strong defense or free trade. Few Democrats protested when Trump left the Trans-Pacific Partnership, thereby handing China a big foreign policy win. Even Hillary Clinton, who must know better, felt compelled to oppose TPP to appease the Democratic base.
Still, if the Clintons' moderate views represented the Democrats' center of gravity, I would feel comfortable becoming a Democrat. But increasingly it's obvious that it's more Bernie's party than Hillary's. There are Democrats I admire and support—particularly young centrists, many of them with military backgrounds, such as Representatives Seth Moulton of Massachusetts, Conor Lamb of Pennsylvania, and Stephanie Dang Murphy of Florida, and Mayor Pete Buttigieg of South Bend, Indiana. Virginia senators Tim Kaine and Mark Warner also fit the bill. But they are, sad to say, marginal figures, even if Kaine was a vice presidential nominee. But then so was Joe Lieberman—and he was driven out of the Democratic Party for being too conservative.
It's hard to name a single prominent centrist leader in the Democratic Party. Where is the Scoop Jackson of today? He or she simply doesn't exist. The heart of the party lies with figures of the left such as Senators Elizabeth Warren and Bernie Sanders, the heirs to such firebrands of the sixties and seventies as Eugene McCarthy and George McGovern. All of these "progressives" had great youth appeal because they offered sweeping and idealistic, if impractical, proposals. In the case of Sanders, he proposes free medical care and free education, along with federal jobs, without calculating the price tag of this vast experiment with socialism.
In 2017 Sanders introduced a Medicare for All bill that sounds great—who wouldn't want free health care for all?—but that would necessitate a sweeping, government-driven restructuring of 17 percent of the economy. Among the consequences: 155 million Americans who currently have employer-funded health insurance would have to move to Medicare. Presumably all of the insurance companies that currently provide health coverage would go bankrupt unless they could find a new business model. That is the kind of ambitious social engineering that governments can never pull off without causing massive problems and disruptions.
Sanders has not bothered to calculate the price tag of his single-payer plan, but it would obviously be a budget-buster in spite of his claims to squeeze out unnecessary overhead. His bill is, in fact, considerably more generous than Canada's health system, which he cites as a model. The best estimate is that the Sanders bill would cost $1.4 trillion a year. Keep in mind that the 2018 federal budget already calls for spending $4 trillion and that soon the _annual_ budget deficit will exceed $1 trillion. Through such profligacy, we have accumulated more than $21 trillion in federal debt—and climbing. The Sanders bill, by increasing federal spending 35 percent, would either add to the mountain of debt or cause tax rates to spike. Either way, our economic prospects would be put in peril. The Medicare for All plan would also endanger our national security by further crowding out the defense budget, scientific research, the arts, education, foreign aid, environmental protection, infrastructure, law enforcement, and all the other spending programs that are classified as "discretionary."
And yet at least a third of all Democratic senators have endorsed his legislation, including potential presidential candidates such as Elizabeth Warren, Cory Booker, Kirstin Gillibrand, and Kamala Harris. Other Democrats support other versions of the same idea—of opening up Medicare to any American who wants it. The _Washington Post_ 's liberal blogger Paul Waldman writes that "we're getting awfully close to a consensus among Democratic politicians, on that one basic idea." This is an idea far more radical and costly than President Obama's Affordable Care Act, which simply subsidized private insurance for people who can't afford it and was based on a plan first implemented by Mitt Romney in Massachusetts.
Free health care is only part of the basket of benefits that the progressive wing of the Democratic Party is now promising to provide at no charge. Sanders has also introduced, with Senator Warren's support, a College for All Act that would provide free tuition at public universities for students from families that make less than $125,000 a year and that would make community colleges tuition-free for everyone. The estimated cost of this subsidy: $47 billion a year. In addition, Sanders advocates offering a $15-an-hour job with the federal government to every worker "who wants or needs one," as well as providing full health care benefits. Senators Gillibrand and Booker have joined in supporting this make-work plan even though Sanders has no idea of how much it will cost or how to pay for it.
Even Kevin Drum, a blogger for _Mother Jones_ , describes Sanders's make-work scheme as "damn close to insane": "It's about 3–10 percent of the labor force effectively nationalized forever by the federal government, which makes it roughly comparable to the emergency labor force employed for a few years by the WPA during the depths of the Depression. This is why even our lefty comrades in social democratic Europe don't guarantee jobs for everyone. It would cost a fortune; it would massively disrupt the private labor market; it would almost certainly tank productivity; and it's unlikely in the extreme that the millions of workers in this program could ever be made fully competent at their jobs."
Yet progressive ideologues want to go even further. The influential writer Jedediah Purdy advocates taking advantage of an anti-Trump backlash to push for a socialist wish list, including raising tax rates to 70 percent, creating universal family leave and child care, mandating compulsory unionization, and giving noncitizens the right to vote. None of Purdy's ideas will be enacted anytime soon, but his writing may be indicative of where the Democratic Party is heading. Senator Sanders's proposals for government-run health care, federal jobs, and free education are radical enough, and they are attracting mainstream support. When I raise questions about the practicality of Sanders's schemes, I am often met with withering scorn from progressives claiming that I'm in favor of people dying in the streets for lack of health care. This is the mirror image of the kind of invective, questioning the motives of political opponents, that Trump and other right-wing populists specialize in.
Note, also, that Sanders and many other progressives actually agree with Trump on many of the issues—they, too, are sympathetic to protectionism and isolationism or at least noninterventionism. A Warren/Sanders administration could be nearly as hostile to American global leadership—save on issues such as global warming and LBGT rights—as the Trump administration has been. Don't get me wrong: we need to lead on climate change and LBGT rights. But we also need to lead on other issues as well.
All of this suggests that the Democratic Party is drifting leftward as the Republican Party is drifting rightward. The polarization of US politics leaves anyone who is not a socialist or a far-right populist feeling increasingly disenfranchised.
WHAT AMERICA NEEDS IS a center-right party—either a reborn GOP or an entirely new party—free of any taint of Trumpism. Such a party would not necessarily reflect all of my policy preferences, as outlined at the start of this chapter: socially liberal, fiscally conservative, pro-environment, internationalist, pro-defense, pro–gun control, pro-immigration. While I would love it if a political party would espouse as many of these views as possible, I am willing to compromise on some issues because that's the messy, unsatisfying nature of democratic politics. For example, I could make common cause with more socially conservative or more fiscally profligate voters—and in fact I have done so in the past because the Republican Party has long included both factions. But I could not possibly compromise one inch on support for the rule of law or on opposition to racism, sexism, xenophobia, and other forms of prejudice. Above all, the party I envision would need to be optimistic and inclusive rather than hateful and divisive. It would need to speak for what Arthur Schlesinger Jr. in 1949 called "the vital center."
Unfortunately, the last time a successful third party was created in America was 1854, when the Republican Party was started by disenchanted members of the Whig Party. Our entire political system is designed to entrench the Republican-Democratic duopoly and marginalize third parties. It may be possible to create a new third party in California, where, at least in statewide races, the Republican Party has all but ceased to exist. But in the rest of the country it is simply too difficult to start a third-party movement from the grassroots level, as libertarians can attest. There may be, however, an outside chance of doing so from the top down. Admittedly it would take a moonshot to succeed—but we really did reach the moon once.
The example of Emmanuel Macron could point the way. France is another country that is riven between the far left and far right—a situation that is increasingly common across the democratic world. In the first round of presidential balloting in 2017, the neo-fascist Marine Le Pen and the neo-Marxist Jean-Luc Mélenchon together won more than 40 percent of the vote. But the leading vote-getter, who was eventually elected in a second round of balloting over Le Pen, was Macron—a former Socialist who had left his old party to launch an independent bid on a centrist platform. Once elected, Macron selected a slate of candidates to support his agenda in the National Assembly. Republic on the March, as his party is called, won a landslide victory, enabling Macron to push his reformist agenda through the legislature, notwithstanding obstructionism from labor unions that object to his attempts to loosen work rules so as to get the economy moving.
We could use an American Macron—someone who can make centrism sexy. Such a candidate would need name recognition, funding, charisma, and above all, policy knowledge. If such a candidate were to win the presidency on a third-party ticket, he or she could shatter the two-party system from the top down by leading Democratic and Republican officeholders to abandon their old parties while drawing fresh faces into politics. Alternatively, such an outsider could capture the Republican nomination much as Trump himself did despite his lack of party credentials—or, in a much more optimistic vein, as Eisenhower did in 1952. If the Republican Party's shameless embrace of Trump shows anything, it is how malleable its agenda is. Having embraced white nationalism, protectionism, deficit spending, immorality, obstruction of justice, and isolationism under Trump's leadership, Republicans could just as easily reverse themselves again should a charismatic centrist win the presidency under their banner. The GOP tribe, seemingly, will do whatever its chief instructs. That's a big problem under Trump but could be a big advantage under a more reasonable successor.
Given how entrenched the two-party system is in America, rehabilitating the GOP, following a series of cleansing defeats, seems the more likely path toward establishing a principled center-right party. The way to start the process would be with a credible challenge to Trump's renomination in 2020. An insurgent campaign would be almost certain to lose but, as historian Matthew Dallek argues, it could severely harm the incumbent and change the direction of the party. That's what previous challengers did: Eugene McCarthy's 1968 run against Lyndon Johnson made the Democratic Party dovish on foreign policy and forced Johnson out of the race, Ronald Reagan's 1976 run helped defeat Gerald Ford and led the Republican Party to embrace his brand of conservatism, Ted Kennedy's 1980 run helped defeat Jimmy Carter and made the Democratic Party more liberal and coastal, and Pat Buchanan's 1992 run helped defeat George H. W. Bush and advanced the "culture war" agenda that Trump later picked up.
By comparison, the third-party option is less likely—but not impossible. Look at how much support H. Ross Perot won as a third-party candidate running on a deficit-reduction platform in 1992. He garnered 18.9 percent of the vote even though he left the campaign in the middle, knew little about politics, and was more than a touch screwy. Of course, we don't need another Perot—in Trump, we have an even more noxious version of the conspiracy-mongering, know-nothing tycoon. What we could use is someone like Dwight D. Eisenhower. The more I study him, the more I like Ike. As a young man, I was critical of Eisenhower for being a plodding general and president who disdained bold gestures whether on the battlefield or in the political process: his forces did not cut off the escape of German troops from Normandy in 1944 and, as president in the 1950s, he did not take the lead in fighting segregation or McCarthyism. But as I have matured, I have come to appreciate Eisenhower's virtues. He wasn't a military genius like Douglas MacArthur or a political genius like Franklin Roosevelt, but he was smart and steady. Above all, he had a first-class temperament that allowed him to grapple with the most difficult problems while maintaining an air of equanimity—an emotional state far removed from Trump's manic rage. Trump is disorganized and chaotic; Eisenhower was a consummate manager who ran the White House more effectively than anyone before or since. Trump is a divider; Eisenhower was a uniter.
We don't necessarily need a president to be a retired general officer, although there are potential candidates such as former Admirals William McRaven and James Stavridis and former Generals Stanley McChrystal and Jim Mattis (now, of course, the secretary of defense) who might fill that role. We do need someone who can appeal to the forgotten middle, where I find myself stranded along with other voters. Oprah Winfrey, as bright as she is, Mark Cuban, Dwayne Johnson, or some other celebrity won't cut it: they don't know enough about government to be a competent president, and after Trump we can't afford another amateur in the Oval Office. The ideal candidate would be a younger, more charismatic Michael Bloomberg. John Kasich, the Ohio governor, congressman, and presidential candidate, who has become much too moderate for the Republican Party, could fit the bill. So might Senators Ben Sasse or Jeff Flake. Any of them could either challenge Trump for the nomination or run as a third-party candidate and name a centrist Democrat such as Seth Moulton as a running mate. Of course, the danger of a third-party candidacy is that it could split the anti-Trump vote and reelect the president—but it could also drain away conservative votes that Trump needs to win.
Perhaps this is simply a fantasy. Maybe it's impossible for a centrist candidate to emerge out of our increasingly polarized politics. If that's the case, I fear the Democrats will continue to drift left and the Republicans right—a trend that will do more to imperil our country's future than any external threat because it will make it impossible to solve our problems on a bipartisan basis. As Pogo put it, we have met the enemy and he is us.
Of one thing I am sure: the Republican Party as currently constituted does not deserve to survive. The more quickly it expires, the faster it can be replaced with a more responsible center-right party. That day is not as far off as you might imagine, notwithstanding Republicans' electoral success in 2016. Given the extent to which the Trumpified GOP is alienating young people and minorities, it seems to have a death wish. Indeed, the national party may be following the example of the California GOP, which can no longer win statewide races, and even has trouble qualifying candidates for the ballot, because it became so identified with anti-immigrant bias under Governor Pete Wilson in the 1990s. By 2015, Latinos in California had come to outnumber non-Hispanic whites. If California is a trendsetter, as it has been in the past, Republicans in the rest of the country have good cause to fear for the future.
Republicans can continue to win elections nationally for a few more years by relying on a base of older, less educated white voters, but that strategy will prove less and less successful as the number of people of color continues to grow. By 2044, whites are expected to be a minority in America—very bad news for a party that has alienated everyone who isn't white. The GOP is turning off the young just as quickly: a 2017 poll found that only 19 percent of respondents between the ages of 18 and 34 approved of Trump's presidency while 67 percent disapproved.
I grew up in the 1980s when Reagan-style conservatism was cool. Young people today are growing up in a world where Trump-style conservatism is deservedly unfashionable. I am not surprised that my own kids—ranging in age from sixteen to twenty—are not becoming conservatives. I hardly blame them. Indeed, I would wonder what was wrong with them if they embraced conservatism at a time when the movement has become virtually synonymous with Trumpism.
HAVING LEFT THE Republican Party and even the conservative movement behind, I will not eagerly or quickly embrace _any_ party or movement in the future. I have spent most of my life as part of a political movement that has revealed itself to be morally and intellectually bankrupt. That is a chastening lesson about the price of loyalty.
I am happy, then, to make common cause with like-minded individuals in the struggle to maintain democracy in the face of a populist onslaught. At a time of extreme crisis such as this one, the normal policy differences between the center-left and center-right fade into insignificance. All people of goodwill must come together to defend liberal democracy from the populist threat. And, indeed, one of the most salutary developments of the Trump era is how it has brought the president's critics, from both the left and the right, together. I find myself retweeting _Mother Jones_ contributors—and they me (often with the preface, "I can't believe I'm retweeting Max Boot"). I had to laugh when I saw this tweet from someone who goes by the handle Digitalist: "It is deeply disturbing that the current US political climate is so abnormal that Max Boot and I have ended up on the same side of this many issues." It's funny—and true.
I have also become involved in starting a centrist group called the Renew Democracy Initiative, along with former world chess champion Garry Kasparov, historian Anne Applebaum, financier Richard Hurowitz, novelist and columnist Richard North Patterson, former German defense minister Karl-Theodor zu Guttenberg, and many others. This is one of a panoply of civil-society groups on both the center-left and center-right that have arisen to protest Trump and other populists who are an affront to democracy and decency. Many of these organizations were represented at the National Summit for Democracy held in a Washington office building in February 2018. It was an invigorating event where civil-society activists came together to figure out how to protect our institutions from the threat of Trumpism. I spent my time, for example, in a working group brainstorming ideas for protecting the United States from the kind of influence operation that Russia mounted in 2016. For someone like me, who feels aghast at the direction of the Trumpified Republican Party, it was reassuring to meet so many on the left and right who share my horror of what the president is doing to the country we love.
But, after leaving my old tribe, I am in no hurry to join a new one. For now, I am content to remain a party of one. Having become first an American and then a conservative, I was eager in the past to adopt a group identity. It wasn't easy to do: it required first learning English and then the language of "movement conservatism." I spent decades immersed in conservative thought and society, advancing from a lowly high school volunteer for a Senate campaign in California to a _Wall Street Journal_ editor, an adviser to Republican presidential candidates, and a regular contributor to leading conservative publications. Now I prefer to think for myself rather than subscribe to the groupthink that previously defined my existence.
I did not even realize until the crisis of 2016, when I left the Republican Party, the extent to which my thinking in the past had been circumscribed by my allegiance to the conservative movement. Like oxygen, the pressure to conform is something that you only notice when it's gone. For the first time since I was twenty-four years old—a quarter of a century ago—I do not draw any pay from any conservative organization. There is something frightening but also energizing about becoming a free agent at last. Having escaped the corrosion of conservatism, I am a political Ronin, and will swear allegiance to no master in the future. I will fight for my principles wherever they may lead me.
## **ACKNOWLEDGMENTS**
THIS BOOK IS THE BRAINCHILD OF MY EXTRAORDINARY editor, Robert Weil, to whom it is dedicated. This is the third book in a row I have worked on with Bob, and by this point I cannot imagine working with another editor. He is quite simply the best—not only at editing but also at coming up with book ideas. This volume was his inspiration; I hope I have done justice to his vision. I am also grateful to the entire team at Norton/Liveright, including Marie Pantojan, Nick Curley, Bill Rusin, Peter Miller, Cordelia Calvert, Steven Pace, Brendan Curry, Anna Oler, and Steve Attardo—all of whom are as dedicated to books as Bob is.
I am blessed to have in my corner not only the world's greatest editor but also the greatest literary agent—Tina Bennett of WME. She has shepherded this book from conception to publication and offered valuable advice and support along the way. Her assistant Svetlana Katz has also been extraordinarily helpful.
My friends Gary Rosen, Bill Kristol, and Richard North Patterson read the manuscript and helped me to improve the book in all sorts of ways. I cannot thank them enough for taking time from their own labors to assist me.
Some of the arguments advanced in this book first appeared in my articles, in somewhat different form, and I am grateful to my editors for running them: Fred Hiatt, Jackson Diehl, and Ruth Marcus at the _Washington Post_ , where I now have the privilege of being a columnist; David Rothkopf and Jonathan Tepperman at _Foreign Policy_ , where I was previously a columnist; Susan Brenneman at the _Los Angeles Times_ , where I have written off and on since I was in college; Jill Lawrence at _USA Today_ ; and John Podhoretz at _Commentary_ , where I was a regular blogger for a decade. At all those publications, extraordinary line editors improved my prose; I am particularly grateful to Christian Caryl at the _Post_. Not long after signing to write this book, I agreed to become a global affairs analyst at CNN, and I am grateful to my colleagues there—especially to CNN president Jeff Zucker, Vice President Rebecca Kutler, and the producers and hosts of _Anderson Cooper 360_ , _New Day_ , and _CNN Tonight_ —for allowing me to comment on the news. Previously I was a regular commentator at MSNBC and greatly appreciate the opportunity that so many of its hosts and producers provided to me.
The Council on Foreign Relations remains my professional home, as it has been since 2002, and I cannot imagine a more congenial place to work or better bosses than Council president Richard Haass and Director of Studies James Lindsay. They have always supported me and never told me what to say or write. The Council's mission of fostering America's engagement with the world is needed now more than ever, and I am proud to be a small part of this distinguished organization. My research associate, Sherry Cho, provided invaluable support throughout.
Finally I am grateful to my partner, Sue Mi Terry, not only for her trenchant comments on the manuscript, which helped to improve it, but also for everything else that she has done to help keep me on an even keel amid the choppy currents of our contemporary politics.
## **NOTES**
**A BBREVIATIONS**
_FP: Foreign Policy_
_LAT: Los Angeles Times_
MB: Max Boot
_NYT: New York Times_
_WP: Washington Post_
_USAT: USA Today_
**P ROLOGUE: NOVEMBER 8, 2016**
. "Here's Donald Trump's Presidential Announcement Speech," _Time_ , June 16, 2015, http://time.com/3923128/donald-trump-announcement-speech/.
. Ben Schreckinger, "Trump attacks McCain: 'I like people who weren't captured,'" _Politico_ , July 18, 2015, https://www.politico.com/story/2015/07/trump-attacks-mccain-i-like-people-who-werent-captured-120317.
. "Donald Trump Criticized for Mocking Disabled Reporter," _Snopes_ , Jan. 11, 2017, https://www.snopes.com/2016/07/28/donald-trump-criticized-for-mocking-disabled-reporter/.
. Barry Goldwater, _Conscience of a Conservative_ (LaVergne, TN: Bottom of the Hill Publishing, 2010).
**1. T HE EDUCATION OF A CONSERVATIVE**
. Gal Beckerman, _When They Come for Us We'll Be Gone: The Epic Struggle to Save Soviet Jewry_ (Boston: Houghton Mifflin Harcourt, 2010).
. William F. Buckley Jr., "On Donald Trump and Demagoguery," _National Review_ , Jan. 22, 2016, https://www.nationalreview.com/2016/01/william-f-buckley-donald-trump-demagoguery-cigar-aficionado/.
. "On the Meaning of Life," _Letters of Note_ , Jan. 31, 2012, http://www.lettersofnote.com/2012/01/on-meaning-of-life.html.
. German Lopez, "America's Unique Gun Violence Problem, Explained in 17 Maps and Charts," _Vox_ , Oct. 2, 2017, https://www.vox.com/policy-and-politics/2017/10/2/16399418/us-gun-violence-statistics-maps-charts.
. David Corn, "Remember How Dinesh D'Souza Outed Gay Classmates—and Thought It Was Awesome?," _Mother Jones_ , Jan. 24, 2014, https://www.motherjones.com/politics/2014/01/dinesh-dsouza-indictment-dartmouth-outed-gay-classmates/.
**2. T HE CAREER OF A CONSERVATIVE**
. MB, "Americans Need to Overcome Partisan Enmity—But Our President Is Stoking It," _WP_ , May 29, 2018, https://www.washingtonpost.com/news/global-opinions/wp/2018/05/29/americans-need-to-overcome-partisan-enmity-but-our-president-is-stoking-it/?utm_term=.686bc7bf396e.
. MB, "The Case for American Empire," _Weekly Standard_ , Oct. 15, 2001, https://www.weeklystandard.com/max-boot/the-case-for-american-empire.
. MB, "To Help Restore U.S. Standing, Rumsfeld Must Take the Fall," _LAT_ , May 13, 2004, http://articles.latimes.com/2004/may/13/opinion/oe-boot13.
**3. T HE SURRENDER**
. MB, "Donald Trump's Remarkably Consistent Inconsistency," _Commentary_ , Aug. 18, 2015, http://maxboot.net/donald-trumps-remarkably-consistent-inconsistency/.
. Philip Elliott, "Donald Trump Brags His Way through New Hampshire," _Time_ , Feb. 5, 2016, http://time.com/4208870/donald-trump-brags/.
. MB, "Does Donald Trump Know the Right General Can't Save a Failed Foreign Policy?," _Commentary_ , Sept. 9, 2015, http://maxboot.net/does-donald-trump-know-the-right-general-cant-save-a-failed-foreign-policy/.
. Chris Cillizza, "A Fact Checker Looked into 158 Things Donald Trump Said. 78 Percent Were False," _WP_ , July 1, 2016, https://www.washingtonpost.com/news/the-fix/wp/2016/07/01/donald-trump-has-been-wrong-way-more-often-than-all-the-other-2016-candidates-combined/?utm_term=.6df3fedaf473.
. MB, "Max Boot: Trump Is a Character Test for the GOP," _USAT_ , Feb. 29, 2016, https://www.usatoday.com/story/opinion/2016/02/29/trump-tests-republicans-max-boot/81123934/.
. MB, "'Sheriff Joe' and Donald Trump Are Emblems of Racism and Lawlessness," _LAT_ , Aug. 28, 2017, http://www.latimes.com/opinion/op-ed/la-oe-boot-arpaio-pardon-20170827-story.html.
. MB, "Trump Bows to Russia Again: Max Boot," _USAT_ , Feb. 6, 2017, https://www.usatoday.com/story/opinion/2017/02/06/trump-interview-putin-killer-innocent-thug-moral-relativism-max-boot-column/97525480/.
. Jeremy Diamond, "Donald Trump on Protester: 'I'd Like to Punch Him in the Face,'" _CNN_ , Feb. 23, 2016, https://www.cnn.com/2016/02/23/politics/donald-trump-nevada-rally-punch/index.html.
. Gordon F. Sander, "When Nazis Filled Madison Square Garden," _Politico_ , Aug. 23, 2017, https://www.politico.com/magazine/story/2017/08/23/nazi-german-american-bund-rally-madison-square-garden-215522.
. MB (@MaxBoot), Twitter, Nov. 22, 2015, https://twitter.com/maxboot/status/668447756512456705?lang=en.
. John McNeill, "How Fascist Is Donald Trump? There's Actually a Formula for That," _WP_ , Oct. 21, 2016, https://www.washingtonpost.com/posteverything/wp/2016/10/21/how-fascist-is-donald-trump-theres-actually-a-formula-for-that/?utm_term=.4b86d9a226e9.
. MB, "Donald Trump's Honesty Problem," _Commentary_ , Feb. 22, 2016, http://maxboot.net/donald-trumps-honesty-problem/.
. MB, "There Is No Escape from Trump," _Commentary_ , Mar. 3, 2016, http://maxboot.net/there-is-no-escape-from-trump/.
. MB and Benn Steil, "Selling America Short," _Weekly Standard_ , Feb. 26, 2016, https://www.weeklystandard.com/max-boot-benn-steil/selling-america-short.
. Publius Decius Mus [Michael Anton], "The Flight 93 Election," _Claremont Review of Books_ , Sept. 5, 2016, http://www.claremont.org/crb/basicpage/the-flight-93-election/.
. "Open Letter on Donald Trump from GOP National Security Leaders," _War on the Rocks_ , Mar. 2, 2016, https://warontherocks.com/2016/03/open-letter-on-donald-trump-from-gop-national-security-leaders/.
. "Against Trump," _National Review_ , Jan. 22, 2016, https://www.national review.com/2016/01/donald-trump-conservative-movement-menace/.
. MB, "Trump vs. Mueller is a Battle for America's Soul," _WP_ , Feb. 26, 2018, https://www.washingtonpost.com/opinions/trump-vs-mueller-is-a-battle-for-americas-soul/2018/02/26/0979904c-1b19-11e8-9de1-147dd2df3829_story.html?utm_term=.5ab69d72bd8a.
. Alexander Burns, "Anti-Trump Republicans Call for a Third-Party Option," _NYT_ , Mar. 3, 2016, https://www.nytimes.com/2016/03/03/us/politics/anti-donald-trump-republicans-call-for-a-third-party-option.html.
. Hadas Gold and Oliver Darcy, "Salem Executives Pressured Radio Hosts to Cover Trump More Positively, Emails Show," _CNN_ , May 9, 2018, http://money.cnn.com/2018/05/09/media/salem-radio-executives-trump/index.html.
. David Frum, "Conservatism Can't Survive Donald Trump Intact," _The Atlantic_ , Dec. 19, 2017, https://www.theatlantic.com/politics/archive/2017/12/conservatism-is-what-conservatives-think-say-and-do/548738/.
. MB, "Max Boot: Trump Is a Character Test for the GOP," _USAT_ , Feb. 29, 2016, https://www.usatoday.com/story/opinion/2016/02/29/trump-tests-republicans-max-boot/81123934/.
. David A. Graham, "Which Republicans Oppose Donald Trump? A Cheat Sheet," _The Atlantic_ , Nov. 6, 2016, https://www.theatlantic.com/politics/archive/2016/11/where-republicans-stand-on-donald-trump-a-cheat-sheet/481449/.
. Richard Brookhiser, "WFB Today," _National Review_ , Feb. 17, 2018, https://www.nationalreview.com/magazine/2018/02/17/william-f-buckley-trump-conservatism-needs-rebuilding/.
. MB, "The Republican Party Is Dead," _LAT_ , May 8, 2016, http://www.latimes.com/opinion/op-ed/la-oe-boot-republicans-in-exile-20160508-story.html.
. Matthew Yglesias, "New Report Details Trump-Inspired Surge in Anti-Semitism," _Vox_ , Oct. 19, 2016, https://www.vox.com/policy-and-politics/2016/10/19/13326336/trump-antisemitism.
. "Anti-Semitic Incidents in US More than Doubled within 2 Years—Report," _Times of Israel_ , Feb. 27, 2018, https://www.timesofisrael.com/anti-semitic-incidents-in-us-more-than-doubled-within-2-years-report/?utm_source=The+Times+of+Israel+Daily+Edition&utm_campaign=fe43d6c8aa-EMAIL_CAMPAIGN_2018_02_27&utm_medium=email&utm_term=0_adb46cec92-fe43d6c8aa-55237393.
. Yochi Dreazen, "It's Time to Acknowledge Reality: Donald Trump Talks Like an Anti-Semite," _Vox_ , Oct. 14, 2016, https://www.vox.com/world/2016/10/14/13288138/donald-trump-anti-semite-israel-david-duke-racism-misogny-clinton.
. MB, "Trump's Opposition Research Firm: Russia's Intelligence Agencies," _LAT_ , July 25, 2016, http://www.latimes.com/opinion/op-ed/la-oe-boot-trump-russian-connection-20160725-snap-story.html.
. Eugene Scott, Ashley Killough, and Daniel Burke, "Evangelicals 'Disgusted' by Trump's Remarks, But Still Backing Him," _CNN_ , Oct. 21, 2016, https://www.cnn.com/2016/10/07/politics/donald-trump-evangelical-leaders/index.html.
. MB, "This Lifetime GOP Voter Is with Her," _FP_ , Nov. 6, 2016, http://foreignpolicy.com/2016/11/06/this-lifetime-gop-voter-is-with-her-why-republicans-should-vote-for-hillary-clinton/.
. Philip Bump, "Donald Trump Will Be President Thanks to 80,000 People in Three States," _WP_ , Dec. 1, 2016, https://www.washingtonpost.com/news/the-fix/wp/2016/12/01/donald-trump-will-be-president-thanks-to-80000-people-in-three-states/?utm_term=.e836b07cbe2e.
. Heather Boushey, "The Appealing Logic that Underlies Trump's Economic Ideas," _The Atlantic_ , March 15, 2017, https://www.theatlantic.com/business/archive/2017/03/trumps-economic-logic/519381/.
. Ian Bremmer, _Us vs. Them: The Failure of Globalism_ (New York: Portfolio/Penguin, 2018), 17.
. The bipartisan Financial Crisis Inquiry Commission concluded that "it was the collapse of the housing bubble—fueled by low interest rates, easy and available credit, scant regulation, and toxic mortgages—that was the spark that ignited a string of events, which led to a full-blown crisis in the fall of 2008." _Conclusions of the Financial Crisis Inquiry Commission_ , http://fcic-static.law.stanford.edu/cdn_media/fcic-reports/fcic_final_report_conclusions.pdf.
. Donald J. Trump (@realDonaldTrump), Twitter, Jan. 20, 2017, https://twitter.com/realdonaldtrump/status/822502450007515137.
. Jonah Goldberg, Max Boot, Michael Brendan Dougherty, William Voegeli, and Emily Ekins, "Conservatives Ponder the Future of the GOP under Trump," _LAT_ , Nov. 13, 2016, http://www.latimes.com/opinion/op-ed/la-oe-gop-future-roundtable-20161113-story.html.
**4. T HE CHAOS PRESIDENT**
. MB, "Why a Trump Presidency Might Not Be as Awful as We Fear," _FP_ , Nov. 9, 2016, http://foreignpolicy.com/2016/11/09/why-a-trump-presidency-might-not-be-as-awful-as-we-fear/.
. MB, "NeverTrumpers Should Not Shun Trump: Max Boot," _USAT_ , Nov. 13, 2016, https://www.usatoday.com/story/opinion/2016/11/13/never-trumpers-should-not-shun-trump-national-security-checks-balances-max-boot/93767170/.
. MB, "The Grave Dangers and Deep Sadness of 'America First,'" _FP_ , Jan. 23, 2017, http://foreignpolicy.com/2017/01/23/the-grave-dangers-and-deep-sadness-of-america-first-donald-trump/.
. MB, "Slapdash Trump Order Ignores Real Danger: Max Boot," _USAT_ , Jan. 29, 2017, https://www.usatoday.com/story/opinion/2017/01/29/slapdash-trump-order-ignores-real-danger-max-boot-column/97211882/.
. Kathryn Dunn Tenpas, "Why Is Trump's Staff Turnover Higher than the 5 Most Recent Presidents?," _Brookings_ , Jan. 19, 2018, https://www.brookings.edu/research/why-is-trumps-staff-turnover-higher-than-the-5-most-recent-presidents/.
. Philip Bump, A Quarter of Trump's 'Highest IQ' Cabinet Has Been Replaced," _WP_ , Mar. 28, 2018, https://www.washingtonpost.com/news/politics/wp/2018/03/28/a-quarter-of-trumps-highest-iq-cabinet-has-been-replaced/?utm_term=.9e981e387e1f.
. Aaron Blake, "'It's Chaos. . . . It's Not Good for Anything': After Rejecting Trump's Offer, Ted Olson Admonishes Him," _WP_ , Mar. 26, 2018, https://www.washingtonpost.com/news/the-fix/wp/2018/03/26/its-chaos-its-not-good-for-anything-after-snubbing-trump-ted-olson-admonishes-him/?utm_term=.284eb538f93c.
. Trump Golf Count, http://trumpgolfcount.com/.
. Patrick Radden Keefe, "McMaster and Commander," _New Yorker_ , Apr. 30, 2018, https://www.newyorker.com/magazine/2018/04/30/mcmaster-and-commander.
. Susan B. Glasser, "The Price of Getting Inside Trump's Head," _New Yorker_ , May 11, 2018, https://www.newyorker.com/news/news-desk/the-price-of-getting-inside-trumps-head.
. Arelis R. Hernandez, "New Puerto Rico Data Shows Deaths Increased by 1,400 after Hurricane Maria," _WP_ , June 1, 2018, https://www.washingtonpost.com/national/new-puerto-rico-data-shows-deaths-increased-by-1400-after-hurricane-maria/2018/06/01/43bb4278-65e2-11e8-99d2-0d678ec08c2f_story.html?utm_term=.
. John Harwood (@JohnJHarwood), Twitter, Apr. 2, 2018, https://twitter.com/johnjharwood/status/980900978789675008?s=11.
. Brian Klaas (@brianklaas), Twitter, Apr. 25, 2018, https://twitter.com/brianklaas/status/989127820420026369.
. Robert Shapiro, "Trump Lags behind His Predecessors on Economic Growth," _Brookings_ , May 17, 2018, https://www.brookings.edu/blog/fixgov/2018/05/17/trump-lags-behind-his-predecessors-on-economic-growth/?utm_campaign=Brookings%20Brief&utm_source=hs_email&utm_medium=email&utm_content=63022866.
. Julian E. Barnes, "Pompeo Says Iran Nuclear Deal Could Only Be Preserved with a 'Substantial Fix,'" _WSJ_ , Apr. 28, 2018, https://www.wsj.com/articles/pompeo-begins-first-official-foreign-trip-amid-uncertainty-over-iran-deal-1524821386.
. Alan Levin and Jesse Hamilton, "Trump Takes Credit for Killing Hundreds of Regulations That Were Already Dead," _Bloomberg_ , Dec. 11, 2017, https://www.bloomberg.com/news/features/2017-12-11/trump-takes-credit-for-killing-hundreds-of-regulations-that-were-already-dead.
. Jeffry Bartash, "Trump's Regulatory Rollback for the U.S. Economy Is a Dud—So Far," _MarketWatch_ , Feb. 12, 2018, https://www.marketwatch.com/story/trumps-regulatory-rollback-for-the-us-economy-is-a-dud-so-far-2018-02-12.
. Eliana Johnson, Emily Stephenson, and Daniel Lippman, "'Too Inconvenient': Trump Goes Rogue on Phone Security," _Politico_ , May 21, 2018, https://www.politico.com/story/2018/05/21/trump-phone-security-risk-hackers-601903.
**5. T HE COST OF CAPITULATION**
. Marshall Frady, "How George Wallace Harnessed Hate," _Daily Beast_ , Apr. 3, 2016, https://www.thedailybeast.com/how-george-wallace-harnessed-hate.
. James Madison, "The Utility of the Union as a Safeguard against Domestic Faction and Insurrection (continued)," _The Federalist_ , No. 10, Nov. 22, 1787, http://www.constitution.org/fed/federa10.htm.
. Louis Jacobson, "Steele Says GOP Fought Hard for Civil Rights Bills in 1960s," _PolitiFact_ , May 25, 2010, http://www.politifact.com/truth-o-meter/statements/2010/may/25/michael-steele/steele-says-gop-fought-hard-civil-rights-bills-196/.
. MB, "The Difference between George Washington and Robert E. Lee," _FP_ , Aug. 18, 2017, http://foreignpolicy.com/2017/08/18/the-difference-between-george-washington-and-robert-e-lee-trump-sedition-slavery-confederate-monuments/.
. Veronica Stracqualursi, "Trump: NFL Players Who Don't Stand during National Anthem Maybe 'Shouldn't Be in the Country,'" _CNN_ , May 24, 2018, https://www.cnn.com/2018/05/24/politics/trump-nfl-national-anthem/index.html.
. Scott Goldsmith, "Johnstown Never Believed Trump Would Help. They Still Love Him Anyway," _Politico_ , Dec. 11, 2008, https://www.politico.com/magazine/story/2017/11/08/donald-trump-johnstown-pennsylvania-supporters-215800.
. Diana C. Mutz, "Status Threat, Not Economic Hardship, Explains the 2016 Presidential Vote," _Proceedings of the National Academy of Sciences of the United States of America_ , Apr. 23, 2018, http://www.pnas.org/content/early/2018/04/18/1718155115.
. "Ronald Reagan on Immigration," _On the Issues_ , http://www.ontheissues.org/celeb/Ronald_Reagan_Immigration.htm.
. Julie Hirschfeld Davis, "Trump Calls Some Unauthorized Immigrants 'Animals' in Rant," _NYT_ , May 16, 2018, https://www.nytimes.com/2018/05/16/us/politics/trump-undocumented-immigrants-animals.html?smtyp=cur&smid=tw-nytimes.
. Z. Byron Wolf, "Trump Blasts 'Breeding' in Sanctuary Cities. That's a Racist Term," _CNN_ , Apr. 24, 2018, https://www.cnn.com/2018/04/18/politics/donald-trump-immigrants-california/index.html.
. Chris Nichols, "MOSTLY TRUE: Undocumented Immigrants Less Likely to Commit Crimes than U.S. Citizens," _PolitiFact_ , Aug. 3, 2017, http://www.politifact.com/california/statements/2017/aug/03/antonio-villaraigosa/mostly-true-undocumented-immigrants-less-likely-co/.
. "Immigrants from Africa Boast Higher Education Levels than Overall U.S. Population," _New American Economy_ , Jan. 11, 2018, http://www.newamericaneconomy.org/press-release/immigrants-from-africa-boast-higher-education-levels-than-overall-u-s-population/.
. Simone de Beauvoir, _The Works of Simone de Beauvoir: The Second Sex and the Ethics of Ambiguity_ (CreateSpace, 2011).
. Tal Kopan, "Arrests of Immigrants, Especially Non-Criminals, Way Up in Trump's First Year," _CNN_ , Feb. 23, 2018, https://www.cnn.com/2018/02/23/politics/trump-immigration-arrests-deportations/index.html.
. Geneva Sands, "'A Nightmare': Family Bids Goodbye as Undocumented Father of 2 Is Deported to Mexico," _ABC News_ , Jan. 16, 2018, http://abcnews.go.com/US/nightmare-family-bids-goodbye-undocumented-father-deported-mexico/story?id=52367022.
. Seung Min Kim, "Trump Is Blaming Democrats for Separating Migrant Families at the Border. Here's Why This Isn't a Surprise," _WP_ , May 27, 2018, https://www.washingtonpost.com/politics/trump-is-blaming-democrats-for-separating-migrant-families-at-the-border-heres-why-this-isnt-a-surprise/2018/05/27/c07810d8-61d3-11e8-a69c-b944de66d9e7_story.html?utm_term=.b23debbc9f5a.
. "Neocons & Russiagaters Unite! New Think Tank Will Protect Democracy from Russia, Sell Books," _RT_ , Apr. 27, 2018, https://www.rt.com/usa/425303-renew-democracy-initiative-neocons-rdi/.
. MB, "I Came to this Country 41 Years Ago. Now I Feel Like I Don't Belong Here," _WP_ , Sept. 5, 2017, https://www.washingtonpost.com/news/democracy-post/wp/2017/09/05/i-came-to-this-country-41-years-ago-now-trump-is-making-me-feel-like-i-dont-belong-here/?utm_term=.dd6600ee508f.
. Amber Phillips, "The Senate's New Russia Report Just Undercut Trump in Two Big Ways," _WP_ , May 16, 2018, https://www.washingtonpost.com/news/the-fix/wp/2018/05/16/the-russia-report-trump-likes-best-just-took-a-big-credibility-hit/?utm_term=.d98dec7b542a.
. "Trump's Russia Cover-Up By the Numbers—76+ Contacts with Russia-Linked Operatives," _The Moscow Project_ , May 17, 2018, https://themoscowproject.org/explainers/trumps-russia-cover-up-by-the-numbers-70-contacts-with-russia-linked-operatives/.
. Brett Samuels, "Trump: Nobody Has Been Tougher on Russia than Me," _The Hill_ , Apr. 3, 2018, http://thehill.com/homenews/administration/381437-trump-nobody-has-been-tougher-on-russia-than-me.
. Greg Jaffe, John Hudson, and Philip Rucker, "Trump, a Reluctant Hawk, Has Battled His Top Aides on Russia and Lost," _WP_ , Apr. 15, 2018, https://www.washingtonpost.com/amphtml/world/national-security/trump-a-reluctant-hawk-has-battled-his-top-aides-on-russia-and-lost/2018/04/15/a91e850a-3f1b-11e8-974f-aacd97698cef_story.html?__twitter_impression=true.
. MB, "Trump Is Ignoring the Worst Attack on America Since 9/11," _WP_ , Feb. 13, 2017, https://www.washingtonpost.com/opinions/trump-is-ignoring-the-worst-attack-on-america-since-911/2018/02/18/5ad888f2-14f3-11e8-8b08-027a6ccb38eb_story.html.
. Dylan Matthews, "Trump Has Changed How Americans Think about Politics," _Vox_ , Jan. 30, 2018, https://www.vox.com/policy-and-politics/2018/1/30/16943786/trump-changed-public-opinion-russia-immigration-trade.
. Kristen Bialik, "Putin Remains Overwhelmingly Unpopular in the United States," _Pew Research Center_ , Mar. 26, 2018, http://www.pewresearch.org/fact-tank/2018/03/26/putin-remains-overwhelmingly-unpopular-in-the-united-states/.
. Abby Phillip, Robert Barnes, and Ed O'Keefe, "Supreme Court Nominee Gorsuch Says Trump's Attacks on Judiciary Are 'Demoralizing,'" _WP_ , Feb. 8, 2017, https://www.washingtonpost.com/politics/supreme-court-nominee-gorsuch-says-trumps-attacks-on-judiciary-are-demoralizing/2017/02/08/64e03fe2-ee3f-11e6-9662-6eedf1627882_story.html?utm_term=.ad67c8244fc2.
. For a timeline of Trump's obstruction attempts up to early 2018, see Artin Afkhami, "Timeline of Trump and Obstruction of Justice: Key Dates and Events," _Just Security_ , Jan. 25, 2018, https://www.justsecurity.org/45987/timeline-trump-obstruction-justice-key-dates-events/.
. Michael S. Schmidt and Julie Hirschfeld Davis, "Trump Asked Sessions to Retain Control of Russia Inquiry after His Recusal," _NYT_ , May 29, 2018, https://www.nytimes.com/2018/05/29/us/politics/trump-sessions-obstruction.html.
. Donald J. Trump (@realDonaldTrump), Twitter, Dec. 3, 2017, https://twitter.com/realdonaldtrump/status/937305615218696193?lang=en.
. Linda Qiu, "Key Moments in Trump's Interview on 'Fox and Friends,' with Fact Checks," _NYT_ , June 15, 2018, https://www.nytimes.com/2018/06/15/us/politics/trump-fox-and-friends-interview.html.
. Donald J. Trump (@realDonaldTrump), Twitter, Apr. 13, 2018, https://mobile.twitter.com/realDonaldTrump/status/984767560494313472.
. Peter Baker, "Trump Assails Mueller, Drawing Rebukes from Republicans," _NYT_ , Mar. 18, 2018, https://www.nytimes.com/2018/03/18/us/politics/trump-mueller.html.
. MB, "Trump vs. Mueller Is a Battle for America's Soul," _WP_ , Feb. 26, 2018, https://www.washingtonpost.com/opinions/trump-vs-mueller-is-a-battle-for-americas-soul/2018/02/26/0979904c-1b19-11e8-9de1-147dd2df3829_story.html.
. Natasha Bertrand, "How Trumpworld Is Spinning the FBI Report," _The Atlantic_ , June 15, 2018, https://www.theatlantic.com/politics/archive/2018/06/how-trumpworld-is-spinning-the-fbi-report/562922/.
. Donald J. Trump (@realDonaldTrump), Twitter, May 23, 2018, https://twitter.com/realDonaldTrump/status/999246677549768704.
. MB, "Here Are the Political Norms that Trump Violated in Just the Past Week," _WP_ , May 21, 2018, https://www.washingtonpost.com/news/global-opinions/wp/2018/05/21/here-are-the-political-norms-that-trump-violated-in-just-the-past-week/?utm_term=.c083021c230d.
. Donald J. Trump (@realDonaldTrump), Twitter, Apr. 11, 2018, https://twitter.com/realDonaldTrump/status/984020136255541248?ref_src=twsrc%5Etfw&ref_url=https%3A%2F%2Fwww.mediaite.com%2Fonline%2Ftrump-on-fbis-cohen-raid-no-collusion-or-obstruction-other-than-i-fight-back%2F&tfw_creator=KenMeyer91&tfw_site=mediaite.
. Philip Bump, "An Increasing Number of Americans See the FBI as Biased against Trump," _WP_ , Apr. 17, 2018, https://www.washingtonpost.com/news/politics/wp/2018/04/17/an-increasing-number-of-americans-see-the-fbi-as-biased-against-trump/?utm_term=.b82c4635332c.
. Glenn Kessler, Salvador Rizzo, and Meg Kelly, "President Trump Has Made 3,001 False or Misleading Claims So Far," _WP_ , May 1, 2018, https://www.washingtonpost.com/amphtml/news/fact-checker/wp/2018/05/01/president-trump-has-made-3001-false-or-misleading-claims-so-far/?utm_term=.086753215f1b&__twitter_impression=true.
. Office of the United States Trade Representative, "Canada: U.S.-Canada Trade Facts," https://ustr.gov/countries-regions/americas/canada.
. Donald J. Trump (@realDonaldTrump), Twitter, May 31, 2018, https://twitter.com/realDonaldTrump/status/1002160516733853696.
. MB, "Trump Is Perfecting the Art of the Big Lie," _WP_ , Mar. 17, 2018, https://www.washingtonpost.com/news/global-opinions/wp/2018/03/17/trump-is-perfecting-the-art-of-the-big-lie/?utm_term=.ffdd48a2b483.
. Anne Gearan and Carol Morello, "Rex Tillerson Says 'Alternative Realities' Are a Threat to Democracy," _WP_ , May 16, 2018, https://www.washingtonpost.com/politics/former-trump-aide-rex-tillerson-says-alternative-realities-are-a-threat-to-democracy/2018/05/16/4d0353f0-594b-11e8-8836-a4a123c359ab_story.html?utm_term=.7ec476c7256a.
. Lucia Graves, "How Trump Weaponized 'Fake News' for His Own Political Ends," _Pacific Standard_ , Feb. 26, 2018, https://psmag.com/social-justice/how-trump-weaponized-fake-news-for-his-own-political-ends.
. Damian Paletta and Josh Dawsey, "Trump Personally Pushed Postmaster General to Double Rates on Amazon, Other Firms," _WP_ , May 18, 2018, https://www.washingtonpost.com/business/economy/trump-personally-pushed-postmaster-general-to-double-rates-on-amazon-other-firms/2018/05/18/2b6438d2-5931-11e8-858f-12becb4d6067_story.html?utm_term=.782a0e7dd5a7.
. Brian Stelter, "Giuliani Says Trump 'Denied' the AT&T-Time Warner Deal, then Backtracks," _CNN_ , May 11, 2018, http://money.cnn.com/2018/05/11/media/rudy-giuliani-trump-att-time-warner/index.html.
. MB, "Here Are the Political Norms that Trump Violated in Just the Past Week," _WP_ , May 21, 2018, https://www.washingtonpost.com/news/global-opinions/wp/2018/05/21/here-are-the-political-norms-that-trump-violated-in-just-the-past-week/?utm_term=.932ecf0a5d07.
. Avery Anapol, "Poll: Majority of Trump Supporters Say Media Is 'Enemy of American People,'" _The Hill_ , Dec. 4, 2017, http://thehill.com/blogs/blog-briefing-room/363098-poll-majority-of-trump-backers-say-media-is-enemy-of-american-people.
. John McCain, "Mr. President, Stop Attacking the Press," _WP_ , Jan. 16, 2018, https://www.washingtonpost.com/opinions/mr-president-stop-attacking-the-press/2018/01/16/9438c0ac-faf0-11e7-a46b-a3614530bd87_story.html?utm_term=.da6354624854.
. MB, "The Trump Administration Is in an Unethical League of Its Own," _WP_ , Mar. 1, 2018, https://www.washingtonpost.com/opinions/the-trump-administrations-no-good-very-bad-wednesday/2018/03/01/7dc60fd2-1d69-11e8-ae5a-16e60e4605f3_story.html?utm_term=.a7d55d2d0376.
. Jesse Drucker, Kate Kelly, and Ben Protess, "Kushner's Family Business Received Loans after White House Meetings," _NYT_ , Feb. 28, 2018, https://www.nytimes.com/2018/02/28/business/jared-kushner-apollo-citigroup-loans.html.
. Ana Cerrud and David A. Fahrenthold, "Warning of 'Repercussions,' Trump Company Lawyers Seek Panama President's Help," _WP_ , Apr. 9, 2018, https://www.washingtonpost.com/world/warning-of-repercussions-trump-company-lawyers-seek-panama-presidents-help/2018/04/09/9e3fbb8e-3c2f-11e8-8d53-eba0ed2371cc_story.html.
. "Trump Indonesia Project Is Latest Stop on China's Belt and Road," _South China Morning Post_ , May 11, 2018, http://www.scmp.com/news/asia/southeast-asia/article/2145808/trump-indonesia-project-latest-stop-chinas-belt-and-road.
. Julia Horowitz, "Ivanka Trump Granted Seven New Trademarks in China," _CNN_ , May 28, 2018, http://money.cnn.com/2018/05/28/news/ivanka-trump-china-trademarks/index.html.
. "CREW Releases Report: Profiting from the Presidency: A Year's Worth of President Trump's Conflicts of Interest," _Citizens for Responsibility and Ethics in Washington_ , Jan. 19, 2018, https://www.citizensforethics.org/press-release/crew-releases-report-profiting-presidency-years-worth-president-trumps-conflicts-interest/.
. Jenna Johnson, "As Stormy Daniels Tells Her Story, Six Conservative Americans Debate Whether Trump Is a Role Model," _WP_ , Mar. 3, 2018, https://www.washingtonpost.com/politics/as-stormy-daniels-tells-her-story-six-conservative-americans-debate-whether-trump-is-a-role-model/2018/03/28/bb4f258a-2f86-11e8-b0b0-f706877db618_story.html?utm_term=.b5a8c6b91dde.
. Ashley Merryman, "President Trump's Worst Behaviors Can Infect Us All Just Like the Flu, According to Science," _WP_ , Mar. 3, 2018, https://www.washingtonpost.com/news/inspired-life/wp/2018/03/29/president-trumps-worst-behavior-can-spread-among-us-just-like-the-flu-according-to-science/?utm_term=.e0da8a510bf8.
. Hannah Hartig, "Few Americans See Nation's Political Debate as 'Respectful,'" _Pew Research Center_ , May 1, 2018, http://www.pewresearch.org/fact-tank/2018/05/01/few-americans-see-nations-political-debate-as-respectful/.
. Ian Bremmer (@ianbremmer), Twitter, Apr. 18, 2018, https://twitter.com/ianbremmer/status/986683408611971072.
. "Dwight D. Eisenhower: 12—The President's News Conference, February 17, 1953," _The American Presidency Project_ , http://www.presidency.ucsb.edu/ws/index.php?pid=9623.
. Paul Ryan (@SpeakerRyan), Twitter, Feb. 9, 2012, https://twitter.com/speakerryan/status/167647060957462528.
. Catherine Rampell, "The United States Is Mortgaging Its Future," _WP_ , Apr. 19, 2018, https://www.washingtonpost.com/opinions/the-united-states-is-mortgaging-its-future/2018/04/19/daa35554-4409-11e8-ad8f-27a8c409298b_story.html?utm_term=.9e5905badcbb.
. Eugene Scott, "Rand Paul Calls Out Hypocrisy of GOP in the Trump Era," _WP_ , Feb. 9, 2018, https://www.washingtonpost.com/news/the-fix/wp/2018/02/09/rand-paul-calls-out-hypocrisy-of-gop-in-the-trump-era/?utm_term=.4f98c7af3c92.
. MB, "Republicans Are Making a Mockery of Their Reputations," _WP_ , Feb. 11, 2018, https://www.washingtonpost.com/opinions/republicans-are-making-a-mockery-of-their-reputations/2018/02/10/866aefe0-0eaa-11e8-8890-372e2047c935_story.html?utm_term=.db9678716864.
. Judd Gregg, "Judd Gregg: The GOP Abandons Fiscal Responsibility," _The Hill_ , Apr. 9, 2018, http://thehill.com/opinion/finance/382219-judd-gregg-the-gop-abandons-fiscal-responsibility?amp&__twitter_impression=true.
. Steve LeVine, "Economist: Trump Trade War Will Cost 190,000 Jobs," _Axios_ , Apr. 4, 2018, https://www.axios.com/economist-trump-trade-war-will-already-cost-190k-jobs-1522857360-0d8f5f65-8334-45f7-a2e6-d8251d5c5884.html.
. Alexandra Hutzler, "Trump's Unpopular Tariffs Could Cost U.S. 400,000 Jobs, Economists Estimate," _Newsweek_ , June 6, 2018, http://www.newsweek.com/donald-trump-tariffs-trade-job-loss-961988.
. MB, "Imposing Tariffs Is Stupid Policy," _WP_ , Mar. 5, 2018, https://www.washingtonpost.com/opinions/imposing-tariffs-is-stupid-policy/2018/03/05/b5512ea0-2093-11e8-86f6-54bfff693d2b_story.html.
. Ian Bremmer, _Us vs. Them: The Failure of Globalism_ (New York: Portfolio/Penguin, 2018), 16.
. "Washington Post-ABC News Poll April 8–11, 2018," _WP_ , Apr. 15, 2018, https://www.washingtonpost.com/page/2010-2019/Washington Post/2018/04/15/National-Politics/Polling/question_20289.xml?uuid=DEOE8kCWEeiy3LCkA-RyCg.
. Thomas Wright, "Trump's 19th Century Foreign Policy," _Politico_ , Jan. 20, 2016, https://www.politico.com/magazine/story/2016/01/donald-trump-foreign-policy-213546.
. MB, "Trump Turns the G-7 into the G-6 vs. G-1," _WP_ , June 10, 2018, https://www.washingtonpost.com/opinions/trump-turns-the-g-7-into-the-g-6-vs-g-1/2018/06/10/1d3b7276-6ccc-11e8-afd5-778aca903bbe_story.html?utm_term=.43c3e878d2ba.
. John Wagner, "'He's a Tough Guy': Trump Downplays the Human-Rights Record of Kim Jong Un," _WP_ , June 14, 2018, https://www.washingtonpost.com/politics/hes-a-tough-guy-trump-downplays-the-human-rights-record-of-kim-jong-un/2018/06/14/90ed487e-6fbb-11e8-bf86-a2351b5ece99_story.html?utm_term=.8240 ea13c377.
. Julie Ray, "World's Approval of U.S. Leadership Drops to New Low," _Gallup_ , Jan. 18, 2018, http://news.gallup.com/poll/225761/world-approval-leadership-drops-new-low.aspx.
. Haley Britzky, "Arab Youth See the U.S. as an Enemy," _Axios_ , May 8, 2018, https://www.axios.com/arab-youth-see-us-as-an-enemy-a4415c65-e096-4904-b32e-921ef54651a6.html.
. Donald Tusk (@eucopresident), Twitter, May 16, 2018, https://twitter.com/eucopresident/status/996731038062862336?ref_src=twsrc%5Etfw&ref_url=https%3A%2F%2Fwww.washingtonpost.com%2Fnews%2Fworldviews%2Fwp%2F2018%2F05%2F16%2Fe-u-leader-lights-into-trump-with-friends-like-that-who-needs-enemies%2F&tfw_creator=michaelbirnbaum&tfw_site=WashingtonPost&tid=a_mcntx.
. Greg Jaffe, "Trump Tries to Appear Strong in Syria Even as He Plans to Withdraw," _WP_ , Apr. 14, 2018, https://www.washingtonpost.com/world/national-security/trump-tries-to-appear-strong-in-syria-even-as-he-plans-to-withdraw/2018/04/14/4bb75fe6-400f-11e8-8d53-eba0ed2371cc_story.html?utm_term=.2126ce5bc304.
. Jeffrey Goldberg, "A Senior White House Official Defines the Trump Doctrine: 'We're America, Bitch,'" _The Atlantic_ , June 11, 2018, https://www.theatlantic.com/politics/archive/2018/06/a-senior-white-house-official-defines-the-trump-doctrine-were-america-bitch/562511/.
. MB, "Trump Is the Worst Salesman America Has Ever Had," _FP_ , Jan. 22, 2018, http://foreignpolicy.com/2018/01/22/trump-has-already-destroyed-americas-soft-power/.
**6. T HE TRUMP TOADIES**
. "The President Is Not above the Law," _NYT_ , Apr. 15, 2018, https://www.nytimes.com/interactive/2018/04/15/opinion/editorials/president-above-rule-law.html?action=click&pgtype=Homepage&clickSource=image&module=opinion-c-col-top-region®ion=opinion-c-col-top-region&WT.nav=opinion-c-col-top-region.
. Julie Hirschfeld Davis, "Trump's Cabinet, with a Prod, Extols the 'Blessing' of Serving Him," _NYT_ , June 12, 2017, https://www.nytimes.com/2017/06/12/us/politics/trump-boasts-of-record-setting-pace-of-activity.html.
. Charles Hurt, "Trump the Orator Outlines the Greatness of America to Democrats' Disgust," _Washington Times_ , Feb. 1, 2018, https://www.washingtontimes.com/news/2018/feb/1/donald-trump-highlights-americas-freedom/.
. Chris Cillizza, "Donald Trump Talks Like No Politician Has Ever Talked Before," _CNN_ , Aug. 9, 2017, https://www.cnn.com/2017/08/09/politics/stephen-miller-trump-orator/index.html.
. Fox News, June 1, 2017, http://www.foxnews.com/opinion/2017/06/01/donald-trump-miracle-worker.html.
. Steve Cortes, "Trump's Winning Streak Rivals One of the Greatest Streaks in Sports," _Washington Examiner_ , May 15, 2018, https://www.washingtonexaminer.com/opinion/op-eds/trumps-winning-streak-rivals-one-of-the-greatest-streaks-in-sports.
. Candace Owens (@RealCandaceO), Twitter, Apr. 16, 2018, https://twitter.com/realcandaceo/status/986066422399774720?s=11.
. Mallory Shelbourne, "Trump Praises Conservative Activist Candace Owens as a 'Very Smart Thinker,'" _The Hill_ , May 9, 2018, http://thehill.com/homenews/administration/386857-trump-praises-candace-owens-as-a-very-smart-thinker.
. Fox News (@FoxNews), Twitter, Apr. 15, 2018, https://twitter.com/foxnews/status/985500401976971267?s=11.
. MB, "Donald Trump Is Proving Too Stupid to Be President," _FP_ , June 16, 2017, http://foreignpolicy.com/2017/06/16/donald-trump-is-proving-too-stupid-to-be-president/.
. Rebecca Savransky, "Anderson Cooper Confronts GOP Lawmaker: You Haven't Heard the President Lie?," _The Hill_ , Apr. 17, 2018, http://thehill.com/homenews/media/383471-anderson-cooper-confronts-gop-lawmaker-you-havent-heard-the-president-lie.
. Peter Coy, "After Defeating Cohn, Trump's Trade Warrior Is on the Rise Again," _Bloomberg_ , Mar. 8, 2018, https://www.bloomberg.com/news/articles/2018-03-08/after-defeating-cohn-trump-s-trade-warrior-is-on-the-rise-again.
. Victor Davis Hanson, "Donald Trump, Tragic Hero," _National Review_ , Apr. 30, 2018, https://www.nationalreview.com/magazine/2018/04/30/donald-trump-tragic-hero/.
. Julian Jackson, _France: The Dark Years, 1940–1944_ (New York: Oxford University Press, 2001), 278.
. Zane Anthony, Kathryn Sanders, and David A. Fahrenthold, "Whatever Happened to Trump Neckties? They're Over. So Is Most of Trump's Merchandising Empire," _WP_ , Apr. 13, 2018, https://www.washingtonpost.com/politics/whatever-happened-to-trump-ties-theyre-over-so-is-most-of-trumps-merchandising-empire/2018/04/13/2c32378a-369c-11e8-acd5-35eac230e514_story.html?utm_term=.3592a997c0b9.
. Jackson, _France_ , 279.
. Jon Meacham (@jmeacham), Twitter, May 1, 2018, https://twitter.com/Trump4American1/status/991318857343258624.
. Brian Stelter, "'Mass Firing' at Conservative Site RedState," _CNN_ , Apr. 27, 2018, http://money.cnn.com/2018/04/27/media/redstate-blog-salem-media/index.html?sr=twmoney042718redstate-blog-salem-media1227PMStory.
. Karen Tumulty, "'Republicans Don't Want the Tweet That I Got': Mark Sanford Says Trump Sealed His Loss," _WP_ , June 13, 2018, https://www.washingtonpost.com/opinions/republicans-dont-want-the-tweet-that-i-got-mark-sanford-says-trump-sealed-his-loss/2018/06/13/d36a9fe2-6f0e-11e8-afd5-778aca903bbe_story.html?utm_term=.af88900f3abf.
. Geoffrey R. Stone, _Perilous Times: Free Speech in Wartime_ (New York: W. W. Norton, 2004).
. David M. Oshinsky, _A Conspiracy So Immense: The World of Joe McCarthy_ (New York: Free Press, 1983), 163–65.
. Ben Terris, "Bob Corker Is Free to Speak His Mind about Trump. If He Could Only Make It Up," _WP_ , Apr. 16, 2018, https://www.washingtonpost.com/lifestyle/style/bob-corker-is-free-to-speak-his-mind-about-donald-trump-if-he-could-only-make-it-up/2018/04/16/cc4f2d58-3da0-11e8-974f-aacd97698cef_story.html?utm_term=.8cfe2eb63bcf.
. Erick Erickson, "A Congressman's Profanity Laced Tirade in a Safeway Grocery Store," _The Maven_ , Apr. 11, 2018, https://www.themaven.net/theresurgent/erick-erickson/a-congressman-s-profanity-laced-tirade-in-a-safeway-grocery-store-SeHI2l5bIECGQn4gmnzGaw/?mc_cid=10d8170a1d&mc_eid=3cc50e048e&full=1.
. Cristiano Lima, "Boehner: 'There Is No Republican Party. There Is a Trump Party,'" _Politico_ , May 31, 2018, https://www.politico.com/story/2018/05/31/john-boehner-republican-trump-party-615357.
. Annie Linskey, "In the Era of Donald Trump, New England's Biggest GOP Donor Is Funding Democrats," _Boston Globe_ , Apr. 14, 2018, https://www.bostonglobe.com/news/politics/2018/04/14/era-donald-trump-new-england-biggest-gop-donor-funding-democrats/QzyFs3i3Yq3o6Ae7QIkhVP/story.html.
. MB, "It's a Disgrace More Republicans Aren't Willing to Choose Country over Partisanship," _WP_ , Apr. 23, 2018, https://www.washingtonpost.com/news/global-opinions/wp/2018/04/23/its-a-disgrace-more-republicans-arent-willing-to-choose-country-over-partisanship/?utm_term=.9e7c7dcd18ff.
. Demetri Sevastopulo (@Dimi), Twitter, Oct. 25, 2018, https://twitter.com/dimi/status/923144727700099074.
. MB, "On Trump, GOP Hasn't Learned Churchill's Lesson: War Is Preferable to Appeasement," _USAT_ , Oct. 26, 2017, https://www.usatoday.com/story/opinion/2017/10/26/trump-gop-hasnt-learned-churchills-lesson-war-preferable-to-appeasement-max-boot-column/799914001/.
**7. T HE ORIGINS OF TRUMPISM**
. "Why the South Must Prevail," _National Review_ , Aug. 24, 1957, https://adamgomez.files.wordpress.com/2012/03/whythesouthmustprevail-1957.pdf.
. Alvin Felzenberg, "How William F. Buckley, Jr., Changed His Mind on Civil Rights," _Politico_ , May 13, 2017, https://www.politico.com/magazine/story/2017/05/13/william-f-buckley-civil-rights-215129.
. Barry Goldwater, _Conscience of a Conservative_ (LaVergne, TN: Bottom of the Hill Publishing, 2010), 20.
. Goldwater, _Conscience_ , 51.
. Goldwater, _Conscience_ , 72.
. Phyllis Schlafly, _A Choice Not an Echo_ (Alton, IL.: Pere Marquette Press, 1964), https://www.metabunk.org/files/Schlafly%20-%20A%20Choice%20Not%20an%20Echo%20-%20The%20Inside%20Story%20of%20How%20American%20Presidents%20are%20Chosen%20(1964).pdf.
. E. J. Dionne, _Why the Right Went Wrong: Conservatism—From Goldwater to Trump and Beyond_ (New York: Simon & Schuster, 2016), loc. 2257, Kindle.
. Dionne, _Why the Right Went Wrong_ , loc. 91, Kindle.
. MB, "How the 'Stupid Party' Created Donald Trump," _NYT_ , July 31, 2016, https://www.nytimes.com/2016/08/01/opinion/how-the-stupid-party-created-donald-trump.html.
. William J. Bennett, "William Bennett: George Will Scorns Pence for the High Crime of Decency," _Fox News_ , May 12, 2018, http://www.foxnews.com/opinion/2018/05/12/william-bennett-george-will-scorns-pence-for-high-crime-decency.html.
. "10 of Sarah Palin's Most Amazingly Stupid Quotes," _Wow247_ , Jan. 20, 2016, http://www.wow247.co.uk/2016/01/20/sarah-palin-quotes/.
. MB, "Down with Populism!," _WSJ_ , Dec. 8, 1994.
. Andrew Kirell, "Dinesh D'Souza Mocked Shooting Survivors. Why Is He Still on the 'National Review' Masthead?," _Daily Beast_ , Feb. 21, 2018, https://www.thedailybeast.com/dinesh-dsouza-national-review-parkland-shooting-survivors-racist-bigot.
. MB, "If This Is What Conservatism Has Become, Count Me Out," _WP_ , Feb. 25, 2018, https://www.washingtonpost.com/opinions/if-this-is-what-conservatism-has-become-count-me-out/2018/02/25/853685c6-19bd-11e8-92c9-376b4fe57ff7_story.html.
. Tim Stelloh, "Sinclair Talk Show Canceled after Host Appears to Threaten Parkland Survivor David Hogg," _NBC News_ , Apr. 9, 2018, https://www.nbcnews.com/news/us-news/sinclair-talk-show-canceled-after-host-appears-threaten-parkland-survivor-n864211.
. Tomi Lahren (@TomiLahren), Twitter, May 14, 2018, https://twitter.com/tomilahren/status/996106261283270656?s=11.
. Derek Hawkins, "Hannity—with His $29 Million Salary and Private Jet—Slams 'Overpaid' Media Elites," _WP_ , Sept. 28, 2016, https://www.washingtonpost.com/news/morning-mix/wp/2016/09/28/hannity-slams-overpaid-media-elites-then-journalists-respond-noting-his-29m-salary-and-private-jet/?utm_term=.91607e317873.
. MB, "If This Is What Conservatism Has Become, Count Me Out," _WP_ , Feb. 25, 2018, https://www.washingtonpost.com/opinions/if-this-is-what-conservatism-has-become-count-me-out/2018/02/25/853685c6-19bd-11e8-92c9-376b4fe57ff7_story.html.
. MB, "A Conservative Commentator Revolts against Fox News," _WP_ , Mar. 21, 2018, https://www.washingtonpost.com/news/global-opinions/wp/2018/03/21/a-conservative-commentator-revolts-against-fox-news/.
. Matthew Gertz, "I've Studied the Trump-Fox Feedback Loop for Months. It's Crazier than You Think," _Politico_ , Jan. 5, 2018, https://www.politico.com/magazine/story/2018/01/05/trump-media-feedback-loop-216248.
. Robert Costa, Sarah Ellison, and Josh Dawsey, "Hannity's Rising Role in Trump's World: 'He Basically Has a Desk in the Place,'" _WP_ , Apr. 4, 2017, https://www.washingtonpost.com/politics/hannitys-rising-role-in-trumps-world-he-basically-has-a-desk-in-the-place/2018/04/17/e2483018-4260-11e8-8569-26fda6b404c7_story.html?utm_term=.ffe2d91a3d5e.
. MB, "Useful Idiocy," _Commentary_ , July 17, 2017, https://www.commentarymagazine.com/politics-ideas/tucker-carlson-russia-putin-neoconservatism/.
. Tom Namako, "Commentator Just Quit the 'Propaganda Machine,'" _BuzzFeed_ , Mar. 20, 2016, https://www.buzzfeed.com/tomnamako/ralph-peters?utm_term=.jspZ2WnAj#.wdPe7wpMO.
. MB, "2017 Was the Year I Learned About My White Privilege," _FP_ , Dec. 27, 2017, http://foreignpolicy.com/2017/12/27/2017-was-the-year-i-learned-about-my-white-privilege/.
. Tucker Carlson (@TuckerCarlson), Twitter, Dec. 27, 2017, https://twitter.com/TuckerCarlson/status/946158361241968640.
. Caitlin Johnstone, "Iraq-Raping Neocons Are Suddenly Posing as Woke Progressives to Gain Support," _Medium_ , https://medium.com/@caityjohnstone/iraq-raping-neocons-are-suddenly-posing-as-woke-progressives-to-gain-support-289fa527f3f8.
. Soledad O'Brien (@soledadobrien), Twitter, Dec. 28, 2017, https://twitter.com/soledadobrien/status/946268253277577216?lang=en.
**E PILOGUE: THE VITAL CENTER**
. Bruce Bartlett, "A Conservative Case for the Welfare State," _Dissent_ , Apr. 24, 2015, https://www.dissentmagazine.org/online_articles/bruce-bartlett-conservative-case-for-welfare-state.
. MB, "I Would Vote for (a Sane) Donald Trump," _FP_ , Sept. 20, 2017, http://foreignpolicy.com/2017/09/20/i-would-vote-for-a-sane-donald-trump/.
. Bob Fredericks, "This College Professor Is Happy 'Racist' Barbara Bush Is Dead," _New York Post_ , Apr. 18, 2018, https://nypost.com/2018/04/18/this-college-professor-is-happy-racist-barbara-bush-is-dead/?utm_campaign=iosapp&utm_source=mail_app.
. MB, "If This Is What Conservatism Has Become, Count Me Out," _WP_ , Feb. 25, 2018, https://www.washingtonpost.com/opinions/if-this-is-what-conservatism-has-become-count-me-out/2018/02/25/853685c6-19bd-11e8-92c9-376b4fe57ff7_story.html.
. Jonathan Rauch and Benjamin Wittes, "Boycott the Republican Party," _The Atlantic_ , Mar. 15, 2018, https://www.theatlantic.com/magazine/archive/2018/03/boycott-the-gop/550907/.
. Jonathan Chait, "Bernie Sanders's Bill Gets America Zero Percent Closer to Single Payer," _New York_ , Sept. 13, 2017, http://nymag.com/daily/intelligencer/2017/09/sanderss-bill-gets-u-s-zero-percent-closer-to-single-payer.html.
. Tami Luhby, "Sanders' Last 'Medicare for All' Plan Cost Nearly $1.4 Trillion," _CNN_ , Sept. 12, 2017, http://money.cnn.com/2017/09/12/news/economy/sanders-medicare-for-all/index.html.
. MB, "I Would Vote for (a Sane) Donald Trump," _FP_ , Sept. 20, 2017, http://foreignpolicy.com/2017/09/20/i-would-vote-for-a-sane-donald-trump/.
. Paul Waldman, "The Next Big Thing for Democrats: Medicare for All," _WP_ , Apr. 19, 2018, https://www.washingtonpost.com/blogs/plum-line/wp/2018/04/19/the-next-big-thing-for-democrats-medicare-for-all/?utm_term=.cad8ad33eae1.
. Mitchell Wellman, "Here's How Much Bernie Sanders' Free College for All Plan Would Cost," _USAT_ , Apr. 17, 2017, http://college.usatoday.com/2017/04/17/heres-how-much-bernie-sanders-free-college-for-all-plan-would-cost/.
. Mairead Mcardle, "Bernie Sanders Has No Cost Estimate on His Guaranteed-Jobs Plan," _National Review_ , Apr. 24, 2018, https://www.nationalreview.com/news/bernie-sanders-has-no-cost-estimate-on-his-guaranteed-jobs-plan/.
. Kevin Drum, "Need a Job? Just Call Bernie," _Mother Jones_ , Apr. 23, 2018, https://www.motherjones.com/kevin-drum/2018/04/need-a-job-just-call-bernie/.
. Jedediah Purdy, "Normcore," _Dissent_ , Summer 2018, https://www.dissentmagazine.org/article/normcore-trump-resistance-books-crisis-of-democracy.
. Matthew Dallek, "Why the GOP Needs Someone—Anyone—to Challenge Trump in 2020," _WP_ , May 18, 2018, https://www.washingtonpost.com/amphtml/outlook/why-the-gop-needs-someone--anyone--to-challenge-trump-in-2020/2018/05/18/1127f014-59ed-11e8-b656-a5f8c2a9295d_story.html?tid=ss_tw&utm_term=.068d00a0bc86&__twitter_impression=true.
. Matt Lewis, "Donald Trump Is Turning Young Voters Off the GOP—and Maybe Forever," _Daily Beast_ , June 13, 2017, https://www.thedailybeast.com/donald-trump-is-turning-young-voters-off-the-gopand-maybe-forever.
. Digitalist (@SomeDigitalist), Twitter, May 19, 2018, https://twitter.com/somedigitalist/status/997816270056648705?s=11.
. MB, "The Political Center Is Fighting Back," _WP_ , Apr. 25, 2018, https://www.washingtonpost.com/opinions/global-opinions/the-political-center-is-fighting-back/2018/04/25/6170f646-489b-11e8-827e-190efaf1f1ee_story.html.
## **INDEX**
Page numbers listed correspond to the print edition of this book. You can use your device's search function to locate particular terms in the text.
ABC, 130
Abe, Shinzo, 144
Abedin, Huma, 88, 183
Abu Ghraib prisoner-abuse scandal, 52
_Access Hollywood_ tape, xvii, 64, 86–87
Acosta, Alexander, 153
Adams, John, 157
Affordable Care Act. _See_ Obamacare
Afghanistan, 50, 51
African Americans, 65, 112, 115–16, 188–90
African immigrants, 118
Agnew, Spiro, 133, 176
Ailes, Roger, 41
Alien and Sedition Act of 1798, 157
allies, 67, 143–44
disapproval and distrust of US under Trump, 146–47, 148
tariffs imposed on, 123, 142, 143–44
Trump's attacks on, 112, 142–43
_All in the Family_ , 64
Allman, Jamie, 181
"alternative facts," 112. _See also_ Trump, Donald, lies told by
"alternative media," 179–80
Alt Right, 84–85
Amazon, 130–31
"America First" slogan, 68
"American empire," 49–50
American Enterprise Institute, 177
_American Greatness_ , 201
Americanism, 158
American-ness, redefinition in blood-and-soil terms, 120
American power
assertion of, 58
hard power vs. soft power, 147
limits of, 54
Trump presidency as death knell for, 69
waning of, 148
anti-Americanism, 66, 147
anti-Communist foreign policy, 12, 18
Anti-Defamation League, 84
anti-intellectualism, 175–77
anti-Semitism, 3, 18, 53–54, 67, 68, 83–84
Anton, Michael (Publius Decius Mus), 70–71
Apollo-Soyuz space program, 5
Applebaum, Anne, 81, 213
_Apprentice_ , _The_ , 61
Arab Americans, 65
Arab youth, 146
Argentina, 111
Arpaio, Joe, 114–16
Asian Americans, 188–89
Asman, David, 43
Assange, Julian, 85
assimilation, 7
AT&T–Time Warner merger, 131
Atwater, Lee, 170
Australia, 142
authoritarianism, 66, 111–12, 131, 152–53, 155–57
automation, 141
Ayotte, Kelly, 81
Bacevich, Andrew, 49
Baker, Charlie, 81, 200
Baker, Stewart, 125
_Baltimore Sun_ , 34
Bannon, Stephen, 85, 161
Barnes, Fred, 41
Bartley, Robert L., 38–39, 42–45
Basic Books, 179
Bay of Pigs, 97
Beauvoir, Simone de, 118
Beck, Glenn, 72
Beckerman, Gal, 4
Benjamin, René, 156
Bennett, Bill, 78, 177
Berkeley, California. _See also_ University of California, Berkeley
book tour in, 28
City Council of, 26
Berlin Wall, fall of, 27
Bezos, Jeff, 130–31
Biden, Joe, 52
Birchers, John, 18, 166
"birtherism" hoax, 63, 183
Bismarck, Otto von, 198
Blankfein, Lloyd, 84
Bloom, Alan, 179
Bloomberg, Michael, 77, 210
blue-collar workers, 90
Boehner, John, 160, 174
Bolton, John, 55, 98, 121
Booker, Cory, 205
Boot, Max. _See also_ Boot, Max, publications of
accused of "war crimes" by the left, xxiii
adolescence of, 11–16, 24
advocacy for Iraq War, 53–54
ambitions to be a lawyer, 23, 24
ambitions to be a writer, 23–24
anti-Semitic attacks against, 83–84
awakening to "white privilege," 191–92
bar mitzvah of, 13
birth of, 3
career of, 31–58
childhood in US, 5–11
at _Christian Science Monitor_ , 35–36
as columnist for _The Daily Californian_ , 26–28
on debate team, 24–25
on Defense Department advisory board with Hillary Clinton, 71
dismisses early Trump campaign, 61–62
in dual-language household, 9
as editor of high school newspaper, 24
education of, 1–30
as "Eisenhower Republican," 200
ex-friends of, 83
as foreign policy adviser to McCain presidential campaign, 55, 56–57
as foreign policy adviser to Romney presidential campaign, 57, 178
immigration to the US, xvii, 5–8
as an independent, xviii–xix, xxi, 213–14
introduced to conservative movement, 11–12
as journalist, 35–58
as "made guy" of convervative movement, 40–41
meets his future wife, 26
parents of, 5–16
place on the political spectrum, 197–214
political homelessness of, xviii–xix, xxi, 200–201, 212–14
reregisters as independent, xviii–xix, 91, 200, 202
returns to Berkeley during book tour, 28
sees Reagan in person, 20–21
semester abroad at London School of Economics, 26
on staff of _The Daily Californian_ , 26–28
starts "underground" newspaper in high school, 24, 25
support for Iraq war, xxiii, 52
takes job at Council on Foreign Relations, 51–52
on Tucker Carlson's show, 184–86
at UC Berkeley, 25–28, 68
volunteers fro Ed Zschau campaign for US Senate, 21–22
at _Wall Street Journal_ , 38–40, 42–51
wins Eric Breindel Award, 40–41
Boot, Max, publications of
"The Case for American Empire," 49–50
"Down with Populism!," 180
_Out of Order: Arrogance, Corruption and Incompetence on the Bench_ , 47–48
_The Savage Wars of Peace: Small Wars and the Rise of American Power_ , 50, 53, 55
"This Lifetime GOP Voter Is with Her," 89
border wall, 65
Bork, Robert, 44, 48
Bosnia, 4
Boston, Massachusetts, 34, 35–37
_Boston Globe_ , 161
"Bowling Green massacre," 98
Bozell, Brent, 80, 167
_Breitbart_ , 36, 85, 180, 201
Brennan, John, 145
Brennan, Megan, 131
Brezhnev, Leonid, 3, 5
Brinkley, David, 17
broadcast networks. _See also specific networks_.
Trump's threats to revoke licenses of, 130–31
Brooke, Edward, 173
Brookhiser, Richard, 12, 82
Brookings Institution, 101
Brooks, David, 43, 81
_Brown v. Board of Education_ , 47–48, 167
Bryan, William Jennings, 23
Buchanan, James, 62
Buchanan, Pat, 18, 42, 209
Buckley, Pat, 39
Buckley, William F., Jr., 12, 17–18, 39, 167, 176, 177
as gatekeeper of conservative movement, 19
racism of, 167
on Trump, 18
Burke, Arleigh, 58
Burke, Edmund, 13, 58
Bush, Barbara, death of, 201
Bush, George W., 54, 69, 70, 74, 146, 148, 171, 176. _See also_ Bush administration (W.)
Bush, George W. H., 170, 209
Bush, Jeb, 57, 61, 81, 99
Bush administration (W.), 52, 98, 177
Buttigieg, Pete, 203
BuzzFeed, 186
California
as minority-majority state, 211
possibility of third party in, 207
Republican Party in, 22, 211
Caligula, 99–100
Cambridge, Massachusetts, 34
campus speech codes, restrictive, 199
Canada, 129, 143–44
Cantor, Eric, 174
capitalism, 90–91
capital punishment, 28
capitulation, cost of, 107–48
Caribbean immigrants, 118
Carlson, Tucker, 79–80, 184–86, 191–92
Carson, Ben, 61, 132
Carter, Jimmy, 101, 102, 209
Case, Clifford, 173
Castro, Raul, 103
Catholicism, 12
CBS, 130
cell phone cameras, police brutality and, 141
Center for American Progress, Moscow Project, 121
center-right party, need for, 207–9
Central American migrants, "caravan" of, 183
Chaffee, John, 173
"chain migration," 6
_Challenger_ explosion, 20
Chamberlain, Neville, 103
Chambers, Whittaker, 12
Charles, Robert, 154
Charlottesville, Virginia, neo-Nazi demonstration in, 114
Chavez, Hugo, 111
Chavez, Linda, 81
Cheney, Dick, 54
China, 58, 134–35, 139–40, 144–45, 146, 147, 148, 203
Christian Science, 36–37
_Christian Science Monitor_ , 35–36
Christie, Chris, 61, 74
Churchill, Winston, 14
Citizens for Ethics and Responsibility, 135
civic culture, poisoned by Trump, 131
civil rights, 113, 169, 173
Civil Rights Act of 1964, 113
civil rights legislation, 167
civil rights movement, 189, 191
civil service, apolitical, 111
civil society grouops, 213
Clapper, James, 145
_Claremont Review of Books_ , 70
classical liberalism, xxi
Clifford, Stephanie. _See_ Daniels, Stormy
climate change, 102, 198, 203, 206
Clinton, Hillary, xvii, 43–44, 52, 64, 71, 104–5, 123, 159, 182, 203
email investigation and, 125
email scandal and, 88
as First Lady, 104
in Obama State Department, 104–5
presidential campaign of 2016, 76–77, 85, 86, 90
Republican candidates' calls for prosecution of, 128
Russian "dirt" on, 121–22
specious allegations against, 123, 183, 201
as US senator, 71
Clinton, William J., 43–45, 89–90
Gingrich and, 174
impeachment of, 45
Lewinsky scandal and, 87, 133, 137
Club for Growth, 72
CNN, 130
Cockburn, Alexander, 45
Cohen, Eliot, 71
Cohen, Michael, 99
Cohn, Gary, 98, 103–4
Cold War, 16, 27, 65
collective security, 139
college campuses, conservatism on, 29–30
College for All Act, 205, 206
Collins, Susan, 81, 159
collusion, 120–24
color blindness, 114, 199
Comey, James, 88, 125, 128, 129
_Commentary_ magazine, 40, 41, 72, 177, 184
Committee for a Responsible Federal Budget, 138
Communism, 4, 166
"comparative advantage" theory, 139
"compassionate conservatism," 171
Comstock, Barbara, 81
Confederate monuments, removal of, 114
conservatism, 168
American, xix–xx
"blood and soil," xix
conservative principles, 77
conservative Weltanschauung, 33–34
conspiracy theories and, 165
dumbing down of, 175, 178–79, 182
European, xix
extremism and, 165, 168, 192–93
history of, 165
ignorance and, 165
isolationism and, 165
know-nothingism and, 165
late twentieth-century, xxii
meaning of, xix
racism and, 165, 167–68, 169
conservative elite, 72–73
conservative movement, xx–xxi, 12, 19, 40–41. _See also_ conservatism
disallusionment with, 58
free trade and, 67–68
internationalism and, 67–68
as uneasy alliance, 72–73
conservatives, Trump's popularity among, xx
conspiracy theories, 63–64, 76, 112, 165, 171–72, 180–83, 201
Constitution, US, xx
Conway, Kellyanne, 98
Corker, Bob, 141, 159
Cortes, Steve, 154
Coulter, Ann, 180, 182
Council on Foreign Relations, 50–52, 66, 69, 78
counterinsurgency srategy, 53
"Cozy Bear," 85
Cranston, Alan, 22
"creative destruction," costs of, 91
Crédit Mobilier, 132
CrowdStrike, 85
Crozier, Brian, 12
Cruz, Heidi, 78
Cruz, Ted, 61, 73–75, 174
endorsement of Trump, 78–79
opportunism of, 78–79
Cuba, 103
Cuban, Mark, 210
cult of personality, 111, 152–53, 155–57
"cultural appropriatioins," 191
Curiel, Judge Gonzalo Paul, 78
DACA (Deferred Action for Childhood Arrivals), 119
_Daily Beast_ , 181
_Daily Californian_ , _The_ , 26–28, 37
Dallek, Matthew, 209
Daniels, Stormy, 99
Darfur, 4
Dartmouth College, outing of Gay Students Association members at, 29–30
_Dartmouth Review_ , 29–30
Debs, Eugene V., 158
debt-to-GDP ratio, 138
decency, Trump's transgressions against, 113–48
Declaration of Independence, xix–xx
Decter, Midge, 52
"Deep State," 76, 127, 183
defense, 43, 199, 203, 207
deficit, 138
deficit reduction, 197
deficit spending, 208–9
demagogues, 75, 112
democracy, 139
corrosion of, 157
need to defend, 212
Republican Party as danger to, 202–3
Democratic National Committee, 85, 133, 182–83
Democratic Party, 34, 61, 203
civil rights and, 113
donations to, 160–61
lack of centrists in, 204
leftward trajectory of, 204–7, 211
minority rights and, 169
need to support, 202–3
progressive wing of, 204–7
Reagan and, xxi
TTP and, 203
Democrats, 76, 124. _See also_ Democratic Party
Dent, Charlie, 81
Department of Commerce, 141
Department of Health and Human Services, 132
Department of Homeland Security
Immigration and Customs Service, 118–19
Victim of Immigration Crimes Enforcement office, 117–18
Department of Housing and Urban Development, 132
Department of Justice, 64, 88, 124, 127, 128
Department of the Interior, 133
Department of Veterans Affaris, 132
deregulation, White House exaggeration of, 104
desegregation, 47–48, 167–68
despotism, 70–71, 156
despot merchandising operations, 156
détente, 4, 5, 170
Deukmejian, George, 22
Dewey, Thomas, 67
dictatorships, 111, 144–46
Digitalist, 212
Dilulio, John J., 177
Dionne, E. J. Jr., 165, 174
disabilities, 65
discrimination, 187–88, 207
disloyal, purging of the, 68
Dobbs, Lou, 186
dog whistles, 169, 190
Dole, Bob, 173
Douglass, Frederick, 98
Douthat, Ross, 81
Dowd, John, 98
Dowd, Matthew, 81
Dow Jones Industrial Average, 101
Dreamers, Trump's withdrawal of legal protections from, 119
Drudge Report, 180
Drum, Kevin, 205–6
D'Souza, Dinesh, 29, 180–81, 182, 201
Trump's pardoning of, 124
Dukakis, Michael, 170
Duke, David, 64
Duterte, Rodrigo, 112, 145
economy, US, 101–2
economic power, 147
economic stagnation, 90
tax cuts and, 103–4
unaffected by Trump's deregulatory efforts, 104
education, of a conservative, 1–30
Edwards, Edwin, 87
Edwards, John, 135
Egypt, 111, 131
Eisenhower, Dwight D., 18, 137, 158–59, 166–67, 168, 175, 176, 177, 209–10
Emoluments Clause, 135
entitlement reform, 138–39, 197
"entrepreneurs," 72–73
environmental protection, 173, 198, 207
Environmental Protection Agency, 132
Erdogan, Recep Tayyip, 111, 131, 145
Eric Breindel Awared for Excellence in Opinion Journalism, 40–41
Erickson, Erick, 80, 160
Espionage Act of 1917, 157–58
ethics, 132–37
European Commission, 147
European Union, 16, 123, 144
evangelical Christian leaders, 87–88, 155
evangelical Christians, 87–88, 136–37
Executive Order No. 9066, 158
extremism
conservatism and, 165, 168, 192–93
Republican Party and, 201
Facebook, 112, 121
Factor, Mallory, 40
"fairness doctrine," 179
"fake news," 97, 128–32
adopted as mantra by authoritarian leaders, 131
"Fancy Bear," 85
far-right populism, in Europe, 111, 112
fascism, 70–71, 111
fascist rallies, 1930s, 66–67
traits of, 68
FBI (Federal Bureau of Investigation), 88–89, 124
Flynn and, 97–98
Nunes's attacks on, 127–28
raids offices of Michael Cohen, 99
Trump's attacks on, 125–26, 127–28
Watergate scandal and, 133
federal debt, 137
_Federalist 10_ , 112
Federalists, 157
Feith, Douglas, 54
Ferguson, Niall, 49
Finn, Mindy, 81
Fiornina, Carly, 64
_Firing Line_ , 17
First Amendment, 65, 131. _See also_ free press; free speech
fiscal conservatism, 173, 197, 207
fiscal irresponsibility, 137–39
Flake, Jeff, 81, 159, 161, 210
Florida, xviii
Flynn, Michael, 97–98, 122, 125, 133
Ford, Gerald, 170, 209
foreign policy. _See also_ interventionism; noninterventionism
anti-Communist, 12, 18
internationalist, 43, 173, 199–200, 207
militaristic, 54–55
moral concerns and, 5
neoconservative, 3–5
_Foreign Policy_ magazine, xvii–xviii, 89, 95, 187
_Forum_ , _The_ , 24
Fosdick, Dorothy, 4
Foster, Vince, 44
Founding Fathers, 112
Fox News Channel, 17, 33, 36, 41, 43, 100, 112, 126, 129, 157, 160
conspiracy theories peddled by, 182–84
hypocrisy of, 129
narrative of the loony left, 201
pernicious influence of, 41
popularity of, 180
Trump and, 79–80, 183–87
Fox News website, 154
Frady, Marshall, 109–10
France, 144, 155–56, 208
Francis, Samuel, 18
freedom, 139
Freedom Caucus, 155
Freedom House, downgrades US in annual "Freedom in the World" report, 145–46
freedom of religion, 65
free markets, 12, 90, 197–98
Free Press, 179, 181
free press, 111, 112, 146. _See also_ press, the
free speech, 191, 199
free trade, 43, 67–68, 139–42, 198, 203
free-trade policy, 140
French, David, 81
_FrontPage_ magazine, 201
Frum, David, 80, 81
Fund, John, 39
fusionism, 12
G-20 summit, Hamburg, Germany, 144
Galbraith, John Kenneth, 18
Gallup, 146
Garcia, Jorge, 119
Gardner, Cory, endorses and then repudiates Trump, 81
Gates, Rick, 122, 133
Gelb, Leslie, 50–51
German American Bund, 67, 68
German Americans, 65
Germany, 67–68, 140, 142, 143, 146
Gerson, Michael, 81
Gertz, Matthew, 183
Gigot, John, 39
Gilder, George, 179
Gillibrand, Kirstin, 205
Gingrich, Newt, 40, 61, 173–74
Giuliani, Rudolph, 70, 98, 131
Glasser, Susan, 100
globalization, 90, 198, 206
Goebbels, Joseph, 84
Goldberg, Jonah, 81
Goldman, Emma, 158
Goldman Sachs, 104
Goldwater, Barry, xx, 167–69, 173, 192, 200
GOP. _See_ Republican Party
Gorbachev, Mikhail, 170
Gorsuch, Neil, 104, 124
Gowdy, Trey, 128
Graham, Franklin, 155
Graham, Lindsay, 81, 159–60
Grant, Ulysses S., 132–33
Grant administration, scandals during, 132–33
Great Britain, 122, 148. _See also_ United Kingdom
Great Depression, 140
Great Recession of 2008–2009, xxi, 54, 56–57, 90, 101
Greece, 138
Gregg, Judd, 138
gross domestic product, 140
Group of Seven (G7), 123, 142, 143
group think, 40, 45, 157–58, 213–14
Grover Cleveland High School, Reseda, California, 24
Guccifer 2.0, 122
gun control, 28–29, 198, 207
Guttenberg, Karl-Theodor zu, 213
Haass, Richard, 51, 55
Hannity, Sean, 79–80, 182, 183–84, 186
Hanson, Victor Davis, 155
harassment, 188, 190–91
"hard hats," 176
Harding adminstration, 133
hard power, vs. soft power, 147
Harris, Kamala, 205
Hart, Gary, 136–37
Hart, Jeffrey, 12
Haspel, Gina, 55
Hatfield, Mark, 173
hawks, 52, 53–54
Hayden, Michael, 86
Hayek, F. A., 12
Hayes, Stephen, 81
Hebrew Immigrant Aid Society, 8
Henninger, Dan, 42
Heritage Foundation, 177
Hertog, Roger, 40
Hewitt, Hugh, 63, 79
Hicks, Hope, 98
Higgins, James and Heather, 40
Hindenburg, Paul von, 74
history, 13–14
Hitchens, Christopher, 45
Hitler, Adolf, 68, 74, 111, 130
Hofstadter, Richard, 175
Hogan, Larry, 81, 200
Hogg, David, 181
Holt, Lester, 125
homophobia, 30
Horowitz, David, 201
Horton, Willie, 170
House Intelligence Committtee, 126
House of Representatives, 173–74
House Republican caucus, ungovernability of, 174–75
House Un-American Affairs Committee, 167
Houston, Texas, 5
Huckabee, Mike, 129
Hughes, Charles Evans, 175
_Human Events_ , 179
human rights, 4, 139
Hungary, 111, 112
Hurowitz, Richard, 213
Hurricane Katrina, 54
Hurricane Maria, 100
Hurt, Charles, 152–54
Hussein, Saddam, 52, 54
hypernationalism, 68
identity politics, 188–89, 199
ignorance, conservatism and, 165
immigrants
deportation of undocumented, 119
Reagan's celebration of, 117, 120
Trump's demonization of, 9, 117–20
immigration, xv, 43, 141, 198–99, 207
anti-immigrant sentiments, 117–20
"chain migration," 6
Reagan's celebration of, 117
refuseniks, 5–7
Immigration and Customs Service, increase in arrests by, 118–19
immorality, Republican Party's embrace of, 208–9
incivility, 201
indoctrination, 33
Indonesia, 134–35
_Infowars_ , 36, 180
Ingraham, Laura, 29, 79, 181, 182, 186, 201
injustice, 188
integration, 199
internationalism, 67–68
internationalist foreign policy, 43, 173, 199–200, 207
international law, 139
internet, politics and, 19
internships, 34
interventionism, 48–50
intolerance, 201
Iowa caucuses, 73–74
Iran, 58, 98, 102, 191–92. _See also_ Iran nuclear deal
Iran–Contra scandal, 133
Iran nuclear deal, Trump's withdrawal from, 98, 102, 103, 142, 144, 147
Iraq, 51–54
war in, xxi, xxiii, 51–54, 57, 90, 146
withdrawal of troops from, 53, 57
Islamic State of Syria and Iraq (ISIS), 53, 57, 102, 106
isolationism, 67–68, 165, 173, 206, 208–9
Israel, 7, 13, 84, 102, 134
Italy, 111, 138
Jackson, Andrew, 169
Jackson, Henry "Scoop," 4, 204
Jackson, Ronny, 100
"Jacksonians," 169
Jackson–Vanik Amendment, 5
Japan, 67–68, 140, 142, 144
Japanese Internment Camps, 158
Jarrar, Randa, 200–201
Javits, Jacob, 173
Jay Z, 64
Jefferson, Thomas, 114
Jerusalem, Israel, US embassy in, 102, 106
Jews
American, 4, 42–43, 80, 84
Jewish conservatives, 42–43
Soviet, 3–5
Jindal, Bobby, xvi
job loss, 141
John Birch Society, 168
John Paul II, Pope, 154
Johnson, Dwayne, 210
Johnson, Lyndon, 171, 209
Jones, Alex, 182
Jordan, Jim, 155
journalism, 35–58
judges, 47–48
judiciary
blocks Trump's Muslim ban, 97
independence of, 111
"judicial activism," 47–48
nominations to Supreme Court, 104
Trump's appointment of conservative judges, 124
Trump's attacks on, 78
Kabaservice, Geoffrey, 165
Kagan, Robert, 81
Kaine, Tim, 203
Karsh, Yousuf, 14
Kasich, John, 75, 81, 210
Kasparov, Garry, 213
Kavanaugh, Brett, 104
Kelly, John, 98, 99
Kelly, Megyn, 64
Kemp, Jack, 29, 78
Kennedy, John F., 34, 96, 97, 175
Kennedy, Ted, 209
Kerry, John, 52
Kessler, Andy, 46
KGB, 5
Khan, Sadiq, 143
Kim Jong Un, 55, 103, 106, 129, 142, 144, 145, 155, 183
King, Don, 64
King, Martin Luther, Jr., 114, 154
Kinsley, Michael, 39
Kinzinger, Adam, 81
Kirk, Mark, 81
Kirk, Russell, 12–13
Kirkpatrick, Jeane, 66, 177
Kislyak, Sergey, 122
Kissinger, Henry, 4, 5, 177
Klarman, Seth, 160–61
know-nothingism, xxi, 165, 175
Korea–US Free Trade Agreement, 140
Kremlin, 71, 86, 120–24. _See also_ Russia
Kristol, Irving, 41
Kristol, William, 41, 42, 72, 81, 177, 201
Krugman, Paul, 45
Kruse, Michael, 115
Kudlow, Larry, 144
Kuehnelt-Leddihn, Erik von, 12
Kuhn, Fritz Julius, 67
Ku Klux Klan, 64, 168, 170, 189
Kushner, Jared, 64, 84
Trump Tower meeting and, 121–22
unethical behavior of, 133–35
Kuwait, 131
Laffer, Arthur, 39
LaGuardia, Fiorello, 67
Lahren, Tomi, 181–82
Lamb, Conor, 203
Las Vegas, Nevada, February 2016 rally at, 66
Latinos, 65, 188–89, 211
Law and Justice Party (Poland), 112
leader cults, 68. _See also_ cult of personality
Lee, Robert E., 114
leftists, xxii–xxiii
Le Pen, Marine, 208
_Le Sabre_ , 24
Levin, Mark, 80, 180
Lewinsky, Monica, 44–45, 133, 137
LGBTQ rights, 197, 203, 206
libel laws, 66
liberal elite, 72–73
"triggering" of, 105, 181–82
libertarianism, 25, 72, 176, 192, 207
Libya, 131
Lieberman, Joe, 178, 203
Limbaugh, Rush, 80, 179, 180
Lincoln, Abraham, 175
Lindbergh, Charles, 68
Lindsay, John, 173
Livingston, Bob, 137
Loesch, Dana, 80
London School of Economics, 26
Long, Huey, 109, 112
Los Angeles, California, 5–6, 24
_Los Angeles Times_ , 16, 34, 91
lost-golden-age syndrome, 68
Love, Mia, 81
loyalty, 71–72
Macron, Emmanuel, 144, 208
Magnin, Rabbi Edgar, 13
Maistre, Joseph de, 16
"Make America Great Again" slogan, 67, 90, 116–17, 172
Manafort, Paul, 133
Trump Tower meeting and, 121–22
mandatory unionization, 206
Manhattan Institute, 40
Mano, D. Keith, 12
Mar-a-Lago, 135
Marino, Gordon, 46
Marshall, George C., 159
masculinity, fetishization of, 68
_Masses_ , _The_ , 158
mass media. _See specific outlets_
mass mobilization, 68
Mattis, James, 99, 210
May, Theresa, 143
McAuliffe, Christa, 20
McCabe, Andrew, 125
McCain, John, 55–56
endorses and then repudiates Trump, 81
presidential campaign of 2008, 55, 56, 178
Senate career of, 56–57
speech on dangers of Trumpism, 159
support for journalists, 132
Trump's attacks on, xv–xvi
in Vietnam War, xvi, 55–56
McCarthy, Eugene, 204, 209
McCarthy, Joseph, 109, 112, 158–61, 166, 167, 171, 192
McCarthyism, 158–59, 166, 210
McChrystal, Stanley, 210
McConnell, Mitch
endorsement of Trump, 77
puts party above country, 86
refuses to cooperate with Obama in confronting Kremlin, 86
McGovern, George, 204
McGrath, Bryan, 71
McIntosh, David, 72
McMaster, H. R., 98, 99, 120–21, 123, 127
McMullin, Evan, 81
McNeill, John, 68
McRaven, William, 210
Meacham, Jon, 157
media, 109. _See also_ broadcast networks; press, the
Trump as beneficial to media business, 131
Trump's attacks on, 130–32
Media Matters for America, 183
Media Research Center, 80
Medicare for All, 204–5, 206
Medved, Michael, 72
Meese, Edwin, III, 72
Mehlman, Ken, refuses to endorse Trump, 81
Mélenchon, Jean-Luc, 208
melting pot, 199
Mencken, H. L., 22–23, 88, 180
Merkel, Angela, 143
#MeToo movement, xxi, 191
Mexican Americans, 78
Mexican immigrants, xv, 65, 112
Mexico, 65, 134, 142
Meyer, Frank, 12
"microaggresions," 191
"Middle America," 72–73
Midwest, 89–90
militarism, 54–55, 68
military culture, 51–52
military power, 147
Miller, Stephen, 120, 154
minorities, 188–90. _See also specific groups_
demonization of, 203
Trump's attacks on, 112
minority-majority states, 211
Mnuchin, Steven, 153
mob rule, 180
Montenegro, 142
Moody's, 140
Moore, Russell, 72, 135
moral relativism, 66
Mormons, 80
_Morning Joe_ , 65–66
Morrell, Michael, 86
_Mother Jones_ , 29, 205–6, 212
Moulton, Seth, 203, 210
movement conservatism, 173, 200, 213. _See also_ conservative movement
Moynihan, Daniel Patrick, 177
MS-13 gang, 117
Mueller, Robert S., III, 120, 125, 183
public service of, 126
Trump's attacks on, 126
Mukasey, Michael, 72
Mundell, Robert, 39
Munich Security Conference, 56, 103
Murdoch, Rupert, 41
Murkowski, Lisa, 81, 159
Murphy, Michael, 81
Murphy, Stephanie Dang, 203
Murray, Charles, 179
Muslim ban, 65, 97, 124
Muslims, xvi, 65, 118, 143
Mussolini, Benito, 111
Myanmar, 131
_Nation_ , _The_ , 26, 158
National Academy of Sciences, 116
National Empowerment Televsion, 180
_National Review_ , 11–12, 16–19, 24, 30, 37, 39, 41, 72, 82, 167, 177, 179
"Against Trump" issue, 80
National Rifle Association (NRA), 80
National Security Council, 70
National Summit for Democracy, 213
nation-building, 50
nativism, 65, 116–20
NATO allies, 123. _See also_ allies
Navarro, Ana, 81
Navarro, Peter, 144, 155
Nazi Germany, 117, 140
Nazism, 67, 111
NBC News, 130
neoconservative foreign policy, 3–5
neoconservatives, 41–42, 53–54
neo-Nazis, 114
Nero, 100
#NeverTrump conservatives, xxii, 46, 72, 75–77, 80–81, 95–96, 105, 201
_New Criterion_ , 40
New Deal, 166
New Deal coalition, 89
New Hampshire primary, presidential campaign of 2016, 74
New Left, 188
news, vs. opinion, 36
Newsmax, 180
news networks, coverage of Trump, 73
_Newsweek_ , 16–17
New York, New York, conservative society in, 39–40
_New Yorker_ , 100
_New York Post_ , 41
_New York Times_ , 35, 45, 130, 131, 133–34, 152, 176
NFL (National Football League), 115, 199
Nicaragua, 145
Nichols, Tom, 81, 201
nicknames, derogatory, 67
Niven, David, 18
Nixon, Richard, 89, 124, 129, 133, 170, 176–77
Nixon administration, 4, 133
noninterventionism, 57, 148, 206
Noonan, Peggy, 20
North American Free Trade Agreement (NAFTA), 140
North Atlantic Treaty Organization (NATO), Article V mutual-defense provision, 142
North Korea, 55, 58, 98, 129–30, 145, 152
efforts to denuclearize, 103, 106
sanctions on, 106
summit with, 103, 142, 155
NowThis, 129
Nunes, Devin, 126–28, 201
Nuremberg rallies, 67, 84
Obama, Barack, 56, 66, 69, 89–90, 96, 129, 182, 205. _See also_ Obama adminstration
"birtherism" hoax and, 63, 183
Castro and, 103
deficit and, 138
economic expansion under, 101
Iran nuclear deal and, 103
ISIS and, 102
job creation under, 101
"lead from behind" foreign policy, 57
noninterventionism of, 57, 148
pleads for bipartisan unity to confront Kremlin, 86
presidential campaign of 2008, 57
presidential campaign of 2012, 57
stock market under, 101
withdrawal of troops from Iraq, 53, 57
Obama adminstration, 98, 174
Obamacare, 159, 174, 183, 205
O'Brien, Soledaad, 192
obstruction of justice, 125–26, 160
Republican Party's embrace of, 208–9
Olson, Ted, 99
opinion, vs. news, 36
opioid epidemic, 90
opposition, self-definition by, 68
oppression, 188
O'Quinn, John, 46–47
Orban, Viktor, 112, 131
O'Reilly, Bill, 79
orthodoxy, 40, 45–46
Orwell, George, 192
Other, stigmatization of the, 118
Owens, Candace, 154–55
Page, Carter, 71, 127
Page, Lisa, 127
Palin, Sarah, 56, 178–79
Panama, 134
Papadopoulos, George, 71, 121
Papen, Franz von, 74
Paris Climate Accord, Trump's withdrawal from, 102, 142, 143
partisanism, 44
party line. _See_ group think
Patrice Lumumba University, 8
Patterson, Richard North, 213
Patton, George S., 155
Paul, Rand
endorsement of Trump, xvi
hypocrisy of, 138
presidential campaign of 2016 and, 61
Pax Americana, end of, 139–48
Pence, Mike, 152–53, 177
people of color, 90, 211. _See also specific groups_
people with disabilities, 65
Perdue, Sonny, 153
Perle, Richard, 4, 54
Perlstein, Rick, 165
Peron, Juan, 111
Perot, H. Ross, 209
Perry, Rick, xvi
Pershing, John J., 129
personal insults, 67, 136
Pétain, Marshal Philippe, 155–57
Peters, Ralph, 184, 186–87
Petraeus, David, 53, 57
Pew survey, 136
Philippines, 111, 112, 131, 145
Pirro, Jeanine, 79, 186
Podhoretz, John, 41, 42, 72, 81
Podhoretz, Norman, 41, 52, 83, 177
Pogo, 211
Poland, 111, 112, 131
police misconduct, xxi, 189–90
political correctness, 187, 191
political courage, rarity of, 158
political discourse, debasement of, 186
political opponents, attempts to annihilate, 44
politicians, misconduct and, 135–37
_Politico_ , 115
politics
democratization of, 19
the internet and, 19
tribal, 77
Politifact, 63–64
Pompeo, Mike, 98
populism, 47, 111, 112, 169, 172, 175, 177, 179–80, 199, 212
Porter, Rob, 135
Portman, Rob, 81
Posobiec, Jack, 201
Powell, Colin, 54
presidency, respect for, 68–69
presidential election of 1976, 209
presidential election of 1988, 136–37
presidential election of 2008, 57, 178
presidential election of 2012, 57
presidential election of 2015, 58
presidential election of 2020, 209
presidential election of 2016, 58, 61, 62–63, 70–71, 86, 88–89. _See also specific candidates_
aftermath of, 95–96
December 2015 Republican debate, 63
DNC and, 85–86
election night, xvii–xviii
Electoral College vs. popular vote results, 120
exit polls, xvii–xviii
Iowa caucuses, 73–74
New Hampshire primary, 74
Republican primaries, xvi, 73–76
Russian interference in, 85–86, 120–21
South Carolina primary, 74–75
presidents and presidential candidates, deference to, 68–69
press, the. _See also specific publications_ 146
First Amendment protection of, 131 ( _see also_ free speech)
Trump's attacks on, 130–32
pretectionism, 67–68
Price, Tom, 132
Priebus, Reince, 98, 153
pro-choice, 197
progressives, 204–6
pro-gun orthodoxy, 28–29
propaganda, 183
protectionism, 139–42, 206, 208–9
Pruitt, Scott, 132
_Public Interest_ , 41
publishing, 179, 182
Puerto Rico, 100
punditry, 37–38
Purdy, Jedediah, 206
"pursuit of Happiness," xix–xx
Putin, Vladimir, 97, 111, 120, 131, 144, 183, 203
regime of, 6, 58
Republicans cozying up to, 123–24
rigged reelection victory of, 123
Trump as useful idiot for, 86
Trump's admiration for, 58, 66, 85, 145, 203
Putin regime, 6
Rabinowitz, Dorothy, 39
racism, xxi, 84–85, 113–16, 187–88, 190, 199, 201, 207
of Buckley, 167
conservatism and, 165, 167–68, 169
Republican Party and, 165, 167–68, 169
Trump and, 113–16, 169
radio, 109, 179, 180
Rauch, Jonathan, 202
Reagan, Nancy, 134
Reagan, Ronald, xxi, 19–21, 41, 55, 58, 66, 89, 96, 134, 154, 167, 177. _See also_ Reagan administration
assassination attempt on, 19–20
celebration of immigrants, 117, 120
_Challenger_ explosion and, 20
Cold War policies of, 27
conservative movement and, 29
deregulation achieved by, 104
"Make America Great Again" slogan and, 116–17
as moderate, 170, 176
"Morning in America" slogan, 21
as movement conservative, 19
outstretched-hand approach of, 30
presidential campaign of 1976, 209
rejects endorsement by Ku Klux Klan, 64
speech at Pierce Community College, 21
supply-side economics and, 37
tax reform act of 1986, 137
"time for choosing" speech, 176
Reagan administration, 177
Federal Communications Commission under, 179
Iran–Contra scandal and, 133
Reagan Democrats, 89–90
Realpolitik, 4, 5
Red State, 157
Reed, Ralph, 87
refugees, 7, 119, 143. _See also_ immigrants
refuseniks, 3–5
Regional Comprehensive Economic Partnership, 140
Regnery Publishing, 179
Renew Democracy Initiative, 212–13
Reporters Without Borders, 146
Republican Congress, lack of principles, 138–39
Republican Convention of 1964, 168–69
Republican donors, 160–61
Republican Party, xxi, 61, 77
alienation of young people and minorities, 211–12
anti-intellectualism and, 175–77
ascendancy of Trump supporters in, 82–83
as a "big tent" party, 173
in California, 22
capitulation of, 59–91
civil rights and, 113, 167, 169
control of all three branches of federal government in 2016, 91
control of House of Representatives in 1994, 173–74, 180
cowardice of, 161
as cult, 80–81
as danger to rule of law and democracy, 202–3
death of, 82
disallusionment with, 54–55, 58
driving out of moderates, 173
embrace of Trump's protectionism, 141
embrace of white nationalism, xxi
endorsement of Trump, xvi
expiration of, 211
extremism and, 192–93, 201
failure to call out Trump on his lies, 132
free trade and, 67–68
groupthink in, 157–58
history of, 172–73, 175
hypocrisy of, 138
intellectual ruin of, 138
internationalism and, 67–68
isolationism and, 173
knee-jerk loyalty to, 75
know-nothingism and, xxi
need to vote against, 202–3
obstruction of justice and, 160
as "party of ideas" under Reagan, 177
as party of Lincoln, 113, 169
as party of southern whites, 169
as party of white privilege, 169
populism and, 177
primaries in 2016 election, xvi
racism and, 165, 167–68, 169
redefined by Trump, 175
rehabilitation of, 209
reputation for fiscal austerity, 137
rightward trajectory of, 172–74, 207, 211
segregationism and, 173
as "stupid party," 175, 178–79, 182–83
surrender to Trumpism, 59–91, 202–3, 208–9, 211
Trump as character test for, 80–81
"win at all costs" mindset, 86
Republican presidents, as moderates, 166–71, 176. _See also specific presidents_
Republican primaries, presidential campaign of 2016, 73–76
Republicans, 166–70. _See also_ Republican donors; Republican Party; Republican voters
animus toward, 104–5
cozying up to Putin, 123–24
endorsements of Trump, xvi, xvii
failure to resist Trump's authoritarianism, 157
groupthink and, 40, 45, 157–58, 213–14
hypocrisy of, 123, 160
ignorance of history, 68
ignore Trump's fascist and white-nationalist tendencies, 68
ignore Trump's misconduct and ethics violations, 135–36
as moderate, 166–73, 176
under pressure to support Trump, 79–80
pretend not to notice Trump's behavior, 66
Russia and, 57–58
on Senate Intelligence Committee, 121
situational ethics practiced by, 159
statements about Trump before his election, 152
Trump's popularity among, xx
unwillingness to defend Mueller, 126
unwillingness to speak out against Trump, 151–52
view of Russia, 124
view of Trump as role model for their kids, 136
who endorsed Trump but later repudiated him, 81
who refused to endorse Trump, 81
Republican voters, 72–73
retraining, 141
Ricardo, David, 139
Rice, Condoleezza, 54, 177
Rice, Susan E., 127
Rich, Seth, 183
right-wing talk radio, 179–80
RINOs (Republicans in Name Only), 200
Riverside, California, 6
Roberts, Cokie, 17
Robin, Corey, 165
Rockefeller, Nelson, 168, 172–73
"Rockefeller Republicans," 173, 200
Roman emperors, excesses of, 99–101
Romney, George, 172–73
Romney, Mitt, 205
presidential campaign of 2012, 57
refuses to endorse Trump, 81
warning about dangers of Russia, 57–58
Roosevelt, Franklin D., 67, 145, 153, 158
Roosevelt, Theodore, 175
Root, Elihu, 175
Rose, Charlie, 135
Rosenstein, Rod, 125
RT, 36, 182, 183
Rubin, Jennifer, 81
Rubio, Marco, 58, 61, 62–63, 74, 75
endorsement of Trump, xvi, 75–76
opportunism of, 76
rule of law, 124–28, 207. _See also_ obstruction of justice
attacks on, 203
Republican Party as danger to, 202–3
Trump's transgressions against, 112, 113
Rumsfeld, Donald, 52, 54
Russia, 57–58, 71, 146, 147. _See also_ Kremlin; Putin regime
dissidents killed in, 6
expansion of power by, 123
Group of Seven and, 123
hacking of DNC by, 182–83
influence operations of, 213
murder of former Russian agent in Britain, 122
presidential campaign of 2016 and, 85–86, 88–89, 120–21, 182–85
sanctions on, 122
special counsel investigation into Russian meddling, 123, 128
transformation to dictatorship, 111
Trump campaign and, 85–86, 88–89, 184–85
Russia investigation, 123, 128
Russian Americans, 65
Russian diplomats, expulsion from US, 122–23
Russian embassy, 122
Russian oligarchs, 99
Russians
indicted in Special Counsel investigation, 120
Trump campaign and, 120–24
Rust Belt, 34, 89
Rwanda, dehumanizing language used in, 117
Ryan, Paul, 160
accused of being insufficiently conservative, 174–75
announcement of retirement, 138
endorsement of Trump, xvi, 77–78
fiscal irresponsibility and, 137
hypocrisy of, 137, 138–39
on moral relativism, 66
Safire, William, 176
Salem Media Group, 79
Salter, Mrk, 81
San Bernadino, California, terrorist attack in, 65
Sander, Gordon F., 67
Sanders, Bernie, 71, 85, 203, 204–6
Sanford, Mark, 157
Sasse, Ben, 81, 159–60, 210
Saudis, 144
Savage, Michael, 180
Scarborough, Joe, 66, 81
Schiller, Keith, 98
Schlafly, Phyllis, 171–72, 192
Schmidt, Steve, 81
Schumer, Chuck, 52
Schwarzenegger, Arnold, xv
Scranton, William, 173
Second Amendment, fetishization of, 29 ( _see also_ gun control)
Secret Service, 66
Sedition Act of 1918, 158
Seeley, J. R., 148
segregation, 187
segregationism, 4, 109–11, 165, 167–68, 169, 173, 189
Sejanus, Lucius, 99
self-indoctrination, 33
Senate Intelligence Committee, Republican majority of, 121
September 11, 2001, terrorist attacks, 48–49, 65
Sessions, Jefferson, III, 125
sexism, 187–88, 189, 190–91, 201, 207
sexual harassment, 188, 190–91
"shithole countreis," 3, 118
Shlaes, Amity, 43
Shulkin, David, 132
Shultz, George P., 177
"silent majority," 176
Simon, John, 12
Simpson-Bowles Commission, 197
Sinclair Broadcast Group, 181
Singapore, summit in, 103, 142
Singer, Paul, 40
single-payer health care, 204–5, 206
Sisi, Abdel Fattah-el, 111
situational ethics, 159
slavery, 187
Smith, Adam, 95
Smith, Margaret Chase, 158
Smoot-Hawley Tariff Act of 1930, 140
Sobran, Joseph, 12, 18
socialism, 204, 206
social issues, 43
social liberalism, 197, 207
social media, 112, 121. _See also specific platforms_
social safety net, 197
Social Security Administration, Supplemental Security Income payments, 6
social welfare benefits, 141
soft power, vs. hard power, 147
Somoza, Anastasio, 145
Soros, George, 84
South Carolina primary, presidential campaign of 2016, 74–75
"Southern Strategy," 170
South Korea, 67–68, 142
Soviet Jews, 3–5, 9
Soviet Union, 3–4
anti-Semitism in, 3–5
collapse of, 7
Jewish emigration from, 5–6
Sowell, Thomas, 72
_Spy_ magazine, 30
Stalin, Josef, 76–77, 130
states' rights, 169, 170
Statue of Liberty, 120
Stavridis, James, 210
Steele, Christopher, 127
Steele, Michael, refuses to endorse Trump, 81
Steele dossier, 127
Steffens, Lincoln, 174
Steil, Benn, 69
Stein, Jill, 85
Stephens, Bret, 46, 81
Stevens, Stuart, 81
Stevenson, Adlai, 175
Steyn, Mark, 184–85
stock market, 101
Stone, Roger, 122
Strassel, Kimberly, 46
strongmen, 111, 131
Strzok, Peter, 127
Sullivan, Dan, endorses and then repudiates Trump, 81
superpower equivalence, 66
superpowers, downfall of, 147
supply-side economics, 37, 38–39
Supreme Court
confirmation hearings, 1987, 44
nomination of Gorsuch to, 104
Trump's appointees to, 104, 124
swamp, the, 79
"Swastika Nation," 67
Sykes, Charlie, 80, 81
Syria, 57, 102, 122, 123, 131
Taft, Robert, Jr., 173
tariffs, 68, 139, 140, 141, 142, 143–44
tax cuts, 43, 103–4, 137, 161
cut in corporate tax rate, 106
negative impact of trade wars on, 104
tax increases, 206
Tea Party, 61, 174
Teapot Dome scandal, 133
technology, 190
television, 109, 112, 179, 180–87. _See also_ broadcast networks; _specific outlets and programs_
Terry, Sue Mi, xviii, 26, 34, 43, 48
Thatcher, Margaret, 154
theatricality, 68
TheBlaze, 180
think tanks, 177
third parties, 207–8, 209–11
_This Week with David Brinkley_ , 17
Thomas, Cal, 72
Thurmond, Strom, 173, 192
Tiberius, 99
Tillerson, Rex, 96, 98, 130
Time Warner, 131
toadyism, 149–61
Toledano, Ralph de, 12
totalitarianism, 140
tough-on-crime policy, 173
trade partners, 68
trade surpluses, 139
trade wars, 104, 123, 139, 140
traditional social views, 12
Trans-Pacific Partnership, Trump's withdrawal from, 97, 139–40, 142, 203
transparency, as guarantor of freedom, 132
treaties, Trump's withdrawal from, 102
tribal politics, 77
Trudeau, Justin, 129, 143–44
Truman, Harry S., 96, 139
Trump, Donald. _See also_ Trump administration; Trump campaign; Trump Organization
abandonment of free trade, 139–42
on _Access Hollywood_ tape, 86–87
admiration for dictators and strongmen, 144–46
admiration for Putin, 58, 66, 85, 145, 203
Alt Right and, 84–85
anger at expulsion of Russian diplomats, 123
anti-immigrant policies of, 117–20
anti-intellectualism of, 63
anti-Semitism and, 84
appeasement of Kim Jong Un, 103, 130, 142, 144
approval ratings of, 101
attacks "fake news media," 97
attacks on FBI, 125–26, 127–28
attacks on immigrants, 117–18
attacks on judiciary, 78, 124
attacks on McCain, 55–56
attacks on Mexican immigrants, xv
attacks on political opponents as "traitors," 44
attacks on protesting NFL players, 115–16
attacks on Special Counsel Mueller, 126
authoritarian impulses of, 111–12
aversion to traditional allies, 142–43
"birtherism" hoax and, 63
Buckley on, 18
calls African and Caribbean nations "shithole countries," 118
calls for Hillary Clinton to be locked up, 86
calls neo-Nazis "very fine people," 114
challenge to 2020 renomination of, 209
as chaos president, 93–106
as character test for Republican Party, 80–81
compared to Forrest Gump, 160
compared to George Wallace, 109–11
comparisons to other leaders, 153–55
congratulates Putin for rigged reelection victory, 123
conservative case for, 101–6
conservative opposition to, 72
conspiracy theories and, 63–64, 112, 183
at December 2015 Republican debate, 63
deficit and, 138
dehumanizing language used by, 117
demagoguery of, 91
desire to change libel laws, 66
disrespect for rule of law, 124–28
donations to Democrats, 61
draft deferments of, xvi, 55–56
economic record of, 101–4
election of, xv, xvii ( _see also_ presidential election of 2016)
emboldens authoritarian leaders worldwide, 131
emerges as Republican nominee, 75
end of Pax American and, 139–48
ethics violations by, 133–35
exaggeration of deregulatory achievements, 104
fails to reject Duke's endorsement, 64
failure to understand soft power, 147
"fake news" and, 128–32
fascism of, 68
firing of James Comey, 125, 129
fiscal irresponsibility of, 137–39
flatterers of, 152–56
flip-flopping between parties, 61
focus on hard power, 147
Fox News Channel and, 183–86
functional illiteracy of, 100
Helsinki summit, behavior at, 123, 145
ignorance of, 63, 192–93
immigrant heritage of, 65
inaugural address of, 96–97
incoherent record on Russia, 122
indifference to Russian election meddling, 123, 201
instigation of violence at rallies, 66
inveighing against "globalist" elites, 54
Iraq War and, 54–55
ISIS and, 102
Jewish commentators critical of, 80, 84
lack of education, 51–52
lack of qualifications, 63, 68
lack of restraint, 68–69
lack of understanding of military, 51–52
as latest manifestation of populist derangement, 172
lawsuits threatened by, 47
lies told by, xvi, 63–64, 73, 97, 112, 128–29, 155
loses in Iowa caucuses, 73–74
loses popular vote, 89
loyalty and, 71–72
malevolence tempered by incompetence, 192–93
manipulation of followers by, 73
merchandising by, 156
Middle East trip, 154
misconduct of, 135
mocked as "short-fingered vulgarian" by _Spy_ magazine, 30
on _Morning Joe_ , 65–66
Muslim ban, 65, 97, 124
name-calling by, 75, 105 ( _see also specific targets_ )
narcissism of, 18
narrow Electoral College victory in 2016, 120
nativism of, 65, 116–20
nepotism and, 133–34
news coverage of, 73
nominated by followers for Nobel Peace Prize, 155
obsession with Amazon, 130–31
obstruction of justice and, 125–26, 160
opposes removal of Confederate monuments, 114
pardon power of, 124
policy proposals of, 67–68
popularity among Republicans and conservatives, xx
presidential campaign of 2016 and, 61
pro-Israel stance of, 84, 102, 106
protectionism of, 139–42
racism and, 64–65, 78, 113–16, 169
reading ability of, 56, 100
reading habits of, 13
receptivity to, 54–55
recreational pursuits of, 100
redefinition of reality by, 129–30
redefinition of Republican Party by, 175
refusal to clearly acknowledge Russian interference in US elections, 85–86, 123
refusal to divest himself of business holdings, 134
refusal to reveal tax returns, 134
refusal to say that he'd accept election loss, 86
Republican case for supporting, 101–6
retweets of white supremacists, 64
Russia and, 58, 123, 145
scapegoating of minorities by, 65–66
self-praise by, 153
self-promotion by, 73
selling skills of, 146
sexist comments by, 64, 86–87
sexual affairs and payoffs and, 135
size of inaugural audience and, 97
State of the Union address 2018, 153
stereotyping and stigmatizing of minority groups by, 64–65
stock market under, 101
summit with Kim Jong Un, 103, 142
summit with Putin, 123, 145
supporters of, 115–16
support for Iraq war, 52
susceptibility to own propaganda, 183
tax-and-spend policies of, 103–4
television and, 100, 183
threats to pull US troops out of Germany, South Korea, and Japan, 67–68
threats to revoke licenses of broadcast networks, 130–31
threat to US security, 69
trade negotiations with China, 134–35
traduces basic norms of American democracy, 86–87
transgressions against common decency, 113–48
tweets by, 64, 183
unilateralism of, 148
as "useful idiot" of Putin, 86
violations of Emoluments Clause, 135
volatility of, 112–13, 192–93
war on truth by, 128–32
wealthy lifestyle of, 91
white identity politics and, 188–89
white nationalists and, 84–85
WikiLeaks and, 121
willingness to appoint conservative judges, 124
wins Republican nomination, xvi, 208
withdrawal from Iran nuclear deal, 102, 103, 142, 144, 147
withdrawal from Paris Climate Accord, 102, 142, 143
withdrawal from treaties, 102 ( _see also specific agreements_ )
withdrawal from TTP, 97, 139–40, 142, 203
withdrawal of legal protections from Dreamers, 119
women and, 64
xenophobia of, xv
youth disapproval of, 211–12
Trump, Donald, Jr., 121–22, 134, 185
Trump, Eric, 134
Trump, Ivanka, 64, 133–35, 143
Trump administration. _See also specific members_
ethics violations by, 97–98, 132–37
incompetence of, 97–98
toadies in, 152–53
turnover rate in, 98–99
Trump campaign, 61
"America First" slogan, 68
anti-Semitic propaganda used by, 84
collusion and, 120–24
dubious policy advisers hired by, 71
first day of, xv, 65
incompetence of, 62–63
Kremlin and, 120–24
Russia and, 85–86, 88–89, 120–24, 184–85
Trump rallies, 21, 66, 73, 168
"Trump derangement syndrome," 82
"Trump Doctrine," 147
Trump effect, 82, 136
Trump International Hotel, 135
Trumpism, xx, 213
origins of, 163–93
Republican Party's surrender to, 59–91, 202–3, 208–9, 211
rise of, xxi
Trump Organization, 134, 156
Trump rallies, 62–63
Trump supporters, 82–83, 149–61
evangelical Christians, 87–88, 136–37
the press and, 131
Russia investigation and, 121, 128
Trump merchandise and, 156
Trump toadies, 149–61
white anxiety and, 116
Trump Tower
campaign rollout at, xv
Trump Tower meeting, June 9, 2016, 121–22
Trump University, 61
truth, as guarantor of freedom, 132
Turkey, 111, 131, 145
Turning Point USA, 154–55
Tusk, Donald, 147
Twitter, 112, 183
two-party system, entrenchment of, 207–11
Tyson, Mike, 64
Ukraine, 123
US weapons sales to, 122
unalienable rights, 191
United Arab Emirates, 134
United Kingdom, 143
United States
as benign superpower, 147
checks and balances in, 111–12
damage to standing in the world, 146–47
debt-to-GDP ratio in, 138
demoted in annual "Freedom in the World" report, 145–46
as a minority-majority country in near future, 211
press freedom downgraded in, 146
security threatened by Trump, 69
waning power of, 148
world approval for leadership rating, 146
Universal Declaration of Human Rights, Article 13, 4
universal family leave and child care, 206
University of California, Berkeley, 25–28, 68
ur-conservatives, 166
_USA Today_ , 95–96
US intelligence community, 120. _See also specific agencies_
US military intervention, 52
US Postal Service, 131
van den Haag, Ernest, 12
Venezuela, 111
Ventura, Jesse "The Body," 62
Vichy France, 155–56
Vietnam War, 54
violence, glorification of, 68
"virtue signaling," 82, 191–92
vital center, the, 195–214
voting rights, 113, 206
Voting Rights Act of 1965, 113
Waldman, Paul, 205
Wallace, George, 89, 109–11, 112
Wallace, Nicolle, 81
_Wall Street Journal_ , 35, 37–39, 42–51, 141, 174, 177, 180, 185, 213
Wanniski, Jude, 39
Warner, Mark, 203
War of the Rocks website, pledge of opposition to Trump, 71–72
Warren, Earl, 166
Warren, Elizabeth, 204, 205, 206
"wars of choice," 55
Washington, George, 114
_Washington Examiner_ , 154
_Washington Post_ , 128–29, 130–31, 134, 184, 201, 205
_Washington Times_ , 152–54
Wasserman Schulz, Debbie, 85
Watergate scandal, 124, 133
_Weekly Standard_ , 41, 49–50, 69, 72
Weiner, Anthony, 88
Weinstein, Harvey, 135
welfare state, 197–98
Whig Party, 207
Whiskey Ring, 132
white anxiety, 116, 199
white ethnic, working-class politics, 34
white identity politics, 188–89
white male privilege, 187–88. _See also_ white privilege
white nationalism, Republican Party's embrace of, xxi, 208–9
white nationalists, 84–85, 189
white privilege, 169, 191–92
white supremacists, 64, 68, 189
Whitewater, 43–44
white working-class voters, 89–90
WikiLeaks, 85, 121, 122
Will, George F., 17, 19, 81, 177
Wilson, Edmund, 171
Wilson, James Q,, 179
Wilson, Pete, 22, 211
Wilson, Rick, 81
Wilson, Woodrow, 158
Winfrey, Oprah, 210
Wittes, Benjamin, 202
Wolfowitz, Paul, 54, 177
women, rights of, 188, 190–91
women's rights movement, 191
Wooden, John, 154
World Trade Organization, 140
World War I, anti-immigrant sentiments during, 65
World War II, 140, 158
Wray, Christopher, 128
Wright, Jim, 173
xenophobia, 207
Xi Jinping, 144–45
Yale University, 34
Yellen, Janet, 84
Yiannopoulos, Milo, 180
Zamora, Jim Herron, 27
Zschau, Ed, US Senate campaign of, 21–22
ZTE, 135
## **A LSO BY MAX BOOT**
_The Road Not Taken: Edward Lansdale and the American Tragedy in Vietnam_
_Invisible Armies: An Epic History of Guerrilla Warfare from Ancient Times to the Present_
_War Made New: Technology, Warfare, and the Course of History, 1500 to Today_
_The Savage Wars of Peace: Small Wars and the Rise of American Power_
## **ABOUT THE AUTHOR**
Max Boot is a historian, best-selling author, and foreign policy analyst who has been called one of the "world's leading authorities on armed conflict" by the International Institute for Strategic Studies. He is the Jeane J. Kirkpatrick Senior Fellow in National Security Studies at the Council on Foreign Relations, a columnist for the _Washington Post_ , and a global affairs analyst for CNN. His most recent books are the _New York Times_ bestsellers T _he Road Not Taken: Edward Lansdale and the American Tragedy in Vietnam_ and _Invisible Armies: An Epic History of Guerrilla Warfare from Ancient Times to the Present Day_.
Copyright © 2018 by Max Boot
All rights reserved
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## Contents
1. Cover
2. Title
3. Contents
4. Prologue: November 8, 2016
5. 1. The Education of a Conservative
6. 2. The Career of a Conservative
7. 3. The Surrender
8. 4. The Chaos President
9. 5. The Cost of Capitulation
1. I. Racism
2. II. Nativism
3. III. Collusion
4. IV. The Rule of Law
5. V. "Fake News"
6. VI. Ethics
7. VII. Fiscal Irresponsibility
8. VIII. The End of the Pax Americana
10. 6. The Trump Toadies
11. 7. The Origins of Trumpism
12. Epilogue: The Vital Center
13. Acknowledgments
14. Notes
15. Index
16. About the Author
17. Copyright
## Guide
1. Cover
2. Contents
3. Title
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| {
"redpajama_set_name": "RedPajamaBook"
} | 1,606 |
\section{\label{sec:introduction}Introduction}
One of the most exciting frontiers within the field of quantum computing is the topic of quantum advantage, which aims to design experiments that demonstrate the ability of current quantum devices to significantly outperform classical computers at a well defined computational task~\cite{harrow2017quantum,hangleiter2022computational}. The theoretical design and real-world implementation of such experiments has been the focus of significant effort in the superconducting circuit~\cite{arute2019quantum,wu2021strong} and quantum photonics~\cite{zhong2020quantum,zhong2021phase,madsen2022quantum} communities in the form of Random Circuit Sampling (RCS)~\cite{boixo2018characterizing,bouland2019complexity} and Gaussian Boson Sampling (GBS)~\cite{hamilton2017gaussian,kruse2019detailed,deshpande2022quantum,grier2021complexity}.
GBS consists of sending a set of input squeezed states into an interferometer and measuring its output using photon-number~\cite{hamilton2017gaussian,kruse2019detailed} or threshold~\cite{quesada2018gaussian} detectors. The former directly sample the photon number distribution while the later sample binary patterns of clicks that indicate whether light has been detected or not. It has been shown that sampling from the theoretical probability distribution of the resulting detection patterns, i.e., the ground truth distribution of the ideal experiment, is a computationally hard task~\cite{hamilton2017gaussian,deshpande2022quantum,grier2021complexity}.
These results have important caveats, for example, the known proofs of hardness for GBS require the photon number density (the average number of photons per mode) to be small so that the probability of two or more photons being measured in the same detectors is very small.
On the algorithmic side, the best known methods to classically simulate these quantum sampling problems scale exponentially in the number of detected photons or counted clicks~\cite{bulmer2022boundary,quesada2022quadratic,quesada2020exact}.
Verifying that quantum samplers are operating correctly remains an active area of research (cf. the review paper by Hangleiter and Eisert~\cite{hangleiter2022computational} and references therein). For RCS it was identified early on that an estimate of the cross entropy between the samples generated by the physical device and the probability distribution associated with the ideal computation served as a witness of quantum advantage~\cite{boixo2018characterizing,bouland2019complexity}.
For GBS the situation is more complex, as the development of a proper figure of merit that allows to readily verify a claim of quantum computational advantage is still an open challenge~\cite{hangleiter2022computational,deshpande2022quantum}.
On this account, the validation of GBS usually relies on a series of tests that rule out possible classical hypotheses or that compare the quality of the samples generated by the quantum machine against samples generated by classically efficient methods.
Two recent landmark threshold GBS experiments by Zhong et al. using 100-mode~~\cite{zhong2020quantum} and 144-mode~\cite{zhong2021phase} interferometers have claimed to achieve quantum computational advantage.
The authors use three different validation tests: the comparison of truncated first to fourth order correlation functions (i.e., the first to fourth order click cumulants of the probability distribution), a Bayesian test and the Heavy Output Generation (HOG) test.
The first examines how well the correlations in the observed data match the correlations predicted by the squeezed state hypothesis.
The second test compares how good the squeezed state hypothesis is at explaining the observed data relative to other hypotheses such as thermal states, coherent states, distinguishable squeezed states and uniform probability distributions.
The third test looks at how well the samples generated by the experiment have ``heavy outputs'' (i.e., correspond to events with high probability) in the ideal distribution relative to samples generated by classically efficient methods. These classically efficient methods can be physically motivated as considered by Zhong et al.~\cite{zhong2020quantum,zhong2021phase} but need not be~\cite{villalonga2021efficient}.
We propose an alternative classical hypothesis to explain the experiments of Zhong et al. Our hypothesis is based on the probability distribution that is obtained from using classical mixtures of coherent states (resulting in Gaussian states that we term \emph{squashed states}~\cite{qi2020regimes,jahangiri2020point}) as inputs of the interferometers in GBS setups. These states are classical, possessing a non-negative Glauber-Sudarshan $P$ function~\cite{drummond2014quantum,rahimi2016sufficient,rahimi2015can} and upon interacting on an interferometer generate fully separable (i.e. having zero entanglement) multimode states, by virtue of having a positive multimode Glauber-Sudarshan $P$ function.
We find that, in the high photon number density regime where the quantum advantage is claimed to exist, our hypothesis is better at explaining the data observed by Zhong et al. than their ground truth hypothesis of quantum squeezed states interacting in an interferometer.
To compare different hypotheses, we investigate two of the tests (correlation functions and Bayesian test) that the authors of the experiment employed.
We find that for configurations in the high photon number density regime, on which most of the quantum computational advantage claim relies, the truncated correlation functions predicted by the squashed states distributions are as consistent with the experimental results as those predicted by the ground truth of the experiment given by pure squeezed states. Moreover, the Bayesian test shows that in this regime the squashed states hypothesis is more likely to describe the experimental samples than the squeezed ground truth.
Thus, the hypothesis introduced in this paper forms an alternative classical explanation for current threshold GBS experiments that should be weighted in future experiments.
After considering hypothesis testing, we generate samples from our squashed states hypothesis (which can be done in polynomial time and space in the number of modes/clicks~\cite{rahimi2016sufficient,qi2020regimes,gupt2019thewalrus}) and compare against the experimental samples from the Jiuzhang experiments, to see if, like the Boltzmann machine and greedy methods from Villalonga et al.~\cite{villalonga2021efficient}, we can spoof the HOG test. Surprisingly, despite the squashed states being a more likely explanation of the data observed in Jiuzhang 1.0/2.0, their samples are unable to spoof the HOG test.
This paper is organized as follows: In Sec.~\ref{sec:distributions} we obtain the ground truth distribution of the Zhong et al. experiments and define the squashed states hypothesis. In Sec.~\ref{sec:validation} we specify the different tests used in the validation of the experimental samples against the squashed states hypothesis and show the corresponding results. Finally, we conclude and give a possible explanation for the behavior of the squashed states hypothesis in the different regimes of photon number density in Sec.~\ref{sec:discussion}.
\section{\label{sec:distributions}Ground truth and squashed states distributions}
A threshold GBS experiment has three stages: preparation of $K$ (single-mode) squeezed states, evolution in an interferometer with $M\geq K$ output modes, and sampling of the output statistics using threshold detectors.
These detectors do not resolve the incoming number of photons, they can only indicate if light has arrived to the detector or not.
Therefore, the outcomes of the measurement can be expressed as $M$-bit strings, patterns consisting only of zeros and ones, where one indicates that a detector has been triggered (the detector has `clicked') and zero indicates that no light has been detected.
Jiuzhang 1.0 (2.0) uses 25 two-mode squeezed states (TMSS), corresponding to 50 squeezed states, as inputs to a 100-mode (144-mode) interferometer for the first (second) setup.
The specification of the different experimental configurations, along with the theoretical photon number density , $\nu = \bar{N} / M$ (the ratio between the mean photon number and the number of output modes), mean number of clicks, $\bar{C}$, and standard deviation of the number of clicks, $\sigma(C)$, of the Jiuzhang 1.0 and Jiuzhang 2.0 experiments are shown in Table~\ref{tab:experiment_specifics}. The values of $\bar{C}$ and $\sigma(C)$ were computed using the methods of Refs.~\cite{grier2021complexity, gupt2019thewalrus}.
\begin{table}[!t]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\rule{0pt}{11pt} \textbf{Experiment} & $\bm{P}(\text{W})$ & $\bm{w}(\mu\text{m})$ & $\bm{\nu}$ & $\bm{\bar{C}}$ & $\bm{\sigma(C)}$\\ \hline
Jiuzhang 1.0& - & - & $0.786$ & $41.042$ & $6.509$\\ \hline
\multirow{7}{*}{Jiuzhang 2.0}
& $0.5$ & $125$ & $0.055$ & $7.273$ & $3.391$\\ \cline{2-6}
& $1.412$ & $125$ & $0.161$ & $19.256$ & $5.596$\\ \cline{2-6}
& $0.15$ & $65$ & $0.044$ & $5.976$ & $3.016$\\ \cline{2-6}
& $0.3$ & $65$ & $0.093$ & $11.941$ & $4.325$\\ \cline{2-6}
& $0.6$ & $65$ & $0.218$ & $24.664$ & $6.269$\\ \cline{2-6}
& $1.0$ & $65$ & $0.442$ & $41.794$ & $7.95$\\ \cline{2-6}
& $1.65$ & $65$ & $0.975$ & $66.866$ & $9.02$\\ \hline
\end{tabular}
\caption{Available specifications (power $P$ and focus waist $w$) of the light source and theoretical photon density $\nu=\frac{\bar{N}}{M} = \frac{1}{M} \sum_{i=1}^M \langle a_i^\dagger a_i \rangle$, theoretical mean number of clicks $\bar{C}$, and theoretical standard deviation of the number of clicks $\sigma(C)$ of the different configurations of Jiuzhang 1.0 (with $M=100$ modes) and Jiuzhang 2.0 experiments (with $M=144$ modes).
}
\label{tab:experiment_specifics}
\end{table}
\begin{figure*}[ht]
\centering
\subfloat[]{\label{fig:scheme}
\includegraphics[scale=0.41]{ustc_gbs_setup.pdf}
}
\hfill
\subfloat[]{\label{fig:hypotheses}
\includegraphics[scale=0.41]{hypotheses.pdf}
}
\caption{(a) Description of the Jiuzhang 1.0/2.0 GBS hypotheses. Pairs of adjacent input squeezed or squashed states are sent into 50:50 beamsplitters represented by the vertical lines with dots at the end. The alternating orientation of the input states together with the 50:50 beamsplitters generate pairs of two-mode squeezed or squashed states. These states are then sent into a lossy interferometer, represented by the rectangle, whose action is described by a rectangular sub-unitary matrix $\bm{T}$. The output state of the interferometer is sampled using threshold detectors. (b) Representation of the noise ellipse of several Gaussian states used for the validation of the Jiuzhang 1.0/2.0 experiments. The ground truth of the experiments corresponds to the use of squeezed states corresponding to the blue line. Jiuzhang 1.0 and Jiuzhang 2.0 have been validated against hypotheses using coherent states and thermal states represented by the red-dotted and greed-dotted lines, respectively. In this work we validate the experimental results against a hypothesis using the squashed states represented by the purple line. For reference, we also show the ellipse corresponding to the vacuum state with a black-dotted line.}
\label{fig:experiment}
\end{figure*}
The ground truth of the Jiuzhang 1.0 and Jiuzhang 2.0 experiments is described as 25 input two-mode squeezed states that interfere in a linear lossy interferometer. The output state is then measured using ideal threshold detectors. The interferometer in this model contains the information about the losses in the real experiment (collection efficiency, propagation loss, and finite detector efficiency).
Two-mode squeezed states are mathematically defined as
\begin{equation}
|\zeta_{kl}\rangle=\exp\left(-\zeta a_k^\dagger a_l^\dagger+\zeta ^{*} a_l a_k\right)|0_k,0_l\rangle,
\label{eq:TMSS_def}
\end{equation}
where $|0_k,0_l\rangle$ is the vacuum state of modes $k$ and $l$, $a_k, a_l$ and $a_k^\dagger, a_l^\dagger$ are the annihilation and creation operators of these modes, and $\zeta=re^{i\phi}$ is a complex squeezing parameter. The annihilation and creation operators of the modes satisfy the usual bosonic canonical commutation relations $[a_i,a_j] = [a_i^\dagger, a_j^\dagger] = 0 $ and $[a_i, a_j^\dagger] = \delta_{i,j}$.
The state $|\zeta_{kl}\rangle$ can also be expressed in the form~\cite{serafini2017quantum, barnett2002methods}
\begin{equation}
|\zeta_{kl}\rangle=U_\text{BS}\left[S_k(\zeta)|0_k\rangle\otimes S_l(-\zeta)|0_l\rangle\right],
\label{eq:TMSS_to_SMSS}
\end{equation}
where $U_\text{BS} = \exp[\frac{\pi}{4}(a_l^\dagger a_k - a_k^\dagger a_l)]$ is a unitary transformation representing the action of a $50:50$ real beamsplitter, and $S_k(\zeta)=\exp[-\frac{1}{2}(\zeta a_k^{\dagger2} - \zeta^* a_k^2)]$ is the single-mode squeezing operator of mode $k$. This expression indicates that a TMSS can be physically generated by interfering two single-mode squeezed states (SMSS) compressed along orthogonal directions into a $50:50$ real beamsplitter.
For Jiuzhang 1.0 and Jiuzhang 2.0, the squeezing parameters of the input states $\zeta_i=r_i,$ $i=1,\cdots,25,$ are considered to be real and positive.
The information about the phases $\phi_i$ is absorbed into the rectangular sub-unitary matrix $\bm{T}$ that describes the action of the lossy interferometer.
The size of $\bm{T}$ is $M \times 50$, where $M$ is the number of output modes.
Data about the squeezing parameters and the matrix $\bm{T}$ is available in~\cite{ustc2020experimental} for Jiuzhang 1.0, and in~\cite{ustc2021raw} for Jiuzhang 2.0.
In order to verify the results of the Jiuzhang experiments, we need to determine the theoretical probability distribution of the experimental samples, which is commonly known as the \textit{ground truth} distribution of the experiment. The specification of this distribution will also allow us to motivate the definition of the probability distribution of the squashed states.
The ground truth distributions of the Jiuzhang experiments are completely characterized by the output state of the interferometer.
Since the input states are Gaussian, and the interferometer is linear, the output state is also Gaussian.
Gaussian states of $K$ modes are characterized by their vector of first moments and their covariance matrix.
If we write the annihilation operators as $a_k = (x_k + i p_k)/\sqrt{2\hbar},$ in terms of the hermitian in- (out-~) phase quadratures $x_k$ $(p_k)$, we can define the vector $\bm{r}^\text{T} = (x_1, x_2, \cdots, x_K,p_1, p_2, \cdots, p_K)$. Then, assuming a vanishing vector of first moments $\overline{\bm{r}} = \operatorname{Tr} (\rho_G \bm{r}) = \bm{0},$ (as is the case for the Jiuzhang experiments) the covariance matrix can be written as $\bm{\sigma} = \frac{1}{2}\operatorname{Tr} \left(\rho_G \{ \bm{r}, \bm{r}^T\}\right).$
Here, $\rho_G$ denotes the density matrix of the Gaussian state and $\{A,B\} = AB + BA$ stands for the anticommutator of operators $A$ and $B$. On this account, the ground truth distribution of the Jiuzhang experiments will be completely specified by the covariance matrix of the output states. In what follows, we will explain how to compute this matrix.
We first specify the covariance matrix of the input states. To do this, we follow the procedure indicated by Eq.~\ref{eq:TMSS_to_SMSS}: we generate 25 TMSS by interfering 50 pairs of SMSS with squeezing parameters $\{-r_1, r_1,\dots,-r_{25}, r_{25}\}$ into 25 beamsplitters that act on consecutive modes. The covariance matrix of one SMSS with squeezing parameter $r$ is
\begin{equation}
\bm{\sigma}_\text{SMSS}^{(1)}=\frac{\hbar}{2}
\begin{pmatrix}
e^{-2r}&0\\
0&e^{2r}
\end{pmatrix}.
\label{eq:one_SMSS}
\end{equation}
Note that in this covariance matrix, the variances of the two quadratures are inversely related, thus saturating the uncertainty relation~\cite{serafini2017quantum}.
We can now construct the $100\times100$ covariance matrix of the 50 SMSS (in the ordering indicated by $\bm{r}^\text{T}$) as
\begin{align}
\begin{split}
\bm{\sigma}_\text{SMSS} = \frac{\hbar}{2}\,& \text{diag}\left(e^{2r_1},\, e^{-2r_1},\dots,\,e^{2r_{25}},\, e^{-2r_{25}},\right.\\
&\left.\,e^{-2r_1},\, e^{2r_1},\dots,\,e^{-2r_{25}},\, e^{2r_{25}}\right).
\end{split}
\label{eq:single_squeezed_cov}
\end{align}
where $\text{diag}(v_1,\ldots,v_m)$ forms a diagonal matrix of size $m$ with entries $v_1,\ldots,v_m$.
A $50:50$ real beamsplitter interfering two adjacent modes (see Fig.~\ref{fig:scheme}) is represented by the matrix
\begin{equation}
\bm{H}=
\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & -1 \\
1 & 1
\end{pmatrix}.
\label{eq:beam_splitter}
\end{equation}
The corresponding matrix for the 25 beamsplitters has the form
\begin{equation}
\bm{B}=\bigoplus_{k=1}^{50}\bm{H},
\label{eq:beam_splitters}
\end{equation}
The covariance matrix of the input TMSS can then be computed as
\begin{equation}
\bm{\sigma}_\text{TMSS} = \bm{B}\bm{\sigma}_\text{SMSS}\bm{B}^\text{T}.
\label{eq:input_TMSS_cov}
\end{equation}
Given a $K$-mode input Gaussian state with covariance matrix $\bm{\sigma}_{\text{IN}}$, an interferometer channel with transmission matrix $\bm{T}$ maps it to an $M$-mode Gaussian state with covariance matrix $\bm{\sigma}_{\text{OUT}}$ given by~\cite{serafini2017quantum}
\begin{align}
\bm{\sigma}_{\text{OUT}} &= \mathcal{L}_{\bm{T}}(\bm{\sigma}_{\text{IN}}) = \frac{\hbar}{2}\left(\mathbb{I}_{2M} - \bm{V}\bm{V}^{T}\right) + \bm{V}\bm{\sigma}_\text{IN}\bm{V}^\text{T}, \\
\bm{V}&=
\begin{pmatrix}
\text{Re}(\bm{T}) & -\text{Im}(\bm{T})\\
\text{Im}(\bm{T}) & \text{Re}(\bm{T})
\end{pmatrix}.
\end{align}
The complex transmission matrix in the last equation is generally rectangular and of dimensions $M \times K$.
With this notation in mind, we can then simply write the covariance matrix of the squeezed ground truth hypothesis as
$ \bm{\sigma}_{(\text{SQUE})} = \mathcal{\bm{L}}_{\bm{T}}(\bm{\sigma}_{\text{TMSS}})$.
Having the covariance matrix $\bm{\sigma}$ of a given Gaussian state we can calculate click-probabilities as~\cite{quesada2018gaussian}
\begin{align}\label{eq:ground_truth_distribution}
\mathrm{Pr}(\bm{s}) &= \frac{\text{Tor}\left[\bm{O}_{(\bm{s})}\right]}{\sqrt{\text{det}(\bm{\Sigma})}},
\text{ with } \bm{O} = \mathbb{I}_{2M} - \bm{\Sigma}^{-1} \text{, } \\
\bm{\Sigma} &= \frac{\mathbb{I}_{2M}}{2} + \frac{1}{\hbar} \bm{R} \bm{\sigma} \bm{R}^\dagger \text{ and } \bm{R} = \frac{1}{\sqrt{2}}
\begin{pmatrix}
\mathbb{I}_M & i\mathbb{I}_M\\
\mathbb{I}_M & -i\mathbb{I}_M
\end{pmatrix}.
\end{align}
Here, $\text{Tor}(\bm{A})$ is the Torontonian of matrix $\bm{A}$ (see Appendix~\ref{app:tor} for a definition). The operation $\bm{A}_{(\bm{s})}$ is explained as follows: suppose that the detection pattern $\bm{s}$ contains $C$ clicks ($C$ ones) observed in the modes $\{j_1,\dots,j_C\}$, then matrix $\bm{A}_{(\bm{s})}$ is obtained by keeping only the rows and columns $\{j_1,\dots,j_C,j_1 + M,\dots,j_C + M\}$ of matrix $\bm{A}$, whose size is $2M \times 2M$.
To obtain probabilities associated with the ground truth we simply let $\bm{\sigma} \to \bm{\sigma}_{(\text{SQUE})}$ in the last equation. Note that the results presented here can be extended to Gaussian states with displacements~\cite{thekkadath2022experimental} using loop Torontonians~\cite{bulmer2022threshold}.
We now turn to the definition of the squashed states distribution. When losses are incorporated in the input states, squeezed states become squeezed thermal states~\cite{qi2020regimes}.
These states still have a quadrature with noise lower than that of the vacuum state. We define the squashed states as squeezed thermal states with vacuum fluctuations in one quadrature and larger fluctuations in the other. The output states obtained by interfering squashed states are classical, which means that we can efficiently sample from their probability distribution using classical computers. As can be seen in Fig.~\ref{fig:hypotheses}, the noise ellipse of the squashed states suggests that they are better approximations to the squeezed states than the classical states that Zhong et al. used for the validation of the experimental results which have circular noise ellipses. Indeed, squashed states are the classical Gaussian states with the highest fidelity to thermal squeezed states~\cite{qi2020regimes}.
The definition of the probability distribution associated with the squashed distribution is now straightforward: we need to apply the interferometer channel with transmission $\bm{T}$ to 25 two-mode squashed states. In turn, the input covariance matrix to the interferometer is obtained by replacing $\bm{\sigma}_\text{TMSS}$ for the input 25 two-mode squashed states covariance matrix, $\bm{\sigma}_0'$. This matrix is obtained in the same way we constructed $\bm{\sigma}_\text{TMSS}$: we interfere 50 single-mode squashed states into 25 real $50:50$ beamsplitters. The covariance matrix of a single-mode squashed state with mean photon number $\bar{n}=\sinh^2r$ is
\begin{equation}
\bm{\sigma}_\text{0}^{(1)}=\frac{\hbar}{2}
\begin{pmatrix}
1&0\\
0&1+4\bar{n}
\end{pmatrix}.
\label{eq:one_squashed}
\end{equation}
This covariance matrix has no squeezing as one of the quadratures is at the vacuum level while the other has excess (classical) noise proportional to $\bar{n}$, thus they can be described as classical mixtures of coherent states as shown in Ref.~\cite{jahangiri2020point}.
Notice that the matrix above can be obtained by replacing $e^{-2r}$ and $e^{2r}$ in Eq.~\eqref{eq:one_SMSS} by 1 and $1+4\bar{n}$, respectively. This same procedure can be used to construct the covariance matrix of the 50 single-mode squashed states, $\bm{\sigma}_0$: we replace the $e^{-2r_k}$ and $e^{2r_k}$ terms in Eq.~\eqref{eq:single_squeezed_cov} by 1 and $1+4\bar{n}_k
, respectively. Then, $\bm{\sigma}_0'$ can be written as
\begin{equation}
\bm{\sigma}_0' = \bm{B}\bm{\sigma}_0\bm{B}^\text{T}.
\label{eq:input_squashed_cov}
\end{equation}
It is worth mentioning that the squashed states need not have the same mean photon numbers as the squeezed states used in the definition of the covariance matrix in Eq.~\eqref{eq:input_TMSS_cov}. We make this choice in order to have a photon number distribution with lower total variation distance to the distribution predicted by the ground truth. However, the squashed states hypothesis allows the modification of the $\bar{n}_k$. In particular, we may slightly change the input photon numbers so that instead of being $\bar{n}_k = \sinh^2 r_k$ they are $\bar{n}_k'=(1-\epsilon_k) \sinh^2 r_k$ with $\epsilon_k$ small. This modification of the squashed states hypothesis will prove useful when validating the experimental results.
We can write the covariance matrix associated with the squashed states hypothesis as $\bm{\sigma}_{\text{(SQUA)}} = \mathcal{L}_{\bm{T}}(\bm{\sigma_0}')$ .
The covariance matrix of this state satisfies $\bm{\sigma}_{\text{(SQUA)}} \ge \frac{\hbar}{2} \mathbb{I}_{2M}$ and thus the Gaussian state associated with it can be written as a mixture of products of single-mode coherent states~\cite{jahangiri2020point,rahimi2015can,rahimi2016sufficient}, implying that this Gaussian state is separable across any partition of its modes.
Finally, having the covariance matrix of the state, we can obtain probabilities associated with it by letting $\bm{\sigma} \to \bm{\sigma}_{(\text{SQUA})}$ in Eq.~\eqref{eq:ground_truth_distribution}.
\section{\label{sec:validation}Validation tests}
The results of the Jiuzhang experiments have been validated against a number of hypotheses and adversaries. These validations generally made use of three different tests: a Bayesian test, the Heavy Output Generation (HOG) test, and the comparison of the click cumulants of the different possible distributions with those of the experimental samples. Here, we will study how well the squashed states hypothesis explains the experimental data relative to the squeezed ground truth hypothesis and, moreover, will use samples generated from the squashed states hypothesis to perform the HOG test.
\subsection{\label{sec:cumulants}Click cumulants of the distribution}
The comparison of click cumulants was first used in the validation of the Jiuzhang 2.0 experiment~\cite{zhong2021phase}. The authors used this method to investigate how robust the experimental samples are against classical simulation schemes based on marginal distributions. They claim that the presence of ``non-trivial genuine high-order correlation in the
GBS samples are evidence of robustness against possible classical simulation schemes''~\cite{zhong2021phase}. These high-order correlation functions are given by cumulants (also called Ursell functions) defined in terms of moments of a multidimensional random variable $\bm{X} = (X_1,X_2,\ldots,X_M)$ as
\begin{align}
\kappa(X_1,\dots,X_n) =\sum_\pi (|\pi|-1)!(-1)^{|\pi|-1}\prod_{B\in\pi} \left\langle \prod_{i\in B}X_i \right\rangle,
\end{align}
where $\pi$ runs through the list of all partitions of $\{ 1, ..., n \}$, $B$ runs through the list of all blocks of the partition $\pi$, and $|\pi|$ is the number of parts in the partition.
Note that the first order cumulants are simply the means $\kappa(X_i) = \braket{X_i}$ and that the second order cumulants are the covariances $\kappa(X_i, X_j) = \braket{X_i X_j} - \braket{X_i} \braket{X_j}$.
Since the probability distribution associated with a threshold detector experiment has binary outcomes $0,1$ in each mode, it is straightforward to see that moments of the distribution correspond to marginal probabilitites
\begin{align}
\braket{X_{i_1} X_{i_2} \ldots X_{i_n}} = \Pr\left(i_1=1,i_2=1,\ldots i_n = 1 \right).
\end{align}
The latter in turn can be computed by constructing the marginal covariance matrix of the modes $i_1,i_2,\ldots,i_n$ and using Eq.~\eqref{eq:ground_truth_distribution}.
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.39]{click_cumulants_1.pdf}
\caption{Comparison between the experimental click cumulants (up to fourth order) and those predicted by the squashed states (violet) and ground truth (blue) distributions for Jiuzhang 1.0 ($\nu=0.786$) and two configurations of Jiuzhang 2.0 ($\nu=0.044$ and $\nu=0.975$). For a hypothesis that perfectly describes the experimental samples, all the cumulants would lie in the dashed straight lines shown in the figure. The Pearson and Spearman correlation coefficients between theoretical and experimental cumulants are shown in Fig.~\ref{fig:coeff}.
}
\label{fig:click_cumulants}
\end{figure*}
\begin{figure*}[ht]
\centering
\subfloat[]{\label{fig:pearson}
\includegraphics[scale=0.39]{pearson_ccoeff.pdf}
}
\hfill
\subfloat[]{\label{fig:spearman}
\includegraphics[scale=0.39]{spearman_ccoeff.pdf}
}
\caption{Pearson (a) and Spearman (b) correlation coefficients between experimental click cumulants and those predicted by the theoretical squashed states (violet) and ground truth (blue) distributions as functions of the photon number density. The uncertainties in the correlation coefficients (represented by the error bars) were obtained using bootstrapping. Notice that, with increasing photon number density, the correlation coefficients obtained using the squashed states distribution become progressively closer to those obtained with the ground truth distribution.}
\label{fig:coeff}
\end{figure*}
Following Ref.~\cite{zhong2021phase}, we compute the cumulants up to fourth order of the squashed states distribution of the different setups of the Jiuzhang experiment and compare them with those of the experimental samples.
We also make the corresponding comparison with the ground truth distributions. For these computations, we used $10^7$ of the $\sim5\times10^7$ samples available in Ref.~\cite{ustc2020experimental}. Note that the number of cumulants increases sharply with the order; for a system with $M$ modes there are $M!/(\ell!(M-\ell)!)$ possible combinations of $\ell$ different modes (without repetitions), which is precisely the number of cumulants of order $\ell$. Considering this fact, we randomly select $10^5$ sets of three and four different modes for the computation of the third and fourth order cumulants, respectively. We compute all the possible cumulants of first and second order.
Fig.~\ref{fig:click_cumulants} shows the results for Jiuzhang 1.0 ($\nu = 0.786$) and for two configurations of Jiuzhang 2.0 ($\nu = 0.044$ and $\nu = 0.975$).
Fig.~\ref{fig:coeff} shows the Pearson and Spearman correlation coefficients between experimental and theoretical cumulants as functions of $\nu$ (for all the configurations of Jiuzhang 1.0/2.0). The corresponding $p$-values, computed using the function \texttt{cor.test} from the R stats package~\cite{Rmanual}, are lower than \verb|machine_epsilon|$=2.2\times10^{-16}$ for every configuration, indicating a negligible probability of obtaining these correlation coefficients using uncorrelated data sets (i.e. a negligible probability of obtaining these correlation coefficients by chance). The same computation using the \texttt{pearsonr} and \texttt{spearmanr} functions from the SciPy stats (\texttt{scipy.stats}) module~\cite{2020SciPy-NMeth}, results in $p$-values that are effectively zero for all $\nu$.
It is clear that for low densities, as exemplified by the $\nu = 0.044$ configuration in Fig.~\ref{fig:coeff}, the cumulants predicted by the squashed states distribution are not consistent with the experimental results. In this case we can readily admit the ground truth of the experiment as the better hypothesis. This observation also holds for configurations with photon number densities between $\nu=0.055$ and $\nu=0.218$ as seen by looking at the Pearson and Spearman correlation coefficients in Fig.~\ref{fig:pearson}.
However, we note that by increasing $\nu$, the cumulants of the squashed states distribution become progressively consistent with the experimental samples. Indeed, for setups with $\nu=0.786$ and $\nu=0.975$ (also for $\nu=0.442$), the experimental results are as consistent with the squashed states distribution as they are with the ground truth distribution of the experiment.
In these cases, the comparison of cumulants does not allow to directly determine which of these two distributions better describes the experiment. This suggests that for configurations with high photon number density the presence of higher order cumulants in the experimental samples can be efficiently reproduced by a classical hypothesis, which implies that the idea of having high-order correlations in the data does not necessarily shield the experiment from classical simulations.
This result is of particular importance even if it does not hold for all $\nu$. From Table~\ref{tab:experiment_specifics} (and also from Fig.~\ref{fig:click_probabilities} in Appendix~\ref{app:drummond}) it can be inferred that configurations with low $\nu$ have click number distributions that are mostly located below 40 clicks, that is, these configurations mostly generate samples with less than 40 clicks. This upper limit in the number of clicks makes the simulation of these setups a feasible task for a classical computer~\cite{quesada2018gaussian, kaposi2021polynomial, bulmer2022boundary}. It is the configurations of the Jiuzhang experiments with high photon number density that become increasingly difficult to simulate. The comparison of click cumulants shown here indicates that a simulation scheme using the squashed states hypothesis efficiently reproduces the results of the experiment in this regime.
\subsection{\label{sec:entropies}Bayesian test}
The Bayesian test that was used in the validation of the Jiuzhang 1.0/2.0 experiments compares two different hypotheses regarding the true probability distribution of the experimental samples~\cite{bentivegna2014bayesian}. This test gives the degree of confidence of one hypothesis over another. In this case, we compute the degree of confidence of the ground truth of the experiment over the squashed states hypothesis. The test relies on the computation of probabilities of individual samples, which is a computationally hard task for patterns with a high number of clicks. For this reason, we use sets of samples with fixed click numbers (following Refs.~\cite{zhong2020quantum, zhong2021phase, villalonga2021efficient}) between 5 and 25. This upper limit in the number of clicks takes into account that the computation of the corresponding probabilities requires quadruple precision of complex type numbers, thus increasing the computational cost. Indeed, obtaining the probability for a single pattern of 25 clicks takes $\sim3.5$ hours using a 64-core CPU with two AMD Rome 7532 processors with $2.4\,\text{GHz}$ clock speed, using a custom implementation of the Torontonian function which uses Quadruple-precision (128 bits for each real number) provided by the library \texttt{DoubleFloats.jl}~\cite{sarnoffDoubleFloats2022} in the Julia Programming language~\cite{bezanson2017julia}.
Consider a set $\bm{S} = \{\bm{s}_1,\dots,\bm{s}_N\}$ of $N$ experimental samples, each of them containing $C$ clicks. The probability of obtaining one of these samples, given that it has $C$ clicks, under the hypothesis $\text{HYP} \in \{\text{SQUA}, \text{\text{SQUE}}\}$ is given by
\begin{align}
\mathrm{Pr}_{(\text{HYP})}(\bm{s}_k|C) = \mathrm{Pr}_{(\text{HYP})}(\bm{s}_k) / \mathrm{Pr}_{(\text{HYP})}(C)
\end{align}
where $\mathrm{Pr}_{(\text{HYP})}(\bm{s}_k)$ is the probability of sample $\bm{s}_k$ under hypothesis $\text{HYP}$ given in terms of a Torontonian (cf. Eq.~\eqref{eq:ground_truth_distribution}) and $\mathrm{Pr}_{(\text{HYP})}(C)$ is the grouped probability of obtaining $C$ clicks in total, again under the hypothesis $\text{HYP}$. The probability of obtaining the set of samples $\bm{S}$ under a given hypothesis $\text{HYP}$ takes the form
\begin{equation}
\mathrm{Pr}_{(\text{HYP})}(\bm{S}|C) = \prod_{k=1}^N \mathrm{Pr}_{(\text{HYP})}(\bm{s}_k|C).
\label{eq:joint_gt_probability}
\end{equation}
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.41]{pbay_entropies.pdf}
\caption{Bayesian test for the different setups of the Jiuzhang experiments in terms of $\Delta H(C)$. Error bars are obtained from the computed uncertainty in the simulation of the grouped click probabilities. To compute $\Delta H(C)$ for configurations with $\nu\leq 0.442$, we used $4000$ experimental samples for each $C$. For the $\nu=0.786$ setup we used $1668$ samples for $C=21$, $2885$ for $C=22$, and $4000$ for the remaining click numbers. The results with the label ``$\nu=0.442$ (Modified)'' correspond to the configuration with $\nu=0.442$ using a modified squashed states hypothesis that slightly changes the input mean number of photons.}
\label{fig:bay_entropies}
\end{figure*}
We define the Bayesian ratio, $r_\text{B}(C)$, which can be interpreted as the probability assigned to the ground truth hypothesis for a given number of clicks, as
\begin{align}
\begin{split}
r_\text{B}(C)&=\frac{\mathrm{Pr}_{(\text{SQUE})}(\bm{S}|C)}{\mathrm{Pr}_{(\text{SQUE})}(\bm{S}|C) + \mathrm{Pr}_{(\text{SQUA})}(\bm{S}|C)} \\
&= \frac{1}{1 + \chi_\text{B}(C)},
\end{split}
\label{eq:bayesian_ratio}
\end{align}
where $\chi_\text{B}(C) = \mathrm{Pr}_{(\text{SQUA})}(\bm{S}|C)/ \mathrm{Pr}_{(\text{SQUE})}(\bm{S}|C)$. The Bayesian test consists in checking the convergence of $r_\text{B}(C)$ when the number of samples is increased: if $r_B(C) \rightarrow 1$ for any $C$, we conclude that the ground truth hypothesis is more likely to describe the experimental samples. Conversely, if $r_B(C) \rightarrow 0$ for any $C$, the squashed states hypothesis becomes more likely.
An alternative way to express this test is obtained by writing $\chi_\text{B}= \exp(N\Delta H (C))$, where
\begin{align}
\begin{split}
\Delta H(C) =& -\frac{1}{N}\sum_{k=1}^N\ln\left[\mathrm{Pr}_{(\text{SQUE})}(\bm{s}_k|C)\right]\\
&+\frac{1}{N}\sum_{k=1}^N\ln\left[\mathrm{Pr}_{(\text{SQUA})}(\bm{s}_k|C)\right]\\
=& H_{(\text{SQUE})}(C) - H_{(\text{SQUA})}(C).
\end{split}
\label{eq:bay_cross_ent_diff}
\end{align}
The quantities $H_{(\text{SQUE})}$ and $H_{(\text{SQUA})}$ are estimators of the cross-entropy, for a given number of counts, of the ground truth and squashed states distributions relative to the real probability distribution of the experimental samples. In terms of the cross-entropy difference $\Delta H(C)$, $r_\text{B}(C) = \left[1+\exp(N\Delta H(C))\right]^{-1}$ and, for a increasing number of samples, the condition $r_B(C) \rightarrow 1$ is equivalent to $\Delta H(C)<0$, while $r_B(C) \rightarrow 0$ is equivalent to $\Delta H(C)>0$.
An important step for the computation of the Bayesian test is the determination of the grouped click probability distributions $\mathrm{Pr}_{(\text{SQUE})}(C)$ and $\mathrm{Pr}_{(\text{SQUA})}(C)$. To do this, we use the simulation method introduced in Ref.~\cite{drummond2022simulating}. The definition of this method, as well as the resulting click probability distributions and the parameters used in the simulation, are shown in Appendix~\ref{app:drummond}. It is worth mentioning that this method allows the determination of $\mathrm{Pr}_{(\text{SQUE})}(C)$ and $\mathrm{Pr}_{(\text{SQUA})}(C)$ for all click numbers $C$, not only those for which the probabilities of individual samples are easily computed.
Fig.~\ref{fig:bay_entropies} shows the results of the Bayesian test for the Jiuzhang 1.0 and Jiuzhang 2.0 experiments in terms of cross-entropy differences. For configurations with $\nu\leq0.218$, we considered experimental samples with click numbers between 5 and 20. In these cases, we used $4000$ samples for each $C$. For the $\nu = 0.786$ and $\nu = 0.442$ setups, we considered click numbers between $21$ and $25$ due to the lack of sufficient samples with $C$ lower than $20$. In the case of $\nu = 0.786$, we used $1668$ samples for $C=21$, $2885$ for $C=22$, and $4000$ for the remaining click numbers. For $\nu = 0.442$, we used $4000$ samples for each $C$. The results for $\nu=0.975$ are not addressed on account of the insufficient number of experimental samples within the range of click numbers considered.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.45]{photon_number_mod.pdf}
\caption{Comparison of the input mean number of photons between the modified squashed states hypothesis and the ground truth of the experiment (which also correspond to the original squashed states hypothesis) for the Jiuzhnag 2.0 configuration with $\nu=0.442$. The values of $\epsilon_k$ are chosen in order to minimize the difference between the first order cumulants predicted by the squashed states hypothesis and those obtained from the experimental samples.}
\label{fig:photon_number}
\end{figure}
For low photon number density, the Bayesian test consistently indicates that the ground truth is more likely than the squashed states hypothesis (in agreement with the comparison of click cumulants). However, as was the case with the click cumulants comparison, by increasing $\nu$ the values of $\Delta H(C)$ become progressively closer to zero, indicating that the degree of confidence in the squashed states hypothesis increases. This trend is verified by the results for the $\nu=0.786$ configuration of Jiuzhang 1.0, which indicate that the squashed states hypothesis is more likely to describe the experimental samples than the ground truth of the experiment.
For $\nu=0.442$ in Jiuzhang 2.0, we observe that $\Delta H(C)$ remains negative for the range of click numbers considered. Nevertheless, it is possible to slightly modify the squashed states hypothesis as was mentioned earlier in Sec.~\ref{sec:distributions}: we replace the mean number of photons of the input squashed states $\bar{n}_k$ by $\bar{n}_k'=(1-\epsilon_k)\bar{n}_k$, where the $\epsilon_k$ are chosen in order to minimize the difference between the first order cumulants predicted by the squashed states distribution and those obtained from the experimental samples.
With this modification, we obtain the results shown in Fig.~\ref{fig:bay_entropies} under the label ``$\nu=0.442$ (Modified)''. The comparison between the modified mean number of photons and the ones originally used in the squashed states hypothesis is shown in Fig.~\ref{fig:photon_number}. Notice that $\Delta H(C)$ changes sign for all the values of $C$ considered, implying that the modified squashed states hypothesis becomes more likely than the ground truth of the experiment.
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.41]{phog_entropies.pdf}
\caption{HOG test for the different setups of the Jiuzhang experiments in terms of $\Delta E(C)$. Error bars are obtained from the computed uncertainty in the simulation of the grouped click probabilities. To compute $\Delta E(C)$ for configurations with $\nu\leq 0.442$, we used $4000$ experimental and squashed states samples for each $C$. For the $\nu=0.786$ setup we used $1668$ samples for $C=21$, $2885$ for $C=22$, and $4000$ for the remaining click numbers. The results with the label ``$\nu=0.442$ (Modified)'' correspond to the configuration with $\nu=0.442$ using a modified squashed states hypothesis that slightly changes the input mean number of photons.}
\label{fig:hog_entropies}
\end{figure*}
We explain the higher likelihood of the squashed states hypothesis for high $\nu$ by noticing that these experiments use input states with different squeezing parameters and threshold detectors. For a general GBS setup, it can be shown~\cite{grier2021complexity} that the expected mean number of clicks, $\bar{C}$, when using threshold detectors follows the relation
\begin{equation}
\frac{1}{\bar{C}} = \frac{1}{\bar{N}} + \frac{1}{M},
\label{eq:empirical_relation}
\end{equation}
where $\bar{N}$ is the expected mean number of photons and $M$ is the number of output modes. In terms of the photon number density $\nu = \bar{N} / M$ we have
\begin{equation}
\frac{\bar{N}}{\bar{C}} = 1 + \nu.
\label{eq:empirical_relation_2}
\end{equation}
As can be seen from the equation above, $\bar{N}\sim \bar{C}$ for $\nu \ll 1$. This suggests that in this regime we can interpret each click detected as corresponding to approximately one single photon. On the other hand, for increasing $\nu$, $\bar{N} >\bar{C}$, implying that a single click corresponds to more than one photon (there is an increasing number of collisions). These observations suggest that in the low $\nu$ regime we can interpret a measurement using threshold detectors as an approximate photon number resolving (PNR) measurement. For increasing $\nu$ we can no longer state that the use of threshold detectors is similar to the use of PNR detectors. When the squeezing parameters of the input states are all the same, a PNR measurement can readily distinguish between squeezed and squashed states (this can be checked by comparing their second order cumulants~\cite{madsen2022quantum}). This does not necessarily hold when the squeezing parameters are different, $\nu$ is high or threshold detectors are used.
\subsection{\label{sec:hog}HOG test}
The Heavy Output Generation (HOG) test was first introduced in Ref.~\cite{zhong2020quantum} for the validation of the Jiuzhang 1.0 experiment. This test compares the probabilities with respect to the ground truth distribution of two sets of samples: one corresponding to experimental samples, and the other corresponding to samples from an alternative distribution, which in our case corresponds to the squashed states distribution. This test verifies if a sampler using the alternative distribution can generate samples with higher ground truth probability than the experimental samples.
The definition of the HOG test is made in a similar fashion to the Bayesian test. We define the HOG ratio, $r_\text{HOG}(C)$, as
\begin{align}
\begin{split}
r_\text{HOG}(C)&=\frac{\mathrm{Pr}_{(\text{SQUE})}(\bm{S}|C)}{\mathrm{Pr}_{(\text{SQUE})}(\bm{S}|C) + \mathrm{Pr}_{(\text{SQUE})}(\bm{S'}|C)} \\
&= \frac{1}{1 + \chi_\text{HOG}(C)},
\end{split}
\label{eq:hog_ratio}
\end{align}
where $\chi_\text{HOG}(C) = \mathrm{Pr}_{(\text{SQUE})}(\bm{S'}|C)/ \mathrm{Pr}_{(\text{SQUE})}(\bm{S}|C)$ and $\bm{S'}=\{\bm{s'}_1,\dots,\bm{s'}_N\}$ is a set of $N$ samples obtained from the squashed states distribution (each one with $C$ clicks). $\mathrm{Pr}_{(\text{SQUE})}(\bm{S}|C)$ is computed according to Eq.~\eqref{eq:joint_gt_probability}. We emphasize that in the last equation all the probabilitites are taken with respect to the squeezing SQUE distribution. The test consists in checking the convergence of $r_\text{HOG}(C)$ for an increasing number of samples: $r_\text{HOG}(C)\rightarrow 1$ if we generally find that the experimental samples have higher ground truth probability for any $C$, while $r_\text{HOG}(C)\rightarrow 0$ when the samples from the squashed states distribution have higher ground truth probability for any $C$.
As in the case of the Bayesian test, the ratio $\chi_\text{HOG}(C)$ may be rewritten as $\chi_\text{HOG}(C) = \exp(N\Delta E(C))$, where
\begin{align}
\begin{split}
\Delta E(C) &= -\frac{1}{N}\sum_{k=1}^N\ln\left[\mathrm{Pr}_{(\text{SQUE})}(\bm{s}_k|C)\right]\\
&+\frac{1}{N}\sum_{k=1}^N\ln\left[\mathrm{Pr}_{(\text{SQUE})}(\bm{s'}_k|C)\right]\\
&= E_{(\text{SQUE})}(C) - E'_{(\text{SQUE})}(C).
\end{split}
\label{eq:hog_cross_ent_diff} \end{align}
In terms of the cross-entropy difference $\Delta E(C)$~\cite{villalonga2021efficient}, $r_\text{HOG}(C) = \left[1 + \exp(N\Delta E(C))\right]^{-1}$ and, for increasing $N$, the conditions $r_\text{HOG}(C)\rightarrow 1$ and $r_\text{HOG}(C)\rightarrow 0$ are equivalent to $\Delta E(C)<0$ and $\Delta E(C)>0$, respectively.
In Fig.~\ref{fig:hog_entropies} we show the results for the HOG test, in terms of $\Delta E(C)$, for the Juizhang 1.0 and Juizhang 2.0 experiments. The parameters used in the calculation of the grouped click probabilities were the same as those used in the computation of the Bayesian test (see Appendix~\ref{app:drummond}). The ranges of click numbers considered here, as well as the corresponding $N$, were also the same as in the case of the Bayesian test. The results for the $\nu =0.975$ configuration of Juizhang 2.0 are not shown due to the lack of experimental samples in the range of click numbers considered.
As can be seen in Fig.~\ref{fig:hog_entropies}, for the Juizhang 1.0 experiment, we get inconclusive results as $\Delta E(C)$ changes sign depending on the total click number sector $C$. For all the configurations of Juizhang 2.0, we consistently find that the experimental samples have higher ground truth probability than the squashed states samples, even for setups in the high photon number density regime. Additionally, contrary to the case of the Bayesian test, the cross-entropy difference $\Delta E(C)$ does not consistently approach zero with increasing $\nu$.
Adjusting the mean number of photons of the input states in the squashed states hypothesis for the $\nu =0.442$ setup, we obtain the results shown in Fig.~\ref{fig:hog_entropies} under the label ``$\nu=0.442$ (Modified)''. This modification in the mean number of photons is the same as the one used in the computation of the Bayesian test. In opposition to the Bayesian test, the modification of the squashed states hypothesis does not significantly change the behavior of $\Delta E(C)$; the HOG test still indicates that the experimental samples have higher ground truth probability.
\section{\label{sec:discussion}Discussion}
In this work we proposed an alternative hypothesis for the validation of the Jiuzhang GBS experiments. This hypothesis is based on the probability distribution of classical mixtures of coherent states that we call squashed states.
For the validation of the experimental results against this alternative hypothesis, we used the same methods as the authors of the experiments.
The results for the click cumulants comparison show that the theoretical cumulants predicted by the squashed states distribution are not consistent with the experimental results for setups with low photon number density, $\nu$. However, it is noticeable that with increasing $\nu$ the squashed states cumulants become progressively consistent with those obtained from the experimental samples. This trend is confirmed by the results for the configurations with high photon number density, for which the theoretical cumulants predicted by the squashed states distribution are as compatible with the experimental results as those predicted by the ground truth of the experiment. These results suggest that the presence of high-order cumulants in the experimental samples can be efficiently reproduced by classical hypotheses lacking any quantum correlation.
The Bayesian test follows a similar behavior as the comparison of click cumulants. By analyzing the cross-entropy difference $\Delta H(C)$ as function of the number of clicks, we find that, for low $\nu$, $\Delta H(C)<0$, indicating that the ground truth of the experiment is more likely to describe the experimental samples than the squashed states hypothesis. Nevertheless, for increasing photon number density, $\Delta H(C)$ approaches zero for all the number of clicks considered. In the high $\nu$ regime, $\Delta H(C)>0$, indicating that the squashed states hypothesis is a better explanation of the experimental samples than the ground truth of the experiment.
Even though the results of the click cumulants comparison and the Bayesian test favor the ground truth distribution in the low density ($\nu$) regime, it is important to note that setups in this regime can be efficiently simulated using classical algorithms~\cite{bulmer2022boundary}, while those with high $\nu$ are progressively harder for classical computers. Thus, the quantum computational advantage claim of the Jiuzhang experiments is mostly supported by the implementation of configurations with high $\nu$. Our results show that it is precisely for this regime that the squashed states hypothesis works best, indicating that it is a suitable candidate for a classical explanation of the Jiuzhang 1.0/2.0 experiments.
Finally, we generated samples from the squashed states distribution and compared them against the experimental samples at the task of heavy output generation (HOG) from the squeezed states distribution. In this case we found that, surprisingly, except for a subset of the data in Jiuzhang 1.0, the experiments perform better at the HOG task than the samples from the squashed states hypothesis.
Our results provide a more nuanced picture of the different tools employed in arguing about quantum advantage in threshold GBS experiments. On the one hand we found that the experimental samples from Jiuzhang 1.0 and 2.0 in the high photon number density regime are better explained by states lacking any quantum feature than by genuine squeezed states of light.
On the other hand the samples generated using this alternative hypothesis are not able to spoof the HOG test when compared to the samples of the experiment. However, note that other efficient classical methods have already outperformed the Jiuzhang 1.0 and 2.0 at this task~\cite{villalonga2021efficient}. However, in absence of rigorous results about the significance of the HOG test for GBS (as opposed to RCS), these negative results must be taken with a grain of salt, especially when alternative hypotheses are more probable than the ground truth.
This work thus provides a new adversary that should be considered against future GBS experiments and, perhaps more importantly, further motivates the need to identify proper metrics and optimal classical adversaries for quantum advantage in the context of threshold GBS.
\section*{Acknowledgements}
N.Q. acknowledges support from the Ministère de l'Économie et de l'Innovation du Québec and the Natural Sciences and Engineering Research Council of Canada. This research was enabled in part by support provided by Calcul Québec and Compute Canada. N.Q. and J.M.-C. thank B. Villalonga and J.F.F. Bulmer for helpful pointers in parsing the data in Refs. ~\cite{ustc2020experimental,ustc2021raw}, P. Drummond and C.-Y. Lu for insightful correspondence and M. Houde for a critical reading of the manuscript.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 673 |
\section{Introduction}
Multi-band superconductors such as MgB$_2$ and the iron pnictides have been attracted much attention because of its
high critical temperature.
Although MgB$_2$ is a phonon-mediated superconductor, it has a large critical temperature $T_c \sim 40$K, originating from the multi-band effects.
Multi-band effects are recognized as one of the ways to increase the critical temperature.
The discovery of the iron-pnictides had a striking impact on many researchers in condensed matter physics.
Many kinds of phenomena with multi-band effects have been proposed and confirmed in iron-based superconductors\cite{Kobayashi,Kontani,Kuroki}.
Furthermore, recently found superconductors characterized by topological invariants, {\it i.e.} topological superconductors, are also multi-band superconductors,
since the internal degrees of freedom ({\it e.g.} spins, orbitals or particle-hole spaces) in a multi-band system induce topological twists in wave functions\cite{Hasan;Kane:2010,Ando:2013,Schnyder;Ludwig:2008,Fu;Berg:2009}.
The huge computational cost originating from the multiple degrees of freedom prevents theorists from understanding the
physical properties in multi-band superconductors.
For example, in the iron-based superconductor LaFeAsO, the five-orbital two-dimensional tight-binding model has been used as the effective model\cite{Kuroki}.
There are also ten-orbital three-dimensional tight-binding models as effective models to analyze experiments in another iron-pnictides.\cite{NagaiPro}
In addition, when dealing with vortices and surfaces in multi-band systems, the computational cost becomes huger, since the momentum is not a good quantum number in inhomogeneous systems.
For example, in the topological superconductors, it is important to study the quasiparticle excitations so-called the Majorana fermions around surfaces and vortices, in terms of the bulk-edge correspondence\cite{Hasan;Kane:2010}.
The Ginzburg-Landau framework, which is usually used to examine the distribution of the order parameter in the inhomogeneous superconductors,
is not suitable for dealing with the quasiparticle excitations.
Even if we use the mean-field framework such as the Bogoliubov-de Gennes framework, the simulations of the nano-size multi-band superconductors needs enormous computational costs.
We point out that the effective models ({\it e.g.}, derived by the first principles calculations) might have too many bands to describe the low energy physics of superconductivity.
We should note that the number of the bands crossing the Fermi level in normal states is {\it less than} four even in models for iron-pnictides.
The low energy physics in superconducting state are characterized by the quasiparticles on the bands crossing the Fermi level in normal states,
since a characteristic energy scale of the superconducting gap ($\sim$meV) is much smaller than that of the bands far from the Fermi level ($\sim$ eV).
In the single-band weak-coupling Bardeen-Cooper-Schrieffer (BCS) framework, the theory using information only at the Fermi surface, called the quasiclassical Eilenberger theory, has many successes\cite{KopninText}.
In multi-band superconductors,
eliminating the high-energy bands not crossing the Fermi level can reduce the number of the bands in a low energy effective theory as shown in Fig.\ref{fig:Feband}, since
the high-energy bands can not affect the physical quantities in superconducting state.
\begin{figure}[t]
\begin{center}
\resizebox{ 0.7\columnwidth}{!}{\includegraphics{Fig1.eps}}
\end{center}
\caption{\label{fig:Feband}(Color online) Schematic figure of the multi-band Eilenberger theory. The band dispersions are calculated in the five-orbital effective model for LaOFeAs\cite{Kuroki,NagaiNJP}. The bands in the shaded regions are neglected in the multi-band Eilenberger theory.
}
\end{figure}
The quasiclassical Eilenberger theory is successful in the BCS model of superconductivity.
The theoretical framework is based on the fact that the coherence length $\xi$ is sufficiently greater than the Fermi wavelength $1/k_{\rm F}$.
Various kinds of analytical and numerical techniques on the quasiclassical Eilenberger theory have been developed and
successfully applied to the studies of a large number of conventional and unconventional superconductors\cite{KopninText,Volovik,NagaiJPSJ:2006,Eilenberger,NagaiMeso,Miranovic,Melnikov,NagaiPRL,NagaiCe,Graser,Iniotakis}.
The Eilenberger theory was applied into the two-band superconductor MgB$_{2}$.
In the conventional models for MgB$_{2}$, one neglects the off-diagonal inter-band elements in Green's function.
In this case, two decoupled Eilenberger equations can describe the quasiparticle excitations, and the multi-band effects are included only through solving the gap equations\cite{Mugikura,KTanaka,Gumann,Kogan}.
There are two kinds of bases to consider the multi-band systems.
First one is the basis which is orthogonal in the momentum space (so-called ``band''-basis).
Other is the basis which is orthogonal in the real space, such as the $d$-orbitals in models for iron-based superconductors and spins in a model with the spin-orbit coupling term.
We call these real-space orthogonal basis ``orbital'' basis in this paper.
In the quasiclassical Eilenberger theory, the quasiclassical Green's functions depend on the momentum of the relative motion in momentum space and the center-of-mass coordinate in real space.
In the previous multi-band quasiclassical theories\cite{Prozorov,Silaev},
the decoupled Eilenberger equations are used by neglecting the off-diagonal elements of the Hamiltonian in both band basis in momentums space and orbital basis in real space.
In the case of the two-band model for MgB$_{2}$, this assumption is valid to describe the quasiparticle excitations.
There is, however, no Eilenberger theory which includes the off-diagonal inter-band elements, except for a perturbative approach\cite{Moor}.
The inter-band elements of Green's function are important in the complicated multi-band systems, such as the iron-pnictides and the topological superconductors.
For example,
the iron-based superconductors have many entangled Fe-$3d$ orbitals at the Fermi level.
The Hamiltonian proposed in the iron-based superconductors is diagonalized in momentum space by the momentum dependent unitary matrix.
The ratio of which orbital is dominant originates from this unitary matrix and depends on the momentum.
This ``orbital'' character, how the orbitals are entangled, at the each Fermi wave number is important to understand the physics in iron-based superconductors\cite{Kemper}.
The off-diagonal inter-band elements of Green's functions are induced by those of the unitary matrix.
These off-diagonal elements become important when a self-energy is induced by an inter-band scattering, which is important in a system with impurities or vortices.
A model of topological superconductors usually have the off-diagonal elements in spin-space due to a spin-orbit coupling.
The spins rotate in momentum space, originating from the spin-orbit coupling so that
the Hamiltonian is not diagonal with the use of spin basis in real space.
The ``spin'' character, how the spins are entangled, induces topological superconductivity even in system with $s$-wave on-site pairing interaction in two dimension\cite{Sato,Nagai2D}.
Therefore, the information about ``orbital'' characters in momentum space is important factor to describe multi-band effects.
In this paper, we propose a quasiclassical Eilenberger framework in multi-band superconductors with a systematic low-energy projection.
By eliminating the high energy bands far from the Fermi level, we derive the multi-band Andreev equations and quasiclassical Eilenberger equations
in the projected space constructed from the bands crossing the Fermi level.
We show that the resultant multi-band Eilenberger equations are similar to the single-band ones, except for some corrections to describe multi-band effects.
The quasiclassical framework uses the fact that the coherence length $\xi$ is usually longer than the Fermi wave length $1/k_{\rm F}$ in a lot of superconductors.
The orbital characters on the Fermi surfaces in normal states are naturally included in our theory.
This paper is organized as follows.
In Sec.~\ref{sec:model}, we introduce the model of the multi-band superconductors.
The mean-field multi-band Bogoliubov-de Gennes (BdG) Hamiltonian is proposed.
We introduce the multi-band BdG equations and the gap equations, which is the starting point of the quasiclassical theory.
In Sec.~\ref{sec:dec}, we discuss the decoupled Eilenberger theory used in past studies.
We show that orbital characters can not be included in this theory.
In Sec.~\ref{sec:wave}, we derive the multi-band quasiclassical Andreev equations, starting with the multi-band BdG Hamiltonian.
The Andreev equations describe the wave functions in the quasiclassical approach.
In Sec.~\ref{sec:dec}, the multi-band quasiclassical Eilenberger equations are derived.
The Eilenberger equations are the equations of motion of the quasiclassical Green's function.
We discuss the difference between the previous theory and our theory.
In Sec.~\ref{sec:multi}, we discuss the physical meanings of our multi-band Eilenberger theory.
In Sec.~\ref{sec:single}, we apply the multi-band Eilenberger theory in the various kinds of systems as examples.
We show that the previous theoretical results are reproduced by our theory and the corrections originating from orbital characters are important in multi-band systems.
In Sec.~\ref{sec:sum}, the summary is given.
\section{Model}
\label{sec:model}
\subsection{Multi-band BdG equations}
Let us start with the mean-field BdG Hamiltonian in the $2 N \times 2 N$ Nambu-Gor'kov space,
\begin{align}
{\cal H} &= \frac{1}{2} \int d \Vec{r}_1 d \Vec{r}_2 \Vec{\Psi}(\Vec{r}_1)^{\dagger}
\check{H}(\Vec{r}_1,\Vec{r}_2)
\Vec{\Psi}(\Vec{r}_2). \label{eq:1}
\end{align}
Here, the column vector $\Vec{\Psi}(\Vec{r})$ is composed of $N$ fermionic annihilation $\psi_{\alpha}$ and creation operators $\psi_{\alpha}^{\dagger}$ at the position $\Vec{r}$ ($\alpha = 1,\cdots,N$), $\Vec{\Psi}(\Vec{r}) = (\{ \psi_{\alpha}(\Vec{r}) \}, \{ \psi_{\alpha}^{\dagger}(\Vec{r}) \})^{\rm T}$, where $\{ \psi_{\alpha}(\Vec{r}) \} = (\psi_{1}(\Vec{r}),\cdots, \psi_{N}(\Vec{r}))^{\rm T}$ and $\{ \psi_{\alpha}^{\dagger}(\Vec{r}) \} = (\psi_{1}^{\dagger}(\Vec{r}),\cdots, \psi_{N}^{\dagger}(\Vec{r}))^{\rm T}$.
The subscript $\alpha$ in $\psi_{\alpha}(\Vec{r})$ or $\psi_{\alpha}^{\dagger}(\Vec{r})$ indicates a quantum index depending on spin or
orbital, etc.
These quantum indices are labeled by the orthogonal basis in real space, which we call orbital basis.
The Bogoliubov-de Gennes Hamiltonian with a matrix form is composed of
\begin{align}
\check{H}(\Vec{r}_1,\Vec{r}_2) \equiv
\check{H}^{\rm N}(\Vec{r}_1,- i \Vec{\nabla}_{1} )\delta(\Vec{r}_1-\Vec{r}_2) + \check{\Delta}(\Vec{r}_1,\Vec{r}_2).
\end{align}
Throughout the paper, {\it hat} $\hat{a}$ denotes a $N \times N$ matrix and {\it check} $\check{a}$ denotes a $2N \times 2N$ matrix.
The $2 N \times 2 N$ normal-state Hamiltonian matrix $\check{H}(\Vec{r}_1,\Vec{r}_2)$ and superconducting order parameter matrix $\check{\Delta}(\Vec{r}_1,\Vec{r}_2)$ are respectively defined by
\begin{align}
&\check{H}^{\rm N}(\Vec{r}_1,- i \Vec{\nabla}_{1} ) \nonumber \\
&\equiv
\left(\begin{array}{cc}
\hat{H}^{\rm N}(\Vec{r}_1,- i \Vec{\nabla}_{1} ) & 0 \\
0 & -\hat{H}^{\ast {\rm N}}(\Vec{r}_1,- i \Vec{\nabla}_{1} )
\end{array}\right),
\end{align}
\begin{align}
\check{\Delta}(\Vec{r}_1,\Vec{r}_2) &\equiv
\left(\begin{array}{cc}
0 & \hat{\Delta}(\Vec{r}_1,\Vec{r}_2) \\
\hat{\Delta}^{\dagger}(\Vec{r}_2,\Vec{r}_1) & 0
\end{array}\right).
\end{align}
The order parameter matrix is given by (so-called the gap equations) \cite{SigristUeda}
\begin{align}
\hat{\Delta}_{\alpha \beta}(\Vec{r}_1,\Vec{r}_2) &= \sum_{\Vec{k},\Vec{q}} e^{i \Vec{k} \cdot (\Vec{r}_{1} - \Vec{r}_{2})}
e^{i \Vec{q} \cdot (\Vec{r}_{1} + \Vec{r}_{2})/2} \hat{\Delta}_{\alpha \beta} (\Vec{k},\Vec{q}), \\
\hat{\Delta}_{\alpha \beta} (\Vec{k},\Vec{q}) &= -
\sum_{\Vec{k}', \gamma \gamma'} V_{\beta \alpha ; \gamma \gamma'}(\Vec{k},\Vec{k}')
\langle
\psi_{\Vec{q}/2+\Vec{k}',\gamma} \psi_{\Vec{q}/2 - \Vec{k}', \gamma'}
\rangle, \label{eq:gapeq}
\end{align}
with the multi-orbital interaction matrix $V_{\alpha \beta;\gamma \gamma'}(\Vec{k},\Vec{k}')$ and
$\psi_{\alpha}(\Vec{r}) = \sum_{\Vec{k}} \psi_{\Vec{k},\alpha} \exp (i \Vec{k} \cdot \Vec{r})$.
The multi-band BdG equations are then expressed as
\begin{align}
\int d \Vec{r}_{2} \check{H}(\Vec{r}_{1},\Vec{r}_{2}) \Vec{\phi}(\Vec{r}_{2}) &= E \Vec{\phi}(\Vec{r}_{1}). \label{eq:BdGeq}
\end{align}
With the use of the eigenvectors $\Vec{\phi}(\Vec{r}_{1})$,
the mean-field BdG Hamiltonian (\ref{eq:1}) is diagonalized.
\subsection{Multi-band Gor'kov equations}
The Dyson equation in Nambu-Gor'kov space (Gor'kov equation) is obtained by
\begin{align}
& \int d\Vec{r}' \left(i \omega_{n} \check{1}\delta(\Vec{r}_{1}- \Vec{r'}) - \check{H}^{\rm N}(\Vec{r}_1,\Vec{r}') \right. \nonumber \\
& \left. - \check{\Delta}(\Vec{r}_{1},\Vec{r}') -\check{\Sigma}(\Vec{r}_1,\Vec{r}',i \omega_n) \right)
\check{G}(\Vec{r}',\Vec{r}_2, i \omega_n)
= \delta(\Vec{r}_1-\Vec{r}_2) \check{1} ,
\end{align}
with
\begin{align}
\check{H}^{\rm N}(\Vec{r}_1,\Vec{r}_2) \equiv
\left(\check{H}^{\rm N 0}(\Vec{r}_{1}) + \check{H}^{\rm N 1}(- i \Vec{\nabla}_{\Vec{r}_{1}}) \right) \delta(\Vec{r}_{1} - \Vec{r}_{2}) ,
\end{align}
where, $\omega_{n} = (2 n + 1) \pi T$ is the fermionic matsubara frequency,
$\check{\Sigma}(\Vec{r}_1,\Vec{r}',i \omega_n)$ denotes the self-energy.
Here, the $2 N \times 2 N$ Green's function is determined by
\begin{align}
\check{G}(\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2) &\equiv - \langle {\rm T}_{\tau}
\Psi(\Vec{r}_1,\tau_1) \Psi^{\dagger}(\Vec{r}_2,\tau_2)
\rangle, \\
&=
\left(\begin{array}{cc}
\hat{G}(\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2) & \hat{F}(\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2) \\
\hat{\bar{F}}(\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2) & \hat{\bar{G}}(\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2)
\end{array}\right),
\end{align}
\begin{align}
G_{\alpha \beta} (\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2) &\equiv
- \langle {\rm T}_{\tau}
\psi_{\alpha}(\Vec{r}_1,\tau_1) \psi^{\dagger}_{\beta}(\Vec{r}_2,\tau_2)\rangle,\\
F_{\alpha \beta} (\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2) &\equiv
- \langle {\rm T}_{\tau}
\psi_{\alpha}(\Vec{r}_1,\tau_1) \psi_{\beta}(\Vec{r}_2,\tau_2)\rangle,\\
\bar{F}_{\alpha \beta} (\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2) &\equiv
- \langle {\rm T}_{\tau}
\psi_{\alpha}^{\dagger}(\Vec{r}_1,\tau_1) \psi^{\dagger}_{\beta}(\Vec{r}_2,\tau_2)\rangle, \\
\bar{G}_{\alpha \beta} (\Vec{r}_1,\Vec{r}_2,\tau_1-\tau_2) &\equiv
- \langle {\rm T}_{\tau}
\psi_{\alpha}^{\dagger}(\Vec{r}_1,\tau_1) \psi_{\beta}(\Vec{r}_2,\tau_2)
\rangle,
\end{align}
with the imaginary time $\tau$.
The local density of states is given by
\begin{align}
N(\Vec{r},E) &=\frac{-1}{\pi} {\rm Im} \: \left[ \lim_{\eta \rightarrow +0} {\rm Tr} \: \hat{G}(\Vec{r},\Vec{r},i \omega_{n} \rightarrow E + i \eta) \right].
\end{align}
\section{Decoupled Eilenberger equations}
\label{sec:dec}
\subsection{Model and assumptions}
Let us discuss the decoupled multi-band quasiclassical theory, which is appropriate in the conventional two-band models for MgB$_{2}$\cite{Mugikura}.
We assume that all $N \times N$ matrices in the BdG Hamiltonian are diagonal in the ``band'' basis, expressed as
\begin{align}
\left[ \hat{H}^{\rm N} (\Vec{r}_1,- i \Vec{\nabla}_{1} ) \right]_{\alpha \beta} \sim H^{\rm N}_{\alpha}(\Vec{r}_1,- i \Vec{\nabla}_{1} ) \delta_{\alpha \beta}, \\
\left[ \hat{\Delta}(\Vec{r}_1,\Vec{r}_2) \right]_{\alpha \beta} \sim \Delta_{\alpha}(\Vec{r}_1,\Vec{r}_2) \delta_{\alpha \beta}.
\end{align}
This assumption is equivalent to that off-diagonal inter-band elements are neglected.
In this case, the normal state Hamiltonian in momentum space is expressed as
\begin{align}
\hat{H}^{\rm N} (\Vec{k}) &= \left(\begin{array}{ccc}
\lambda_{1}(\Vec{k}) & 0 & 0 \\
0 & \ddots & 0 \\
0 & 0 & \lambda_{N}(\Vec{k})
\end{array}\right),
\end{align}
with the eigenvalues $\lambda_{i}(\Vec{k})$.
In real space, the Fourier transformation on each band makes the band-diagonal Hamiltonian.
The mean-field BdG Hamiltonian in Eq.~(\ref{eq:1}) is rewritten as
\begin{align}
{\cal H} &= \frac{1}{2} \sum_{\alpha} \int d \Vec{r}_1 d \Vec{r}_2 \Vec{\Psi}_{\alpha}(\Vec{r}_1)^{\dagger}
\check{H}_{\alpha}(\Vec{r}_1,\Vec{r}_2)
\Vec{\Psi}_{\alpha}(\Vec{r}_2),
\end{align}
with
\begin{align}
&\check{H}_{\alpha}(\Vec{r}_1,\Vec{r}_2) \equiv \nonumber \\
&\left(\begin{array}{cc}
H^{\rm N}_{\alpha}(\Vec{r}_1,- i \Vec{\nabla}_{1} )\delta(\Vec{r}_1-\Vec{r}_2) &
\Delta_{\alpha}(\Vec{r}_1,\Vec{r}_2)
\\
\Delta_{\alpha}^{\ast}(\Vec{r}_2,\Vec{r}_1) & -H^{{\rm N} \ast}_{\alpha}(\Vec{r}_1,- i \Vec{\nabla}_{1})\delta(\Vec{r}_1-\Vec{r}_2)
\end{array}\right).
\end{align}
Here, the column vector $\Vec{\Psi}_{\alpha}(\Vec{r})$ is composed of fermionic annihilation $\psi_{\alpha}$ and creation operators $\psi_{\alpha}^{\dagger}$ on the $\alpha$ band at the position $\Vec{r}$ ($\alpha = 1,\cdots,N$), $\Vec{\Psi}_{\alpha}(\Vec{r}) = (\psi_{\alpha}(\Vec{r}), \psi_{\alpha}^{\dagger}(\Vec{r}))^{\rm T}$.
There is no inter-band effect in the Hamiltonian, since the BdG Hamiltonian $\check{H}_{\alpha}$ is determined on each band.
The gap equations in Eq.~(\ref{eq:gapeq}) are rewritten as
\begin{align}
\Delta_{\alpha}(\Vec{k},\Vec{q}) &= - \sum_{\Vec{k},\gamma} V_{\alpha \gamma}(\Vec{k},\Vec{k}')
\langle
\psi_{\Vec{q}/2+\Vec{k}',\gamma} \psi_{\Vec{q}/2 - \Vec{k}', \gamma}
\rangle.
\end{align}
In this approximation, the multi-band effects are included as the inter-band pairing interactions only in the pairing $V_{\alpha \gamma}$.
Thus, the quasiclassical decoupled Eilenberger equations are easily obtained as
\begin{align}
& i \Vec{v}_{\rm F,\alpha} \Vec{\nabla}_{\Vec{R}} \check{g}_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F},z) +
\left[z \sigma_{z} - \check{\Delta}_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F}) \sigma_{z}
, \check{g}_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F},z) \right]_{-} = 0, \label{eq:mgb2type} \\
& \check{\Delta}_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F}) =
- \pi T \sum_{n} \sum_{\beta} \int \frac{d S_{F}}{|\Vec{v}_{\rm F}|} V_{\alpha \beta}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}')
f_{\Vec{R}}^{\beta}(\Vec{k}_{\rm F}',i \omega_{n}),
\end{align}
with quasiclassical Green's function on the $\alpha$ band expressed as
\begin{align}
\check{g}_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F},z) &\equiv \oint d \xi_{\Vec{k}}^{\alpha} \sigma_{z} \check{G}^{\alpha}_{\Vec{R}}(\Vec{k},z), \\
&\equiv \left(\begin{array}{cc}
g_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F}',i \omega_{n}) & f_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F}',i \omega_{n}) \\
\bar{f}_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F}',i \omega_{n}) & \bar{g}_{\Vec{R}}^{\alpha}(\Vec{k}_{\rm F}',i \omega_{n})
\end{array}\right).
\end{align}
Here, we neglect the self energies and the vector potentials for simplicity, $z$ denotes a complex energy, $\sigma_{z}$ is the Pauli matrix in $2 \times 2$ Nambu-Gor'kov space, $[\check{A},\check{B}]_{-} = \check{A} \check{B} - \check{B} \check{A}$ is used, and
$G^{\alpha}_{\Vec{R}}(\Vec{k},z)$ is the Green's function of the $\alpha$ band.
It should be noted that the self-energy must be diagonalized in this basis and $\check{A}$ in this section is $2 \times 2$ matrix in Nambu space.
The above equations were used in the simple multi-band superconductors such as MgB$_{2}$.
For MgB$_{2}$, there are two bands (so-called $\sigma$- and $\pi$-bands) and the gap equations connect information on each band.
\subsection{Appropriate region of the decoupled Eilenberger equations}
Now, we discuss the appropriate region of the decoupled Eilenberger equations in the previous section.
We introduce the Hamiltonians $\hat{H}^{\rm orbital}(\Vec{k})$ and $\hat{H}^{\rm band}(\Vec{k})$ in normal states with the orbital basis and the band basis in momentum space, respectively.
Generally, the Hamiltonian $\hat{H}^{\rm orbital}(\Vec{k})$ has the off-diagonal elements, since the orbital basis is a basis which is diagonal in real space.
With the use of the momentum dependent unitary matrix $\hat{U}(\Vec{k})$,
one can obtain the diagonal Hamiltonian expressed as
\begin{align}
\hat{U}^{\dagger}(\Vec{k}) \hat{H}^{\rm orbital}(\Vec{k}) \hat{U}(\Vec{k})
&= \left(\begin{array}{ccc}
\lambda_{1}(\Vec{k}) & 0 & 0 \\
0 & \ddots & 0 \\
0 & 0 & \lambda_{N}(\Vec{k})
\end{array}\right), \\
&\equiv \hat{H}^{\rm band}(\Vec{k}).
\end{align}
The ``band'' basis is determined by the unitary transformation of the orbital basis in momentum space.
If the unitary matrix does not depend on momentum, one can choose the basis which simultaneously diagonalizes the Hamiltonian in both real and momentum spaces.
Generally, the decoupled Eilenberger equations are derived by neglecting the off-diagonal elements.
We show three examples that the decoupled Eilenberger theory is not appropriate as follows.
The first example is an impurity problem in the multi-band superconductors.
In the previous study\cite{Belova}, they assumed that the impurity-induced self energy was described as a momentum-independent band-diagonal matrix, which lead to the decoupled Eilenberger equations.
We point out that this assumption induces non-local impurities in real space, as follows.
In the band basis with this assumption, the Gor'kov equations become
\begin{align}
\check{G}_{\Vec{k}}^{\rm band}(z)&= \check{G}_{\Vec{k}}^{0 {\rm band}}(z) + \check{G}_{\Vec{k}}^{0 {\rm band}}(z) \check{\Sigma}^{\rm band}(z) \check{G}_{\Vec{k}}^{\rm band}(z).
\end{align}
Here, $\check{A}^{\rm band}$ denotes the matrix defined by the basis which diagonalizes the normal-state Hamiltonian $\check{H}_{\Vec{k}}^{\rm orbital}$.
To describe impurities in real space, one has to use the orbital basis in real space.
With use of the unitary transformation from the band basis to the orbital basis,
the impurity-induced self-energy in the orbital basis $\check{\Sigma}^{{\rm orbital}}$ should have a momentum dependence expressed as
\begin{align}
\check{\Sigma}^{{\rm orbital}}(\Vec{k},z) &= \check{U}(\Vec{k}) \check{\Sigma}^{\rm band}(z)\check{U}^{\dagger}(\Vec{k}),
\end{align}
with
\begin{align}
\check{U}(\Vec{p}) &= \left(\begin{array}{cc}\hat{U}(\Vec{p}) & 0 \\0 & \hat{U}^{\ast}(-\Vec{p})\end{array}\right).
\end{align}
If the self-energy in the band basis is obtained by the $T$-matrix approximation for randomly distributed point impurities given as
\begin{align}
\check{\Sigma}^{{\rm band}}(z) &=n_{\rm imp}\check{V} + n_{\rm imp} \sum_{\Vec{p}} \check{V} \check{G}_{\Vec{p}}^{0 \: {\rm orbital}}(z) \check{V},
\label{eq:sigma2}
\end{align}
the self-energy in the orbital basis becomes
\begin{align}
&\check{\Sigma}^{\rm orbital}(\Vec{k}, z) =n_{\rm imp} \check{V}^{\rm orbital}(\Vec{k},\Vec{k}) \nonumber \\
&+ n_{\rm imp} \sum_{\Vec{p}}
\check{V}^{\rm orbital}(\Vec{k},\Vec{p})
\check{G}_{\Vec{p}}^{0 {\rm orbital}}(z) \check{V}^{\rm orbital}(\Vec{p},\Vec{k}),
\end{align}
with the effective ``non-local'' impurity potential defined as
\begin{align}
\hat{V}^{\rm orbital}(\Vec{k},\Vec{p}) &\equiv \check{U}(\Vec{k}) \check{V} \check{U}^{\dagger}(\Vec{p}).
\end{align}
Therefore, the decoupled Eilenberger equations does not describe the local impurity potentials.
The second example is the proximity-induced impurity-robust $p$-wave effective order parameter on the surface of a topological insulator, as discussed later in Sec.~\ref{sec:rp}.
With the use of the band basis, an effective chiral $p$-wave order parameter can be derived by the previous quasiclassical framework.
This previous framework, however, can not describe the robustness against non-magnetic impurities, which was proposed by directly solving the BdG equations\cite{Ito:2011ct}.
The impurity robust $p$-wave superconductor is naturally introduced in our framework.
The third example is the appearance condition of the Andreev bound states at a surface.
In a single band model, the Andreev bound states occur when the sign of the gap function changes through the scattering process\cite{Kashiwaya:2000ic}.
In a multi band model, an ambiguity of the "sign" of the order parameter makes the above statement unclear.
This ambiguity can not be resolved by the previous quasiclassical framework.
In our multi-band quasiclassical Eilenberger approach, we can overcome this difficulty by deriving the most appropriate effective order parameter,
which obeys the statement of the Andreev bound states as discussed later in Sec.~\ref{sec:arbit}.
We can use the decoupled Eilenberger equations if we assume the momentum-independent unitary matrix $(\check{U}(\Vec{k}) = \check{U})$.
In this assumption, however, we can not treat the ``orbital'' character.
Therefore, we propose the general multi-band Eilenberger theory.
\section{Quasiclassical treatment I: wave-function approach}
\label{sec:wave}
In this section, we derive the quasiclassical equations on the basis of the BdG equations.
The quasiclassical theory is founded on an assumption that the coherence length $\xi$ is much longer than the
Fermi wave length $1/k_{\rm F}$ (i.e. $\xi k_{\rm F} \ll 1$)\cite{Volovik}.
This assumption is valid, if the order parameter amplitude is much smaller than the Fermi energy, and
this condition is fully fulfilled in BCS weak-coupling superconductivity.
In this theory, the wave function is expressed by a product of the fast oscillating one characterized by the Fermi momentum
$p_{\rm F}$ and the slowly varying one by the coherence length.
We proposed the quasiclassical theory for the multi-orbital topological superconductor\cite{NagaiTopo}.
The generalization of this theory is proposed in this section.
\subsection{Assumptions}
We assume that the eigen vector $\Vec{\phi}(\Vec{r})$ in Eq. (\ref{eq:BdGeq}) is
expressed by a product of the fast oscillating one characterized by the Fermi momentum and the slowly varying one by the coherence length expressed as
\begin{align}
\Vec{\phi}(\Vec{r}) &= \sum_{\Vec{k}_{\rm F}} e^{i \Vec{k}_{\rm F} \Vec{r}}
\Vec{\phi}_{\Vec{k}_{\rm F}}'(\Vec{r}), \label{eq:wave}
\end{align}
where
\begin{align}
\Vec{\phi}_{\Vec{k}_{\rm F}}'(\Vec{r}) &\equiv
\sum_{l=1}^{M}
\left(\begin{array}{c}
\Vec{u}^{\rm N}_{l}(\Vec{k}_{\rm F}) f_{l}^{\Vec{k}_{\rm F}}(\Vec{r}) \\
\Vec{v}^{\rm N}_{l}(\Vec{k}_{\rm F}) g_{l}^{\Vec{k}_{\rm F}}(\Vec{r})
\end{array}\right).
\end{align}
Here, $f_{l}^{\Vec{k}_{\rm F}}(\Vec{r})$ and $g_{l}^{\Vec{k}_{\rm F}}(\Vec{r})$ correspond to slowly varying components,
$\Vec{u}^{\rm N}_{l}(\Vec{k}_{\rm F})$, $\Vec{v}^{\rm N}_{l}(\Vec{k}_{\rm F})$ are
the fast oscillating functions adopted as normal-state uniform eigenvectors satisfying the eigen-equations,
\begin{align}
\check{H}^{\rm N1}(\Vec{k})
\left(\begin{array}{c}
\Vec{u}^{\rm N}_{l}(\Vec{k})\\
\Vec{v}^{\rm N}_{l}(\Vec{k})
\end{array}\right) &= \epsilon_{l}(\Vec{k})
\left(\begin{array}{c}
\Vec{u}^{\rm N}_{l}(\Vec{k}) \\
\Vec{v}^{\rm N}_{l}(\Vec{k})
\end{array}\right),
\end{align}
where
\begin{align}
\check{H}^{\rm N 1}(\Vec{k}) &= \left(\begin{array}{cc}
\hat{H}^{\rm N 1}(\Vec{k}) & 0 \\
0 & \hat{H}^{\rm N 1}(-\Vec{k})^{\ast}
\end{array}\right).
\end{align}
The Fermi surfaces in normal states are expressed by the set of the zero-energy eigenvalues of $\hat{H}^{\rm N 1}(\Vec{k})$.
The $M$-eigenvalues $\epsilon_{l}(\Vec{k})$ cross the Fermi level (i.~e.~ $\epsilon_{l} (\Vec{k}_{\rm F}) = 0$).
We assume that the eigenvalues near the Fermi level are same ($\epsilon_{1}(\Vec{k}), \cdots, \epsilon_{M}(\Vec{k}) = \xi(\Vec{k})$) expressed as
\begin{align}
\check{H}^{\rm N1}(\Vec{k})
\left(\begin{array}{c}
\Vec{u}^{\rm N}_{l}(\Vec{k})\\
\Vec{v}^{\rm N}_{l}(\Vec{k})
\end{array}\right) &= \xi(\Vec{k})
\left(\begin{array}{c}
\Vec{u}^{\rm N}_{l}(\Vec{k}) \\
\Vec{v}^{\rm N}_{l}(\Vec{k})
\end{array}\right). \label{eq:eigen}
\end{align}
\begin{figure}[t]
\begin{center}
\resizebox{1 \columnwidth}{!}{\includegraphics{Fig2.eps}}
\end{center}
\caption{\label{fig:cross}The schematic figures of the electron bands in the multi-orbital system characterized by $M = 2$ at the Fermi wave momentum $\Vec{k}_{\rm F}$.
}
\end{figure}
This assumption is appropriate when $M$ denotes the internal degrees of freedom at the Fermi level as shown in Fig.~\ref{fig:cross}(a).
We note that there is an exception as shown in Fig.~\ref{fig:cross}(b).
For example, this exception occurs when the Fermi level is located at the center of the Dirac cone.
However, the assumption is usually appropriate in many realistic materials.
We should note that there is the relation between $\Vec{u}_{l}^{\rm N}(\Vec{k})$ and $\Vec{v}_{l}^{\rm N}(\Vec{k})$ at the Fermi energy expressed as
\begin{align}
\Vec{v}_{l}^{\rm N}(\Vec{k}_{\rm F}) &= \Vec{u}_{l}^{\rm N \ast}(-\Vec{k}_{\rm F}).
\end{align}
\subsection{Andreev-type equations}
\begin{widetext}
By substituting Eq.(\ref{eq:wave}) into the BdG equations (\ref{eq:BdGeq}),
we obtain the Andreev-type quasiclassical equations.
Eventually, we have the $2 M \times 2 M$ quasiclassical BdG equations represented as (in detail, see Appendix \ref{app:and}),
\begin{align}
\left(\begin{array}{cc}
- i \Vec{v}_{\rm F} \cdot \Vec{\nabla} + V_{0}(\Vec{r},\Vec{k}_{\rm F}) & \Delta_{\rm eff}(\Vec{r},\Vec{k}_{\rm F}) \\
\Delta_{\rm eff}^{\dagger}(\Vec{r},\Vec{k}_{\rm F}) & i \Vec{v}_{\rm F} \cdot \Vec{\nabla} - V_{0}^{\ast}(\Vec{r},-\Vec{k}_{\rm F})
\end{array}\right)
\left(\begin{array}{c}
\Vec{f}^{\Vec{k}_{\rm F}}(\Vec{r}) \\
\Vec{g}^{\Vec{k}_{\rm F}}(\Vec{r})
\end{array}\right)
&=
E
\left(\begin{array}{c}
\Vec{f}^{\Vec{k}_{\rm F}}(\Vec{r}) \\
\Vec{g}^{\Vec{k}_{\rm F}}(\Vec{r})
\end{array}\right),\label{eq:and}
\end{align}
\end{widetext}
with introducing $M \times M$ matrices $V_{0}$ and $\Delta_{\rm eff}$ defined by
\begin{align}
V_{0}(\Vec{r},\Vec{k}_{\rm F}) &\equiv \tilde{U}_{\Vec{k}_{\rm F}}^{M \dagger} \hat{H}^{\rm N 0}(\Vec{r}) \tilde{U}_{\Vec{k}_{\rm F}}^{M}, \\
\Delta_{\rm eff}(\Vec{r},\Vec{k}_{\rm F}) &\equiv \tilde{U}_{\Vec{k}_{\rm F}}^{M \dagger} \hat{\Delta}(\Vec{r},\Vec{k}_{\rm F}) \tilde{U}_{-\Vec{k}_{\rm F}}^{M \ast}, \label{eq:gapeff}
\end{align}
with the $N \times M$ matrix given by
\begin{align}
\tilde{U}_{\Vec{k}_{\rm F}}^{M} = (\Vec{u}_{1}^{\rm N}(\Vec{k}_{\rm F}), \cdots, \Vec{u}_{M}^{\rm N}(\Vec{k}_{\rm F})). \label{eq:ukm}
\end{align}
Here, the $M$-component vectors $\Vec{f}^{\Vec{k}_{\rm F}}(\Vec{r})$ and $\Vec{g}^{\Vec{k}_{\rm F}}(\Vec{r})$ denote $\Vec{f}^{\Vec{k}_{\rm F}}(\Vec{r})^{\rm T} = ( f_{1}^{\Vec{k}_{\rm F}}(\Vec{r}), \cdots, f_{M}^{\Vec{k}_{\rm F}}(\Vec{r}) )$ and $\Vec{g}^{\Vec{k}_{\rm F}}(\Vec{r})^{\rm T} = (g_{1}^{\Vec{k}_{\rm F}}(\Vec{r}), \cdots, g_{M}^{\Vec{k}_{\rm F}}(\Vec{r}) )$, respectively.
The $N \times M$ matrix has the relation expressed as
\begin{align}
\tilde{U}_{\Vec{k}}^{M \dagger} \tilde{U}_{\Vec{k}}^{M} &= 1_{M \times M}.
\end{align}
Note that $\tilde{U}_{\Vec{k}}^{M} \tilde{U}_{\Vec{k}}^{M \dagger} \neq 1_{N \times N}$ if $M$ is not equal to $N$.
With the use of the $N \times M$ matrix $ \tilde{U}_{\Vec{k}}^{M}$, the eigenvector $\Vec{\phi}(\Vec{r})$ in the BdG equations (\ref{eq:BdGeq}) is approximated as
\begin{align}
\Vec{\phi}(\Vec{r}) \sim \sum_{\Vec{k} = \Vec{k}_{\rm F}} e^{i \Vec{k}_{\rm F} \Vec{r}}
\left(\begin{array}{c} \tilde{U}_{\Vec{k}}^{M} \Vec{f}^{\Vec{k}_{\rm F}}(\Vec{r})\\
\tilde{U}_{-\Vec{k}}^{M \ast} \Vec{g}^{\Vec{k}_{\rm F}}(\Vec{r})
\end{array}\right).
\end{align}
The resultant quasiclassical BdG equations (\ref{eq:and}) are equivalent to the linearized BdG (Andreev) equations if we consider the
single band system ($N = 1$)\cite{Volovik}.
Thus, we successfully reduce the matrix dimension from the number of the bands $N$ to the number of the degenerated Fermi levels $M$ in this quasiclassical treatment.
\subsection{Boundary condition at a specular surface}
Let us discuss the boundary condition at a specular surface.
For simplicity, we consider that the material is filled in the region $z > 0$.
By assuming the translational symmetry along the $x$ and $y$ axes which conserves the momentum $\Vec{k}_{{\rm F} \parallel} = (k_{{\rm F}x},k_{{\rm F}y})$, the boundary condition is given by
\begin{align}
\Vec{\phi}(\Vec{k}_{{\rm F} \parallel},z=0) &= 0. \label{eq:bound}
\end{align}
First, we find the solutions which satisfy the above boundary condition in normal states at the Fermi energy, expressed as
\begin{align}
\Vec{u}^{\rm N} (\Vec{k}_{{\rm F} \parallel},z=0)&= 0.
\end{align}
By solving the eigenvalue equations with the normal-state $N \times N$ Hamiltonian $\hat{H}^{\rm N 1}(\Vec{k})$:
\begin{align}
\hat{H}^{\rm N 1}(\Vec{k}_{{\rm F} \parallel},k_{z}^{i}) \Vec{u}_{l,(\Vec{k}_{{\rm F} \parallel},k_{z}^{i})}^{{\rm N}} &= 0, \label{eq:boundnh}
\end{align}
the boundary condition becomes
\begin{align}
\sum_{i}^{K} \sum_{l}^{M} c_{i}^{l}(\Vec{k}_{{\rm F} \parallel}) \Vec{u}_{l,(\Vec{k}_{{\rm F} \parallel},k_{z}^{i})}^{{\rm N}} &= 0. \label{eq:nbound}
\end{align}
Note that $k_{z}^{i}$ is a complex number and satisfies ${\rm Im} \: k_{z}^{i} \geq 0 $.
Here, $K$ denotes the number of $k_{z}^{i}$ with the same conserved momentum ($\Vec{k}_{{\rm F} \parallel}$).
For example,
in the case of $K = 2$ and $M = 1$, the boundary condition is satisfied only when the vector $\Vec{u}_{2,(\Vec{k}_{{\rm F} \parallel},k_{z})}^{{\rm N}}$
is parallel to the vector $\Vec{u}_{1,(\Vec{k}_{{\rm F} \parallel},k_{z})}^{{\rm N}}$ expressed as
\begin{align}
\Vec{u}_{2,(\Vec{k}_{{\rm F} \parallel},k_{z})}^{{\rm N}} &= e^{i \Phi_{12}} \Vec{u}_{1,(\Vec{k}_{{\rm F} \parallel},k_{z})}^{{\rm N}}. \label{eq:u12}
\end{align}
Here $\Phi_{12}$ is the overall phase difference between these two vectors.
Thus, we obtain
\begin{align}
c_{2} &= - e^{- i \Phi_{12}} c_{1} \label{eq:k2m1}.
\end{align}
In a superconducting state, we use the quasiclassically-approximated wavefunctions.
In the quasiclassical treatment, the eigenvector $\Vec{\phi}(\Vec{k}_{{\rm F} \parallel}, \Vec{r})$ in the BdG equations is approximated as
\begin{align}
\Vec{\phi}(\Vec{k}_{{\rm F} \parallel},\Vec{r}) \sim \sum_{i}^{K} e^{i k_{z}^{i} z}
\left(\begin{array}{c}
\tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{i})}^{M} \Vec{f}^{k_{z}^{i}}(\Vec{k}_{{\rm F} \parallel},z)
\\
\tilde{U}_{(-\Vec{k}_{{\rm F} \parallel},-k_{z}^{i})}^{M \ast} \Vec{g}^{k_{z}^{i}}(\Vec{k}_{{\rm F} \parallel},z)
\end{array}\right),
\end{align}
with the boundary conditions
\begin{align}
\sum_{i}^{K} \sum_{l}^{M} c_{i}^{l}(\Vec{k}_{{\rm F} \parallel}) f_{l}^{k_{{\rm F}z}^{i}}(\Vec{k}_{{\rm F} \parallel},z=0) &= 0, \label{eq:boundf} \\
\sum_{i}^{K} \sum_{l}^{M} c_{i}^{l \ast}(-\Vec{k}_{{\rm F} \parallel}) g_{l}^{k_{{\rm F}z}^{i}}(\Vec{k}_{{\rm F} \parallel},z=0) &= 0.\label{eq:boundg}
\end{align}
Here, we define the $N \times M$ matrix given by
\begin{align}
\tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{i})}^{M} = (\Vec{u}_{1,(\Vec{k}_{{\rm F} \parallel},k_{z}^{i})}^{{\rm N}}, \cdots, \Vec{u}_{M,(\Vec{k}_{{\rm F} \parallel},k_{z}^{i})}^{{\rm N}}).
\end{align}
In order to find the coefficients $c_{i}^{l}(\Vec{k}_{{\rm F} \parallel})$, we have to solve the boundary condition (\ref{eq:nbound}) in normal states.
For example, in the case of $K = 2$, the boundary condition in Eq.~(\ref{eq:nbound}) is expressed as
\begin{align}
\tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}^{M} \Vec{c}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}^{M} +
\tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}^{M} \Vec{c}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}^{M}
&= 0, \label{eq:k2}
\end{align}
with $\Vec{c}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{i})}^{M T} = (c_{i}^{1}(\Vec{k}_{{\rm F} \parallel}),\cdots,c_{i}^{M}(\Vec{k}_{{\rm F} \parallel}))$.
We show the boundary conditions of two superconducting systems with $K = 2$, which can be easily obtained, as examples.
In the case of $N = M$, the solution of the boundary condition (\ref{eq:k2}) is
\begin{align}
\Vec{c}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}^{M} &=
-\tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}^{M \dagger} \tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}^{M} \Vec{c}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}^{M}, \label{eq:bbk2}
\end{align}
since the relation $\tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}^{M} \tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}^{M \dagger} = 1_{N \times N}$
is satisfied.
Note that Eq.~(\ref{eq:bbk2}) is not the solution in Eq.~(\ref{eq:k2}) if $N \neq M$.
By substituting Eq.~(\ref{eq:bbk2}) into Eqs.~(\ref{eq:boundf}) and (\ref{eq:boundg}), we obtain
\begin{align}
\Vec{f}^{k_{z}^{2}}(z=0) &= - \tilde{V}^{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})} \Vec{f}^{k_{z}^{1}}(z=0), \label{eq:fnm} \\
\Vec{g}^{k_{z}^{2}}(z=0) &= -\tilde{V}^{(-\Vec{k}_{{\rm F} \parallel},-k_{z}^{2}) \ast}_{(-\Vec{k}_{{\rm F} \parallel},-k_{z}^{1})} \Vec{g}^{k_{z}^{1}}(z=0), \label{eq:gnm}
\end{align}
where the ``transfer matrix'' $\tilde{V}^{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}$ is determined as
\begin{align}
\tilde{V}^{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}
&\equiv
\tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}^{M \dagger} \tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}^{M}. \label{eq:transfer}
\end{align}
Next, we consider the case of $M = 1$ and $K = 2$.
The boundary condition (\ref{eq:k2m1}) becomes
\begin{align}
f^{k_{z}^{2}}(\Vec{k}_{{\rm F} \parallel},z=0) &= - e^{- i \Phi_{12}} f^{k_{z}^{1}}(\Vec{k}_{{\rm F} \parallel},z=0), \label{eq:fgbf} \\
g^{k_{z}^{2}}(\Vec{k}_{{\rm F} \parallel},z=0) &= - e^{i \Phi_{12}} g^{k_{z}^{1}}(\Vec{k}_{{\rm F} \parallel},z=0), \label{eq:fgb}
\end{align}
which is equivalent to the boundary condition in the single band case when $\Phi_{12} = 0$.
We should note again that the above boundary condition can be only used when the normal-state eigenvectors with different momenta $k_{z}^{1}$ and $k_{z}^{2}$ are parallel
to each other shown in Eq.~(\ref{eq:u12}), since we have to find the correct boundary condition for $f^{k_{z}^{2}}(\Vec{k}_{{\rm F} \parallel},z=0)$ which satisfies Eq.~(\ref{eq:boundf}).
The characteristic momentum $k_{z}^{i}$ is obtained by solving a normal-state eigenvalue equation (\ref{eq:boundnh}) with the boundary condition (\ref{eq:nbound}) at the Fermi energy.
This momentum $k_{z}^{i}$ is usually a real number since wavefunctions with a Fermi wave number can usually satisfy the boundary condition.
In this case, we solve the Andreev equations (\ref{eq:and}) with the real-number momentum $(\Vec{k}_{{\rm F} \parallel},k_{z}^{i})$.
However, the momentum $k_{z}^{i}$ is not the Fermi wave number, when there are bound states in normal states.
The normal-state wavefunction localized at a boundary is described by a complex momentum.
We will show the system which needs the complex wave number to describe the bound states, as discussed in Sec.~\ref{sec:single}.
We discuss the existence condition of solutions in Eq.~(\ref{eq:nbound}).
With the use of the matrix representation, the equation (\ref{eq:nbound}) is rewritten as
\begin{align}
{\cal U} \Vec{C} &= 0,
\end{align}
where the $N \times M K$ matrix ${\cal U}$ and $M K$-dimension vector $\Vec{C}$ are defined as
\begin{align}
{\cal U} &\equiv (\tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}^{M},\cdots, \tilde{U}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{K})}^{M} ), \\
\Vec{C} &\equiv \left(\begin{array}{c}
\Vec{c}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}^{M} \\
\vdots \\
\Vec{c}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{K})}^{M}
\end{array}\right).
\end{align}
The above equations are linear homogeneous equations with $M K$ unknowns.
These equations can have the solutions when $M K > N$.
Finally, we note that the general boundary condition for the conventional quasiclassical equations has been discussed by several groups\cite{Shelankov:2000jq,Eschrig:2009bz}.
The incoming and outgoing wavefunctions are connected by the scattering matrix $\hat{S}$ expressed as
\begin{align}
\Vec{\phi}(\Vec{k}_{\rm out}) &= \hat{S}_{\Vec{k}_{\rm out} \Vec{k}_{\rm in}} \Vec{\phi}(\Vec{k}_{\rm in}),
\end{align}
where $\Vec{k}_{\rm in(out)}$ is the wave number of the incoming (outgoing) quasiparticles.
Here, $\hat{S}$ is the scattering matrix defined at the Fermi energy in normal states\cite{Eschrig:2009bz}.
The above relation can be used to develop the general boundary condition in the multi-band superconductors.
The development of the general boundary condition for the multi-band quasiclassical equations is a future issue.
\section{Quasiclassical treatment II: Green's function approach}
\label{sec:green}
In this section, we derive the equation of motion of the quasiclassical Green's function so-called Eilenberger equations in a multi-band system.
The size of the matrices of the Green's function is reduced by the low-energy projection.
\subsection{Wigner representation}
\label{sec:wigner}
The Wigner representation is usually introduced in terms of the derivation of the quantum Boltzmann equations.
The transport-like equations of motions of the quasiclassical Green's functions in the single band system
are systematically derived with the use of the Wigner representations.
We introduce the Wigner representation defined by
\begin{align}
\check{A}(\Vec{R},\Vec{k}) &\equiv \int d\Vec{\bar{r}} e^{-i \Vec{k} \cdot \Vec{\bar{r}}}
\check{A}(\Vec{r}_{1},\Vec{r}_{2}) \Bigl|_{\Vec{r}_{1}=\Vec{R}+\bar{\Vec{r}}/2,\Vec{r}_{2}=\Vec{R}-\bar{\Vec{r}}/2}.
\end{align}
Here, $\Vec{R} = (\Vec{r}_{1}+\Vec{r}_{2})/2$ and $\Vec{\bar{r}} = \Vec{r}_{1} - \Vec{r}_{2}$ are the center-of-mass coordinate and the relative
coordinate, respectively.
The Gor'kov equations in the Wigner representation are expressed by
\begin{align}
& \left(
i \omega_{n} - \check{H}^{\rm N 0}(\Vec{R}) - \check{H}^{\rm N 1}(\Vec{k}) - \check{\Delta}(\Vec{R},\Vec{k}) - \check{\Sigma}(\Vec{R},\Vec{k},i \omega_{n})
\right) \nonumber \\
&\star \check{G}(\Vec{R},\Vec{k},i \omega_{n}) = \check{1}. \label{eq:wig}
\end{align}
Here, we introduce the $\star$-product (Moyal product) determined by
\begin{align}
\check{A}(\Vec{R},\Vec{k}) \star \check{B}(\Vec{R},\Vec{k}) &\equiv
\exp \left[ \frac{i}{2 } (\Vec{\nabla}_{\Vec{k}'} \cdot \Vec{\nabla}_{\Vec{R}}-\Vec{\nabla}_{\Vec{k}} \cdot \Vec{\nabla}_{\Vec{R}'})\right] \nonumber \\
&\times \check{A}(\Vec{R},\Vec{k}) \check{B}(\Vec{R}',\Vec{k}') \Big|_{\Vec{k}' = \Vec{k}, \Vec{R}' = \Vec{R}}.
\end{align}
We note that there is another Gor'kov equation called the ``right-hand Gor'kov'' equation.
In terms of the Wigner representation, the right-hand equation is expressed as
\begin{align}
& \check{G}(\Vec{R},\Vec{k},i \omega_{n}) \star \nonumber \\
& \left(
i \omega_{n} - \check{H}^{\rm N 0}(\Vec{R}) - \check{H}^{\rm N 1}(\Vec{k}) - \check{\Delta}(\Vec{R},\Vec{k}) - \check{\Sigma}(\Vec{R},\Vec{k},i \omega_{n})
\right)
= \check{1}.
\end{align}
The local density of states with the Wigner representation is expressed as
\begin{align}
N(\Vec{r},E) &= \frac{-1}{\pi} {\rm Im} \: \left[ \lim_{\eta \rightarrow +0} \sum_{\Vec{k}} {\rm Tr} \: \hat{G}(\Vec{r},\Vec{k},i \omega_{n} \rightarrow E + i \eta) \right]. \label{eq:wigldos}
\end{align}
Let us derive the quasiclassical equations from Eq.~(\ref{eq:wig}) as follows.
In superconductors, the characteristic length of the center-of-mass coordinates is the coherence length $\xi$, which
is much longer than that of the relative coordinates characterized by $1/k_{\rm F}$.
Assuming that the characteristic coherence length is long, the Moyal product in the first order of $\Vec{\nabla}_{\Vec{R}}$ is given as
\begin{align}
& \check{A}(\Vec{R},\Vec{k}) \star \check{B}(\Vec{R},\Vec{k}) \sim
\check{A}(\Vec{R},\Vec{k}) \check{B}(\Vec{R},\Vec{k}) \nonumber \\
&+ \frac{i}{2}
\left(
\Vec{\nabla}_{\Vec{R}} \check{A}(\Vec{R},\Vec{k}) \cdot \Vec{\nabla}_{\Vec{k}} \check{B}(\Vec{R},\Vec{k})
- \Vec{\nabla}_{\Vec{k}} \check{A}(\Vec{R},\Vec{k}) \cdot \Vec{\nabla}_{\Vec{R}} \check{B}(\Vec{R},\Vec{k})
\right).
\end{align}
Then, the Gor'kov equations are expressed as (see, Appendix \ref{sec:expansion})
\begin{widetext}
\begin{align}
& \left(
i \omega_{n} -
\check{H}^{\rm N 0}(\Vec{R})
- \check{H}^{\rm N 1}(\Vec{k})
- \check{\Delta}(\Vec{R},\Vec{k}) - \check{\Sigma}(\Vec{R},\Vec{k},i \omega_{n})
\right)
\check{G}(\Vec{R},\Vec{k},i \omega_{n}) \nonumber \\
& - \frac{i}{2 }
\Vec{\nabla}_{\Vec{R}} \left[ \check{H}^{\rm N 0}(\Vec{R}) + \check{\Delta}(\Vec{R},\Vec{k}) + \check{\Sigma}(\Vec{R},\Vec{k},i \omega_{n})\right] \cdot \Vec{\nabla}_{\Vec{k}} \check{G}(\Vec{R},\Vec{k},i \omega_{n})
+ \frac{i}{2 }
\Vec{\nabla}_{\Vec{k}} \check{H}^{\rm N 1}(\Vec{k}) \cdot \Vec{\nabla}_{\Vec{R}} \check{G}(\Vec{R},\Vec{k},i \omega_{n})
= \check{1}. \label{eq:wig1}
\end{align}
\end{widetext}
The above equations are simultaneous differential equations with $\Vec{k}$ and $\Vec{R}$.
In a single-band system, the above equations becomes the differential equations with respect to $\Vec{R}$ with a parameter $\Vec{k}_{\rm F}$ (i.e. the quasiclassical Eilenberger equations),
since we can eliminate the $\Vec{k}$-mixing term $\Vec{\nabla}_{\Vec{k}} \check{G}(\Vec{R},\Vec{k},i \omega_{n})$, with the use of the contour integration with respect to the energy $\xi_{\Vec{k}}$ and the relation $\Vec{\nabla}_{\Vec{k}} \propto \partial/\partial \xi_{\Vec{k}}$.
Thus, in order to derive the multi-band quasiclassical Eilenberger equations,
the $\Vec{k}$-mixing term $\Vec{\nabla}_{\Vec{k}} \check{G}(\Vec{R},\Vec{k},i \omega_{n})$ has to be eliminated.
However, we can not simply integrate the above equations with respect to $\xi_{\Vec{k}}$,
since $\xi_{\Vec{k}}$ is the energy obtained by diagonalizing the Hamiltonian $\hat{H}^{\rm N 1}(\Vec{k})$
and the Gor'kov equations are determined in the orbital basis.
One has to characterize the equation by the Fermi wave momentum in the low-energy region.
\subsection{Projection to the effective low-energy model}
We introduce the projection matrix to develop the multi-band quasiclassical theory.
The projection matrix eliminates the degree of freedom about the high energy region.
We define the $2 N \times 2 N$ projection matrix $\check{P}_{\Vec{k}}$ as
\begin{align}
\check{P}_{\Vec{k}} &\equiv U_{\Vec{k}}^{M} U_{\Vec{k}}^{M \dagger}. \label{eq:projection}
\end{align}
Here, the $2 N \times 2 M$ matrix $U_{\Vec{k}}^{M}$ is defined by
\begin{align}
U_{\Vec{k}}^{M}&\equiv
\left(\begin{array}{cc}
\tilde{U}_{\Vec{k}}^{M} & 0 \\
0 & \tilde{U}_{-\Vec{k}}^{M \ast}
\end{array}\right), \label{eq:matu}
\end{align}
where the $N \times M$ matrix is defined by Eq.~(\ref{eq:ukm}).
The projection operator satisfies the relation
\begin{align}
\check{P}_{\Vec{k}}\check{P}_{\Vec{k}} &= \check{P}_{\Vec{k}},
\end{align}
since the matrix $U_{\Vec{k}}^{M}$ always satisfies the relation
\begin{align}
U_{\Vec{k}}^{M \dagger} U_{\Vec{k}}^{M} &= \bar{1}_{2 M \times 2 M}. \label{eq:uu}
\end{align}
In order to show the physical meaning of the projection matrix, we operate $\check{P}_{\Vec{k}}$ on the homogeneous $N$-band Green's function in normal states determined by
\begin{align}
\check{G}(\Vec{k}, i \omega_{n}) &= (i \omega_{n}\check{1} - \check{H}^{\rm N 1}(\Vec{k}))^{-1}.
\end{align}
The component of $\check{G}(\Vec{k}, i \omega_{n})$ is
expressed as
\begin{align}
G_{\alpha \beta}(\Vec{k}, i \omega_{n}) &=
\sum_{\gamma=1}^{N}
\frac{\tilde{U}_{\alpha \gamma}(\Vec{k}) \tilde{U}_{\beta \gamma}^{\ast}(\Vec{k})}
{i \omega_{n} - \epsilon_{\gamma}(\Vec{k})}, &
(1 \leq \alpha,\beta \leq N)
\end{align}
Here, the $N \times N$ unitary matrix $\tilde{U}(\Vec{k})$ diagonalizes $\hat{H}^{\rm N 1}(\Vec{k})$
and $\epsilon_{\gamma}(\Vec{k})$ is the $\gamma$-th eigenvalue of $\hat{H}^{\rm N 1}(\Vec{k})$.
Operating $\check{P}_{\Vec{k}}$ on $\check{G}(\Vec{k}, i \omega_{n})$, we obtain
\begin{align}
\left[ \check{P}_{\Vec{k}} \check{G}(\Vec{k}, i \omega_{n})\right]_{\alpha \beta} &=
\sum_{\gamma=1}^{M}
\frac{\tilde{U}_{\Vec{k}\alpha \gamma}^{M}\tilde{U}_{\Vec{k} \beta \gamma}^{M \ast}}
{i \omega_{n} - \xi_{\Vec{k}}},
\end{align}
with $1 \leq \alpha,\beta \leq N$.
Here, we use the assumption in (\ref{eq:eigen}).
The above equation means that the projection operator $\check{P}_{\Vec{k}}$ eliminates
the information of large eigenvalues from $\check{G}(\Vec{k}, i \omega_{n})$.
The difference $\delta \check{G}(\Vec{k}, i \omega_{n}) \equiv \check{G}(\Vec{k}, i \omega_{n})- \check{P}_{\Vec{k}} \check{G}(\Vec{k}, i \omega_{n})$
becomes
\begin{align}
\left[ \delta \check{G}(\Vec{k}, i \omega_{n}) \right]_{\alpha \beta}&=
\sum_{\gamma, \epsilon_{\gamma} \neq 0}^{N-M}
\frac{\tilde{U}_{\alpha \gamma}(\Vec{k}) \tilde{U}_{\beta \gamma}^{\ast}(\Vec{k})}
{i \omega_{n} - \epsilon_{\gamma}(\Vec{k})} ,
\end{align}
with $1 \leq \alpha,\beta \leq N$.
If the eigenvalues are located far from the Fermi energy $(|\omega_{n}| \ll \epsilon_{\gamma}(\Vec{k}_{\rm F}))$,
$\delta \check{G}(\Vec{k}_{\rm F}, i \omega_{n})$ becomes negligible small so that
we can obtain the relation expressed as
\begin{align}
\check{P}_{\Vec{k}} \check{G}(\Vec{k}, i \omega_{n}) &\sim \check{G}(\Vec{k}, i \omega_{n}), \label{eq:pg}
\end{align}
which is appropriate in the low-energy region.
We introduce the $2 M \times 2 M$ reduced matrix $\bar{A}$
expressed as
\begin{align}
\bar{A}(\Vec{k}) &\equiv U_{\Vec{k}}^{M \dagger} \check{A}(\Vec{k}) U_{\Vec{k}}^{M},
\end{align}
where $\check{A}$ is the $2 N \times 2 N$ matrix used in the Gor'kov equation (\ref{eq:wig}).
With the use of Eqs.~(\ref{eq:uu}) and (\ref{eq:pg}),
$\check{A}$ can be expressed by
\begin{align}
\check{A}(\Vec{k}) &\sim U_{\Vec{k}}^{M} \bar{A}(\Vec{k}) U_{\Vec{k}}^{M \dagger}, \label{eq:areduced}
\end{align}
in the low-energy region ($\Vec{k} \sim \Vec{k}_{\rm F}$).
\subsection{Multi-band quasiclassical Green's function}
Let us construct the $2 M \times 2 M$
quasiclassical multi-band Eilenberger equations in the projected space.
By multiplying the the both sides in Eq.~(\ref{eq:wig1}) by the matrices $U_{\Vec{k}}^{M \dagger}$ and $U_{\Vec{k}}^{M}$, subtracting the right-hand Gor'kov equation, and integrating over $\xi_{\Vec{k}}$ (in detail, see Appendix \ref{sec:quasieilen}),
we obtain $2 M \times 2 M$ quasiclassical multi-band Eilenberger equation
expressed as
\begin{widetext}
\begin{align}
i \Vec{v}_{\rm F}(\Vec{k}_{\rm F}) \cdot \Vec{\nabla}_{\Vec{R}}
\bar{g}_{\Vec{R}}(\Vec{k}_{\rm F},z) +
\left[
z \bar{\sigma}_{z}-
\bar{V}_{0 \Vec{R}}(\Vec{k}_{\rm F}) \bar{\sigma}_{z}
-
\bar{\Delta}_{\Vec{R}}(\Vec{k}_{\rm F})\bar{\sigma}_{z} -
\bar{\Sigma}_{\Vec{R}} (\Vec{k}_{\rm F},z) \bar{\sigma}_{z}
,
\bar{g}_{\Vec{R}}(\Vec{k}_{\rm F},z)
\right]_{-}
&= 0, \label{eq:multi}
\end{align}
\end{widetext}
where we introduce the $2 M \times 2 M$ Green's function, non-local potentials, order parameters, and self-energies determined by
\begin{align}
\bar{G}_{\Vec{R}} (\Vec{k},z)
&\equiv U_{\Vec{k}}^{M \dagger} \check{G} (\Vec{R},\Vec{k},z) U_{\Vec{k}}^{M} \label{eq:greenu} \\
\bar{V}_{0 \Vec{R}}(\Vec{k}) &\equiv U_{\Vec{k}}^{M \dagger} \check{H}^{\rm N 0}(\Vec{R})U_{\Vec{k}}^{M}, \label{eq:vu}\\
\bar{\Delta}_{\Vec{R}}(\Vec{k}) &\equiv U_{\Vec{k}}^{M \dagger}
\check{\Delta}(\Vec{R},\Vec{k})
U_{\Vec{k}}^{M}, \label{eq:deltau} \\
\bar{\Sigma}_{\Vec{R}}(\Vec{k}, z) &\equiv U_{\Vec{k}}^{M \dagger}
\check{\Sigma}(\Vec{R},\Vec{k}, z)
U_{\Vec{k}}^{M}. \label{eq:sigmau}
\end{align}
Here,
$\bar{\sigma}_{z}$ denotes the Pauli matrix in the Nambu-Gor'kov space and
we define the $2 M \times 2 M$ quasiclassical Green's function expressed as
\begin{align}
\bar{g}_{\Vec{R}}(\hat{\Vec{k}}_{\rm F},z) &\equiv \oint d\xi_{\Vec{k}}\bar{\sigma}_{z} \bar{G} (\Vec{R},\Vec{k},z), \label{eq:quasi}
\end{align}
where $\oint$ means taking the contributions from poles close to the Fermi surface.
The matrix structure of the above equation is equivalent to that of the single-band Eilenberger equation.
When the eigenvalue is not degenerated ($M = 1$), the $2 \times 2$ equation (\ref{eq:multi}) can be regarded as that in a spin-singlet single band superconductor\cite{Serene,Choi,NagaiJPSJ}.
Therefore, we call the $2 M \times 2 M$ matrix representation the ``single-band description''.
The band index $\alpha$ is useful to compare with the present multi-band theory by
replacing $\bar{g}_{\Vec{R}}^{\alpha}(\hat{\Vec{k}}_{\rm F},z)$ as $\bar{g}_{\Vec{R}}(\hat{\Vec{k}}_{\rm F},z)$.
The multi-band Eilenberger equations (\ref{eq:multi}) are similar to the decoupled Eilenberger equations Eq.~(\ref{eq:mgb2type}).
We should note that our multi-band theory includes the orbital characters, since all matrices are determined in the projected space.
For example, the self-energy with the $T$-matrix approximation depends on the momentum as discussed in Eq.~(\ref{eq:sigma2}).
We should note that the normalization condition is equivalent to that in the single-band system expressed as (in detail, see Appendix \ref{sec:normalization})
\begin{align}
\bar{g} \bar{g} &= - \pi^{2} \bar{1}. \label{eq:normalization}
\end{align}
\subsection{Relations in the projected space}
Let us discuss the relations satisfied in the projected space.
$M \times M$ order parameter is defined in Eq.~(\ref{eq:gapeff}).
If the original order parameter have the relation $\hat{\Delta}(\Vec{R},\Vec{k})^{\dagger} \hat{\Delta}(\Vec{R},\Vec{k}) = \Delta_{0}(\Vec{R},\Vec{k}) 1_{N \times N}$,
the projected order parameter have the similar relation:
\begin{align}
\Delta_{\rm eff}(\Vec{R},\Vec{k})^{\dagger} \Delta_{\rm eff}(\Vec{R},\Vec{k}) &\sim \Delta_{0}(\Vec{R},\Vec{k}) 1_{M \times M},
\end{align}
because of $\tilde{U}_{\Vec{k}}^{M } \tilde{U}_{\Vec{k}}^{M \dagger} \hat{\Delta}(\Vec{R},\Vec{k}) \sim \hat{\Delta}(\Vec{R},\Vec{k}) $ near the Fermi energy.
Here, $\Delta_{\rm eff}(\Vec{R},\Vec{k})$ is determined in Eq.~(\ref{eq:gapeff}).
This indicates that the unitarity of the order parameter is conserved in the projected space.
\subsection{Physical quantities and Gap equations}
Let us express physical quantities with the use of the multi-band quasiclassical Green's function.
By substituting Eq.~(\ref{eq:areduced}) into Eq.~(\ref{eq:wigldos}),
the local density of states is expressed as
\begin{align}
N(\Vec{r},E) &= \frac{-1}{\pi} {\rm Im} \: \left[ \lim_{\eta \rightarrow +0} \sum_{\Vec{k}} {\rm Tr} \:
\bar{G}^{11}(\Vec{r},\Vec{k},i \omega_{n} \rightarrow E + i \eta) \right] , \\
&= \frac{-1}{\pi} {\rm Im} \: \left[ \lim_{\eta \rightarrow +0} \langle {\rm Tr} \:
\bar{g}^{11}(\Vec{r},\Vec{k},i \omega_{n} \rightarrow E + i \eta) \rangle_{\hat{\Vec{k}}_{\rm F}} \right],
\end{align}
where $\bar{G}^{11}$ and $\bar{g}^{11}$ denote $(1,1)$-element in the particle-hole space,
the bracket denotes the Fermi-surface average $\langle \cdots \rangle_{\hat{\Vec{k}}_{\rm F}} = \int \cdots d S_{\rm F}(\hat{\Vec{k}}_{\rm F})|\Vec{v}_{\rm F}(\hat{\Vec{k}}_{\rm F})|^{-1}/\int d S_{\rm F}(\hat{\Vec{k}}_{\rm F})|\Vec{v}_{\rm F}(\hat{\Vec{k}}_{\rm F})|^{-1}$.
Other physical quantities can be expressed by the multi-band quasiclassical Green's function in the same manner.
Finally, we complete the multi-band quasiclassical theory by giving the self-consistent equations for the order parameters.
The gap equations in the Wigner representation is given as
\begin{align}
\hat{\Delta}_{ \alpha \beta} (\Vec{R},\Vec{k}) &= - T
\sum_{\Vec{k}', \gamma \gamma'}^{N} \sum_{n} V_{\beta \alpha ; \gamma \gamma'}(\Vec{k},\Vec{k}')
\hat{F}_{\gamma \gamma'}(\Vec{R},\Vec{k}', i \omega_{n})
\end{align}
By using Eq.~(\ref{eq:deltau}),
the $M \times M$ order parameter matrix is expressed by
\begin{align}
&\Delta_{{\rm eff} l_{1} l_{2} \Vec{R}}(\Vec{k}_{\rm F}) = \nonumber \\
&
- T \sum_{n} \sum_{l_{3} l_{4}}^{M}
\langle
\bar{V}^{l_{1} l_{2}}_{l_{3} l_{4}}(\hat{\Vec{k}}_{\rm F},\hat{\Vec{k}}_{\rm F}')
\bar{g}^{12}_{l_{3} l_{4} \Vec{R}}( \hat{\Vec{k}}_{\rm F}',i \omega_{n})
\rangle_{\hat{\Vec{k}}_{\rm F}'},
\end{align}
where $\bar{g}^{12}$ denotes $(1,2)$-element in the particle-hole space
and
$\bar{V}^{l_{1} l_{2}}_{l_{3} l_{4}}(\hat{\Vec{k}}_{\rm F},\hat{\Vec{k}}_{\rm F}')$ is the effective interaction expressed as
\begin{align}
&\bar{V}^{l_{1} l_{2}}_{l_{3} l_{4}}(\hat{\Vec{k}}_{\rm F},\hat{\Vec{k}}_{\rm F}') \equiv
\sum_{\alpha \beta \gamma \gamma'}
V_{\beta \alpha ; \gamma \gamma'}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}') \nonumber \\
&\times
\tilde{U}_{\Vec{k}_{\rm F} \alpha l_{1}}^{M \ast} \tilde{U}_{-\Vec{k}_{\rm F} \beta l_{2}}^{M \ast}
\tilde{U}_{\Vec{k}_{\rm F}' \gamma l_{3}}^{M} \tilde{U}_{-\Vec{k}_{\rm F}' \gamma' l_{4}}^{M \ast}.
\end{align}
We can simplify the above gap equations if the pairing interaction has a separable form expressed as\cite{Allen}
\begin{align}
V_{\beta \alpha ; \gamma \gamma'}(\Vec{k},\Vec{k}') &=
V_{\alpha \beta}(\Vec{k}) V_{\gamma \gamma'}(\Vec{k}').
\end{align}
Here, we use the relation $V_{\beta \alpha ; \gamma \gamma'}(\Vec{k},\Vec{k}') =
V_{\gamma' \gamma ; \alpha \beta}(\Vec{k}',\Vec{k})$\cite{SigristUeda}.
The gap equations in this case are given as
\begin{align}
\Delta_{\Vec{R}} &=
- T \sum_{n} \sum_{l_{3} l_{4}}^{M}
\langle
\tilde{V}_{l_{3} l_{4}}^{\ast}(\hat{\Vec{k}}_{\rm F}')
\bar{g}^{12}_{l_{3} l_{4} \Vec{R}}( \hat{\Vec{k}}_{\rm F}',i \omega_{n})
\rangle_{\hat{\Vec{k}}_{\rm F}'},
\end{align}
with
\begin{align}
\Delta_{{\rm eff}\Vec{R}}(\Vec{k}_{\rm F}) &\equiv \Delta_{\Vec{R}} \tilde{V}(\hat{\Vec{k}}_{\rm F}), \\
\tilde{V}(\hat{\Vec{k}}_{\rm F}) &\equiv \tilde{U}_{\Vec{k}_{\rm F}}^{M \dagger} V(\Vec{k}_{\rm F})
\tilde{U}_{-\Vec{k}_{\rm F}}^{M \ast},
\end{align}
where we assume that $V_{\gamma \gamma'}(\Vec{k}')^{\ast} = V_{\gamma \gamma'}(\Vec{k}')$.
\subsection{Perturbative approach in the quasiclassical theory: Zeeman and spin-orbit couplings}
In this section, we discuss the method to treat the Zeeman and spin-orbit couplings.
In the previous studies\cite{Alexander:1985jh,Vorontsov:2010it,Ichioka:2007dp,Hayashi,Rieck},
they used the $4 \times 4$ matrix quasiclassical Eilenberger equations in spin and Nambu spaces expressed as
\begin{align}
&i \Vec{v}_{\rm F} \cdot \Vec{\nabla}_{\Vec{R}} \check{g}_{\Vec{R}}(\Vec{k}_{\rm F},z) \nonumber \\
&+\left[
z \sigma_{z} -\check{\Delta}\sigma_{z} - \check{H}_{1}(\Vec{k}_{\rm F}) \sigma_{z},\check{g}_{\Vec{R}}(\Vec{k}_{\rm F},z)
\right]_{-} = 0.
\end{align}
Here, $\sigma_{i}$ denotes the Pauli matrix in the Nambu space and $\check{H}_{1}(\Vec{k}_{\rm F})$
includes the spin-orbit and/or Zeeman coupling terms.
These quasiclassical equations can describe a vortex state in a Fulde-Ferrell-Larkin-Ovchinnikov superconductor\cite{Ichioka:2007dp}.
In terms of our theory, the above equations are obtained by assuming that the number of degenerated Fermi surfaces is two ($M = 2$).
In general, however, the Zeeman and spin-orbit interactions split the degenerated bands (i.~e.~ $M = 2 \rightarrow M = 1$ ).
Thus, it is found that the previous studies assume that the Zeeman and spin-orbit interactions are weak.
With the use of the perturbative approach in the multi-band quasiclassical theory, we can derive the above equations as follows.
Let us divide $\hat{H}^{\rm N1}(\Vec{k})$ in Eq.~(\ref{eq:eigen}) into the two terms as
\begin{align}
\hat{H}^{\rm N1}(\Vec{k}) &= \hat{H}^{\rm N1}_{0}(\Vec{k}) + \hat{H}_{1}(\Vec{k}).
\end{align}
Thus, in order to construct the projection operator $\check{P}_{\Vec{k}}$, we can use the eigenvectors obtained by
\begin{align}
\hat{H}^{\rm N1}_{0}(\Vec{k}) u_{\Vec{k}}^{i} &= \xi_{\Vec{k}} u_{\Vec{k}}^{i}.
\end{align}
This approach is appropriate for the case that the inter-band pairing between the different Fermi surfaces is important.
The perturbative Zeeman field enables us to treat the Pauli paramagnetic depairing in the quasiclassical framework.
\subsection{Riccati-type equations}
It is known that it is difficult to numerically solve the quasiclassical Eilenberger equations\cite{NagaiMeso}, since
the equations have a divergent solution as a particular solution.
A careful computational treatment is required for integrating the Eilenberger equations with the use of the so-called explosion method\cite{Thuneberg:1982fp}.
To avoid this difficulty, the Riccati-type equations, which are obtained by a special parametrization form of the quasiclassical Green's function, are used\cite{YKato,Nagato,Higashitani,NagatoLow,SchopohlMaki,Schopohl}.
In addition, to solve the Riccati equation stably, we have proposed the efficient numerical method for obtaining unique solutions in the single-band Eilenberger
framework\cite{NagaiMeso}.
We show that it is easy to expand this method into the multi-band systems.
For simplicity, we neglect the self-energy $\bar{\Sigma} = 0$.
We use a special parametrization form of the quasiclassical Green's function to solve Eq.~(\ref{eq:multi}).
The solution $\bar{g}$ of Eq.~(\ref{eq:multi}) can be written as,
\begin{align}
\bar{g} &= - i \pi \bar{N} \left(\begin{array}{cc}
(\tilde{1} - \tilde{a} \tilde{b}) & 2 i \tilde{a} \\
-2 i \tilde{b} & -(\tilde{1}- \tilde{b} \tilde{a})
\end{array}\right), \\
\bar{N} &= \left(\begin{array}{cc}
(\tilde{1} + \tilde{a} \tilde{b})^{-1} & 0 \\
0 & (\tilde{1}+ \tilde{b} \tilde{a})^{-1}
\end{array}\right),
\end{align}
where $M \times M$ matrices $\tilde{a}$ and $\tilde{b}$ are the solutions of the
following matrix-type Riccati differential equations:
\begin{align}
\Vec{v}_{\rm F} \cdot \Vec{\nabla} \tilde{a} &= - 2 \omega_{n} \tilde{a} - \tilde{a} \tilde{\Delta}^{\dagger} \tilde{a} + \tilde{\Delta}. \\
\Vec{v}_{\rm F} \cdot \Vec{\nabla} \tilde{b} &= 2 \omega_{n} \tilde{b} + \tilde{b} \tilde{\Delta} \tilde{b} - \tilde{\Delta}^{\dagger}.
\end{align}
Since the above equations contain $\Vec{\nabla}$ only through $\Vec{v}_{\rm F} \cdot \Vec{\nabla}$,
these can be reduced to a one-dimensional problem on a straight line in the direction of the
Fermi velocity $\Vec{v}_{\rm F}$:
\begin{align}
v_{\rm F} \frac{\partial \tilde{a} }{\partial s} &= - 2 \omega_{n} \tilde{a} - \tilde{a} \tilde{\Delta}^{\dagger} \tilde{a} + \tilde{\Delta}. \\
v_{\rm F} \frac{\partial \tilde{b} }{\partial s} &= 2 \omega_{n} \tilde{b} + \tilde{b} \tilde{\Delta} \tilde{b} - \tilde{\Delta}^{\dagger}.
\end{align}
In a bulk system with $\tilde{\Delta}^{\dagger} \tilde{\Delta} \propto \tilde{1}$, the solutions of the Riccati equations are
\begin{align}
\tilde{a}(\omega_{n}) &= \frac{\tilde{\Delta}}{\omega_{n} + \sqrt{\omega_{n}^{2} + \frac{1}{2} {\rm Tr} \: \tilde{\Delta} \tilde{\Delta}^{\dagger}}}, \label{eq:riccatia}\\
\tilde{b}(\omega_{n}) &= \frac{\tilde{\Delta}^{\dagger}}{\omega_{n} + \sqrt{\omega_{n}^{2} + \frac{1}{2} {\rm Tr} \: \tilde{\Delta} \tilde{\Delta}^{\dagger}}}. \label{eq:riccatib}
\end{align}
According to the previous paper\cite{NagaiMeso}, putting $\tilde{a}(s_{a}) = 0$ and $\tilde{b}(s_{b}) =0$ as initial values,
one can obtain physical solutions by integrating the Riccati equations.
Here, $s_{a}$ and $s_{b}$ are initial spatial points.
\subsection{Boundary condition for the Riccati parameters at a specular surface}
To solve Riccati-type differential equations, one has to consider the boundary condition for Riccati parameters $\tilde{a}$ and $\tilde{b}$.
In this section, we consider a specular surface.
In the case of $M = 1$ or $M = 2$, we can show the transformation of the linearized BdG equations to the matrix
Riccati equations.
Thus, the boundary condition for the Riccati parameters can be determined explicitly.
We note that, in many materials, the number of the degenerated Fermi levels $M$ is not larger than $M = 2$.
If the Fermi level with a certain momentum $\Vec{k}_{\rm F}$ is not degenerated ($M = 1$), the Riccati equations are derived by the relation:
\begin{align}
\tilde{a}(\Vec{r},\Vec{k}_{\rm F}) = i \frac{f^{\Vec{k}_{\rm F}}(\Vec{r})}{g^{\Vec{k}_{\rm F}}(\Vec{r})}.
\end{align}
With the use of the boundary condition for wavefunctions in Eqs.~(\ref{eq:fgbf}) and (\ref{eq:fgb}), the boundary condition at a surface $z = 0$ with $K = 2$ is given by
\begin{align}
\tilde{a}(k_{{\rm F}z}^{2}) &= e^{- 2 i \Phi_{12}} \tilde{a}(k_{{\rm F}z}^{1}).
\end{align}
If the Fermi level is doubly degenerated ($M = 2$), we find the relation expressed as
\begin{align}
\tilde{a}(\Vec{r},\Vec{k}_{\rm F}) &= i U(\Vec{k}_{\rm F},\Vec{r}) V(\Vec{k}_{\rm F},\Vec{r})^{-1}
\end{align}
where the $2 \times 2$ matrices $U(\Vec{k}_{\rm F},\Vec{r})$ and $V(\Vec{k}_{\rm F},\Vec{r})$ are determined by
\begin{align}
U(\Vec{k}_{\rm F},\Vec{r}) &\equiv
\left(\begin{array}{cc}
\Vec{f}^{\Vec{k}_{\rm F}}(\Vec{r}) & - \Vec{g}^{-\Vec{k}_{\rm F}}(\Vec{r})^{\ast}
\end{array}\right), \\
V(\Vec{k}_{\rm F},\Vec{r}) &\equiv
\left(\begin{array}{cc}
\Vec{g}^{\Vec{k}_{\rm F}}(\Vec{r}) & \Vec{f}^{-\Vec{k}_{\rm F}}(\Vec{r})^{\ast}
\end{array}\right).
\end{align}
We obtain the boundary condition at a surface $z = 0$ with $K = 2$ and $N = M = 2$ expressed as
\begin{align}
\tilde{a}(k_{{\rm F}z}^{2}) &= \tilde{V}^{(k_{{\rm F}x},k_{{\rm F}y},k_{{\rm F}z}^{2})}_{(k_{{\rm F}x},k_{{\rm F}y},k_{{\rm F}z}^{1})}
\tilde{a}(k_{{\rm F}z}^{1}) \left[ \tilde{V}^{-(k_{{\rm F}x},k_{{\rm F}y},k_{{\rm F}z}^{2}) \ast}_{-(k_{{\rm F}x},k_{{\rm F}y},k_{{\rm F}z}^{1})} \right]^{-1}. \label{eq:ricboundary}
\end{align}
Note that $\tilde{V}^{(k_{{\rm F}x},k_{{\rm F}y},k_{{\rm F}z}^{2})}_{(k_{{\rm F}x},k_{{\rm F}y},k_{{\rm F}z}^{1})}$ is determined in Eq.~(\ref{eq:transfer}).
\subsection{Arbitrary transformation about normal-state eigenvectors: Appearance condition of the Andreev bound states}\label{sec:arbit}
We discuss an arbitrariness of the $N \times M$ matrix $\tilde{U}_{\Vec{k}_{\rm F}}^{\rm M}$.
With the use of this arbitrariness, one can discuss the appearance condition of the Andreev bound states in multi-band superconductors.
As shown in Eq.~(\ref{eq:ukm}), the $N \times M$ matrix $\tilde{U}_{\Vec{k}_{\rm F}}^{\rm M}$ consists of the zero-energy degenerated eigen vectors about
the normal-state Hamiltonian $\hat{H}^{\rm N 1}(\Vec{k})$.
Because of the degeneracy, it should be noted that the $N \times M$ matrix $\tilde{U}_{\Vec{k}_{\rm F}}^{\rm M}$ has additional degrees of freedom expressed as
\begin{align}
\tilde{U}_{\Vec{k}_{\rm F}}^{\rm M'} &= \tilde{U}_{\Vec{k}_{\rm F}}^{\rm M} \hat{A}_{\Vec{k}_{\rm F}},
\end{align}
with a $M \times M$ arbitrary unitary matrix $\hat{A}_{\Vec{k}_{\rm F}}$.
Although this matrix does not change the $2 N \times 2 M$ projection matrix $\check{P}_{\Vec{k}_{\rm F}}$,
the representation of the effective order parameters $\Delta_{\rm eff}(\Vec{r},\Vec{k}_{\rm F})$ determined in
Eq.~(\ref{eq:gapeff}) depends on the matrix $\hat{A}_{\Vec{k}_{\rm F}}$, expressed as
\begin{align}
\Delta_{\rm eff}(\Vec{r},\Vec{k}_{\rm F}) &= \hat{A}^{\dagger}_{\Vec{k}_{\rm F}} \tilde{U}_{\Vec{k}_{\rm F}}^{M \dagger} \hat{\Delta}(\Vec{r},\Vec{k}_{\rm F}) \tilde{U}_{-\Vec{k}_{\rm F}}^{M \ast} \hat{A}^{\ast}_{-\Vec{k}_{\rm F}}.
\end{align}
It should be noted that this arbitrary transformation does not change any physical quantities, since the matrix $\hat{A}_{\Vec{k}_{\rm F}}$ changes the boundary condition.
With the use of the unitary matrix $\hat{A}_{\Vec{k}_{\rm F}}$, we can simplify the boundary condition as follows.
In the case of $K = 2$ and $N = M$, the boundary conditions (\ref{eq:fnm}) and (\ref{eq:gnm}) become
\begin{align}
\Vec{f}^{k_{z}^{2}}(z=0) &= - \tilde{V'}^{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}
\Vec{f}^{k_{z}^{1}}(z=0), \\
\Vec{g}^{k_{z}^{2}}(z=0) &= -\tilde{V'}^{(-\Vec{k}_{{\rm F} \parallel},-k_{z}^{2}) \ast}_{(-\Vec{k}_{{\rm F} \parallel},-k_{z}^{1})}
\Vec{g}^{k_{z}^{1}}(z=0),
\end{align}
where
\begin{align}
\tilde{V'}^{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})} &\equiv
\hat{A}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})} \tilde{V}^{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}
\hat{A}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})}^{\dagger}.
\end{align}
By using the matrix $\hat{A}_{\Vec{k}_{\rm F}}$ satisfying the relation
\begin{align}
\hat{A}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})} &= \hat{A}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})} (\tilde{V}^{(\Vec{k}_{{\rm F} \parallel},k_{z}^{2})}_{(\Vec{k}_{{\rm F} \parallel},k_{z}^{1})} )^{\dagger},
\end{align}
we obtain the simplified boundary condition expressed as
\begin{align}
\Vec{f}^{k_{z}^{2}}(z=0) &= -
\Vec{f}^{k_{z}^{1}}(z=0), \\
\Vec{g}^{k_{z}^{2}}(z=0) &= -
\Vec{g}^{k_{z}^{1}}(z=0).
\end{align}
In addition, in the case of $K = M = N = 2$, the boundary condition for the Riccati amplitudes (\ref{eq:ricboundary}) becomes
\begin{align}
\tilde{a}(k_{{\rm F}z}^{2}) &= \tilde{a}(k_{{\rm F}z}^{1}),
\end{align}
The above boundary condition is equivalent to that for spin-triplet superconductivity in the past quasiclassical treatment.
In the case of an effective one-band system ($M = 1$) with $K = 2$, the $1 \times 1$ unitary matrix $\hat{A}_{\Vec{k}_{\rm F}}$ is rewritten as $\hat{A}_{\Vec{k}_{\rm F}} = e^{i \phi_{\Vec{k}_{\rm F}}}$.
Thus, we can erase the overall phase $\Phi_{12}$ in Eq.~(\ref{eq:k2m1}),
The boundary condition becomes
\begin{align}
f^{k_{z}^{2}}(z=0) &= -
f^{k_{z}^{1}}(z=0), \\
g^{k_{z}^{2}}(z=0) &= - g^{k_{z}^{1}}(z=0).
\end{align}
We obtain the boundary condition for the Riccati amplitude expressed as
\begin{align}
\tilde{a}(k_{{\rm F}z}^{2}) &= \tilde{a}(k_{{\rm F}z}^{1}), \label{eq:m1r}
\end{align}
which is completely equivalent to the boundary condition in the single-band quasiclassical Eilenberger theory.
Now, we can discuss the appearance condition of the Andreev bound state at a surface in multi-band superconductors.
In a single band model, the Andreev bound states occur when the sign of the gap function changes through the scattering process.
In the case of $K = 2$ and $M = 1$, we can easily discuss the appearance condition of the Andreev bound states.
Note that the bound states appear if the condition (\ref{eq:u12}) is satisfied in this case.
To use the above boundary conditions (\ref{eq:m1r}), the order parameter matrix after the scattering process should be
\begin{align}
\Delta_{\rm eff}(\Vec{r},\Vec{k}_{{\rm F}}^{2}) &=
e^{ - i \left( \Phi^{\Vec{k}_{{\rm F}}^{2}}_{\Vec{k}_{{\rm F}}^{1}}
+ \Phi^{-\Vec{k}_{{\rm F}}^{2}}_{-\Vec{k}_{{\rm F}}^{1}}
\right)
}
\tilde{U}_{\Vec{k}_{\rm F}^{2}}^{M \dagger} \hat{\Delta}(\Vec{r},\Vec{k}_{\rm F}^{2}) \tilde{U}_{-\Vec{k}_{\rm F}^{2}}^{M \ast}
\end{align}%
with
\begin{align}
e^{i \Phi^{\Vec{k}_{{\rm F}}^{2}}_{\Vec{k}_{{\rm F}}^{1}}
}
&\equiv
\tilde{U}_{\Vec{k}_{{\rm F}}^{2}}^{M \dagger} \tilde{U}_{\Vec{k}_{{\rm F}}^{1}}^{M }.
\end{align}
The quasiclassical Green's function at a surface diverges when the relation
\begin{align}
1 + a(i \omega_{n} \rightarrow \epsilon + i \eta, \Vec{k}_{{\rm F}}^{1}) b(i \omega_{n} \rightarrow \epsilon + i \eta, \Vec{k}_{{\rm F}}^{2}) &= 0
\end{align}
is satisfied.\cite{NagaiJPSJ}
With the use of the bulk solutions in Eqs.~(\ref{eq:riccatia}) and (\ref{eq:riccatib}),
the appearance condition of the zero-energy Andreev bound states becomes
\begin{align}
|\Delta_{\rm eff}(\Vec{k}_{{\rm F}}^{1})| |\Delta_{\rm eff}(\Vec{k}_{{\rm F}}^{2})| + \Delta_{\rm eff}(\Vec{k}_{{\rm F}}^{1}) \Delta_{\rm eff}(\Vec{k}_{{\rm F}}^{2})^{\ast} = 0.
\end{align}
Thus, the sign of the order parameter is important for the appearance condition of the Andreev bound state even in multi-band superconductors.
\section{Multi-band effects}
\label{sec:multi}
We discuss the physical meanings of the multi-band quasiclassical theory described by Eq.~(\ref{eq:multi}).
In our theory, the ``multi-band'' effect is characterized by two factors.
The first factor is how many solutions are in Eq.~(\ref{eq:eigen}) (i.e. the information of the eigenvalues).
The second one is how the orbitals are mixed in Eq.~(\ref{eq:eigen}) (i.e. the information of the eigenvectors).
\subsection{Eigenvalues}
We discuss the eigenvalues obtained in Eq.~(\ref{eq:eigen}).
In the quasiclassical theory, the number of the solutions at the Fermi surface $M$ characterizes the multi-band effect.
For example, in the three-orbital spin-singlet superconductors, the superconducting order parameter is described by
a $3 \times 3$ matrix ($N = 3$) in Eq.~(\ref{eq:gapeq}).
Let us consider that the only one band crosses the Fermi energy at the Fermi surface with the Fermi wave number $\Vec{k}_{\rm F}$ ($M = 1$) as shown in Fig.~\ref{fig:moshiki}.
\begin{figure}[tb]
\begin{center}
\resizebox{0.7 \columnwidth}{!}{\includegraphics{Fig3.eps}}
\end{center}
\caption{\label{fig:moshiki}The schematic figure of the electron bands in the three-orbital superconductors.
}
\end{figure}
In this case, the other bands must have the much higher or lower energies.
Thus, the superconducting order parameters on or between these bands (e.~g. $\Delta_{11}(\Vec{k}_{\rm F})$ or $\Delta_{12}(\Vec{k}_{\rm F})$) can not affects the physical quantities, since
these order parameters ($\sim$ meV) are much smaller than the energy scales of electron bands ($\sim$ eV).
Therefore, the $1 \times 1$ order parameter matrix on the band crossing the Fermi energy (e.~g. $\Delta_{22}(\Vec{k}_{\rm F})$ in Fig.~\ref{fig:moshiki}) is only effective for the superconductivity.
In terms of the above point,
many multi-band superconductors such as MgB$_{2}$ or iron-pnictides can be described by the single-band superconducting gap because of $M = 1$ in these materials.
\subsection{Eigenvectors}
We discuss the eigenvectors in Eq.~(\ref{eq:eigen}), which describes the ratio of the hybridization of the orbital characters at the Fermi wave number $\Vec{k}_{\rm F}$.
The multi-band effects are described by these eigenvectors.
For example, we consider the impurity self-energy with the Born approximation.
In our framework, $2 M \times 2 M$ self-energy becomes
\begin{align}
& \bar{\Sigma}(\Vec{k}_{\rm F},i \omega_{n}) \bar{\sigma}_{z} =n_{\rm imp} \bar{V}(\Vec{k}_{\rm F},\Vec{k}_{\rm F})\nonumber \\
& + n_{\rm imp} \langle \bar{V}(\Vec{k}_{\rm F}, \Vec{k}_{\rm F}') \bar{g}(\Vec{k}_{\rm F}',i \omega_{n}) \bar{V}(\Vec{k}_{\rm F}', \Vec{k}_{\rm F}) \rangle_{\Vec{k}_{\rm F}'}, \label{eq:sigma}
\end{align}
where we determine
\begin{align}
\bar{V}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}') &\equiv U_{\Vec{k}_{\rm F}}^{M \dagger}
\check{V}_{0} \check{\sigma}_{z} U_{\Vec{k}_{\rm F}'}^{M}
. \label{eq:vmulti}
\end{align}
In the case of an effective single band system ($M = 1$), $\bar{V}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}') $ becomes the $2 \times 2$ matrix and
its $(1,1)$-component is expressed as
\begin{align}
\bar{V}^{11}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}') &= \Vec{u}_{1}^{\dagger}(\Vec{k}_{\rm F})
\hat{V}_{0} \Vec{u}_{1}(\Vec{k}_{\rm F}')
\end{align}
with
\begin{align}
\hat{H}^{\rm N 1}(\Vec{k}_{\rm F}) \Vec{u}_{1}(\Vec{k}_{\rm F}) &= 0.
\end{align}
In the case of $\hat{V}_{0} \propto \hat{1}$, the strength of the multi-band effects can be determined by the orthogonality between the eigenvectors at the different Fermi momenta.
We note the case of the $T$-matrix approximation for randomly distributed impurities.
The $2 N \times 2 N$ matrix self-energy is written as
\begin{align}
\check{\Sigma}(i \omega_{n})\bar{\sigma}_{z} &= n_{\rm imp} \check{T}(i \omega_{n}),
\end{align}
where
\begin{align}
\check{T}(i \omega_{n}) &= \check{V}_{0} + \sum_{\Vec{k'}} \check{V}_{0} \check{G}_{\Vec{k}'}(i \omega_{n}) \check{T}(i \omega_{n}).
\end{align}
In our framework, the self-energy with $2 M \times 2 M$ matrix form is obtained as
\begin{align}
\bar{\Sigma}(\Vec{k}_{\rm F}, i \omega_{n}) &= n_{\rm imp} \bar{T}_{\Vec{k}_{\rm F} \Vec{k}_{\rm F}}( i \omega_{n}),
\end{align}
where
\begin{align}
\bar{T}_{\Vec{k}_{\rm F} \Vec{k}_{\rm F}'}( i \omega_{n}) &= \bar{V}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}') +
\langle
\bar{V}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}'') \bar{g}(\Vec{k}_{\rm F}'',i \omega_{n}) \bar{T}_{\Vec{k}_{\rm F}'',\Vec{k}_{\rm F}'}
\rangle_{\Vec{k}_{\rm F}''}. \label{eq:selfT}
\end{align}
Thus, the eigenvectors of the normal state Hamiltonian are important to describe the impurity effects.
\subsection{Non-local anisotropic potentials in the projected space}
As shown in the previous sections, the multi-band Eilenberger equations (\ref{eq:multi}) have the orbital characters through the matrix $\tilde{U}^{M}(\Vec{k}_{\rm F})$.
It should be noted that the matrix $\tilde{U}^{M}(\Vec{k}_{\rm F})$ makes the potential $\check{H}^{\rm N 0}(\Vec{R})$ {\it non-local} and anisotropic.
The non-locality originates from the non-unitary transformation which erases the information.
Non-local potentials have been used as pseudo-potentials in the first-principles calculations.
In the first-principle calculations, the pseudo-potential method which treats valence electrons only is commonly used
to erase the degrees of inner-shell electrons.
In our theory, the non-local potentials are used to erase the fast oscillations characterized by $\Vec{k}_{\rm F}$.
The multi-band effects are understood by the non-locality and anisotropy in the projected space.
The effective potential $\bar{V}_{0 \Vec{R}}(\Vec{k}_{\rm F})$ in Eq.~(\ref{eq:multi}) is non-local and anisotropic,
since the potential depends on the center-of-mass coordinate $\Vec{R}$ and the relative coordinate $\Vec{k}_{\rm F}$.
The non-locality and anisotropy can be easily understood through the example of the self-energy with the Born equation expressed as Eq.~(\ref{eq:sigma}).
The above self-energy can be regarded as that made from the non-local potential $\bar{V}(\Vec{r},\Vec{r}')$.
We should note that the non-locality and anisotropy are strong in the iron-based superconductors, since $d$-orbitals are strongly entangled at the Fermi level.
\subsection{Differences between the present and previous quasiclassical multi-band frameworks}
We discuss the differences between the present and previous quasiclassical multi-band treatments.
Fundamentally, note that our framework is an extension of a previous single-band Eilenberger framework.
Thus, we make the derivation similar to the previous single-band one.
Our main point is the low-energy projection which systematically reduces a $N$-band system to a $M$-band system.
There are two kinds of the reductions in our paper.
The first one is same as that in the previous paper\cite{Serene}, which uses Fermi velocities and Green's functions at the Fermi level in normal states.
The second one is the reduction of the band-degree of freedom, which was not mentioned in Ref.~{\it et al}.
Our equations can treat the off-diagonal elements of a Green's function between bands.
A previous theory treated inter-band effects only through gap-equations not the Eilenberger equations.
Note that the results in the previous papers using the quasiclassical theory were obtained only in the problem which the previous framework was available.
Our framework extends the applicable region of the quasiclassical theory.
We claim that there was no multi-band Eilenberger equation which can treat inter-band effects correctly in inhomogeneous systems.
In the previous framework, for example, it is hard to study the vortex bound states with impurities in complicated multi-band superconductors, such as iron-based superconductors.
One usually considers the five-orbital model for the iron-based superconductors whose Fermi surfaces are constructed by only three-bands.
In this case, the decoupled Eilenberger equations have three band indices.
When one introduces a self energy ({\it e.g.} the impurity-induced self energy), the off-diagonal elements of the self energy can not be described by the decoupled Eilenberger equations.
On the other hand, such a self-energy matrix should be defined by the Dyson equation with Feynman-diagram techniques in five-orbital model.
In addition,
band-coupled Eilenberger equations except for our framework can be constructed only in the case that the Fermi velocities are the same in the different bands,
since the decoupled Eilenberger equations with a band index are characterized by a Fermi velocity on the band.
Finally, we show the example which makes clear difference between our and previous frameworks, qualitatively.
The corrections in our framework describe the multi-orbital effects.
Multi-band effects in the present framework are necessary to correctly calculate a reduction of the critical temperature caused by impurities.
While the self energy with T-matrix approximation induced by the randomly-distributed impurities does not depend on the momentum in the
previous decoupled Eilenberger theory, the self-energy in our Eilenberger equation depends on momentum.
If the momentum dependence is neglected, the quasiclassical theory can not correctly calculate the reduction of the critical temperature.
We show the qualitative difference between the present and previous frameworks in the system on the topological insulator in Sec.~\ref{sec:rp}.
The theoretical calculation by directly solving the BdG equations suggested that the proximity-induced superconductivity on the surface of the topological insulator
is robust against non-magnetic impurities\cite{Ito:2011ct}.
The present multi-band framework can correctly describe the robustness.
The previous quasiclassical framework, however, can not reproduce this robustness.
\section{Multi-band quasiclassical approximations in various kinds of systems: Examples}
\label{sec:single}
In this section, we apply the multi-band quasiclassical theory in the various kinds of systems as examples.
\subsection{Noncentrosymmetric Superconductors: CePt$_{3}$Si}
\label{sec:CePt}
We show that our multi-band quasiclassical theory makes the past debates clear.
The noncentrosymmetric superconductor CePt$_{3}$Si has the Rashba-type spin-orbit interaction due to the
lack of the inversion symmetry\cite{HayashiCe,NagaiCe}.
The mixed spin-singlet-triplet model has been used to study this material.
By assuming that the spatial variations of the $s$-wave pairing component of the pair potential are
the same as those of the $p$-wave pairing component, the gap function is expressed as
\begin{align}
\hat{\Delta}(\Vec{k}) &= \left[\Psi \sigma_{0} + \Vec{d}_{\Vec{k}} \cdot \Vec{\sigma} \right] i \sigma_{y}.
\end{align}
Here, $\sigma_{i}$ is the Pauli matrix in spin space.
The normal-state Hamiltonian with the Rashba-type spin-orbit coupling is written as
\begin{align}
\hat{H}(\Vec{k}) &= (\lambda_{\Vec{k}} - \mu) \sigma_{0} + \Vec{g}_{\Vec{k}} \cdot \Vec{\sigma}, \label{eq:ceham}
\end{align}
where the spin-orbit interaction satisfies the relation $\Vec{g}_{- \Vec{k}} = - \Vec{g}_{\Vec{k}}$.
Here, $\lambda_{\Vec{k}}$ is the dispersion without the spin-orbit interaction.
We determine $\Vec{g}_{\Vec{k}}$ as\cite{HayashiCe,NagaiCe}
\begin{align}
\Vec{g}_{\Vec{k}}^{\rm T} &\equiv
g(-\sin \phi \sin \theta, \cos \phi \sin \theta,0).
\end{align}
We assume that the $\Vec{d}$-vector is parallel to $\Vec{g}_{\Vec{k}}^{\rm T} $ expressed as
\begin{align}
\Vec{d}_{\Vec{k}}^{\rm T} &\equiv \Delta_{d} (-\sin \phi \sin \theta, \cos \phi \sin \theta,0).
\end{align}
In the previous papers\cite{HayashiCe}, from the original Eilenberger equation for noncentrosymmetric superconductivity\cite{HayashiNMR,SchopohlLow},
they have obtained two equations corresponding to these split Fermi surface I and II in the case
of the $s$+$p$-wave pairing state,\cite{HayashiPRB}
\begin{align}
i \Vec{v}_{\rm I,II} \cdot \Vec{\nabla} \check{g}_{\rm I,II} + [i \omega_{n} \check{\tau}_{3} - \check{\Delta}_{\rm I, II},\check{g}_{\rm I, II}]&= 0. \label{eq:ce}
\end{align}
where $\check{\Delta}_{\rm I, II} = [(\check{\tau}_{1}+ i \check{\tau}_{2})\Delta_{\rm I, II} - (\check{\tau}_{1}- i \check{\tau}_{2} )\Delta_{\rm I,II}^{\ast}]/2$, $\Delta_{\rm I, II} = \psi \pm \Delta_{d} \sin \theta$ are the order parameters on the Fermi surface I and II, $(\check{\tau}_{1},\check{\tau}_{2},\check{\tau}_{3})$ are the Pauli matrices in the particle-hole space, and the commutator $[\check{a},\check{b}]= \check{a}\check{b}- \check{b}\check{a}$.
There are many successes with the use of the above decoupled equations.
We should note that there are some debates about the appropriate region of the above approach\cite{Hayashipri}.
In the real material such as CePt$_{3}$Si, there is the strong spin-orbit coupling ($\sim eV$).
It has been not clear whether this approach is the weak-spin-orbit coupling approach, since the two same-size spherical Fermi surfaces
are assumed.
The difference of the size of the each Fermi surface depends on the strength of the spin-orbit coupling.
On the other hand, the size of the Fermi surfaces is naturally considered in our multi-band theory.
Let us apply the multi-band quasiclassical theory to the noncentrosymmetric superconductors in order to derive the
decoupled Eilenberger equations Eq.~(\ref{eq:ce}) directly from the Hamiltonian (\ref{eq:ceham}).
The eigenvalues of the $2 \times 2$ matrix in Eq.~(\ref{eq:ceham}) is given by
\begin{align}
\epsilon_{\pm}(\Vec{k}) &= \lambda_{\Vec{k}} - \mu \pm |\Vec{g}_{\Vec{k}}|.
\end{align}
Although there are two Fermi surfaces, the eigenvalue is not degenerated so that we obtain $M = 1$.
The eigenvectors associated with $\epsilon_{\pm}(\Vec{k})$ are expressed as
\begin{align}
u_{\Vec{k}}^{+} &= \frac{1}{\sqrt{2}} \left(\begin{array}{c}
- i e^{- i \phi} \\
1
\end{array}\right),\\
u_{\Vec{k}}^{-} &= \frac{1}{\sqrt{2}} \left(\begin{array}{c}
1\\
-i e^{i \phi}
\end{array}\right).
\end{align}
The $1 \times 1$ effective gap function is given as
\begin{align}
\Delta_{\pm} &= \pm i e^{\pm i \phi} (\Psi \pm \Delta_{d} \sin \theta).
\end{align}
The above effective gap function is not equivalent to $\Delta_{\rm I,II}$ in the previous paper.
We should note that a representation of the effective gap function has
an arbitrary degree of freedom expressed as
\begin{align}
\Delta_{\rm eff}(\phi,\theta)' &= A(\phi,\theta)^{\dagger} \Delta_{\rm eff}(\phi,\theta) A(\phi+\pi,\theta + \pi)^{\ast},
\end{align}
as discussed in Sec.~\ref{sec:arbit}.
Thus, we can use a $1 \times 1$ arbitrary unitary matrix in order to change
a representation of the effective gap.
With the use of the $1 \times 1$ unitary matrix $A_{\pm}(\phi,\theta)$ defined as
\begin{align}
A_{\pm}(\phi,\theta) &= \pm e^{\pm i \frac{\phi}{2}},
\end{align}
the effective gap function can be rewritten as
\begin{align}
\Delta_{\pm}' &=\Psi \pm \Delta_{d} \sin \theta,
\end{align}
which is equivalent to that in the previous papers.\cite{NagaiCe,HayashiCe}
Next, we assume the degenerated Fermi surface ($M = N = 2$), which is appropriate when $|\Vec{g}_{\Vec{k}}| \ll 1$.
With the use of the unitary matrix $A(\phi,\theta)$ defined as
\begin{align}
A(\phi,\theta) &\equiv \left(\begin{array}{cc}
e^{i \frac{\phi}{2}} & 0 \\
0 & -e^{-i \frac{\phi}{2}}
\end{array}\right), \label{eq:aphi}
\end{align}
the effective gap function can be rewritten as
\begin{align}
\Delta_{\rm eff}(\phi,\theta)' &= \left(\begin{array}{cc}
\Psi + \Delta_{d} \sin \theta & 0 \\
0 & \Psi - \Delta_{d} \sin \theta
\end{array}\right). \label{eq:effectiveDelta}
\end{align}
In terms of the multi-band quasiclassical theory, we clarify that the decoupled equations (\ref{eq:ce}) are valid with the arbitrary strength spin-orbit coupling.
Finally, we consider a specular reflection at a surface perpendicular to $x-y$ plane.
We consider that the quasiparticles before and after a scattering have momentum $\Vec{k}_{1} = (\phi_{1},\theta)$, $\Vec{k}_{2} = (\phi_{2},\theta)$, respectively.
By assuming the degenerated Fermi surface ($M = N = K = 2$),
the unitary matrix with $A(\phi,\theta)$ in Eq.~(\ref{eq:aphi}) becomes
\begin{align}
\tilde{U}_{\Vec{k}}^{M} &=\frac{1}{\sqrt{2}} \left(\begin{array}{cc}
-i e^{- i \phi/2} & -e^{- i \phi/2} \\
e^{i \phi/2} & i e^{ i \phi/2}
\end{array}\right),
\end{align}
whose effective order parameter is given in Eq.~(\ref{eq:effectiveDelta}).
The transfer matrix $\tilde{V}_{\Vec{k}_{1}}^{\Vec{k}_{2}} \equiv \tilde{U}_{\Vec{k}_{2}}^{M \dagger}\tilde{U}_{\Vec{k}_{}}^{M}$ is expressed as
\begin{align}
\tilde{V}_{\Vec{k}_{1}}^{\Vec{k}_{2}} &=
\left(\begin{array}{cc}
\cos \Delta \phi & -\sin \Delta \phi \\
\sin \Delta \phi & \cos \Delta \phi
\end{array}\right),
\end{align}
with $\Delta \phi \equiv(\phi_{1}-\phi_{2})/2$.
This transfer matrix suggests that both intra- and inter-band scatterings occur at a specular surface.
The surface bound states and spin currents discussed in the previous paper\cite{VV}
can be explained in terms of this band-active surface.
\subsection{Three-orbital model: Sr$_{2}$RuO$_{4}$}
Let us apply our theory to a multi-band superconductor.
In this section, we consider Sr$_{2}$RuO$_{4}$ as the three-band superconductor.
The many tight-binding models for Sr$_{2}$RuO$_{4}$ have been proposed by several authors\cite{Zabolotnyy, Kee,Ng,PuetterPRB,WCLee}.
According to Ref.~\onlinecite{Zabolotnyy}, the effective tight-binding Hamiltonian is expressed by the three-orbital model characterized
by $d^{yz}$-, $d^{xz}$-, and $d^{xy}$- orbitals ($N = 3$) expressed as
\begin{align}
\check{H}(\Vec{k}) &= \left(\begin{array}{ccc}
\epsilon_{\Vec{k}}^{yz} - \mu & \epsilon_{\Vec{k}}^{\rm off} + i \lambda & - \lambda \\
\epsilon_{\Vec{k}}^{\rm off}- i \lambda & \epsilon_{\Vec{k}}^{xz} - \mu & i \lambda \\
-\lambda & - i \lambda & \epsilon_{\Vec{k}}^{xy} - \mu
\end{array}\right),
\end{align}
where
\begin{align}
\epsilon^{yz}_{\Vec{k}} &= -2 t_{2} \cos(k_{x}) - 2 t_{1} \cos (k_{y}), \nonumber \\
\epsilon^{xz}_{\Vec{k}} &= -2 t_{1} \cos(k_{x}) - 2 t_{2} \cos (k_{y}), \nonumber \\
\epsilon^{xy}_{\Vec{k}} &= -2 t_{3} (\cos(k_{x})+\cos(k_{y})) - 4 t_{4} \cos (k_{x}) \cos(k_{y}) \nonumber \\
&- 2 t_{5} (\cos (2 k_{x}) + \cos(2 k_{y})), \nonumber \\
\epsilon_{\Vec{k}}^{\rm off} &= -4 t_{6} \sin (k_{x}) \sin (k_{y}),
\end{align}
with $\lambda = 0.032$, $t_{1} = 0.145$, $t_{2} = 0.016$, $t_{3} = 0.081$, $t_{4} = 0.039$, $t_{5} = 0.005$, $t_{6} = 0$, and
$\mu = 0.122$.
Here, we adopt the material parameters in Ref.~\onlinecite{Zabolotnyy}, which can successfully describe the three Fermi surfaces for
Sr$_{2}$RuO$_{4}$ as shown in Fig.\ref{fig:fermi}.
\begin{figure}[t]
\begin{center}
\resizebox{0.6 \columnwidth}{!}{\includegraphics{Fig4.eps}}
\end{center}
\caption{\label{fig:fermi}(Color online) Fermi surfaces for the three-band superconductor Sr$_{2}$RuO$_{4}$.
}
\end{figure}
We call each band as band I, band II, and band III in ascending order of the eigenvalues.
We consider the non-magnetic impurities to discuss the non-local anisotropic effective potential.
We introduce the in-plane anisotropy of the effective potential defined as
\begin{align}
V(\theta,\theta') \equiv \Vec{u}_{\Vec{k}_{\rm F}(\theta)}^{\dagger} \Vec{u}_{\Vec{k}_{\rm F}(\theta')},
\end{align}
Here, $\Vec{k}_{\rm F}(\theta)$ denotes the position of the most inner Fermi surface (i.e. band III) in momentum space ($\Vec{k}_{\rm F}(\theta) =k_{\rm F}(\theta)(\cos \theta,\sin \theta)$).
As shown in Fig.~\ref{fig:aniso}, the right-angled scatterings suppress in the most inner Fermi surface.
This suppression originates from the fact that the eigenvector associated with $\Vec{k}_{\rm F}(\theta=0)$ mainly consists of $d_{xz}$ orbital and the eigenvector associated with $\Vec{k}_{\rm F}(\theta'=0)$ mainly consists of $d_{yz}$ orbital.
\begin{figure}[t]
\begin{center}
\resizebox{0.9 \columnwidth}{!}{\includegraphics{Fig5.eps}}
\end{center}
\caption{\label{fig:aniso}(Color online) The in-plane anisotropy of the effective potential $|V(\theta,\theta')|$ at the most inner Fermi surface in the three-band superconductor Sr$_{2}$RuO$_{4}$.
}
\end{figure}
\subsection{Heavy fermion CeCoIn$_{5}$/YbCoIn$_{5}$ superlattice: The perturbative approach}
In this section, we consider the system with both the spin-orbit coupling and the Zeeman interaction.
The locally noncentrosymmetric systems are realized in the heavy fermion CeCoIn$_{5}$/YbCoIn$_{5}$ superlattice\cite{Mizukami}.
In these systems, the layer-dependent spin-orbit coupling induces the exotic superconducting states.
In the $N$-layer spin-singlet $s$-wave superconductor,
the multi-band Eilenberger equations with $4 N \times 4 N$ matrix quasiclassical Green's function are
written as
\begin{align}
i \Vec{v}_{\rm F}\cdot \Vec{\nabla}
\check{g} +
\left[
z \check{\tau}_{z}
-
\check{\Delta}
-
\check{K} (\Vec{k}_{\rm F})
,
\bar{g}
\right]_{-}
&= 0,
\end{align}
with
\begin{align}
\check{K} (\Vec{k}_{\rm F}) &\equiv
\left( t_{\perp} \hat{H}_{\rm inter}+\mu_{\rm B} h \hat{H}_{Z}\right) \check{\tau}_{0} \nonumber \\
&+ \left(\begin{array}{cc}
\hat{H}_{\rm SO}(\Vec{k}_{\rm F}) & 0 \\
0 & \hat{H}_{\rm SO}^{\ast}(-\Vec{k}_{\rm F})
\end{array}\right), \\
\check{\tau}_{z} &= \left(\begin{array}{cc}
\sigma_{0} \otimes I_{N \times N} & 0 \\
0 & -\sigma_{0} \otimes I_{N \times N}
\end{array}\right) \\
\check{\Delta} &= \left(\begin{array}{cc}
0 & \hat{\Delta} \\
- \hat{\Delta}^{\dagger} & 0
\end{array}\right), \\
\hat{H}_{\rm inter} &= \sigma_{0} \otimes T_{\perp}, \\
\hat{H}_{Z} &= - \sigma_{z} \otimes I_{N \times N}, \\
\hat{H}_{\rm SO}(\Vec{k}_{\rm F}) &= \Vec{g}(\Vec{k}_{\rm F}) \cdot \Vec{\sigma} \otimes S_{d}.
\end{align}
Here, $\sigma_{z}$ is the $2 \times 2$ Pauli matrix, $I_{N \times N}$ is the unit matrix, $T_{\perp}$ is the hopping matrix between layers, and
$S_{d} = {\rm diag} (\alpha_{i},\cdots,\alpha_{N})$ denotes the layer-dependent spin-orbit interaction.
In the above equations, the hopping, the Zeeman, and the spin-orbit coupling terms are regarded as the perturbations with
$2N$-degenerated Fermi surfaces.
With the use of this perturbation theory, one can treat the inhomogeneous system with vortices\cite{HigashiLT}.
\subsection{Topological superconductors with the strong spin-orbit coupling: Cu$_{x}$Bi$_{2}$Se$_{3}$}
We discuss the boundary condition in the three-dimensional topological superconductor with the strong spin-orbit coupling in this section.
Cu$_{x}$Bi$_{2}$Se$_{3}$ is the one of the candidates of the topological superconductors where the topologically protected
Majorana bound states form at the boundary.
We have proposed the quasiclassical framework on topological superconductors with strong spin-orbit coupling\cite{NagaiTopo}.
In the previous paper\cite{NagaiTopo}, we have obtained the linearized BdG equations called Andreev equations
by decomposing the slow varying component from the total quasi-particle wave function.
Applying this quasiclassical treatment, the original massive Dirac BdG Hamiltonian
derived from the tight-binding model represented by $8 \times 8$ matrix
is reduced to $4 \times 4$ one.
The resultant Andreev equations become equivalent to those of spin-singlet or triplet superconductors without the spin-orbit coupling.
In this section, we show that the same result is obtained by the Green's function techniques.
The normal-states effective Hamiltonian for Cu$_{x}$Bi$_{2}$Se$_{3}$ is expressed as
\begin{align}
\hat{H}(\Vec{k}) &=
\left(\begin{array}{cc}
(M(\Vec{k}) - \mu ) \sigma_{0} & \Vec{k}\cdot \Vec{\sigma} \\
\Vec{k}\cdot \Vec{\sigma} & (-M(\Vec{k}) - \mu) \sigma_{0} \end{array}\right). \label{eq:topo}
\end{align}
Here $\sigma_{i}$ denotes the Pauli matrix in the spin space.
In the quasiclassical theory, $\hat{H}(\Vec{k})$ is regarded as $\hat{H}^{\rm N 1}(\Vec{k})$ in Eq.~(\ref{eq:eigen}).
The eigenvalues of the $4 \times 4$ matrix are degenerated expressed as
\begin{align}
\epsilon_{i}(\Vec{k}) &= \pm E_{0}(\Vec{k}) - \mu,
\end{align}
where $E_{0}(\Vec{k}) \equiv \sqrt{M(\Vec{k})^{2} + |\Vec{k}|^{2}}$.
In the case of $\mu > 0$, the eigenvectors $u_{\Vec{k}}^{1}$ and $u_{\Vec{k}}^{2}$ ($M=2$), and eigenvalue $\xi_{\Vec{k}}$
in Eq.~(\ref{eq:eigen}) are given as
\begin{align}
\xi_{\Vec{k}} &= E_{0}(\Vec{k}) - \mu \\
u_{\Vec{k}}^{i} &= c \left(\begin{array}{c}
\chi_{i} \\
\frac{\Vec{k} \cdot \Vec{\sigma}}
{E_{0}(\Vec{k})+ M(\Vec{k})}
\chi_{i}
\end{array}\right),
\end{align}
where $\chi^{\rm T}_{1} \equiv (1,0)$, $\chi^{\rm T}_{2} \equiv (0,1)$, $c \equiv \sqrt{(E_{0}+M(\Vec{k}))/2 E_{0}}$.
We consider the $4 \times 4$ odd-parity fully-gapped gap function (so-called $A_{1u}$ state) defined as
\begin{align}
\hat{\Delta} &\equiv \Delta_{0} \left(\begin{array}{cc}
0 & i \sigma_{y} \\
i \sigma_{y} & 0
\end{array}\right). \label{eq:gappseudo}
\end{align}
By substituting this gap function into Eq.~(\ref{eq:gapeff}), we obtain the $2 \times 2$ effective gap function
written as
\begin{align}
\Delta_{\rm eff}(\Vec{k}) &= \frac{\Delta_{0}}{E_{0}(\Vec{k})} \Vec{k} \cdot \Vec{\sigma} (i \sigma_{y}).
\end{align}
This gap function is completely equivalent to that in the previous paper \cite{NagaiTopo} in terms of the Dirac BdG Hamiltonian.
Let us consider the boundary condition with a specular surface at $z = 0$.
We adopt the boundary condition that all components of the wave-function becomes zero at $z =0$, which is different from that discussed in Ref.~\onlinecite{NagaiQuasi}.
We consider $M(\Vec{k}) = M_{0}(\Vec{k}_{\parallel}) + M_{1} k_{z}^{2}$.
By using Eq.~(\ref{eq:boundnh}), we find that the wave numbers become
\begin{align}
(k_{z \pm})^{2} = \frac{-1}{2 M_{1}^{2}} \left[ 1 + 2 M_{0} M_{1} \pm \xi_{\Vec{k}_{\parallel}}
\right],
\end{align}
with $\xi_{\Vec{k}_{\parallel}} \equiv \sqrt{
(1 + 2 M_{0} M_{1})^{2} + 4 M_{0} (\mu^{2} - M_{0}^{2} - |\Vec{k}_{\parallel}|^{2})}$.
When the condition $\mu >\sqrt{ M_{0}^{2} - |\Vec{k}_{\parallel}|^{2}}$ is satisfied,
there are two real wave numbers and two imaginary wave number, expressed as
\begin{align}
k_{z -}^{1} &= k_{-}
\\
k_{z -}^{2} &= -k_{-
, \\
k_{z +}^{1} &= i \eta_{+
, \\
k_{z +}^{2} &= -i \eta_{+
,
\end{align}
with $k_{-} \equiv \sqrt{-\left[ 1 + 2 M_{0} M_{1} -\xi_{\Vec{k}_{\parallel}} \right]/(2 M_{1}^{2})}$ and
$\eta_{+} \equiv \sqrt{\left[ 1 + 2 M_{0} M_{1} + \xi_{\Vec{k}_{\parallel}} \right]/(2 M_{1}^{2})}$.
By assuming that the material is filled in the region $z > 0$, the coefficient of the wavefunction with $k_{z +}^{2}$ is zero.
Thus, we obtain $K = 3$.
The boundary condition (\ref{eq:nbound}) is expressed as
\begin{align}
&\left(\begin{array}{ccc}\Vec{u}^{1}_{(\Vec{k}_{\parallel},k_{-})} & \Vec{u}^{1}_{(\Vec{k}_{\parallel},-k_{-})} & \Vec{u}^{1}_{(\Vec{k}_{\parallel},i \eta_{+})}
\end{array}\right) \Vec{c}^{1} \nonumber \\
&+
\left(\begin{array}{ccc}\Vec{u}^{2}_{(\Vec{k}_{\parallel},k_{-})} & \Vec{u}^{2}_{(\Vec{k}_{\parallel},-k_{-})} & \Vec{u}^{2}_{(\Vec{k}_{\parallel},i \eta_{+})}
\end{array}\right) \Vec{c}^{2}
=0.
\end{align}
When we consider $\Vec{k}_{\parallel} = 0$, the coefficients are given as
\begin{align}
\Vec{c}^{1} &= \Vec{c}^{2} = \left(\begin{array}{c}
\frac{1}{2} \left(1 + \frac{\mu + M(k_{-})}{\mu + M(i \eta_{+})} \frac{i \eta_{-}}{k_{-}} \right) \\
\frac{1}{2} \left(1 - \frac{\mu + M(k_{-})}{\mu + M(i \eta_{+})} \frac{i
\eta_{-}}{k_{-}} \right) \\
-1
\end{array}\right) c.
\end{align}
Thus, the boundary condition for the quasiclassical wave function is obtained as
\begin{align}
\Vec{f}^{-k_{-}}(z=0) &= c_{1} \Vec{f}^{k_{-}}(z=0), \\
\Vec{g}^{-k_{-}}(z=0) &= c_{1}^{\ast} \Vec{g}^{k_{-}}(z=0),
\end{align}
where
\begin{align}
c_{1} &=
\frac{
k_{-} (\mu + M(i \eta_{+})) - i \eta_{+} (\mu + M(k_{-}))
}
{
k_{-} (\mu + M(i \eta_{+})) + i \eta_{+} (\mu + M(k_{-}))
}.
\end{align}
In the non-relativistic limit ($|k_{-}| \ll M(k_{-})$),
the boundary condition becomes
\begin{align}
\Vec{f}^{-k_{-}}(z=0) &= -\Vec{f}^{k_{-}}(z=0), \\
\Vec{g}^{-k_{-}}(z=0) &= -\Vec{g}^{k_{-}}(z=0),
\end{align}
which is equivalent to that in a single band quasiclassical framework.
This result is consistent with the fact that this superconductor becomes a $p$-wave superconductor in the non-relativistic limit\cite{Nagai3D}.
\subsection{Robust $p$-wave superconductivity on a surface of topological insulator}
\label{sec:rp}
In this section, we discuss the impurity effect in the $s$-wave gap superconductivity on a surface of topological insulator.
Let us show that the proximity-induced superconductivity on a surface of topological insulator is robust against nonmagnetic impurities\cite{Ito:2011ct}.
We consider that an $s$-wave superconductor is deposited on the surface of the topological insulator\cite{Fu:2008gu}.
The effective two-dimensional Hamiltonian on the surface is described as
\begin{align}
\hat{H}(\Vec{k}) &= \left(\begin{array}{cc}
h_{0}(\Vec{k})& \Delta i \sigma_{y} \\
\Delta^{\ast}(- i \sigma_{y}) &
-h_{0}^{\ast}(-\Vec{k})
\end{array}\right),
\end{align}
with
\begin{align}
h_{0}(\Vec{k}) &\equiv v \Vec{k} \cdot \Vec{\sigma} -\mu \sigma_{0}.
\end{align}
The eigenvalues of the $2 \times 2$ normal state Hamiltonian $h_{0}(\Vec{k})$ are given by
\begin{align}
\epsilon_{\pm}(\Vec{k}) &= - \mu \pm v |\Vec{k}|.
\end{align}
The eigenvectors associated with $\epsilon_{\pm}(\Vec{k})$ are expressed as
\begin{align}
u_{\Vec{k}}^{+} &=
\frac{1}{\sqrt{2}}
\left(\begin{array}{c}
e^{- i \phi} \\
1
\end{array}\right), \\
u_{\Vec{k}}^{-} &=
\frac{1}{\sqrt{2}}
\left(\begin{array}{c}
1 \\
-e^{ i \phi}
\end{array}\right),
\end{align}
where $\Vec{k} = (k_{x},k_{y})= k (\cos \phi, \sin \phi)$.
In the case of $\mu > 0$, the $1 \times 1$ effective gap is given as
\begin{align}
\Delta_{\rm eff}(\Vec{k}) &= \Delta e^{i \phi}. \label{eq:peff}
\end{align}
Thus, the effective $p$-wave superconductivity appears on a surface of the topological insulator.
Let us consider a nonmagnetic impurity effect in this proximity-induced superconductor.
The non-perturbative quasiclassical Green function in a homogeneous system is given as
\begin{align}
\bar{g}(\Vec{k}_{\rm F}, i \omega_{n}) &= \frac{- \pi}{\sqrt{\omega_{n}^{2} + |\Delta|^{2}}}
\left(\begin{array}{cc}
i \omega_{n} & \Delta e^{i \phi} \\
- \Delta e^{-i \phi} & - i \omega_{n}
\end{array}\right).
\end{align}
The effective potential $\bar{V}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}')$ in Eq.~(\ref{eq:vmulti}) is expressed as
\begin{align}
\bar{V}(\Vec{k}_{\rm F},\Vec{k}_{\rm F}') &= V \left(\begin{array}{cc}
e^{i \delta \phi/2} \cos \left( \frac{\delta \phi}{2} \right) & 0 \\
0 & e^{-i \delta \phi /2} \cos \left( \frac{\delta \phi}{2} \right)
\end{array}\right),
\end{align}
with $\delta \phi \equiv \phi - \phi'$.
Here, $V$ is the amplitude of the potentials.
The second order of the impurity self-energy in Eq.~(\ref{eq:sigma}) becomes
\begin{align}
\bar{\Sigma}(\Vec{k}_{\rm F}, i \omega_{n})^{(2)}\bar{\sigma}_{z} &= n_{\rm imp} V^{2}\bar{g}(\Vec{k}_{\rm F}, i \omega_{n})
\int_{0}^{2 \pi} \frac{d\phi' \cos \left(\frac{\phi - \phi'}{2} \right)}{2 \pi}.
\end{align}
Since this self-energy satisfies $[\bar{\Sigma}(\Vec{k}_{\rm F}, i \omega_{n})^{(2)}\bar{\sigma}_{z} ,\bar{g}(\Vec{k}_{\rm F}, i \omega_{n})]_{-}=0$,
the quasiclassical Eilenberger equations with the self-energy are completely equivalent to those without the self-energy.
Therefore, this proximity-induced superconductivity on the surface of a topological insulator is robust against nonmagnetic impurities.
Finally, we point out that the previous decoupled quasiclassical framework can not reproduce the robustness against nonmagnetic impurities proposed in Ref.~\onlinecite{Ito:2011ct}.
With the use of the band basis, the effective gap is given in Eq.~(\ref{eq:peff}) at the Fermi energy.
Since this effective gap means $p$-wave superconductivity, the superconductivity should be
fragile against nonmagnetic impurities in the previous decoupled quasiclassical framework.
\subsection{Surface quasiclassical theory: the partial quasiclassical approximation for topological insulators}
Let us discuss the ``partial'' quasiclassical approximation with considering the topological insulators.
The superconductivity in surface states on topological insulators has been attracted much attention because of the stage of the Majorana Fermion
and a quantum computing.
With the use of the proximity effects from the superconductor on the topological insulator,
the two-dimensional massless Dirac quasiparticles due to the surface bound states on the topological insulator
form the superconducting Cooper pairs.
Therefore, it is important to construct the two-dimensional effective Eilenberger equations originating from the
normal-state surface bound states.
Let us consider the surface at $z = 0$.
By introducing the coordinate $\Vec{r} = (x,y,z) \equiv (\Vec{r}_{\perp},z)$,
we can define the partial Wigner representation expressed as
\begin{align}
& \check{A}_{z_{1}z_{2}}(\Vec{R}_{\perp},\Vec{k}_{\perp}) \equiv \nonumber \\
& \int d\Vec{\bar{r}}_{\perp} e^{-i \Vec{k}_{\perp} \cdot \Vec{\bar{r}}_{\perp}} \check{A}_{z_{1}z_{2}}\left( \Vec{R}_{\perp} + \frac{\Vec{\bar{r}}_{\perp} }{2},\Vec{R}_{\perp} - \frac{\Vec{\bar{r}}_{\perp}}{2} \right).
\end{align}
Here, $\Vec{R}_{\perp} = (\Vec{r}_{1 \perp}+\Vec{r}_{2 \perp})/2$ and $\Vec{\bar{r}}_{\perp}= \Vec{r}_{1 \perp} - \Vec{r}_{2 \perp}$ are the center-of-mass coordinate and the relative
coordinate on the two-dimensional plane parallel to the surface, respectively.
The projection operator is determined as
\begin{align}
\check{P}_{\Vec{k}_{\perp} z_{1}z_{2}} &\equiv U_{\Vec{k}_{\perp}}^{M}(z_{1}) U_{\Vec{k}_{\perp}}^{M \dagger}(z_{2}),
\end{align}
where
\begin{align}
U_{\Vec{k}_{\perp}}^{M}(z)&\equiv
\left(\begin{array}{cc}
\tilde{U}_{\Vec{k}_{\perp}}^{M}(z) & 0 \\
0 & \tilde{U}_{-\Vec{k}_{\perp}}^{M \ast}(z)
\end{array}\right),
\end{align}
with
the matrix $U_{\Vec{k}_{\perp}}^{M}(z) = (u_{\Vec{k}_{\perp}}^{1}(z),\cdots,u_{\Vec{k}_{\perp}}^{M}(z))$.
Here, the vector $u_{\Vec{k}}^{i}(z)$ is the eigenvector expressed as
\begin{align}
\hat{H}^{\rm N 1}(\Vec{k}_{\perp},z) u_{\Vec{k}_{\perp}}^{i}(z) &= \xi_{\Vec{k}_{\perp}} u_{\Vec{k}_{\perp}}^{i}(z).
\end{align}
We note that the Hamiltonian $\hat{H}^{\rm N 1}(\Vec{k},z)$ includes information about a presence of a surface.
The projected effective gap function is given by
\begin{align}
\bar{\Delta}(\Vec{R}_{\perp},\Vec{k}_{\perp}) &\equiv \int dz_{1} dz_{2} U_{\Vec{k}_{\perp}}^{M \dagger}(z_{1}) \check{\Delta}_{z_{1} z_{2}}(\Vec{R}_{\perp},
\Vec{k}_{\perp})U_{-\Vec{k}_{\perp}}^{M \ast}(z_{2}), \label{eq:zgap}
\end{align}
since the projection includes the $z$-integration written as
\begin{align}
\check{A}_{z_{1} z_{2}}' &=
\int dz_{3} \check{P}_{\Vec{k} z_{1}z_{3}} \check{A}_{z_{3} z_{2}}.
\end{align}
Let us consider the three dimensional topological superconductor as an example.
The eigenvector in Eq.~(\ref{eq:topo}) with the boundary condition $u_{\Vec{k}_{\perp}}^{i}(z=0) = 0$ is expressed
as\cite{SCES}
\begin{align}
u_{\Vec{k}_{\perp}}^{i}(z) &= \frac{e^{\frac{z}{2 M_{1}}} \sinh(K z)}{2 \sqrt{A}}
\left(\begin{array}{c}
e^{- i \phi} \\
i \\
i e^{- i \phi} \\
1
\end{array}\right), \label{eq:zu}
\end{align}
where $\xi_{\Vec{k}_{\perp}} = \sqrt{k_{x}^{2}+k_{y}^{2}} - \mu$, $\Vec{k}_{\perp} = (k_{x},k_{y}) =\sqrt{k_{x}^{2}+k_{y}^{2}} (\cos \phi,\sin \phi)$, $M(\Vec{k}) = M_{0}(\Vec{k}_{\perp}) + M_{1} k_{z}^{2} $, $K = (\sqrt{1+4 M_{0} M_{1}})/(2 M_{1})$,
and $A = \int_{0}^{\infty} dz exp(z/M_{1}) |\sinh(Kz)|^{2}$.
Here, we assume $\mu > 0$ and obtain the above solution with the use of the perturbation with respect to $\Vec{k}_{\perp}$.
By substituting the eigenvector in Eq.~(\ref{eq:zu}) and the odd-parity fully-gapped gap function in Eq.~(\ref{eq:gappseudo}) into
Eq.~(\ref{eq:zgap}),
we obtain
\begin{align}
\bar{\Delta}(\Vec{R}_{\perp},\Vec{k}_{\perp}) &= \bar{0}.
\end{align}
Therefore, the proximity-induced odd-parity fully-gapped gap function does not open the spectral gap as shown in Ref.~\onlinecite{SCES}.
With the use of the above method, we directly show that the proximity-induced $s$-wave superconductivity on the topological insulator can be regarded as
a chiral $p$-wave superconductivity, as shown in Sec.~\ref{sec:rp}.
By substituting the eigenvector in Eq.~(\ref{eq:zu}) and the even-parity fully-gapped spin-singlet intra-orbital gap function expressed as
\begin{align}
\hat{\Delta} &\equiv \Delta_{0} \left(\begin{array}{cc}
i \sigma_{y} & 0\\
0 & i \sigma_{y}
\end{array}\right),
\end{align}
into
Eq.~(\ref{eq:zgap}),
we obtain
\begin{align}
\bar{\Delta}(\Vec{R}_{\perp},\Vec{k}_{\perp}) &= \Delta_{\perp} e^{i \phi},
\end{align}
which is equivalent to the chiral $p$-wave superconductivity in Eq.~(\ref{eq:peff}).
\subsection{Others}
Finally, we discuss the advantage of the multi-band quasiclassical theory.
The computational cost drastically decreases with the use of our theory in multi-band systems.
Thus, we can treat the inhomogeneous systems such as those with vortices and surfaces easily, in order to
discuss the magnetic field dependence of the multi-band superconductors.
Since we do not use any assumptions about the electronic structures in normal states, the superconducting system with the arbitrary
tight-binding Hamiltonian derived by the first-principle calculation can be mapped onto the effective low-energy system.
In the theoretical point of view, one might develop the general theory for impurity effects in multi-band superconductors,
since the multi band effects are explicitly included as the non-local and anisotropic potentials.
It should be noted that one can understand what is neglected in the quasiclassical theory in multi-band superconductors.
One might know the difference between the single-band and multi-band superconductors through the study with our multi-band Eilenberger theory.
\section{Summary}
\label{sec:sum}
In summary, we proposed the unified quasiclassical multi-band Eilenberger equations in order to map the multi-band systems onto the effective
systems in the reduced space.
We derived both the Andreev and Eilenberger equations with an arbitrary boundary condition.
We showed that the resultant multi-band Eilenberger equations are similar to the single-band ones, except for some corrections to describe multi-band effects.
The orbital characters on the Fermi surfaces in normal states are included in our theory.
Our theory could describe the past studies with the use of the quasiclassical Eilenberger theory.
Since we do not use any assumptions about the electronic structures in normal states, the superconducting system with the arbitrary
tight-binding Hamiltonian derived by the first-principle calculation can be mapped onto the effective low-energy system.
The potentials, the order-parameters, and self-energies in multi-band systems were mapped onto the non-local ones in the reduced space as shown in Eqs.~(\ref{eq:greenu})-(\ref{eq:sigmau}).
We showed that the self-energy with the $T$-matrix approximation of the non-magnetic impurities becomes non-local and anisotropic.
We pointed out that this non-locality is similar to the pseudo potential in the first-principles calculations.
The multi-band effects can be understood by the non-locality and the anisotropy in the mapped systems.
\section*{Acknowledgment}
The authors would like to acknowledge Masahiko Machida, Susumu Yamada, Yusuke Kato, Yukihiro Ota, Kaori Tanaka and Nobuhiko Hayashi for
helpful discussions and comments.
The calculations have been performed using the supercomputing system PRIMERGY BX900 at the Japan Atomic Energy Agency.
This study was supported by JSPS KAKENHI Grant Number 26800197.
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\section{Introduction}
At tree level, the amplitudes of pure Yang-Mills fields can be written as rational functions of external momenta and polarization vectors in spinor form \cite{Parke:1986gb,Xu:1986xb,Berends:1987me,Kosower,Dixon1,Witten1}. Such rational functions can be analyzed in detail in algebra system. According to this, BCFW recursion relation was proposed and developed in \cite{Britto:2004nj,Britto:2004nc,Britto:2004ap}, and then proved in \cite{Britto:2005fq} using the pole structure of the tree level on shell amplitudes. This has been an exiting progress on the amplitudes in pure Yang-Mills theory. For theory with massive fields \cite{Badger1,Ozeren,Schwinn,Chen1,Chen2}, the amplitudes are also rational functions of external momenta and polarization vectors in spinor form.
At loop level, although the whole amplitudes are no longer rational functions in general, they can be decomposed into some basic scalar integrals with coefficients being rational functions of external spinors \cite{BernD1,BernD2}. The coefficient structures are studied in depth in \cite{Dixon4,Bern,Bern1}. On the other hand, the integrands of the amplitudes are rational functions of the external spinors and integral momenta. For the N=4 planar super Yang-Mills theory, \cite{Nima} gives an explicit recursive formula for the all-loop integrand of scattering amplitudes.
The amplitudes in gauge theory are constrained by gauge symmetry. This leads to Ward identity which constrains the amplitudes at all loop level. Inspired by the BCFW momenta shift, we considered the Ward identity for tree level amplitudes with complexified momenta for a pair of external lines, and then obtained a recursion relation for the boundary terms using BCFW technique in our recent article \cite{Chen}. However, in \cite{Chen}, we chose a particular momenta shift such that the external states of the complexified lines are independent of the complex parameter $z$. Then a natural question is how to obtain a recursion relation for other possible momenta shifts. Furthermore, is it possible to obtain the full amplitudes from the Ward identity, and to extend the technique to one loop amplitudes? In this article, we will give positive answers to all these questions.
In section \ref{TreeRec}, we first give the proof of Ward identity at tree level using Feynman rules directly, and then derive the recursion relation for off shell amplitudes, where the cancellation details in the proof of Ward identity helps to simplify the recursion relations. Section \ref{LoopWard} is parallel to section \ref{TreeRec}. We first extend the proof of Ward identity to one loop level and then derive the recursion relation for one loop off shell amplitudes. Our technique does not rely on the on-shell momenta shifts. Also, in our calculation using the recursion relation, four point vertexes are not used explicitly. We calculate three and four point one loop off shell amplitudes as examples in section \ref{example}.
\section{Ward Identity and Implied Recursion Relation at Tree Level}\label{TreeRec}
In \cite{Chen}, we directly proved complexified Ward identity for pure Yang-Mills fields at tree level, and then used it to deduce a recursion relation for the boundary terms of the complexified amplitudes. Here we generalize the method to deduce a recursion relation for tree level amplitudes with one external off shell line. This section will serve as a basis for our generalization to one loop level in the next section. We will call the external off shell line $L_{\mbox{\tiny off}}$ with momentum $k_{\mbox{\tiny off}}^\mu$, and the corresponding off shell amplitudes $A_\mu$.
\subsection{Proof of Ward Identity at Tree Level}
Although done in our previous paper \cite{Chen}, we briefly summerize some key points in the direct proof of tree level Ward identity, since these points are useful for deriving tree level recursion relation and also will be part of the proof at one loop level in the next section.
The amplitude is complexified by shifting the momenta of a pair of external lines. We choose $L_{\mbox{\tiny off}}$ and one on shell line $L_s$ with momentum $k_s=\lambda_s\td\lambda_s$, and the shift is:
\begin{equation}
k_s \rightarrow k_s-z \eta,\ \ \ \ \ \ \ \ \ \ \ \ \ \ k_{\mbox{\tiny off}}\rightarrow k_{\mbox{\tiny off}}+z\eta,
\label{momshift}
\end{equation}
where z is the complexifing parameter and $\eta$ should satisfy $\eta^2=0$ and $k_s \cdot \eta=0$.
The color ordered Feynman rules of the gauge field are as in \cite{Dixon1}, with outgoing momenta. We also write the Feynman rules for ghost fields here in Figure \ref{ghostFeynrule}, which will be used in the next section.
\begin{figure}[htb]
\centering
\includegraphics[height=3.5cm,width=15cm]{ghostFeynrule.eps}
\caption{Ghost field color ordered Feynman rules. Dashed line for ghost field and solid line for gauge field.}
\label{ghostFeynrule}
\end{figure}
For a three point vertex with line 1, 2 and $L_{\mbox{\tiny off}}$ in anti-clockwise order, we write it in the following form:
\begin{eqnarray}\label{newV3}
V_{\mu_1\mu_2\mu}&\equiv&S_{\mu_1\mu_2\mu}+ R_{\mu_1\mu_2\mu}+M_{\mu_1\mu_2\mu},
\end{eqnarray}
where
\begin{eqnarray}\label{newV3s}
S_{\mu_1\mu_2\mu}&=&\frac{i}{\sqrt 2}\left(\eta_{\mu_1\mu_2}(k_1-k_2)_{\mu}\right) \nonumber\\
R_{\mu_1\mu_2\mu}&=&\frac{i}{\sqrt 2}\left(-2\eta_{\mu_2\mu}(k_{\mbox{\tiny off}})_{\mu_1}+2\eta_{\mu\mu_1}(k_{\mbox{\tiny off}})_{\mu_2}\right) \nonumber\\
M_{\mu_1\mu_2\mu}&=&\frac{i}{\sqrt 2}\left(-\eta_{\mu_2\mu}(k_1)_{\mu_1}+\eta_{\mu\mu_1}(k_2)_{\mu_2}\right).
\end{eqnarray}
We will refer to these terms as S, R and M parts of the vertex. Contracting this vertex with $k_{\mbox{\tiny off}}$, we get:
\begin{equation}
k_{\mbox{\tiny off}}^\mu \cdot V_{\mu_1\mu_2\mu}=\frac{i}{\sqrt{2}}\eta_{\mu_1\mu_2} k_2^2-\frac{i}{\sqrt{2}}\eta_{\mu_1\mu_2} k_1^2+\frac{i}{\sqrt{2}}k_{2\ \mu_2}k_3{}_{\ \mu_1}-\frac{i}{\sqrt{2}}k_{1\ \mu_1}k_3{}_{\ \mu_2},
\label{kedotV}
\end{equation}
and we represent these terms by the symbols in Figure \ref{vertexnotation}. These terms are frequently used throughout the paper, and we will call the terms in the first line of Figure \ref{vertexnotation} as solid triangle terms, and the second line terms as hollow triangle terms.
\begin{figure}[htb]
\centering
\includegraphics[height=4cm,width=12cm]{vertexnotation.eps}
\caption{Notations for \ref{kedotV}. We specialize $L_{\mbox{\tiny off}}$ using photon line.}
\label{vertexnotation}
\end{figure}
Then a proof of tree level Ward identity can be shown in several steps. Assume it holds for N-point and less than N-point amplitudes(for example, 3-point case can be immediately checked), we will show how it holds for (N+1) point amplitudes. We choose $L_{\mbox{\tiny off}}$ as the (N+1)-th line. We can construct an (N+1)-point color ordered diagram from an N-point one by inserting $L_{\mbox{\tiny off}}$ to an N-point diagram between Line 1 and Line N.
First, when $L_{\mbox{\tiny off}}$ is inserted to a propagator or Line 1 or Line N, we denote the vertex as $V_{\mbox{\tiny off}}$, and contract it with $k_{\mbox{\tiny off}}$, the following two hollow triangle terms in Figure \ref{treecancel1} vanish due to less-point Ward Identities or the on-shell conditions of Line 1 or N. The meaning of the symbols are in Figure \ref{vertexnotation}.
\begin{figure}[htb]
\centering
\includegraphics[height=2.5cm,width=10cm]{treecancel1.eps}
\caption{When $L_{\mbox{\tiny off}}$ is inserted to a propagator or Line 1 or Line N, these terms vanish due to less point Ward identity or the on-shell conditions of Line 1 or N. $A_1$ and $A_2$ are sub amplitudes.}
\label{treecancel1}
\end{figure}
Second, $L_{\mbox{\tiny off}}$ is inserted to a three-point vertex in the N-point diagram. These terms and the remaining terms, ie. solid triangle terms, from the above case can be re-combined as in Figure \ref{treecancel2} to cancel each other.
\begin{figure}[htb]
\centering
\includegraphics[height=10cm,width=15cm]{treecancel2.eps}
\caption{A group of diagrams cancel. In (a) and (b), the cancellation is solely due to the vertex structures, not dependent on whether the legs are on shell or off shell. (c) and (d) are due to on shell conditions for Line 1 and Line N: $k_1^2=0$ and $k_N^2$=0.}
\label{treecancel2}
\end{figure}
Figure \ref{treecancel1} and Figure \ref{treecancel2} constitute the proof of Ward identity at tree level.
\subsection{Recursion Relation for Tree Level Off Shell Amplitudes}\label{treerec}
As discussed in \cite{Chen}, from the complexified Ward identity ${{\hat k}_{\mbox{\tiny off}}}^\mu \cdot {\hat A}_\mu=0$, by a derivative over z we get:
\begin{equation}\label{recz}
\hat A_\mu \eta^\mu |_{z\rightarrow 0}=-{d\hat A_\mu\over dz}{\hat k}_{\mbox{\tiny off}}^\mu |_{z\rightarrow 0}.
\end{equation}
The symbol $\hat{}$ represents that the quantity is complexified, ie. depends on the shift parameter z. Here $k_{\mbox{\tiny off}}$ is shifted as in \ref{momshift}: ${\hat k}_{\mbox{\tiny off}}=k_{\mbox{\tiny off}}+z\eta$. Our destination is to calculate $A_\mu$, and we will realize it by calculating the right hand side of \ref{recz}.
We name the vertex which contains $L_{\mbox{\tiny off}}$ as $V_{\mbox{\tiny off}}$. At tree level, we have the following three cases:
\begin{enumerate}
\item the derivative acts on a propagator;
\item the derivative acts on a three point vertex which does not contain $L_{\mbox{\tiny off}}$;
\item the derivative acts on a three point vertex $V_{\mbox{\tiny off}}$.
\end{enumerate}
In the first and second cases, when $V_{\mbox{\tiny off}}$ is a three point vertex, we write $k_{\mbox{\tiny off}}^\mu \cdot V_{\mbox{\tiny off}}{}_{\ \mu}$ as in Figure \ref{vertexnotation}, and take out the hollow triangle terms. These terms, together with the terms from the third case where the derivative acts on the M part of $V_{\mbox{\tiny off}}{}_{\ \mu}$ as written in \ref{newV3s}, add up to be 0 due to Ward identity for some sub amplitudes.
From above we know that in the first and second cases, we only need the solid triangle terms for $k_{\mbox{\tiny off}}^\mu \cdot V_{\mbox{\tiny off}}{}_{\ \mu}$ as represented in Figure \ref{vertexnotation}, when $V_{\mbox{\tiny off}}$ is a three point vertex; in the third case, $\frac{d}{dz}$ only need to act on the S and R part of $V_{\mbox{\tiny off}}{}_{\ \mu}$ as written in \ref{newV3s}. The first two cases can be further simplified. Due to (a) and (b) in Figure \ref{treecancel2}, the terms relevant for the first two cases are reduced to those with $k_{\mbox{\tiny off}}$ neighboring to the three point vertex or the propagator to be differentiated, as depicted in Figure \ref{Tree1}.
\begin{figure}[htb]
\centering
\includegraphics[height=13cm,width=10cm]{Tree1.eps}
\caption{Terms to be calculated for tree-level off-shell amplitudes. Here and following, the dark solid circle symbol $\bullet$ denotes where we act $d\over dz$. We shift $L_{\mbox{\tiny off}}$ and some other line $L_s$. $\{A_i\}$ denote some sub-diagrams with less external states. $\hat A_k$ includes $L_s$. In different diagrams, the same $A_k$ symbols do not mean the same sub amplitudes. They sum over all allowed sub amplitudes.}
\label{Tree1}
\end{figure}
Thus, for the first case, the diagrams are (a) and (b) in Figure \ref{Tree1}. The contributions from (a) and (b) to $-{d\hat A_\mu\over dz}{\hat k}_{\mbox{\tiny off}}^\mu |_{z\rightarrow 0}$ are:
\begin{eqnarray}
&&\mbox{for (a)}\ \ \frac{-i\sqrt{2}}{k_{A_1}^2 k_{A_2}^2}k_{A_1}\cdot \eta\ A_1\cdot A_2,\nonumber\\
&&\mbox{for (b)}\ \ \frac{i\sqrt{2}}{k_{A_1}^2 k_{A_2}^2}k_{A_2}\cdot \eta\ A_1\cdot A_2.
\label{preexptree1}
\end{eqnarray}
As noted in Figure \ref{Tree1}, \{$A_i$\} are some less point amplitudes. $k_{A_i}$ is the total momentum of the external legs contained in the sub amplitude $A_i$. If some $A_i$ just contains one external line $L_m$, we define this $A_i$ to be $i k_m^2 \epsilon_m$, and accompany it with a propagator $\frac{-i}{k_{A_i}^2}=\frac{-i}{k_m^2}$. Another point is that, although $k_s\cdot \eta=0$, we keep it in the evaluations here and below as if it is not 0, as will be explained at the end of this subsection.
The second case corresponds to (c) (d) (e) and (f) in Figure \ref{Tree1}, and the contributions to $-{d\hat A_\mu\over dz}{\hat k}_{\mbox{\tiny off}}^\mu |_{z\rightarrow 0}$ are:
\begin{eqnarray}
&&\mbox{for (c)}\ \ \frac{1}{2 k_{A_1}^2 k_{A_2}^2 k_{A_3}^2}(A_3\cdot \eta\ A_1\cdot A_2+A_1\cdot \eta\ A_2\cdot A_3-2A_2\cdot \eta\ A_1\cdot A_3),\nonumber\\
&&\mbox{for (d)}\ \ \frac{1}{2 k_{A_1}^2 k_{A_2}^2 k_{A_3}^2}(-A_3\cdot \eta\ A_1\cdot A_2+2A_1\cdot \eta\ A_2\cdot A_3-A_2\cdot \eta\ A_1\cdot A_3),\nonumber\\
&&\mbox{for (e)}\ \ \frac{-1}{2 k_{A_1}^2 k_{A_2}^2 k_{A_3}^2}(-2A_3\cdot \eta\ A_1\cdot A_2+A_1\cdot \eta\ A_2\cdot A_3+A_2\cdot \eta\ A_1\cdot A_3),\nonumber\\
&&\mbox{for (f)}\ \ \frac{-1}{2 k_{A_1}^2 k_{A_2}^2 k_{A_3}^2}(-A_1\cdot \eta\ A_2\cdot A_3+2A_2\cdot \eta\ A_1\cdot A_3-A_3\cdot \eta\ A_1\cdot A_2).
\label{preexptree2}
\end{eqnarray}
And the third case corresponds to (g) and (h) in Figure \ref{Tree1}, whose contributions are:
\begin{eqnarray}
&&\mbox{for (g)}\ \ \frac{-i}{\sqrt{2} k_{A_1}^2 k_{A_2}^2}(k_{\mbox{\tiny off}}\cdot \eta \ A_1\cdot A_2+2A_1\cdot \eta\ k_{\mbox{\tiny off}}\cdot A_2-2A_2\cdot \eta\ k_{\mbox{\tiny off}}\cdot A_1),\nonumber\\
&&\mbox{for (h)}\ \ \frac{-i}{\sqrt{2} k_{A_1}^2 k_{A_2}^2}(-k_{\mbox{\tiny off}}\cdot \eta\ A_1\cdot A_2+2A_1\cdot \eta\ k_{\mbox{\tiny off}}\cdot A_2-2A_2\cdot \eta\ k_{\mbox{\tiny off}}\cdot A_1).
\label{preexptree3}
\end{eqnarray}
As explained before \ref{preexptree1}, in this case $\frac{d}{dz}$ only need to act on the S and R part of $V_{\mbox{\tiny off}}{}_{\ \mu}$ as written in \ref{newV3s}.
It can be observed that, \ref{preexptree2} for $L_s$ contained in $A_1$ or $A_2$ or $A_3$, the expressions are the same. In the case when $L_s$ is contained in $A_2$ we should sum (d) and (e) in Figure \ref{Tree1} to see that the expression is the same as when $L_s$ is contained in $A_1$ or $A_3$. The common expression is:
\begin{equation}
\frac{1}{2 k_{A_1}^2 k_{A_2}^2 k_{A_3}^2}(A_3\cdot \eta\ A_1\cdot A_2+A_1\cdot \eta\ A_2\cdot A_3-2A_2\cdot \eta\ A_1\cdot A_3).
\label{finaltree1}
\end{equation}
\ref{preexptree1} and \ref{preexptree2} summed up also give a common expression, regardless of whether $L_s$ is contained in $A_1$ or $A_2$:
\begin{equation}
\frac{-i}{\sqrt{2} k_{A_1}^2 k_{A_2}^2}\left(\ (k_{A_1}-k_{A_2})\cdot \eta\ A_1\cdot A_2+2A_1\cdot \eta\ k_{\mbox{\tiny off}}\cdot A_2-2A_2\cdot \eta\ k_{\mbox{\tiny off}}\cdot A_1\right)
\label{finaltree2}
\end{equation}
The final tree level result for $A_\mu \eta^\mu$ is the sum of \ref{finaltree1} and \ref{finaltree2}, which can be written in the form of $\tilde A_\mu \eta^\mu$. In the expressions we should sum over all allowed allocations of the on shell external legs into $\{A_i\}$. It is easy to show that the off shell amplitude $A_\mu=\tilde A_\mu$. In four dimensional spacetime, we only need to find 4 independent $\eta_i$ such that $A_\mu \eta^\mu_i=\tilde A_\mu \eta^\mu_i$. Since in the shift $\eta^\mu$ is required to satisfy $k_s \cdot \eta=0$ and $\eta^2=0$, the three choices of $\eta_i$ as $\epsilon_s^+$, $\epsilon_s^-$ or $k_s$ satisfy $A_\mu \eta^\mu_i=\tilde A_\mu \eta^\mu_i$. The remaining choice of $\eta_i$ can be chosen as $k_{\mbox{\tiny off}}$. This is not obvious to satisfy $A_\mu \eta^\mu_i=\tilde A_\mu \eta^\mu_i$. However, in our calculations we have kept the terms $k_s\cdot \eta$ as if it is not 0, and by this trick it comes out that $\tilde A_\mu k_{\mbox{\tiny off}}^\mu=0=A_\mu k_{\mbox{\tiny off}}^\mu$. In conclusion $A_\mu=\tilde A_\mu$, where $\tilde A_\mu$ is contained in the sum of \ref{finaltree1} and \ref{finaltree2} in form of $\tilde A_\mu \eta^\mu$.
Compare to Berends-Giele recursion relation \cite{Berends:1987me}, it is seen that \ref{finaltree1} corresponds to $k_{\mbox{\tiny off}}$ contained in a four point vertex, and \ref{finaltree2} is equivalent to the contribution when $k_{\mbox{\tiny off}}$ is contained in a three point vertex. This on one hand supports the correctness of our method, and on the other hand a little undermines the value of our method at tree level. There are also other recursion relations for off shell tree level amplitudes, eg. \cite{Feng}. Yet we are going to extend our method to one loop level, where the situation is much more complicated and our method is new.
\section{Ward Identity and Implied Recursion Relation at 1-loop Level}\label{LoopWard}
In this section we are going to extend our method to 1-loop level. We will show how complexified Ward identity holds at 1-loop level and then we deduce the corresponding recursive calculation of 1-loop off shell amplitude. Using our method, we will calculate three and four point 1-loop off shell amplitudes as examples. In our calculation we use FDH scheme \cite{Bern:1991aq}, in which only the loop momentum is continued to dimensionality different from 4.
We first explain some subtleties at loop level. First, after momentum shifting, some lines on the loop carry complex momenta. This brings ambiguities to the meaning of the loop integral and prevents us from translating the loop momentum $l\rightarrow l+k$ or flip it $l\rightarrow -l$. However, according to equation \ref{recz}, what we need for our technique is the derivative of the integral at the value $z\rightarrow 0$. And it is easy to prove that:
\begin{eqnarray}
&&\int d^D l \frac{d}{dz}f(l^\mu,{\hat k}^\mu)|_{z\rightarrow 0}=\int d^D l \frac{d}{dz}f(-l^\mu,{\hat k}^\mu)|_{z\rightarrow 0},\\
&&\int d^D l \frac{d}{dz}f(l^\mu,{\hat k}^\mu)|_{z\rightarrow 0}=\int d^D l \frac{d}{dz}f(l^\mu+{\hat k}'^\mu,{\hat k}^\mu)|_{z\rightarrow 0}.
\label{complexintegrand}
\end{eqnarray}
Thus for our technique, we can translate or flip the loop momentum even when the integrand is complex.
Second, some attention should be paid to color orderings and symmetry factors. At tree level there is only one color ordering contributing to the the primitive part of the color ordered amplitudes. At one loop level, most diagrams also only have one color ordering. However, for gauge field loop diagrams, there are three kinds of diagrams having two color orderings. Those are diagrams with two vertexes on the loop: two three-point vertexes; two four-point vertexes; a three-point vertex and a four-point vertex. For the first and second cases, the contributions from the two color orderings are the same at integrand level. For the third case, the contributions from the two color orderings at integrand level differ by a translation and flip of the loop momentum, and due to \ref{complexintegrand} the two orderings contribute the same in our method after integration. In a word, these three kinds of diagrams have a factor of 2 from possible color orderings. At the same time, these three kinds of diagrams have symmetry factor $\frac{1}{2}$, just canceling the doubling from color orderings. For ghost loop diagrams, those with two vertexes on the loop also have a doubling from two color orderings, while there is only either clockwise or anti clockwise ghost loop when there are only two vertexes on the ghost loop. We replace the doubling from color orderings by drawing both clockwise and anti-clockwise ghost loop diagrams, which are actually equal when there are only two vertexes on the ghost loop.
Finally, as our convention for the loop momentum for all our loop diagrams, we specify the loop momentum in the following way. For each external leg $L_i$, when we want to make a path from it to the loop, there is one definite vertex $V$ on the loop first encountered in the path, then we say the external leg $L_i$ is associated with this loop vertex $V$. We find the vertex with which $L_{\mbox{\tiny off}}$ is associated, call it $V_0$. Assume all the lines associated with $V_0$ in color ordering are $L_j, L_{j+1}, \cdots, L_N, L_{\mbox{\tiny off}}, L_1, \cdots, L_i$, then we assign the momentum of the first loop propagator on the counter clockwise side of $V_0$ as $l-k_1-\cdots-k_i$, with the loop momentum flowing in counter clockwise direction. $l$ is to be integrated. External leg momenta are outgoing.
\subsection{Proof of Ward Identity at 1-Loop Level}\label{loopWI}
In this section we use $A^l$ for 1-loop amplitudes, and $A^t$ for tree level amplitudes.
Two point and three point 1-loop Ward identity is easy to verify directly. Similar to the proof at tree level, we use induction, assume Ward identity holds for N and less than N point one loop amplitudes, and construct an (N+1) point diagram from an N point one by inserting $k_{\mbox{\tiny off}}$ in different places. We denote the vertex with $k_{\mbox{\tiny off}}$ as $V_{\mbox{\tiny off}}$ and when $V_{\mbox{\tiny off}}$ is a three point vertex, we decompose $k_{\mbox{\tiny off}} \cdot V_{\mbox{\tiny off}}$ as in Figure \ref{vertexnotation}.
{\bf Case 1.} When $k_{\mbox{\tiny off}}$ is linked to a propagator(including gauge field loop propagator) or external line of the N point diagram, the solid triangle terms from $k_{\mbox{\tiny off}} \cdot V_{\mbox{\tiny off}}$ mostly cancel the terms with $k_{\mbox{\tiny off}}$ in a four point vertex, in the manner of Figure \ref{treecancel2}. Only the terms in Figure \ref{loopremain1} remain.
\begin{figure}[htb]
\centering
\includegraphics[height=3.8cm,width=13cm]{loopremain1.eps}
\caption{The remaining terms in the first case that does not cancel in the manner of Figure \ref{treecancel2}. The loop is ghost loop and has two directions.}
\label{loopremain1}
\end{figure}
{\bf Case 2.} We need to consider the hollow triangle terms from $k_{\mbox{\tiny off}} \cdot V_{\mbox{\tiny off}}$ remaining from the above case, and we divide them into two sub cases:
\ \ {\bf \small Sub Case 1.} When $V_{\mbox{\tiny off}}$ is not on the loop, these terms vanish due to Ward identity for less point amplitudes in the induction assumption, similar to the tree level counterpart Figure \ref{treecancel1}.
\ \ {\bf \small Sub Case 2.} The remaining sub case is that $V_{\mbox{\tiny off}}$ is on the gauge field loop. We analyze one of the hollow triangle terms in Figure \ref{goontheloop}. The Figure has considered all the possible cases with the first right side vertex to be three or four point, and different types of second vertex relevant. When the first right side vertex is a three point vertex, acting on it with one of the factor in the hollow triangle term, we can again decompose it as in Figure \ref{vertexnotation} into solid and hollow triangle terms. (a) and (b) in Figure \ref{goontheloop} are in fact the same diagrams as in Figure \ref{treecancel2}. (c) vanishes due to tree level Ward identity, and (d) is due to on shell condition for external legs besides $L_{\mbox{\tiny off}}$. Then the type of term in (e) of Figure \ref{goontheloop} remains, which is a hollow triangle term staying on the loop, and it will act on the next vertex on the loop, repeating the same processes as in (a)-(d) of Figure \ref{goontheloop}, until it meets the final vertex on the loop. For this sub case, the remaining diagrams are in Figure \ref{loopremain2}.
\begin{figure}[htb]
\centering
\includegraphics[height=15cm,width=14cm]{goontheloop.eps}
\caption{Analysis of the action of the hollow triangle terms in {\bf\small Sub Case 2}. The dashed line is not ghost field, but just part of the loop diagram not relevant.}
\label{goontheloop}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[height=8cm,width=12cm]{loopremain2.eps}
\caption{The terms from {\bf \small Sub Case 2}. Except the hollow triangle terms at $V_{\mbox{\tiny off}}$, other hollow and solid triangle terms on the loop are induced from the hollow triangle term of the previous loop vertex, as described in the text of {\bf\small Sub Case 2.} and Figure \ref{goontheloop}.}
\label{loopremain2}
\end{figure}
{\bf Case 3.} The remaining case: $k_{\mbox{\tiny off}}$ is linked to a ghost propagator of the N point diagram, as in Figure \ref{loopremain3}.
\begin{figure}[htb]
\centering
\includegraphics[height=4cm,width=6cm]{loopremain3.eps}
\caption{Diagram for {\bf Case 3.} The ghost loop can be in two directions.}
\label{loopremain3}
\end{figure}
By direct and simple calculations, the terms from Figure \ref{loopremain1}, Figure \ref{loopremain2} and Figure \ref{loopremain3}, with same set of sub amplitudes $A_i$, add up to be 0. Combine {\bf Case 1, 2, 3}, we have proven that Ward identity holds at $N+1$ point one loop level. Thus by induction we have proven Ward identity holds at one loop level using Feynman rules in a direct way.
\subsection{Recursion Relation for Loop Level Off Shell Amplitudes}\label{LoopRec}
Similar to the tree level off shell amplitudes calculation, we can use $\hat A_\mu \eta^\mu |_{z\rightarrow 0}=-{d\hat A_\mu\over dz}{\hat k}_{\mbox{\tiny off}}^\mu |_{z\rightarrow 0}$ to calculate one loop level off shell amplitudes. The experience at tree level, and the details of how Ward identity holds at one loop level discussed in the last subsection, help us to simplify our discussion and calculation of one loop level off shell amplitudes.
When the derivative acts on a gauge field propagator or a vertex which is not on the loop, we can use the expressions derived in section \ref{treerec} directly, ie. \ref{finaltree1} and \ref{finaltree2}, for the contribution to $-{d\hat A_\mu\over dz}{\hat k}_{\mbox{\tiny off}}^\mu |_{z\rightarrow 0}$:
\begin{eqnarray}
&&\frac{1}{2 k_{A_1}^2 k_{A_2}^2 k_{A_3}^2}(A^{l/t}_3\cdot \eta\ A^{l/t}_1\cdot A^{l/t}_2+A^{l/t}_1\cdot \eta\ A^{l/t}_2\cdot A^{l/t}_3-2A^{l/t}_2\cdot \eta\ A^{l/t}_1\cdot A^{l/t}_3),\label{finalloop1}
\\
&&\frac{-i}{\sqrt{2} k_{A_1}^2 k_{A_2}^2}\left(\ (k_{A_1}-k_{A_2})\cdot \eta\ A^{l/t}_1\cdot A^{l/t}_2+2A^{l/t}_1\cdot \eta\ k_{\mbox{\tiny off}}\cdot A^{l/t}_2-2A^{l/t}_2\cdot \eta\ k_{\mbox{\tiny off}}\cdot A^{l/t}_1\right).\nonumber
\end{eqnarray}
In \ref{finalloop1}, we allocate the on shell external legs into $\{A^{l/t}_i\}$ in color ordering, with one and only one $A_i^{l/t}$ being one loop level. As in tree level, in each expression we should sum over all allowed allocations of the on shell external legs into $\{A^{l/t}_i\}$.
When the derivative acts on a gauge field loop propagator or a loop vertex, these are shown in Figure \ref{loop1}. For the same reasons as discussed in tree level recursion calculation, in (a) to (f), we only need to consider $L_{\mbox{\tiny off}}$ next to the propagator or vertex differentiated and only need the solid triangle term. In (g), we only differentiate the S and R terms of the vertex. The M part of the vertex in (g) will be dealt with in the following. In Figure \ref{loop1}, we encounter tree level two line off shell amplitudes $A^t_{\sigma\rho}$. This quantity can also be calculated recursively using our method, but in this paper we will not discuss it, and will use Feynman rules to calculate it in our example. Those $A^t$ without sub indices are tree level one line off shell amplitudes, which can apply our method in the previous section. (a) is 0 due to our convention for the loop momentum, described in the paragraph before section \ref{loopWI}.
Regardless of whether the other shifted line $L_s$ is among $\{L_1,L_2,\cdots,L_j\}$ or among $\{L_{j+1},\cdots,L_N\}$, the contributions to $-{d\hat A_\mu\over dz}{\hat k}_{\mbox{\tiny off}}^\mu |_{z\rightarrow 0}$ from Figure \ref{loop1} are (we use $K_{m,n}$ to represent for $k_m+k_{m+1}+\cdots+k_n$):
\begin{eqnarray}
&&(a)\ \ \ \ \ \ \ \,:0\label{finalloop2}\\
&&(b)+(g): \frac{-i}{\sqrt{2} l^2 (l+k_{\mbox{\tiny off}})^2}(\ (2 l+k_{\mbox{\tiny off}})\cdot \eta\ A^t_{\sigma\rho}(1,2,\cdots,N) \ g^{\sigma\rho}\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ +2\eta^\sigma\ A^t_{\sigma\rho}(1,2,\cdots,N) \ k^\rho-2\eta^\rho\ A^t_{\sigma\rho}(1,2,\cdots,N) \ k^\sigma)\nonumber\\
&&(d)+(e): \frac{1}{2 l^2(l-K_{1,j})^2 K_{j+1,N}^2}(A^t(j+1,\cdots,N)\cdot \eta\ A^t_{\sigma\rho}(1,2,\cdots,j) \ g^{\sigma\rho}\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ -2A^{t\ \sigma}(j+1,\cdots,N) \ \eta^\rho \ A^t_{\sigma\rho}(1,2,\cdots,j) +A^{t\ \rho}(j+1,\cdots,N)\ \eta^\sigma\ A^t_{\sigma\rho}(1,2,\cdots,j)\ )\nonumber\\
&&(c)+(f):\frac{1}{2 (l+k_{\mbox{\tiny off}})^2(l-K_{1,j})^2 K_{1,j}^2}(A^t(1,\cdots,j)\cdot \eta\ A^t_{\sigma\rho}(j+1,\cdots,N)\ g^{\sigma\rho}\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ -2A^{t\ \rho}(1,\cdots,j) \ \eta^\sigma\ A^t_{\sigma\rho}(j+1,\cdots,N) +A^{t\ \sigma}(1,\cdots,j)\ \eta^\rho\ A^t_{\sigma\rho}(j+1,\cdots,N)\ )\nonumber
\end{eqnarray}
\begin{figure}[htb]
\centering
\includegraphics[height=15cm,width=11cm]{loop1.eps}
\caption{Diagrams with derivative acting on the propagator or vertex on the loop, which cannot directly apply the tree level results.}
\label{loop1}
\end{figure}
The final contributions to $-{d\hat A^l_\mu\over dz} {{\hat k}_{\mbox{\tiny off}}}^\mu |_{z\rightarrow 0}$ come from the derivatives in the diagrams of Figure \ref{loopremain1}, Figure \ref{loopremain2} and Figure \ref{loopremain3}. Denoting the diagrams as $D_i$, since $\sum D_i\cdot k_{\mbox{\tiny off}}=0$ from the last subsection, we have $-\sum {d\hat D_{i\ \mu}\over dz}{\hat k}_{\mbox{\tiny off}}^\mu |_{z\rightarrow 0}=\sum D_{i\ \mu}\ \eta^\mu$. This is like an opposite operation compared to the method in the current paper to deal with the set of diagrams $D_i$, but it simplifies the local calculation, eg. the diagrams in Figure \ref{loopremain1} turn out to be not contributing. We use $K_{A_{m,n}}$ to represent for $k_{A_m}+k_{A_{m+1}}+\cdots+k_{A_n}$, with $k_{A_i}$ the total momentum of the external legs contained in the sub amplitude $A_i$. The total momentum conservation is then $K_{A_{1,n}}+k_{\mbox{\tiny off}}=0$. The contributions to $-{d\hat A^l_\mu\over dz} {{\hat k}_{\mbox{\tiny off}}}^\mu |_{z\rightarrow 0}$ from derivatives in Figure \ref{loopremain1}, Figure \ref{loopremain2} and Figure \ref{loopremain3} are:
\begin{eqnarray}
&&\frac{(-i)^n\ (l-K_{A_{1,1}})\cdot A_2\ (l-K_{A_{1,2}})\cdot A_3\ \cdots (l-K_{A_{1,n-2}})\cdot A_{n-1}}{(\sqrt{2})^{n+1}k_{A_1}^2\cdots k_{A_n}^2(l-K_{A_{1,1}})^2\cdots(l-K_{A_{1,n-1}})^2}\nonumber\\
&&(-\frac{2 l\cdot A_1\ (l-K_{A_{1,n-1}})\cdot A_n\ (2l+k_{\mbox{\tiny off}})\cdot \eta}{l^2(l+k_{\mbox{\tiny off}})^2}\nonumber\\
&&+\frac{l\cdot A_1\ A_n \cdot \eta}{l^2}+\frac{(l-K_{A_{1,n-1}})\cdot A_n \ A_1\cdot \eta}{(l+k_{\mbox{\tiny off}})^2}).
\label{finalloop3}
\end{eqnarray}
This expression is well defined when $n\ge 2$. Especially when $n=2$, one should multiply the pre-factor with each term in the bracket to see that it is well defined. When $n=1$, the last two terms in the bracket vanish.
\ref{finalloop1}, \ref{finalloop2} and \ref{finalloop3} constitute our expressions for recursively calculating one loop off shell amplitudes. In each expression, eg. in the above one \ref{finalloop3}, we should sum over all the allowed different allocations of the on shell external legs into $A_1, \cdots, A_n$, with $n=1,2,\cdots,N$. This summation is not written explicitly in the expressions. Similar to our statement in the tree level counterpart, summing over \ref{finalloop1}, \ref{finalloop2} and \ref{finalloop3}, we get a form $A^l_\mu \eta^\mu$, with the $A^l_\mu$ our wanted one loop off shell amplitude.
\subsection{Examples of 1-loop Off Shell Amplitudes}\label{example}
As an application and verification of our method, we have computed three and four point one loop amplitudes with one off shell leg. ie. $A^l_\mu(k_1,k_2)$ and $A^l_\mu(k_1,k_2,k_3)$, by summing up the contributions from \ref{finalloop1}, \ref{finalloop2} and \ref{finalloop3}. We use the integral reduction method in \cite{Ellis} to reduce the integrals to scalar integrals. We use the following notations for the scalar integrations:
\begin{eqnarray*}
&&B0[1,3]=\int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k_1-k_2)^2},\ \ B0[1,4]=\int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k_1-k_2-k_3)^2},\\
&&B0[2,4]=\int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k_2-k_3)^2},\ \ C0[1,2,3]=\int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k_1)^2(l-k_1-k_2)^2},\\
&&C0[1,2,4]=\int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k_1)^2(l-k_1-k_2-k_3)^2},\\
&&C0[1,3,4]=\int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k_1-k_2)^2(l-k_1-k_2-k_3)^2},\\
&&C0[2,3,4]=\int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k_2)^2(l-k_2-k_3)^2},\\
&&D0[1,2,3,4]=\int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k_1)^2(l-k_1-k_2)^2(l-k_1-k_2-k_3)^2}.
\end{eqnarray*}
Other scalar integrations are not needed in this article. The evaluation of the scalar integrals see \cite{BernD2}.
We start from the two point function:
\begin{equation}
A^l_{\mu\nu}(k)=\frac{2-3D}{2(1-D)}(k^2 g_{\mu\nu}-k_\mu k_\nu) \int \frac{d^D l}{(2\pi)^D}\frac{1}{l^2(l-k)^2}.
\label{twopoint1loop}
\end{equation}
Then we can calculate three point one loop off shell amplitude using our method:
\begin{eqnarray}
&&A^l_\mu(k_1,k_2)=\frac{1}{2\sqrt{2}}\left[k_1\cdot \epsilon_2 \ \epsilon_{1\ \mu}-k_2\cdot \epsilon_1\ \epsilon_{2\ \mu}-\frac{2D-5}{D-1}\epsilon_1\cdot \epsilon_2\ (k_1-k_2)_\mu\right.\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left.+\frac{D-4}{D-1}\frac{k_1\cdot \epsilon_2\ k_2 \cdot \epsilon_1}{k_1\cdot k_2}(k_1-k_2)_\mu\right] B0[1,3]\label{threepoint1loop}\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{2\sqrt{2}}k_1\cdot k_2[-3 \epsilon_1\cdot \epsilon_2 \ (k_1-k_2)_\mu+4 k_1\cdot \epsilon_2 \ \epsilon_{1\ \mu}-4 k_2 \cdot \epsilon_1 \ \epsilon_{2\ \mu}]C0[1,2,3].\nonumber
\end{eqnarray}
At four point, the length of the expressions grow very quickly, and we will only give $A^l_\mu(1^+,2^+,3^+)$. Instead of giving this expression directly, we will give $A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu$, $A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu$ and $A^l_\mu(1^+,2^+,3^+) k_1^\mu$. Together with $A^l_\mu(1^+,2^+,3^+) (k_1+k_2+k_3)^\mu=0$, the expressions are enough to determine all the 4 components of $A^l_\mu(1^+,2^+,3^+)$. We choose the spinor representations for $k_{1,2,3}$ and $\epsilon_{1,2,3}$ to be:
\begin{equation}
k_1=\lambda_1 \tilde \lambda_1,\ k_2=\lambda_2 \tilde \lambda_2,\ k_3=\lambda_3 \tilde \lambda_3,\ \epsilon_1=\frac{\lambda_\nu \tilde \lambda_1}{\langle\lambda_\nu \lambda_1\rangle},\ \epsilon_2=\frac{\lambda_\nu \tilde \lambda_2}{\langle\lambda_\nu \lambda_2\rangle},\ \epsilon_3=\frac{\lambda_\nu \tilde \lambda_3}{\langle\lambda_\nu \lambda_3\rangle},
\end{equation}
with $\lambda_\nu$ an arbitrary reference spinor. We will use $\langle\nu 1\rangle$ to stand for $\langle\lambda_\nu \lambda_1\rangle$ and similarly others. We use $(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{D0[1,2,3,4]}$ to denote for the coefficient of $D0[1,2,3,4]$ in $A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu$, and similarly for others. We give the coefficients at $D=4$. The off shell line makes the expressions much more complicated than that with all on shell lines. On one hand, when all lines are on shell, since the amplitudes are gauge invariant, we can choose some specific reference spinor, while in the off shell case we should keep the reference spinor $\lambda_\nu$ arbitrary. On the other hand, there are many terms in the expressions below which is 0 when all lines are on shell. For example, the first coefficient below would be 0 due to $(\langle 12 \rangle [12]+\langle 13 \rangle [13]+\langle 23 \rangle [23])=0$ when all lines were on shell.
Then for $A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu$:
\begin{scriptsize}
\begin{eqnarray*}
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{D0[1,2,3,4]}\\
&&=-\frac{(\langle 13 \rangle \langle 2\nu \rangle-2 \langle 12 \rangle \langle 3\nu \rangle) [12] (\langle 2\nu \rangle [12]+\langle 3\nu \rangle [13]) [23] (\langle 12 \rangle [12]+\langle 13 \rangle [13]+\langle 23 \rangle [23])}{64 \langle 13 \rangle^2 \langle 1\nu \rangle \langle 2\nu \rangle},\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{C0[1,2,4]}\\
&&=-\frac{(\langle 2\nu \rangle [12]+\langle 3\nu \rangle [13])}{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle^2 \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13])} (\langle 1\nu \rangle^2 \langle 23 \rangle^2 (\langle 12 \rangle \langle 13 \rangle \langle 2\nu \rangle [12]^2+2 \langle 12 \rangle \langle 12 \rangle \langle 3\nu \rangle [12]^2\\
&&+4 \langle 12 \rangle \langle 13 \rangle \langle 3\nu \rangle [13] [12]+\langle 13 \rangle^2 \langle 3\nu \rangle [13]^2) [23]-\langle 1\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13]) (2 \langle 3\nu \rangle^2 [12]^2 \langle 12 \rangle^3+\langle 13 \rangle \langle 3\nu \rangle [12] (4 \langle 3\nu \rangle [13]\\
&&-\langle 2\nu \rangle [12]) \langle 12 \rangle^2-2 \langle 13 \rangle^2 \langle 2\nu \rangle [12] (\langle 2\nu \rangle [12]+3 \langle 3\nu \rangle [13]) \langle 12 \rangle-\langle 13 \rangle^3 \langle 2\nu \rangle \langle 3\nu \rangle [13]^2)),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{C0[2,3,4]}\\
&&=-\frac{(\langle 2\nu \rangle [12]+\langle 3\nu \rangle [13]) [23] }{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle^2 \langle 2\nu \rangle}(2 \langle 3\nu \rangle (\langle 1\nu \rangle [12]-\langle 3\nu \rangle [23]) \langle 12 \rangle^2+\langle 13 \rangle (-\langle 1\nu \rangle \langle 2\nu \rangle [12]+2 \langle 1\nu \rangle \langle 3\nu \rangle [13]\\
&&+3 \langle 2\nu \rangle \langle 3\nu \rangle [23]) \langle 12 \rangle+\langle 13 \rangle^2 \langle 2\nu \rangle (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{C0[1,3,4]}\\
&&=\frac{(\langle 2\nu \rangle [12]+\langle 3\nu \rangle [13])}{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle^2 \langle 23 \rangle \langle 2\nu \rangle (\langle 13 \rangle [13]+\langle 23 \rangle [23])} (-2 \langle 3\nu \rangle^2 [12] [23] (2 \langle 13 \rangle [13]+\langle 23 \rangle [23]) \langle 12 \rangle^3+\langle 3\nu \rangle (-2 \langle 23 \rangle^2 \langle 3\nu \rangle [23]^3\\
&&+3 \langle 13 \rangle \langle 23 \rangle (\langle 2\nu \rangle [12]-2 \langle 3\nu \rangle [13]) [23]^2+\langle 13 \rangle^2 [13] (\langle 1\nu \rangle [12] [13]-4 \langle 3\nu \rangle [23] [13]+4 \langle 2\nu \rangle [12] [23])) \langle 12 \rangle^2+\langle 13 \rangle \langle 2\nu \rangle \langle 3\nu \rangle [23] (\langle 13 \rangle^2 [13]^2\\
&&+5 \langle 13 \rangle \langle 23 \rangle [23] [13]+3 \langle 23 \rangle^2 [23]^2) \langle 12 \rangle+\langle 13 \rangle^2 \langle 2\nu \rangle (\langle 23 \rangle^2 \langle 2\nu \rangle [23]^3+3 \langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle [13] [23]^2+\langle 13 \rangle^2 [13]^2 (\langle 1\nu \rangle [13]+3 \langle 2\nu \rangle [23]))),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{C0[1,2,3]}\\
&&=\frac{[12] (\langle 2\nu \rangle [12]+\langle 3\nu \rangle [13])}{64 \langle 13 \rangle^2 \langle 1\nu \rangle^2 \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle} (\langle 2\nu \rangle (2 \langle 1\nu \rangle \langle 2\nu \rangle [12]+3 \langle 1\nu \rangle \langle 3\nu \rangle [13]+\langle 2\nu \rangle \langle 3\nu \rangle [23]) \langle 13 \rangle^2\\
&&+\langle 12 \rangle \langle 3\nu \rangle (\langle 1\nu \rangle \langle 2\nu \rangle [12]-2 \langle 1\nu \rangle \langle 3\nu \rangle [13]-3 \langle 2\nu \rangle \langle 3\nu \rangle [23]) \langle 13 \rangle+2 \langle 12 \rangle^2 \langle 3\nu \rangle^2 (\langle 3\nu \rangle [23]-\langle 1\nu \rangle [12])),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{B0[2,4]}\\
&&=-\frac{(\langle 2\nu \rangle [12]+\langle 3\nu \rangle [13])}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle^2 \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13])^2} (4 \langle 3\nu \rangle^2 [12] [13] (\langle 1\nu \rangle [12]-\langle 3\nu \rangle [23]) \langle 12 \rangle^3+\langle 13 \rangle (\langle 1\nu \rangle [12] (2 \langle 2\nu \rangle^2 [12]^2\\
&&-6 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+5 \langle 3\nu \rangle^2 [13]^2)+\langle 3\nu \rangle (-4 \langle 2\nu \rangle^2 [12]^2+7 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-2 \langle 3\nu \rangle^2 [13]^2) [23]) \langle 12 \rangle^2\\
&&+\langle 13 \rangle^2 (\langle 1\nu \rangle [13] (7 \langle 2\nu \rangle^2 [12]^2-6 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2)+\langle 2\nu \rangle (3 \langle 2\nu \rangle^2 [12]^2-10 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]\\
&&+\langle 3\nu \rangle^2 [13]^2) [23]) \langle 12 \rangle+5 \langle 13 \rangle^3 \langle 2\nu \rangle^2 [12] [13] (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{B0[1,4]}\\
&&=\frac{(\langle 2\nu \rangle [12]+\langle 3\nu \rangle [13])}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle^2 \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13])^2 (\langle 13 \rangle [13]+\langle 23 \rangle [23])^3} (-2 \langle 3\nu \rangle^3 [12]^3 [13] [23] (3 \langle 13 \rangle [13]+2 \langle 23 \rangle [23]) \langle 12 \rangle^5\\
&&+\langle 3\nu \rangle^2 [12]^2 [13] (-8 \langle 23 \rangle^2 \langle 3\nu \rangle [23]^3+\langle 13 \rangle \langle 23 \rangle (5 \langle 2\nu \rangle [12]-28 \langle 3\nu \rangle [13]) [23]^2+2 \langle 13 \rangle^2 [13] (\langle 1\nu \rangle [12] [13]-14 \langle 3\nu \rangle [23] [13]\\
&&+5 \langle 2\nu \rangle [12] [23])) \langle 12 \rangle^4-\langle 3\nu \rangle [12] (4 \langle 23 \rangle^3 \langle 3\nu \rangle^2 [13] [23]^4+4 \langle 13 \rangle \langle 23 \rangle^2 (\langle 2\nu \rangle^2 [12]^2-3 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+6 \langle 3\nu \rangle^2 [13]^2) [23]^3\\
&&+2 \langle 13 \rangle^2 \langle 23 \rangle [13] (6 \langle 2\nu \rangle [12]-11 \langle 3\nu \rangle [13]) (\langle 2\nu \rangle [12]-2 \langle 3\nu \rangle [13]) [23]^2+\langle 13 \rangle^3 [13]^2 (\langle 1\nu \rangle [12] [13] (3 \langle 2\nu \rangle [12]-8 \langle 3\nu \rangle [13])\\
&&+2 (6 \langle 2\nu \rangle^2 [12]^2-28 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+15 \langle 3\nu \rangle^2 [13]^2) [23])) \langle 12 \rangle^3+\langle 13 \rangle (\langle 23 \rangle^3 \langle 3\nu \rangle (-4 \langle 2\nu \rangle^2 [12]^2+7 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]\\
&&-2 \langle 3\nu \rangle^2 [13]^2) [23]^4+\langle 13 \rangle \langle 23 \rangle^2 (\langle 2\nu \rangle^3 [12]^3-32 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2+33 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]-10 \langle 3\nu \rangle^3 [13]^3) [23]^3\\
&&+\langle 13 \rangle^2 \langle 23 \rangle [13] (3 \langle 2\nu \rangle^3 [12]^3-72 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2+62 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]-14 \langle 3\nu \rangle^3 [13]^3) [23]^2+\langle 13 \rangle^3 [13]^2 (\langle 1\nu \rangle [12] [13] (\langle 2\nu \rangle^2 [12]^2\\
&&-14 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+6 \langle 3\nu \rangle^2 [13]^2)+(3 \langle 2\nu \rangle^3 [12]^3-64 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2+53 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]-6 \langle 3\nu \rangle^3 [13]^3) [23])) \langle 12 \rangle^2\\
&&+\langle 13 \rangle^2 \langle 2\nu \rangle (\langle 23 \rangle^3 (3 \langle 2\nu \rangle^2 [12]^2-10 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) [23]^4+\langle 13 \rangle \langle 23 \rangle^2 [13] (\langle 3\nu \rangle [13]-15 \langle 2\nu \rangle [12]) (3 \langle 3\nu \rangle [13]\\
&&-\langle 2\nu \rangle [12]) [23]^3+3 \langle 13 \rangle^2 \langle 23 \rangle [13]^2 (9 \langle 2\nu \rangle^2 [12]^2-26 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) [23]^2+\langle 13 \rangle^3 [13]^3 (\langle 1\nu \rangle [12] [13] (6 \langle 2\nu \rangle [12]\\
&&-11 \langle 3\nu \rangle [13])+(21 \langle 2\nu \rangle^2 [12]^2-58 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) [23])) \langle 12 \rangle+5 \langle 13 \rangle^3 \langle 2\nu \rangle^2 [12] [13] (\langle 23 \rangle^3 \langle 2\nu \rangle [23]^4\\
&&+4 \langle 13 \rangle \langle 23 \rangle^2 \langle 2\nu \rangle [13] [23]^3+6 \langle 13 \rangle^2 \langle 23 \rangle \langle 2\nu \rangle [13]^2 [23]^2+\langle 13 \rangle^3 [13]^3 (\langle 1\nu \rangle [13]+4 \langle 2\nu \rangle [23]))),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_1^\mu)_{B0[1,3]}\\
&&=-\frac{(\langle 2\nu \rangle [12]+\langle 3\nu \rangle [13])}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle^2 \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 13 \rangle [13]+\langle 23 \rangle [23])^3} (\langle 2\nu \rangle [13]^2 (\langle 1\nu \rangle [13] (\langle 2\nu \rangle [12]-\langle 3\nu \rangle [13])+\langle 2\nu \rangle (3 \langle 2\nu \rangle [12]\\
&&-5 \langle 3\nu \rangle [13]) [23]) \langle 13 \rangle^4+[13] (3 \langle 23 \rangle \langle 2\nu \rangle^2 (\langle 2\nu \rangle [12]-3 \langle 3\nu \rangle [13]) [23]^2+\langle 12 \rangle \langle 3\nu \rangle [13] (\langle 1\nu \rangle [13] (\langle 3\nu \rangle [13]-3 \langle 2\nu \rangle [12])\\
&&+6 \langle 2\nu \rangle (\langle 3\nu \rangle [13]-2 \langle 2\nu \rangle [12]) [23])) \langle 13 \rangle^3+(\langle 23 \rangle^2 \langle 2\nu \rangle^2 (\langle 2\nu \rangle [12]-7 \langle 3\nu \rangle [13]) [23]^3+6 \langle 12 \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle [13] (\langle 3\nu \rangle [13]\\
&&-2 \langle 2\nu \rangle [12]) [23]^2+2 \langle 12 \rangle^2 \langle 3\nu \rangle^2 [13]^2 (\langle 1\nu \rangle [12] [13]-3 \langle 3\nu \rangle [23] [13]+5 \langle 2\nu \rangle [12] [23])) \langle 13 \rangle^2-\langle 3\nu \rangle [23] (6 \langle 3\nu \rangle^2 [12] [13]^2 \langle 12 \rangle^3\\
&&+\langle 23 \rangle \langle 3\nu \rangle [13] (9 \langle 3\nu \rangle [13]-5 \langle 2\nu \rangle [12]) [23] \langle 12 \rangle^2+2 \langle 23 \rangle^2 \langle 2\nu \rangle (2 \langle 2\nu \rangle [12]-\langle 3\nu \rangle [13]) [23]^2 \langle 12 \rangle+2 \langle 23 \rangle^3 \langle 2\nu \rangle^2 [23]^3) \langle 13 \rangle\\
&&-4 \langle 12 \rangle^2 \langle 23 \rangle \langle 3\nu \rangle^3 [13] [23]^2 (\langle 12 \rangle [12]+\langle 23 \rangle [23])).
\end{eqnarray*}
\end{scriptsize}
For $A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu$:
\begin{scriptsize}
\begin{eqnarray*}
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu)_{D0[1,2,3,4]}\\
&&=\frac{(\langle 13 \rangle \langle 2\nu \rangle-2 \langle 12 \rangle \langle 3\nu \rangle) [12] [23] (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23]) (\langle 12 \rangle [12]+\langle 13 \rangle [13]+\langle 23 \rangle [23])}{64 \langle 13 \rangle^2 \langle 2\nu \rangle \langle 3\nu \rangle},\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu)_{C0[1,2,4]}\\
&&=\frac{(\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])}{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle^2 (\langle 12 \rangle [12]+\langle 13 \rangle [13])} (\langle 1\nu \rangle^2 \langle 23 \rangle^2 (\langle 12 \rangle (\langle 13 \rangle \langle 2\nu \rangle+2 \langle 12 \rangle \langle 3\nu \rangle) [12]^2+4 \langle 12 \rangle \langle 13 \rangle \langle 3\nu \rangle [13] [12]\\
&&+\langle 13 \rangle^2 \langle 3\nu \rangle [13]^2) [23]-\langle 1\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13]) (2 \langle 3\nu \rangle^2 [12]^2 \langle 12 \rangle^3+\langle 13 \rangle \langle 3\nu \rangle [12] (4 \langle 3\nu \rangle [13]-\langle 2\nu \rangle [12]) \langle 12 \rangle^2\\
&&-2 \langle 13 \rangle^2 \langle 2\nu \rangle [12] (\langle 2\nu \rangle [12]+3 \langle 3\nu \rangle [13]) \langle 12 \rangle-\langle 13 \rangle^3 \langle 2\nu \rangle \langle 3\nu \rangle [13]^2)),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu)_{C0[2,3,4]}\\
&&=\frac{[23] (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])}{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle \langle 2\nu \rangle \langle 3\nu \rangle} (2 \langle 3\nu \rangle (\langle 1\nu \rangle [12]-\langle 3\nu \rangle [23]) \langle 12 \rangle^2+\langle 13 \rangle (-\langle 1\nu \rangle \langle 2\nu \rangle [12]+2 \langle 1\nu \rangle \langle 3\nu \rangle [13]+3 \langle 2\nu \rangle \langle 3\nu \rangle [23]) \langle 12 \rangle\\
&&+\langle 13 \rangle^2 \langle 2\nu \rangle (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])) ,\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu)_{C0[1,3,4]}\\
&&=-\frac{1}{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 13 \rangle [13]+\langle 23 \rangle [23])}(2 \langle 3\nu \rangle^3 [12] [13] [23]^2 \langle 12 \rangle^4+\langle 3\nu \rangle^2 [23] (2 \langle 23 \rangle (\langle 3\nu \rangle [13]-\langle 2\nu \rangle [12]) [23]^2\\
&&+\langle 13 \rangle [13] (-4 \langle 1\nu \rangle [12] [13]+6 \langle 3\nu \rangle [23] [13]-9 \langle 2\nu \rangle [12] [23])) \langle 12 \rangle^3+\langle 3\nu \rangle (-2 \langle 23 \rangle^2 \langle 2\nu \rangle \langle 3\nu \rangle [23]^4+\langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle (3 \langle 2\nu \rangle [12]\\
&&-11 \langle 3\nu \rangle [13]) [23]^3+\langle 13 \rangle^2 [13] (\langle 1\nu \rangle^2 [12] [13]^2+\langle 1\nu \rangle (5 \langle 2\nu \rangle [12]-4 \langle 3\nu \rangle [13]) [23] [13]+\langle 2\nu \rangle (7 \langle 2\nu \rangle [12]-15 \langle 3\nu \rangle [13]) [23]^2)) \langle 12 \rangle^2\\
&&+\langle 13 \rangle \langle 2\nu \rangle \langle 3\nu \rangle [23] (3 \langle 23 \rangle^2 \langle 2\nu \rangle [23]^3+7 \langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle [13] [23]^2+\langle 13 \rangle^2 [13]^2 (\langle 1\nu \rangle [13]+3 \langle 2\nu \rangle [23])) \langle 12 \rangle\\
&&+\langle 13 \rangle^2 \langle 2\nu \rangle (\langle 23 \rangle^2 \langle 2\nu \rangle^2 [23]^4+4 \langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle^2 [13] [23]^3+\langle 13 \rangle^2 [13]^2 (\langle 1\nu \rangle^2 [13]^2+4 \langle 1\nu \rangle \langle 2\nu \rangle [23] [13]+6 \langle 2\nu \rangle^2 [23]^2))) ,\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu)_{C0[1,2,3]}\\
&&=-\frac{[12] (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])}{64 \langle 13 \rangle^2 \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle^2}(\langle 2\nu \rangle (2 \langle 1\nu \rangle \langle 2\nu \rangle [12]+3 \langle 1\nu \rangle \langle 3\nu \rangle [13]+\langle 2\nu \rangle \langle 3\nu \rangle [23]) \langle 13 \rangle^2+\langle 12 \rangle \langle 3\nu \rangle (\langle 1\nu \rangle \langle 2\nu \rangle [12]\\
&&-2 \langle 1\nu \rangle \langle 3\nu \rangle [13]-3 \langle 2\nu \rangle \langle 3\nu \rangle [23]) \langle 13 \rangle+2 \langle 12 \rangle^2 \langle 3\nu \rangle^2 (\langle 3\nu \rangle [23]-\langle 1\nu \rangle [12])) ,\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu)_{B0[2,4]}\\
&&=\frac{(\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle^2 (\langle 12 \rangle [12]+\langle 13 \rangle [13])^2} (4 \langle 3\nu \rangle^2 [12] [13] (\langle 1\nu \rangle [12]-\langle 3\nu \rangle [23]) \langle 12 \rangle^3+\langle 13 \rangle (\langle 1\nu \rangle [12] (2 \langle 2\nu \rangle^2 [12]^2\\
&&-6 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+5 \langle 3\nu \rangle^2 [13]^2)+\langle 3\nu \rangle (-4 \langle 2\nu \rangle^2 [12]^2+7 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-2 \langle 3\nu \rangle^2 [13]^2) [23]) \langle 12 \rangle^2+\langle 13 \rangle^2 (\langle 1\nu \rangle [13] (7 \langle 2\nu \rangle^2 [12]^2\\
&&-6 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2)+\langle 2\nu \rangle (3 \langle 2\nu \rangle^2 [12]^2-10 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) [23]) \langle 12 \rangle\\
&&+5 \langle 13 \rangle^3 \langle 2\nu \rangle^2 [12] [13] (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])) ,\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu)_{B0[1,4]}\\
&&=-\frac{1}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle^2 (\langle 12 \rangle [12]+\langle 13 \rangle [13])^2 (\langle 13 \rangle [13]+\langle 23 \rangle [23])^2}(-4 \langle 3\nu \rangle^3 [12]^2 [13] [23] (\langle 1\nu \rangle [12] [13]-2 \langle 3\nu \rangle [23] [13]\\
&&+\langle 2\nu \rangle [12] [23]) \langle 12 \rangle^5+\langle 3\nu \rangle^2 [12] [13] (4 \langle 23 \rangle \langle 3\nu \rangle (\langle 3\nu \rangle [13]-2 \langle 2\nu \rangle [12]) [23]^3+\langle 13 \rangle (2 \langle 1\nu \rangle^2 [12]^2 [13]^2+\langle 1\nu \rangle [12] (7 \langle 2\nu \rangle [12]\\
&&-20 \langle 3\nu \rangle [13]) [23] [13]+(9 \langle 2\nu \rangle^2 [12]^2-40 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+20 \langle 3\nu \rangle^2 [13]^2) [23]^2)) \langle 12 \rangle^4+\langle 3\nu \rangle (-4 \langle 23 \rangle^2 \langle 2\nu \rangle \langle 3\nu \rangle^2 [12] [13] [23]^4\\
&&+\langle 13 \rangle \langle 23 \rangle (-4 \langle 2\nu \rangle^3 [12]^3+16 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2-31 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]+2 \langle 3\nu \rangle^3 [13]^3) [23]^3+\langle 13 \rangle^2 [13] (\langle 1\nu \rangle^2 [12]^2 (8 \langle 3\nu \rangle [13]\\
&&-3 \langle 2\nu \rangle [12]) [13]^2+\langle 1\nu \rangle [12] (-11 \langle 2\nu \rangle^2 [12]^2+42 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-24 \langle 3\nu \rangle^2 [13]^2) [23] [13]+(-13 \langle 2\nu \rangle^3 [12]^3+74 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2\\
&&-70 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]+8 \langle 3\nu \rangle^3 [13]^3) [23]^2)) \langle 12 \rangle^3+\langle 13 \rangle (\langle 23 \rangle^2 \langle 2\nu \rangle \langle 3\nu \rangle (-4 \langle 2\nu \rangle^2 [12]^2+7 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-2 \langle 3\nu \rangle^2 [13]^2) [23]^4\\
&&+\langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle (\langle 2\nu \rangle^3 [12]^3-35 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2+43 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]-11 \langle 3\nu \rangle^3 [13]^3) [23]^3+\langle 13 \rangle^2 [13] (\langle 1\nu \rangle^2 [12] (\langle 2\nu \rangle^2 [12]^2\\
&&-14 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+6 \langle 3\nu \rangle^2 [13]^2) [13]^2+\langle 1\nu \rangle (3 \langle 2\nu \rangle^3 [12]^3-58 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2+42 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]-6 \langle 3\nu \rangle^3 [13]^3) [23] [13]\\
&&+\langle 2\nu \rangle (3 \langle 2\nu \rangle^3 [12]^3-84 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2+98 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]-16 \langle 3\nu \rangle^3 [13]^3) [23]^2)) \langle 12 \rangle^2+\langle 13 \rangle^2 \langle 2\nu \rangle (\langle 23 \rangle^2 \langle 2\nu \rangle (3 \langle 2\nu \rangle^2 [12]^2\\
&&-10 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) [23]^4+3 \langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle [13] (5 \langle 2\nu \rangle^2 [12]^2-17 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) [23]^3\\
&&+\langle 13 \rangle^2 [13]^2 (\langle 1\nu \rangle^2 [12] (6 \langle 2\nu \rangle [12]-11 \langle 3\nu \rangle [13]) [13]^2+\langle 1\nu \rangle (21 \langle 2\nu \rangle^2 [12]^2-53 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) [23] [13]\\
&&+3 \langle 2\nu \rangle (9 \langle 2\nu \rangle^2 [12]^2-31 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) [23]^2)) \langle 12 \rangle+5 \langle 13 \rangle^3 \langle 2\nu \rangle^2 [12] [13] (\langle 23 \rangle^2 \langle 2\nu \rangle^2 [23]^4\\
&&+4 \langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle^2 [13] [23]^3+\langle 13 \rangle^2 [13]^2 (\langle 1\nu \rangle^2 [13]^2+4 \langle 1\nu \rangle \langle 2\nu \rangle [23] [13]+6 \langle 2\nu \rangle^2 [23]^2))) ,\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) \epsilon_3^\mu)_{B0[1,3]}\\
&&=\frac{1}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle^2 (\langle 13 \rangle [13]+\langle 23 \rangle [23])^2}(\langle 2\nu \rangle [13] (\langle 1\nu \rangle^2 (\langle 2\nu \rangle [12]-\langle 3\nu \rangle [13]) [13]^2+\langle 1\nu \rangle \langle 2\nu \rangle (3 \langle 2\nu \rangle [12]\\
&&-5 \langle 3\nu \rangle [13]) [23] [13]+3 \langle 2\nu \rangle^2 (\langle 2\nu \rangle [12]-3 \langle 3\nu \rangle [13]) [23]^2) \langle 13 \rangle^3+(\langle 23 \rangle \langle 2\nu \rangle^3 (\langle 2\nu \rangle [12]-7 \langle 3\nu \rangle [13]) [23]^3+\langle 12 \rangle \langle 3\nu \rangle [13] (\langle 1\nu \rangle^2 (\langle 3\nu \rangle [13]\\
&&-3 \langle 2\nu \rangle [12]) [13]^2+\langle 1\nu \rangle \langle 2\nu \rangle (5 \langle 3\nu \rangle [13]-11 \langle 2\nu \rangle [12]) [23] [13]+\langle 2\nu \rangle^2 (11 \langle 3\nu \rangle [13]-13 \langle 2\nu \rangle [12]) [23]^2)) \langle 13 \rangle^2+\langle 3\nu \rangle (-2 \langle 23 \rangle^2 \langle 2\nu \rangle^3 [23]^4\\
&&+4 \langle 12 \rangle \langle 23 \rangle \langle 2\nu \rangle^2 (\langle 3\nu \rangle [13]-\langle 2\nu \rangle [12]) [23]^3+\langle 12 \rangle^2 \langle 3\nu \rangle [13] (2 \langle 1\nu \rangle^2 [12] [13]^2+\langle 1\nu \rangle (7 \langle 2\nu \rangle [12]-5 \langle 3\nu \rangle [13]) [23] [13]\\
&&+\langle 2\nu \rangle (9 \langle 2\nu \rangle [12]-11 \langle 3\nu \rangle [13]) [23]^2)) \langle 13 \rangle-4 \langle 12 \rangle^2 \langle 3\nu \rangle^3 [13] [23] (\langle 23 \rangle \langle 2\nu \rangle [23]^2+\langle 12 \rangle (\langle 1\nu \rangle [12] [13]-\langle 3\nu \rangle [23] [13]+\langle 2\nu \rangle [12] [23]))) .
\end{eqnarray*}
\end{scriptsize}
For $A^l_\mu(1^+,2^+,3^+) k_1^\mu$:
\begin{scriptsize}
\begin{eqnarray*}
&&(A^l_\mu(1^+,2^+,3^+) k_1^\mu)_{D0[1,2,3,4]}\\
&&=\frac{[12]}{64 \langle 13 \rangle^2 \langle 1\nu \rangle \langle 2\nu \rangle \langle 3\nu \rangle} (\langle 2\nu \rangle (4 \langle 2\nu \rangle [12]+5 \langle 3\nu \rangle [13]) \langle 13 \rangle^2+3 \langle 12 \rangle \langle 2\nu \rangle \langle 3\nu \rangle [12] \langle 13 \rangle-2 \langle 12 \rangle^2 \langle 3\nu \rangle^2 [12]) [23] (\langle 12 \rangle \langle 1\nu \rangle [12]\\
&&+\langle 13 \rangle \langle 1\nu \rangle [13]+\langle 13 \rangle \langle 2\nu \rangle [23]-\langle 12 \rangle \langle 3\nu \rangle [23]),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) k_1^\mu)_{C0[1,3,4]}\\
&&=\frac{1}{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle}(2 \langle 3\nu \rangle^2 [12]^2 (\langle 3\nu \rangle [23]-\langle 1\nu \rangle [12]) \langle 12 \rangle^4+\langle 13 \rangle \langle 3\nu \rangle [12] (\langle 1\nu \rangle [12] (3 \langle 2\nu \rangle [12]-4 \langle 3\nu \rangle [13])\\
&&+\langle 3\nu \rangle (2 \langle 3\nu \rangle [13]-5 \langle 2\nu \rangle [12]) [23]) \langle 12 \rangle^3+\langle 13 \rangle^2 ([13] (\langle 3\nu \rangle [13]-8 \langle 2\nu \rangle [12]) [23] \langle 3\nu \rangle^2+\langle 1\nu \rangle [12] (2 \langle 2\nu \rangle [12]-\langle 3\nu \rangle [13]) (\langle 2\nu \rangle [12]\\
&&+4 \langle 3\nu \rangle [13])) \langle 12 \rangle^2+\langle 13 \rangle^3 (\langle 1\nu \rangle [13] (6 \langle 2\nu \rangle^2 [12]^2+9 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-2 \langle 3\nu \rangle^2 [13]^2)+\langle 2\nu \rangle (3 \langle 2\nu \rangle^2 [12]^2+2 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]\\
&&-6 \langle 3\nu \rangle^2 [13]^2) [23]) \langle 12 \rangle+\langle 13 \rangle^4 \langle 2\nu \rangle [13] (4 \langle 2\nu \rangle [12]+5 \langle 3\nu \rangle [13]) (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) k_1^\mu)_{C0[2,3,4]}\\
&&=\frac{[23]}{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle \langle 2\nu \rangle \langle 3\nu \rangle}
(2 \langle 3\nu \rangle^2 [12] (\langle 1\nu \rangle [12]-\langle 3\nu \rangle [23]) \langle 12 \rangle^3+\langle 13 \rangle \langle 3\nu \rangle [12] (-3 \langle 1\nu \rangle \langle 2\nu \rangle [12]+2 \langle 1\nu \rangle \langle 3\nu \rangle [13]\\
&&+5 \langle 2\nu \rangle \langle 3\nu \rangle [23]) \langle 12 \rangle^2+\langle 13 \rangle^2 \langle 2\nu \rangle (2 \langle 1\nu \rangle [12] (2 \langle 2\nu \rangle [12]+\langle 3\nu \rangle [13])-\langle 3\nu \rangle (9 \langle 2\nu \rangle [12]+7 \langle 3\nu \rangle [13]) [23]) \langle 12 \rangle\\
&&+\langle 13 \rangle^3 \langle 2\nu \rangle (4 \langle 2\nu \rangle [12]+5 \langle 3\nu \rangle [13]) (\langle 1\nu \rangle [13]+\langle 2\nu \rangle [23])),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) k_1^\mu)_{C0[1,3,4]}\\
&&=\frac{1}{64 \langle 12 \rangle \langle 13 \rangle^2 \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 13 \rangle [13]+\langle 23 \rangle [23])}(2 \langle 3\nu \rangle^3 [12]^2 [23] (2 \langle 13 \rangle [13]+\langle 23 \rangle [23]) \langle 12 \rangle^4+\langle 3\nu \rangle^2 [12] (2 \langle 23 \rangle^2 \langle 3\nu \rangle [23]^3\\
&&+\langle 13 \rangle \langle 23 \rangle (6 \langle 3\nu \rangle [13]-5 \langle 2\nu \rangle [12]) [23]^2-\langle 13 \rangle^2 [13] (\langle 1\nu \rangle [12] [13]-2 \langle 3\nu \rangle [23] [13]+10 \langle 2\nu \rangle [12] [23])) \langle 12 \rangle^3-\langle 13 \rangle \langle 3\nu \rangle (5 \langle 23 \rangle^2 \langle 2\nu \rangle \langle 3\nu \rangle [12] [23]^3\\
&&+2 \langle 13 \rangle \langle 23 \rangle \langle 3\nu \rangle [13] (9 \langle 2\nu \rangle [12]+2 \langle 3\nu \rangle [13]) [23]^2+\langle 13 \rangle^2 [13] ((2 \langle 2\nu \rangle^2 [12]^2+17 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+6 \langle 3\nu \rangle^2 [13]^2) [23]-\langle 1\nu \rangle [12] [13] (2 \langle 2\nu \rangle [12]\\
&&+\langle 3\nu \rangle [13]))) \langle 12 \rangle^2+\langle 13 \rangle^2 (\langle 23 \rangle^2 \langle 2\nu \rangle \langle 3\nu \rangle (\langle 2\nu \rangle [12]-\langle 3\nu \rangle [13]) [23]^3+\langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle (4 \langle 2\nu \rangle^2 [12]^2+3 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-3 \langle 3\nu \rangle^2 [13]^2) [23]^2\\
&&+\langle 13 \rangle^2 [13] (\langle 1\nu \rangle [13] (4 \langle 2\nu \rangle^2 [12]^2+5 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+2 \langle 3\nu \rangle^2 [13]^2)+\langle 2\nu \rangle (8 \langle 2\nu \rangle^2 [12]^2+3 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-3 \langle 3\nu \rangle^2 [13]^2) [23])) \langle 12 \rangle\\
&&+\langle 13 \rangle^3 \langle 2\nu \rangle (4 \langle 2\nu \rangle [12]+3 \langle 3\nu \rangle [13]) (\langle 23 \rangle^2 \langle 2\nu \rangle [23]^3+3 \langle 13 \rangle \langle 23 \rangle \langle 2\nu \rangle [13] [23]^2+\langle 13 \rangle^2 [13]^2 (\langle 1\nu \rangle [13]+3 \langle 2\nu \rangle [23]))),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) k_1^\mu)_{C0[1,2,3]}\\
&&=\frac{[12]}{64 \langle 13 \rangle^2 \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle} (\langle 1\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13]) (\langle 2\nu \rangle (6 \langle 2\nu \rangle [12]+5 \langle 3\nu \rangle [13]) \langle 13 \rangle^2-3 \langle 12 \rangle \langle 2\nu \rangle \langle 3\nu \rangle [12] \langle 13 \rangle+2 \langle 12 \rangle^2 \langle 3\nu \rangle^2 [12])\\
&&+(\langle 13 \rangle \langle 2\nu \rangle-\langle 12 \rangle \langle 3\nu \rangle) (\langle 2\nu \rangle (4 \langle 2\nu \rangle [12]+3 \langle 3\nu \rangle [13]) \langle 13 \rangle^2-3 \langle 12 \rangle \langle 2\nu \rangle \langle 3\nu \rangle [12] \langle 13 \rangle+2 \langle 12 \rangle^2 \langle 3\nu \rangle^2 [12]) [23]),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) k_1^\mu)_{B0[2,4]}\\
&&=\frac{1}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13])}(\langle 1\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13]) (5 \langle 13 \rangle^2 [12] [13] \langle 2\nu \rangle^2+4 \langle 12 \rangle^2 \langle 3\nu \rangle^2 [12] [13]\\
&&+\langle 12 \rangle \langle 13 \rangle (2 \langle 2\nu \rangle^2 [12]^2-6 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2))+(5 \langle 13 \rangle^3 [12] [13] \langle 2\nu \rangle^3+13 \langle 12 \rangle^2 \langle 13 \rangle \langle 3\nu \rangle^2 [12] [13] \langle 2\nu \rangle\\
&&+\langle 12 \rangle \langle 13 \rangle^2 (\langle 2\nu \rangle^2 [12]^2-12 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+\langle 3\nu \rangle^2 [13]^2) \langle 2\nu \rangle-4 \langle 12 \rangle^3 \langle 3\nu \rangle^3 [12] [13]) [23]),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) k_1^\mu)_{B0[1,4]}\\
&&=\frac{1}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 12 \rangle [12]+\langle 13 \rangle [13]) (\langle 13 \rangle [13]+\langle 23 \rangle [23])^3}(2 \langle 3\nu \rangle^3 [12]^3 [13] [23] (3 \langle 13 \rangle [13]+2 \langle 23 \rangle [23]) \langle 12 \rangle^5\\
&&+\langle 3\nu \rangle^2 [12]^2 [13] (8 \langle 23 \rangle^2 \langle 3\nu \rangle [23]^3-11 \langle 13 \rangle \langle 23 \rangle (\langle 2\nu \rangle [12]-2 \langle 3\nu \rangle [13]) [23]^2-2 \langle 13 \rangle^2 [13] (\langle 1\nu \rangle [12] [13]-10 \langle 3\nu \rangle [23] [13]+9 \langle 2\nu \rangle [12] [23])) \langle 12 \rangle^4\\
&&+\langle 3\nu \rangle [12] (4 \langle 23 \rangle^3 \langle 3\nu \rangle^2 [13] [23]^4-4 \langle 13 \rangle \langle 23 \rangle^2 (\langle 2\nu \rangle^2 [12]^2+7 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-4 \langle 3\nu \rangle^2 [13]^2) [23]^3+2 \langle 13 \rangle^2 \langle 23 \rangle [13] (-2 \langle 2\nu \rangle^2 [12]^2\\
&&-42 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+11 \langle 3\nu \rangle^2 [13]^2) [23]^2+\langle 13 \rangle^3 [13]^2 (\langle 1\nu \rangle [12] [13] (5 \langle 2\nu \rangle [12]-6 \langle 3\nu \rangle [13])+2 (2 \langle 2\nu \rangle^2 [12]^2-40 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]\\
&&+7 \langle 3\nu \rangle^2 [13]^2) [23])) \langle 12 \rangle^3+\langle 13 \rangle (-\langle 23 \rangle^3 \langle 2\nu \rangle \langle 3\nu \rangle [12] (4 \langle 2\nu \rangle [12]+17 \langle 3\nu \rangle [13]) [23]^4+\langle 13 \rangle \langle 23 \rangle^2 (\langle 2\nu \rangle^3 [12]^3-2 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2\\
&&-83 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]-4 \langle 3\nu \rangle^3 [13]^3) [23]^3+\langle 13 \rangle^2 \langle 23 \rangle [13] (3 \langle 2\nu \rangle^3 [12]^3+26 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2-136 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]\\
&&-8 \langle 3\nu \rangle^3 [13]^3) [23]^2+\langle 13 \rangle^3 [13]^2 (\langle 1\nu \rangle [12] [13] (\langle 2\nu \rangle^2 [12]^2+18 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-4 \langle 3\nu \rangle^2 [13]^2)+(3 \langle 2\nu \rangle^3 [12]^3+42 \langle 2\nu \rangle^2 \langle 3\nu \rangle [13] [12]^2\\
&&-91 \langle 2\nu \rangle \langle 3\nu \rangle^2 [13]^2 [12]-4 \langle 3\nu \rangle^3 [13]^3) [23])) \langle 12 \rangle^2+\langle 13 \rangle^2 \langle 2\nu \rangle (\langle 23 \rangle^3 (3 \langle 2\nu \rangle^2 [12]^2+12 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-5 \langle 3\nu \rangle^2 [13]^2) [23]^4\\
&&+\langle 13 \rangle \langle 23 \rangle^2 [13] (3 \langle 2\nu \rangle [12]-\langle 3\nu \rangle [13]) (3 \langle 2\nu \rangle [12]+19 \langle 3\nu \rangle [13]) [23]^3+9 \langle 13 \rangle^2 \langle 23 \rangle [13]^2 (\langle 2\nu \rangle^2 [12]^2+10 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]\\
&&-3 \langle 3\nu \rangle^2 [13]^2) [23]^2+\langle 13 \rangle^3 [13]^3 (17 \langle 1\nu \rangle \langle 3\nu \rangle [12] [13]^2+(3 \langle 2\nu \rangle^2 [12]^2+66 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]-17 \langle 3\nu \rangle^2 [13]^2) [23])) \langle 12 \rangle\\
&&-\langle 13 \rangle^3 \langle 2\nu \rangle [13] (\langle 2\nu \rangle [12]-4 \langle 3\nu \rangle [13]) (\langle 23 \rangle^3 \langle 2\nu \rangle [23]^4+4 \langle 13 \rangle \langle 23 \rangle^2 \langle 2\nu \rangle [13] [23]^3+6 \langle 13 \rangle^2 \langle 23 \rangle \langle 2\nu \rangle [13]^2 [23]^2\\
&&+\langle 13 \rangle^3 [13]^3 (\langle 1\nu \rangle [13]+4 \langle 2\nu \rangle [23]))),\\
&&\\
&&(A^l_\mu(1^+,2^+,3^+) k_1^\mu)_{B0[1,3]}\\
&&=-\frac{1}{64 \langle 12 \rangle \langle 13 \rangle \langle 1\nu \rangle \langle 23 \rangle \langle 2\nu \rangle \langle 3\nu \rangle (\langle 13 \rangle [13]+\langle 23 \rangle [23])^3}(2 \langle 3\nu \rangle^3 [12]^2 [13] [23] (3 \langle 13 \rangle [13]+2 \langle 23 \rangle [23]) \langle 12 \rangle^4+\langle 3\nu \rangle^2 [12] [13] (4 \langle 23 \rangle^2 \langle 3\nu \rangle [23]^3\\
&&+\langle 13 \rangle \langle 23 \rangle (7 \langle 3\nu \rangle [13]-11 \langle 2\nu \rangle [12]) [23]^2-2 \langle 13 \rangle^2 [13] (\langle 1\nu \rangle [12] [13]-2 \langle 3\nu \rangle [23] [13]+9 \langle 2\nu \rangle [12] [23])) \langle 12 \rangle^3\\
&&-\langle 13 \rangle \langle 3\nu \rangle [13] (2 \langle 23 \rangle^2 \langle 3\nu \rangle (7 \langle 2\nu \rangle [12]+2 \langle 3\nu \rangle [13]) [23]^3+\langle 13 \rangle \langle 23 \rangle (-8 \langle 2\nu \rangle^2 [12]^2+33 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+9 \langle 3\nu \rangle^2 [13]^2) [23]^2\\
&&+\langle 13 \rangle^2 [13] (\langle 1\nu \rangle [12] [13] (\langle 3\nu \rangle [13]-5 \langle 2\nu \rangle [12])+2 (-8 \langle 2\nu \rangle^2 [12]^2+12 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+3 \langle 3\nu \rangle^2 [13]^2) [23])) \langle 12 \rangle^2\\
&&+\langle 13 \rangle (2 \langle 23 \rangle^3 \langle 2\nu \rangle^2 \langle 3\nu \rangle [12] [23]^4+\langle 13 \rangle \langle 23 \rangle^2 \langle 2\nu \rangle (-3 \langle 2\nu \rangle^2 [12]^2+13 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+2 \langle 3\nu \rangle^2 [13]^2) [23]^3\\
&&+3 \langle 13 \rangle^2 \langle 23 \rangle \langle 2\nu \rangle [13] (-3 \langle 2\nu \rangle^2 [12]^2+9 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+2 \langle 3\nu \rangle^2 [13]^2) [23]^2+\langle 13 \rangle^3 [13]^2 (\langle 1\nu \rangle [13] (-3 \langle 2\nu \rangle^2 [12]^2+4 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]\\
&&+\langle 3\nu \rangle^2 [13]^2)+\langle 2\nu \rangle (-9 \langle 2\nu \rangle^2 [12]^2+23 \langle 2\nu \rangle \langle 3\nu \rangle [13] [12]+6 \langle 3\nu \rangle^2 [13]^2) [23])) \langle 12 \rangle-\langle 13 \rangle^2 \langle 2\nu \rangle (2 \langle 23 \rangle^3 \langle 2\nu \rangle (2 \langle 2\nu \rangle [12]\\
&&+\langle 3\nu \rangle [13]) [23]^4+\langle 13 \rangle \langle 23 \rangle^2 \langle 2\nu \rangle [13] (15 \langle 2\nu \rangle [12]+7 \langle 3\nu \rangle [13]) [23]^3+3 \langle 13 \rangle^2 \langle 23 \rangle \langle 2\nu \rangle [13]^2 (7 \langle 2\nu \rangle [12]+3 \langle 3\nu \rangle [13]) [23]^2\\
&&+\langle 13 \rangle^3 [13]^3 (\langle 1\nu \rangle [13] (3 \langle 2\nu \rangle [12]+\langle 3\nu \rangle [13])+\langle 2\nu \rangle (13 \langle 2\nu \rangle [12]+5 \langle 3\nu \rangle [13]) [23]))).
\end{eqnarray*}
\end{scriptsize}
We checked our four point amplitudes using the known simple results of $A^l(1^+,2^+,3^+,4^+)$ and $A^l(1^+,2^+,3^+,4^-)$ in \cite{Bern:1991aq,Bern2,Kharel,Mahlon:1993si}.
\section{Conclusion}
We have discussed the Ward identity in detail for off shell amplitudes in pure Yang-Mills theory. We explicitly prove that the Ward identity with two complexified external lines holds at tree and one loop level using Feynman rules. Then we use the Ward identity to deduce recursion relations for off shell amplitudes at tree and one loop level. In this technique, three steps are important to simplify the calculation. First, according to the complexfied Ward identity, we can convert the calculation of the amplitudes to the calculation of derivative of the amplitudes. Second, we decompose the three point vertex which contains the off shell line into three terms, which simplifies many steps in our calculation. Thirdly, according to the cancellation details in the proof of complexified Ward identity, we find most terms from different diagrams cancel with each other. The number of remaining effective terms or diagrams are reduced. It turns out that the recursion relation we derive at tree level is equivalent to Berends-Giele recursion relation \cite{Berends:1987me}. However, our expressions at 1-loop level are new. And we present 1-loop off shell three and four point amplitudes as examples of applying our method at 1-loop level.
Comparing with our previous work \cite{Chen}, we find the technique in this article is more universal. Here we can obtain a recursion relation for the total amplitudes instead of just the boundary terms of the amplitudes, and we do not need to use BCFW recursion relation. Furthermore, for this technique, we do not need to avoid the unphysical poles from the polarization vectors of the shifted on shell leg which can also depend on $z$. Hence this technique works well for the amplitudes with any helicity structure and the momenta shifts are more general than the ones in \cite{Chen}. In addition, this technique can be used for calculating one loop off shell amplitudes with any helicity structure.
In principle, it is possible to generalize our method to higher loop levels and to other theories such as QCD. The only obstruct is to classify all the cancellation details for the Ward identity with complexfied external momenta. We leave this to future work. Another extension is to combine our technique with other methods, such as unitary cut, generalized unitary cut, BCFW, OPP \cite{OPP} etc. to further simplify the calculation in pure Yang-Mills theory.
\textbf{Acknowledgement} We thank Edna Cheung, Jens Fjelstad, Konstantin G. Savvidy for helpful discussions. This work is funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), NSFC grant No.~10775067, Research Links Programme of Swedish Research Council under contract No.~348-2008-6049, the Chinese Central Government's 985 Project grants for Nanjing University, the China Science Postdoc grant no. 020400383. the postdoc grants of Nanjing University 0201003020
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,601 |
Maruti Suzuki Swift Special Edition has been launched in India on the occasion of the ongoing festive season. It is based on the entry-level LXi and LDi trims, starting at a price of Rs 4.99 Lakhs.
Maruti Suzuki Swift Special Edition has been launched. It comes at an introductory starting price of Rs 4.99 Lakhs and comes in both petrol and diesel variants. The Limited edition is based on the base LXi and LDi trims. A lot of auto brands are launching special editions, facelifts and more of their models. It is based on the base variant but gets many more features.
The Special Edition gets additional features such as black wheel caps, body-coloured door handles, and body-coloured ORVMs on the exterior.
Inside the cabin, it gets a Bluetooth music system with dual front speakers, power windows for driver and front passenger, central locking with remote and rear parking sensors.
Leaving these features, it does not get any mechanical, cosmetic or feature upgrade. Now, the LXi and LDi variants get more fitments.
Mechanically, the Maruti Suzuki Swift Special Edition continues with the same engines. It gets the 1.2 Litre K12M petrol engine and the 1.3 Litre DDiS diesel engine. The petrol engine produces around 82 BHP of power and 113 Nm of torque. The diesel engine makes 74 BHP of max. power and 190 Nm of peak torque.
On both the engines, 5-speed manual transmission along with an optional Auto Gear Shift (AGS) transmission. The mileage claimed on the petrol variant is around 22 kmpl and that on diesel variant is around 28 kmpl.
The base-variant also comes with Power & Tilt Steering and LED tail lamps. For safety front, it gets dual front airbags, ABS with EBD and brake assist, ISOFIX, Front seat belts with pre-tensioners and force limiter and an engine immobilizer as standard fitments.
The top-spec trim gets a variety of features. On the exteriors, it gets LED DRLs with LED projector headlamps, Electric ORVMs with turn indicators and alloy wheels. Inside the cabin, it receives adjustable front seat headrests, height adjustable driver seat, engine push start-stop button, Auto climate control, and Auto Headlamps.
Also, it gets the SmartPlay infotainment system, steering mounted audio controls, Android Auto, Apple CarPlay and more.
Safety features on the top-spec trim of Swift include the standard fitments with day and night IRVM, Reverse parking sensors with a camera, Front fog lamps, dual front airbags, ABS with EBD and brake assist and Speed-sensitive automatic door lock.
The Maruti Suzuki Swift petrol variant starts from Rs 4.99 lakhs and goes up to Rs 7.76 Lakhs. The diesel variant starts at Rs 5.99 lakhs and goes up to Rs 8.76 Lakhs (all prices ex-showroom New Delhi).
The Swift takes on the likes of Tata Tiago, Hyundai Grand i10, Ford Figo and Honda Brio. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,556 |
Tecno Phantom X2 Pro Review: Impressive, but Not Ideal
By Shumail Ali December 8, 2022 Updated: December 8, 2022 11 Mins Read
Tecno Phantom X2 Pro features a massive 6.8-inch AMOLED display with a 120Hz refresh rate and 500 nits of peak brightness, and 1080 x 2400 pixels resolution for an incredibly sharp viewing experience. Under the hood is the powerhouse MediaTek Dimensity 9000 processor (4nm), which is said to provide lightning-fast performance while also being very power efficient. On top of that, it runs on the Android 12 operating system with HIOS 12 UI for a smooth user interface experience. In our review process, we tested the Tecno Phantom X2 Pro to gauge its performance capabilities.
Tecno Phantom X2 Pro Specifications
Dimensions 6.44 x 2.91 x 0.34 in
Colors Starry Night Blue, Monet Summer
Body Material Glass front (Gorilla Glass 5),
Plastic back
SIMs Dual SIM (Nano-SIM, dual stand-by)
Water & Dust No
Type AMOLED 120Hz capacitive touchscreen, 16M colors
Resolutions 1080 x 2400 pixels
PPI 256 ppi density
Triple Rear 50 MP, f/1.9, (wide), 1/1.3″, 1.2µm, Dual Pixel PDAF, Laser AF
13 MP, 50mm (telephoto), PDAF, 2x optical zoom
8 MP, 120? (ultrawide), 1/4.0″, 1.12µm
Videos [email protected]/60fps, [email protected]
Front Dual 32 MP, f/2.2, (wide), 1/2.0″, 0.8µm
Chipset Mediatek Dimensity 9000
CPU Octa-core (2×2.05 GHz Cortex-A76 & 6×2.0 GHz Cortex-A55)
Fast Charging Fast Chirging 45W
Wireless Charging No
Tecno Phantom X2 Pro Review: Design, Display and Durability
Tecno describes the design of the Phantom X2 Pro as "unibody double-curved", referring to the symmetrical curves of the front and back panels. However, the phone's construction actually consists of three pieces – the front, back, and middle frame. Despite this, the double-curved design adds a unique and visually appealing element to the overall aesthetic of the phone.
The Phantom X2 Pro boasts a large, 6.8 inch AMOLED panel with a full HD 1080 x 2400 resolution and a high refresh rate of 120Hz. The curved front screen provides users with an immersive viewing experience, and the 500 nits of brightness ensures that content is easily visible even in bright outdoor conditions. Although 500 nits is lower than some flagship models, it is still sufficient for most users.
The Phantom X2 Pro's display is protected by Gorilla Glass Victus, known for its durability and toughness in the smartphone industry. This ensures that the display is well-protected against scratches and other damage.
The Phantom X2 Pro features a sleek design with an aluminum frame and plastic back, providing both aesthetic appeal and added strength to the overall structure of the phone. This ensures that the device is durable and able to withstand daily use.
SEE ALSO: Huawei Watch GT Review: Affordable Fitness Tracker
The Phantom X2 Pro has a camera module placed in the top center of the rear. This module features two large camera sensors that may appear unappealing in photographs. However, when viewed up close or with a protective case on the phone, the camera module is less noticeable.
While the design of the Phantom X2 Pro may be a matter of personal preference, the display and durability do not meet industry standards. The display is only 500 nits, which is not particularly bright, especially in direct sunlight. Additionally, the phone is not certified to be water and dustproof, indicating that it may not be as durable as some other smartphones on the market.
Tecno Phantom X2 Pro Review: Performance
Phantom X2 Pro is powered by the MediaTek Dimensity 9000 5G chipset, which is running at its stock CPU clocks. This configuration includes one ARM Cortex-X2 core running at up to 3.05 GHz, three Cortex-A710 cores clocked at up to 2.85 GHz, and four Cortex-A510 cores running at up to 1.8 GHz. These are paired with a 10-core Mali-G710 MC10 GPU, clocked at 850 MHz. The Phantom X2 Pro also features 256GB of internal storage and 12GB of fast LPDDR5X RAM, providing ample space and power for a variety of tasks and activities.
SEE ALSO: Fifine H6 Headset Review: Amplify your Gaming Experience!
The MediaTek Dimensity 9000 5G chipset in the Phantom X2 Pro is a strong performer, capable of handling graphics-intensive games and rendering graphics with ease. When compared to the chipset in the Samsung Galaxy S21 Ultra, the Phantom X2 Pro is a top performer, able to play games using the highest graphics settings without any issues. Overall, users can expect smooth and efficient performance from this phone, with no concerns about its ability to handle demanding tasks.
Performance is a standout feature of the Phantom X2 Pro, thanks to its powerful MediaTek Dimensity 9000 5G chipset. During my time using the phone, I experienced no lag or hanging, and app opening and switching were smooth and fast. Compared to the previously released Tecno Phantom X, the Phantom X2 Pro offers a significant performance boost, thanks to its advanced processor. Overall, users can expect excellent performance from this phone, making it a great choice for those who prioritize speed and efficiency.
In Geekbench 5, the Phantom X2 Pro scores 3984 in the Multi core benchmark and 1248 in the Single core benchmark. In AnTuTu 9, the X2 Pro scored 961576. These impressive scores indicate that the Phantom X2 Pro is a strong performer in terms of speed and efficiency, capable of handling a variety of tasks and activities with ease. Overall, users can expect smooth and reliable performance from this phone.
Phone Multicore score Single core
ROG Phone 6D (X Mode+) 4631 1399
ROG Phone 6D Ultimate (X Mode+) 4575 1360
Motorola Edge 30 Ultra 4265 1276
Xiaomi 12T Pro 4081 1238
Realme GT2 Explorer Master 4021 1336
Tecno Pahntom X2 Pro 3984 1248
Huawei Mate 50 Pro 3839 1277
Geekbench 5 Scores
Tecno Phantom X2 Pro Review: Software and UI
The Phantom X2 Pro is powered by the latest Android 12 operating system and Tecno's custom HiOS 12 interface, providing users with a seamless and efficient user experience. The combination of powerful hardware and advanced software ensures smooth and responsive performance, making the Phantom X2 Pro a top performer in its class.
SEE ALSO: is AliExpress Legit and Safe to Buy from? Our Review – 2023
No Bloatware: The Tecno Phantom X2 Pro offers a clean and pleasant user experience, thanks in part to the reduced presence of bloatware and ads compared to previous models. In the past, Tecno's custom skin has been criticized for these issues, but the Phantom X2 Pro does not annoy users with notifications and offers, resulting in a smooth and enjoyable user experience.
If you are being bothered by notifications from third-party apps on your Phantom X2 Pro, you can easily disable them. This can be done either from within the app itself or from the Notification settings in your phone's settings menu. By disabling notifications, you can enjoy a more peaceful and enjoyable user experience without interruptions from unwanted notifications.
Tecno Phantom X2 Pro Review: Camera
The Phantom X2 Pro boasts major improvements in its camera system compared to the Phantom X. The phone's cameras feature larger sensors and apertures, providing enhanced image quality and performance. The main camera is a 50MP lens, and the phone also includes a 50MP portrait camera with a retractable lens, a 13MP ultra-wide lens, and a 32MP front camera. These powerful cameras provide users with a wide range of shooting options and ensure that every photo is crisp, clear, and detailed. Overall, the Phantom X2 Pro's camera system is a standout feature that sets it apart from its predecessors.
50MP Primary camera
TECNO Phantom X2 Pro's main 50MP camera is based on the Samsung s5kjnv sensor, which is likely the ISOCELL GNV, a custom sensor previously used in the Vivo X80 series. This sensor has a 1/1.3″ size, 1.2µm pixels, and employs an RGBW arrangement, as well as ISOCELL 3.0 technology. The lens's front element is made of glass, providing enhanced image quality and performance. Overall, the Phantom X2 Pro's main camera is a powerful and capable shooter that is sure to impress users with its image quality and versatility.
50MP Portrait camera
The Phantom X2 Pro's 50MP, f/1.49, 65mm portrait camera uses a Samsung s5kjn1tele sensor, commonly known as the Samsung ISOCELL JN1. This sensor has a 1/2.76″ size, 0.64µm pixels, and employs ISOCELL 2.0 technology with Double Super PD autofocus and a Tetrapixel RGB Bayer pixel arrangement. These advanced features ensure that the portrait camera is able to capture detailed and accurate images, with smooth and accurate autofocus for fast-moving subjects. Overall, the portrait camera is a powerful and capable shooter that provides users with enhanced creative options.
13MP Ultrawide camera
The Phantom X2 Pro's 13MP ultrawide camera is based on a Samsung S5K3L6 sensor, which is not commonly found in smartphones. This sensor has a 1/3″ optical format and 1.12µm pixels, providing enhanced image quality and performance. The ultrawide camera allows users to capture more of their surroundings in a single image, providing a wider perspective and greater creative flexibility.
TECNO Phantom X2 Pro's 32MP fixed-focus selfie camera is the same as the one seen on the Tecno Camon 19 Pro. This camera uses a Samsung S5KGD2 sensor, which has a 1/2.8″ size and 0.8µm pixels, as well as ISOCELL PLUS technology and a Tetrapixel RGB Bayer arrangement. These features ensure that the selfie camera is able to capture detailed and accurate images, even in low light conditions.
The Phantom X2 Pro boasts a 50MP main camera that produces sharp and detailed photos. Even in low light conditions, the quality remains impressive with natural colors and adequate saturation. While not the best in its price range, the Phantom X2 Pro's camera produces decent photographs.
While the Phantom X2 Pro's 50MP portrait lens and 13MP ultra-wide lens produce average-quality photos, they are a significant improvement over the previous model, the Phantom X. However, when compared to flagship smartphones like the Galaxy S22 Ultra or iPhone 14 Pro, the X2 Pro's camera falls short and produces only average-quality photos.
The 32MP selfie camera on the Phantom X2 Pro is truly impressive. It captures plenty of detail and sharpness, producing selfies with stunning clarity and natural-looking skin tones. Overall, it is a standout feature of this smartphone and is sure to impress even the most discerning users.
Tecno Phantom X2 Pro: 12.5MP main camera samples – f/1.9, ISO 52, 1/3978s
Tecno Phantom X2 Pro: 50MP main camera samples – f/1.9, ISO 51, 1/3890s
Tecno Phantom X2 Pro: 12.5MP main camera portrait samples – f/1.9, ISO 58, 1/100s
Phantom X2 Pro's impressive camera system is capable of capturing [email protected] video using its two rear 50MP cameras, while the ultrawide and selfie cameras are limited to 1080p. These videos are saved in a standard AVC video stream at 50 Mbps in 4K, with a stable frame rate and a high-quality stereo 48kHz AAC audio track. The videos are packaged in an MP4 container, but unfortunately, the X2 Pro does not offer the option for h.256 (HEVC) video capture.
Phantom X2 Pro's main camera captures impressive 4K videos with plenty of detail and natural-looking colors. However, the dynamic range and contrast could be improved, as the image can sometimes appear overly sharp and overprocessed. Some users may still appreciate the look, however.
In comparison, the 4K capture from the portrait camera generally looks cleaner and more detailed. The colors are just as good as those from the main camera, with natural and mature processing. Overall, the X2 Pro's camera system produces high-quality 4K videos that will satisfy most users.
While the 1080p videos from the Phantom X2 Pro's ultrawide camera are good, they are not particularly impressive. The level of detail is decent, but the colors are a bit more vibrant than those from the main camera, which can be a plus for some users. The contrast is somewhat high, and the dynamic range could be better, but overall the videos are still good quality.
Tecno Phantom X2 Pro Review: Battery
The Tecno Phantom X2 Pro is equipped with a powerful 5,160mAh battery that supports fast 45W wired charging. This allows you to quickly recharge the battery from 0 to 100% in just 55 minutes. On a single charge, the X2 Pro's battery can last for more than 8 hours of use, making it a reliable choice for heavy users. However, it should be noted that the X2 Pro does not support wireless charging or reverse wired or wireless charging.
Tecno Phantom X2 Pro Review: Verdict
Tecno Phantom X2 Pro is a top-of-the-line smartphone that offers a number of impressive features, including a flagship chipset, larger camera sensors, and Gorilla Glass Victus protection. However, for its $920 price tag, the X2 Pro falls short in some areas, such as its lack of IP68 water and dust resistance, OIS stabilization in the camera, wireless charging, and support for HDR10+ and Dolby Vision. Additionally, the X2 Pro only offers a limited software update plan, whereas many competitors offer up to three years of updates. Overall, the X2 Pro is a solid choice, but there are better options available at this price point.
wonderful performance
dependable front glass
120Hz smooth refresh rate
cameras with good sensors
not dust- and water-proof
avoiding wireless charging
There is no OIS stabilization on the cameras.
The display's brightness is lacking.
possibly no software update
Images Courtesy GSMArena
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Shumail Ali
Hi there! My name is Shumail and I love writing about technology. I'm especially interested in phones and fixing guides. I'm only 15 years old but I hope to use my skills to help others learn about technology too. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,118 |
Пиршковень, Пиршковені () — село у повіті Олт в Румунії. Входить до складу комуни Пиршковень.
Село розташоване на відстані 149 км на захід від Бухареста, 18 км на південний захід від Слатіни, 33 км на схід від Крайови.
Населення
За даними перепису населення 2002 року у селі проживали особи, усі — румуни. Усі жителі села рідною мовою назвали румунську.
Примітки
Села повіту Олт | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3 |
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What equipment is needed to extract gold on a small scale for a mine? Update Cancel. The liquid solution containing cyanide and lime is sprayed/mixed with crushed rocks and then drained out. The drained out liquid contains the soluble gold complexes. Could I make money mining gold with a small equipment mine in Canada? | {
"redpajama_set_name": "RedPajamaC4"
} | 8,500 |
Q: Appositives: Should there be a comma with "that" in this sentence? While writing a sentence like the one below, should I insert a comma after "that movie"(to mean one and only film) or withhold the comma as it is?
Her life was a lot like that movie Sleepless in Seattle.
A: Given that you want to refer to that one specific movie, you would omit a comma, just as you have done. The title of the movie is essential information for determining the identity of the movie her life is being compared to.
Reference
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,133 |
I was in desperate need of wireless internet, so Trevor drove me to Queenstown, about 2 hours away and dropped me off. He had some horse business he could take care of around there, so it wasn't a total waste of his time. He wouldn't hear of me taking the bus anyway.
We were socked in a thick fog until about Lumsden, an hour or so away, and then we drove out of it. All the way we'd been slowly gaining altitude, through sheep farms and more farms, and when we got out of the fog, there were big mountains off to our left, and decent sized ones on our right, that they sometimes did endurance rides in.
We reached the bottom tip of Lake Wakatipu, under which I believe there's a sleeping giant, whose breathing causes a rather strange rising and falling of lake level. We drove along the side of it, hugging it along what used to be an old goat track. It's still just a narrow winding 2-lane road, that sometimes squeezed to one lane with road construction. Pretty steep mountains above us, and across the lake, bigger mountains rising sharply out of the lake, no room for even goat tracks over there. If you wanted to head west, there's that mountain range to cross (by bird wing) before you reach another north-south highway (that only goes north to Milford Sound before stopping), and on the other side of that, nothing but hundreds of miles rugged mountains and lakes and Fiordland National Park, with very few trails through it, before you hit the Ocean. I'm sure it's thoroughly been explored at some point, but there can't be many casual visitors. Oh, wouldn't I like to just be dropped off with map and compass and the right gear on the other side of Lake Wakatipu to hike straight across to the ocean!
But I took the easy way, with Trevor driving me and talking horses on the way. We got to Queenstown and took some time finding an affordable hotel with wireless internet. Didn't find either - not so many rooms available (well, maybe not for the likes of me) with a conference in town, none with wireless at the ones we checked, so I settled on a room in a hostel, with wireless internet at cafe in town.
I had the downstairs of a house, with a bit of a view down over the lake. Gee, I'd like to just move in here for a few months! I only had about 2 hours to wander around town before I had to get to work, but, Queenstown seemed to be a nice town in a lovely setting! It's stacked up around the curve of the lake, with a big forested mountain looming right over it (with a gondola ride to the top), and mountains surrounding the lake everywhere. There's a hill on the left as we drove into Queenstown; Trevor said they worked up on top of that on Lord of the Rings, staying on the backside bottom of it, riding up to the top to work every day. I bet there weren't too many whiners on that shoot!
It was quite a busy town, tourists and backpackers strolling everywhere, sitting in outdoor restaurants, riding in vans taking them to adventures, and ducks swimming the lake and strolling the parks. You can do anything outdoorsy you want there (for a fee): bungi jump (no thanks!) sky dive (no thanks!), paraglide (no thanks!), white water raft (no thanks!), take Lord of the Rings tours, water ski, snow ski, steam in a boat up the lake, take scenic flights, go to numerous restaurants (lots of Indian!) and bars and internet cafes. Plenty of upscale hotels and loads of hostels. I'd love to come back spend time just hiking in the surrounding mountains. And riding of course.
Next morning I took a couple of buses to get back to Gore - all day to get 2 hours down the road, but the bus is a great way to see the countryside. One other girl and I had a whole bus to ourselves, and our bus driver gave us a personalized guided tour of the countryside we were passing through.
The driver pointed out the "freezing works" or slaughterhouses where the cattle and sheep are slaughtered (deer are processed in their own plants), the old pulp mill, the big dairy conglomerates that are making a resurge. I said Surely Kiwis didn't consume THAT much milk, he said 97% of the milk products, milk, butter and cheese, are exported.
I also noticed some of the city slogans... If you've noticed, some cities or towns in America have adopted some slogan to put on the sign on the outskirts, such "Winnemucca - City of Paved Streets" (I'm not kidding! Though it may have changed to something more modern by now); Inyokern - Sunshine Capital of the World;" etc - I actually have a little book of them in America. Here, a few of the slogans were "Where Dreams are Possible" and "Northern Southland, Naturally."
I'd say for the whole country: "New Zealand - Come Ride!" | {
"redpajama_set_name": "RedPajamaC4"
} | 8,664 |
[](https://www.npmjs.com/package/jquery.rateit)
[](https://www.nuget.org/packages/jQuery.RateIt)
## Quick start
Several quick start options are available:
* [Download the latest release](https://github.com/gjunge/rateit.js/archive/master.zip).
* Clone the repo: `git clone https://github.com/gjunge/rateit.js.git`.
* Install with [Bower](http://bower.io): `bower install jquery.rateit`.
* Install with [npm](https://www.npmjs.com): `npm install jquery.rateit`.
* Install with [NuGet](https://www.nuget.org): `Install-Package jQuery.RateIt`.
Read the [Wiki documentation](https://github.com/gjunge/rateit.js/wiki) or look at the [examples page](http://gjunge.github.io/rateit.js/examples/).
## About
Fast, Progressive enhancement, touch support, customizable (just swap out the images, or change some CSS), Unobtrusive JavaScript (using HTML5 data-* attributes), RTL support, supports as many stars as you'd like, and also any step size.
Your feedback is more than welcome!
##Why is RateIt different
Although it does the same job as the rest of the jQuery star rating plugins, the main difference is its simplicity.
Most plugins create an element for each (partial) star, be it a div with a star background, or an img tag.
Each of these tags gets a hover event, and a click event. And on these hover/click events it has to go and talk to the other stars, telling them to change their state.
So each star plugin does a lot of DOM alterations (adding the number of stars as elements), and adds lots of events (again the number of stars times 2).
RateIt uses basically three divs.
One background (the inactive state), and two divs on top (the hover, and selected state). In addition it only attaches three event handlers (not counting the use of the reset button aside).
Each of these divs has a x-repeating background, enabling as many stars as you want (or a big image with for example 5 different smilies one next to the other) without adding more elements or event handlers.
Based on the position of the mouse, or the selected value, a certain width is applied to the selection div or the hover div.
##Credits
* Thanks to http://www.fyneworks.com/jquery/star-rating/ for the idea and layout.
* Thanks to http://famfamfam.com for the icon set.
| {
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Q: Cannot get value of checked checkboxes I have implemented this to get value of checked checkboxes in array but dont know where i am wrong i am getting array empty
<input type="checkbox" value="-1" id="-1" /> OPD CONSULTANCY <br />
<input type="checkbox" value="-2" id="-2" /> DOCTOR VISIT <br />
<script>
function allcheckedData() {
var allVals = [];
var checkedVals = [];
$('#all input[type=checkbox]').each(function () {
allVals.push($(this).val());
});
$('#all :checked').each(function () {
checkedVals.push($(this).val());
})
}
</script>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,537 |
/***********************************************************************/
/* */
/* Objective Caml */
/* */
/* Xavier Leroy, projet Cristal, INRIA Rocquencourt */
/* */
/* Copyright 1996 Institut National de Recherche en Informatique et */
/* en Automatique. All rights reserved. This file is distributed */
/* under the terms of the GNU Library General Public License, with */
/* the special exception on linking described in file ../../LICENSE. */
/* */
/***********************************************************************/
/* $Id: utimes.c,v 1.10 2005/03/24 17:20:53 doligez Exp $ */
#include <fail.h>
#include <mlvalues.h>
#include "unixsupport.h"
#ifdef HAS_UTIME
#include <sys/types.h>
#ifndef _WIN32
#include <utime.h>
#else
#include <sys/utime.h>
#endif
CAMLprim value unix_utimes(value path, value atime, value mtime)
{
struct utimbuf times, * t;
times.actime = Double_val(atime);
times.modtime = Double_val(mtime);
if (times.actime || times.modtime)
t = ×
else
t = (struct utimbuf *) NULL;
if (utime(String_val(path), t) == -1) uerror("utimes", path);
return Val_unit;
}
#else
#ifdef HAS_UTIMES
#include <sys/types.h>
#include <sys/time.h>
CAMLprim value unix_utimes(value path, value atime, value mtime)
{
struct timeval tv[2], * t;
double at = Double_val(atime);
double mt = Double_val(mtime);
tv[0].tv_sec = at;
tv[0].tv_usec = (at - tv[0].tv_sec) * 1000000;
tv[1].tv_sec = mt;
tv[1].tv_usec = (mt - tv[1].tv_sec) * 1000000;
if (tv[0].tv_sec || tv[1].tv_sec)
t = tv;
else
t = (struct timeval *) NULL;
if (utimes(String_val(path), t) == -1) uerror("utimes", path);
return Val_unit;
}
#else
CAMLprim value unix_utimes(value path, value atime, value mtime)
{ invalid_argument("utimes not implemented"); }
#endif
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,332 |
\section{Introduction}
Certain aspects of general relativity are well tested. For example, the Schwarzschild metric has been quantitatively verified in the weak-field limit on small scales, e.g., the Solar system \citep{Shapiro64, Bertotti03} and binary radio pulsars \citetext{e.g., \citealp{Hulse75}; \citealp{Taylor79}; \citealp{Weisberg84}}; and on galaxy scales \citep[e.g.,][]{Bolton06}. In another fundamental test of general relativity, the existence of gravity waves has been established \citetext{e.g., \citealp{Taylor82}; \citealp{Weisberg84}, 2003}. General relativity theory, in the form of the Friedmann--Robertson--Walker metric \citetext{\citealp{Friedmann22}, 1924; \citealp{Robertson35}, 1936; \citealp{Walker36}} and the Friedmann equations \citetext{\citealp{Friedmann22, Lemaitre27}} which govern the expansion behavior of the Universe are used extensively in cosmology, and are at the core of this carefully and tightly woven paradigm. However, it is probably fair to say that the Friedmann equations, while providing a very self-consistent and highly successful framework for cosmology, have not been subjected to extensive, independent testing. In this paper we show that one particular aspect of general relativity, i.e., the self-gravity of pressure can be tested quantitatively.
The development of sophisticated big-bang nucleosynthesis (BBN) codes \citep{Wagoner67}, coupled with measurements of the relevant nuclear reaction rates \citep{Caughlan88}, have allowed observations of light elemental abundances to become powerful tools with which to investigate the evolution of the early universe. Computational predictions over a wide range of parameter space, when compared with observations, have yielded constraints on the current-epoch baryon density \citep{Wagoner73}, neutrino physics \citep{Yang79, Kawasaki94}, the fine structure constant \citep{Bergstrom99}, the gravitational constant \citep{Yang79, Copi04}, primordial magnetic fields \citep{Kernan96}, and other parameters of astrophysical interest.
Increasingly exact measurements of elemental abundances, as well as augmented understanding of the processes (i.e., stellar nucleosynthesis) which have altered the original abundances, allow these restrictions to be continually refined. Deuterium abundances \citep{Black73, Omeara06}, helium abundances \citep{Peebles66, Izotov07}, and lithium abundances, including the effects of stellar mixing \citep{Pinsonneault02} have all been well measured. More recently, observations of the cosmic microwave background (CMB) have yielded an independent estimate of $\eta$, the baryon to photon ratio \citep{Spergel07}.
\section{Analysis}
\subsection{Friedmann Equations}
In the standard Friedmann--Robertson--Walker (FRW) cosmology, the expansion of the scale factor, $a$, of the universe as a function of its density and pressure is given by the Friedmann equations. In terms of $\ddot{a}$, the ``acceleration'' of the scale factor is given by:
\begin{equation}
\frac{\ddot{a}}{a} = - \frac{4\pi G}{3} \left(\rho + \frac{3P}{c^2}\right) ~~~.
\end{equation}
where $\rho c^2$ is the energy density and $P$ is the pressure. This is an exact solution of the $\mathcal{G}_{rr}$ Einstein field equation for a homogenous and isotropic universe. Note that the $3P$ term, implying the self-gravity of pressure, has no analog in Newtonian gravity. The other well-used form of the Friedmann equations:
\begin{equation}
\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho ~~~,
\end{equation}
arises from the ${\bf \mathcal{G}}_{tt}$ component of the Einstein field equations (for a flat Universe). For any fluid with a significant pressure, and given its appropriate equation of state, eq.~(1) can be integrated to yield eq.~(2) -- but only, of course, if the $3P$ term is included.
It could be argued that eq.~(2) is adequate to derive the evolution of the scale factor, and yet it depends only on $\rho$, and not explicitly on $P$. Yet, we know that without the $3P$ term in eq.~(1), the constant of proportionality in front of $\rho$ in eq.~(2) will be incorrect, as long as pressure is significant.
The Friedmann equations have been frequently derived from Newtonian theory, in the context of a {\em pressureless} ($P=0$) universe, both for scientific and strictly pedagogical reasons \citetext{e.g., \citealp{Milne34}; \citealp{Uzan01}; see also \citealp{Jordan}}. However, since the concept of self-gravity of pressure does not exist in Newtonian gravity, it follows that eqs.~(1) and (2) cannot be properly derived from any purely Newtonian construction \citep{Rainsford00}. Finally we note that the Friedmann equation given by eq.~(2) (in which $P$ does not explicitly appear) formally ``knows'' about the form of eq.~(1) (which does contain $P$) via the Bianchi identity.
Consider now the radiation dominated epoch of the universe which prevailed for the first few thousand years after the big bang. Let us start with the Friedmann equation in the form of eq.~(1). From special relativistic considerations alone, we know that $P=1/3 \rho c^2$. Therefore the $\ddot a$ equation can be written as:
\begin{equation}
\frac{\ddot{a}}{a} = - \frac{4\pi G(1+\chi)}{3} \rho ~~~,
\end{equation}
where $\chi$ is used as a placemarker for the pressure term (i.e., $\chi = 1$ if the pressure term is present, and $\chi = 0$ if not). Since $\rho \propto a^{-4}$ for radiation and relativistic particles (e.g., neutrinos) undergoing an adiabatic expansion, eq.~(3) can be integrated to yield:
\begin{equation}
\left(\frac{\dot{a}}{a}\right)^2 ~=~~ \frac{8\pi G}{3} \left(\frac{1+\chi}{2} \right)\rho~~~,
\end{equation}
Variation of the value of $\chi$ away from unity would would significantly affect the outcome of BBN, degenerate with a variation in $G$ (the gravitational constant) or $N_{\nu}$ (the number of neutrino families). However, assuming these latter two to be constant at their nominal values of 1 and 3 respectively, nucleosynthetic constraints are able to place limits on $\chi$ and thus test the validity of this aspect of general relativity in this regime.
\subsection{Nucleosynthesis Calculations}
Nucleosynthesis calculations were performed with the canonical Kawano BBN codes \citep{Kawano92} which have been modified and updated with the latest reaction rates. The codes offer the ability to alter relevant early universe parameters such as $\eta$ and $G$. Since the parameter we seek to constrain, i.e., $(1+\chi$)/2, is multiplicative with $G$, we simply vary $G$ as a surrogate. Altering the parameters over repeated runs of these codes yields abundances of light elements as a function of $\eta$ and $\chi$.
In this work $\sim$75000 BBN calculations were performed, varying $\chi$ between $-1/2$ and 7 in steps of 0.02 (or equivalently $G/G_0$ from 1/4 to 4 in linear steps of 0.01) and varying $\log_{10}\eta$ from ${-10}$ to ${-9}$ in steps of 1/200 dex.
The result of these calculations, is a grid of $\{\eta$, $\chi\}$ pairs, each point with a set of associated elemental abundances (see Fig.\,1).
As estimated in \citet{Burles01b} the uncertainties in the nuclear reaction rates leading to deuterium are 3.4\% (95\% confidence limit), and as estimated in \citet{Burles01a} $\gtrsim $0.1\% (95\% confidence limit) for He-4.
\begin{figure*}[t]
\centering
\includegraphics[width = 5in]{f1}
\caption{Combination of the various constraints on $\chi$. The red-hatched region indicates the independent WMAP constraint on $\eta_{10}$ (i.e., the baryon/photon ratio in units of $10^{10}$). The bold contours represent the observed estimates and their $1\,\sigma$ errors. Green contours are deuterium abundances $(D/H)_p$. Blue contours are for the mass ratio of He-4 $(Y_p)$. Purple contours represent lithium abundances $(^7Li/H)_p$. The black ellipse is the 90-percent confidence contour based on the helium-4 and deuterium measurements.}
\end{figure*}
\newpage
\subsection{Light Element Observations}
Deuterium provides an excellent constraint because, as has been understood for the last few decades, the observed amounts necessitate that it must have formed in the big bang rather than in stellar or galactic processes \citep{Reeves73}. It is equally fortunate that it is extremely sensitive to the baryon to photon ratio, $\eta$ \citep{Burles01a}. Deuterium measurements from along the line of sight to high redshift quasars have culminated in the current determination of $\log_{10} (D/H)_p = -4.55 \pm 0.04$ \citetext{$1\,\sigma$ confidence; \citealp{Omeara06}}. Contours of constant deuterium abundance are shown as green curves in Fig.\,1.
Lithium-7 was initially a frontrunner as a nucleosynthetic constraint. However it has somewhat fallen out of favor because of large uncertainties arising from the systematic effects in stellar depletion models. \citet{Pinsonneault02} determine the primordial lithium abundance as $\log_{10} (Li /H)_p = -9.6 \pm 0.2$ ($1\,\sigma$ confidence), which while consistent at the 1-$\sigma$ level with the estimates derived from other light elements, is unable to provide any additional significant constraint due to the relatively large error bars. Until this situation can be improved, lithium-7 will not likely play a strong role in constructing BBN-based parameter limits. Contours of constant lithium abundance are shown as purple curves in Fig.\,1.
Helium-4, though not as sensitive to $\eta$ as deuterium, also serves as a useful constraint on $\chi$. The primordial abundance is determined from present-day observations of HII regions, coupled with models of galactic chemical evolution. The primordial He-4 mass fraction, $Y_p$, is most recently reported to be $0.2472 \pm 0.0012$ \citetext{$1\,\sigma$ confidence; \citealp{Izotov07}}. Contours of constant helium-4 abundance are shown as blue curves in Fig.\,1.
\subsection{WMAP Constraints}
Observations of the cosmic microwave background are able to provide an independent estimate of the baryon/photon ratio $\eta_{10}$ (i.e., $\eta$ in units of $10^{10}$). The {\em WMAP} collaboration \citep{Spergel07} gives as the three-year measurement $\eta_{10} = 6.116_{-0.249}^{+0.197}$. This measurement assumes $\chi = 1$, and therefore does not provide any additional constraint. However, as shown in Figure 1, the {\em WMAP} value of $\eta_{10}$ is in excellent agreement with the light element constraints.
\subsection{Constraints on $\chi$}
Figure 1 displays the results of our analysis. A section of the $\{\eta$, $\chi\}$ parameter space is shown, with {\em number} density (relative to hydrogen) contours for deuterium shown in green, and for {\em mass} fraction of He-4 shown in blue. The purple contours represent the Li-7 number density (relative to hydrogen). The boldest contours indicate the current best abundance determinations, with the adjacent intermediate-width contours indicating the corresponding ranges of uncertainties ($1\,\sigma$ confidence).
Assuming statisically-independent Gaussian errors, one can calculate the probability, via a maximum likelihood analysis, that the abundance determinations agree with the corresponding results of the BBN calculations at a given point in the $\{\eta$, $\chi\}$ parameter space. Only the constraints due to D and He-4 were used to compute the uncertainties in $\chi$ and $\eta$ (see \S 2.3 for an explanation of why Li-7 was not included). The black ellipse is the 90\% probability contour.
\section{Conclusions}
As illustrated in Figure 1, the combined constraints correspond well with the general relativity prediction of $\chi = 1$ and the WMAP constraint on $\eta_{10}$. The 90-percent confidence contour gives a limit of $\chi = 0.97\pm 0.06$. Thus, we have used current light element observations and BBN computations as a test of the general relativistic self-gravity of pressure. The agreement is good to within $\sim$$6\%$. We reiterate that our result can also be expressed as a
constraint on the Newtonian constant, $G$ (during the BBN epoch), the number of light neutrino
families, or degeneracies among neutrinos (see \S1 for references). Assuming that these latter parameters take on their nominally expected values, then the self-gravity associated with the pressure of light and neutrinos during the BBN epoch has been quantitatively measured.
\acknowledgements
JS acknowledges support from the Paul E. Gray (1954) Endowed Fund for UROP. We thank Al Levine, Scott Hughes, and Ed Bertschinger for very helpful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,933 |
Give us your feedback on the DC School Report Card! Take our short survey by Friday, Feb. 15.
Want to know more about what will be on the DC School Report Cards, and how you can use the information to engage more deeply in your child's school? Check out our resources and tools below for families. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,970 |
Feedback – The World Around Me!
Excellent bro! Ur blogs are really fun to read!! keep up the good writing!! but also try to raise some points to the situations happening around the world and specially mangalore k!! All the best!!
Anyway, I just wanted to shoot you this email in case you were interested in giving us any coverage. The event is organized by CARE, which is an international relief organization that focuses on women in poverty.
Its nice to hear from you. Yeah surely I will publish the news. I write for http://www.mangalorean.com on behalf of Mangalore Media Company. The website has more than 2 lakh unique visitors everyday. So I will be able to publish all the material on this site also. Please provide me with the pictures and Press Release/ News, Since I am based in Mangalore, India it is not possible for me to be there. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,412 |
Indianapolis Ballet Incorporated which runs the Indianapolis School of Ballet plans to debut a professional ballet company in the city next year.
Susan Moritz is the interim director of development. She says they are confident in meeting a $1.2 million capital campaign goal by the end of the year.
"We've had eleven years building the solid foundation of the school and really seeing the financial growth, the audience growth, and at this point in time it is just pretty evident that we can move forward," Moritz says.
The company will be called Indianapolis Ballet and audition dancers from across the country and the state.
"Indiana has two of the top-rated collegiate dance programs in the country and there is no resource for those dancers to dance professionally after they graduate," Moritz says. "So we're really focused on keeping that Hoosier talent in the state, in Indianapolis and give them a resource after they are done with school."
Indianapolis Ballet launches in early 2018 with its first performances at the Toby Theater at the Indianapolis Museum of Art. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,969 |
Imperial Motion's Nano Cure Technology is truly something you have to see to believe. Their special blend of nylon rip stop is able to repair puncture holes with just a few rubs of the fingers. We were amazed by how well it worked so we wanted to test the limits and share the results. After poking and prodding the Welder Jacket with a variety of sharp objects, we found that the material was able to quickly seal holes up to about 1/4" in diameter.
Puncture holes were no problem, but the material had a harder time repairing slices and tears. Once the fibers begin to fray, they aren't able to fully return to their original condition. However, we were still thoroughly impressed by its ability to eliminate minor damage. Nano Cure Tech allows you to seal holes quickly and easily, before they get out of hand.
Peep our full collection of Imperial Motion men's apparel and bring home the magic with their assortment of NCT jackets and bags. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,168 |
Increased nucleic acid receptor expression in chronic periodontitis
S. Esra Sahingur, Xia Juan Xia, Stephanie C. Voth, W. Andrew Yeudall, John C. Gunsolley
Background: Nucleic acid sensing has emerged as one of the important components of the immune system triggering inflammation. The aim of this study is to determine the expression of bacterial DNA sensors, including Toll-like receptor 9 (TLR-9), DNA-dependent activator of interferonregulatory factors (DAI), and absent in melanoma 2 (AIM2) in chronic periodontitis (CP versus healthy) (H) tissues. Methods: Thirty-five CP and 27 H gingival biopsies were included. Real-time quantitative polymerase chain reaction was performed to determine mRNA levels of AIM2, DAI, and TLRs (TLR-1 through TLR-9). The difference in gene expression for each sensor between CP and H tissues was calculated using analysis of covariance. The Spearman test was used to determine correlations among innate receptors. The expression of TLR-9, AIM2, and DAI in gingival tissues was further confirmed using immunohistochemistry. Results: The present results reveal statistically significant upregulation of TLR-9 (P <0.006), DAI (P <0.001), and TLR-8 (P <0.01) in CP tissues compared to H sites. Although mRNA expression was not changed significantly between groups for other receptors, the present results reveal significant correlations between receptors (P <0.05), suggesting that cooperation between multiple components of the host immune system may influence the overall response. Immunohistochemistry further confirmed expression of TLR-9, AIM2, and DAI in gingival tissues. Conclusions: This study highlights a possible role for nucleic acid receptors in periodontal inflammation. Future investigations will determine whether cytoplasmic receptors and their ligands can be targeted to improve clinical outcomes in periodontitis. J Periodontol 2013;84:e48-e57.
Journal of periodontology
https://doi.org/10.1902/jop.2013.120739
10.1902/jop.2013.120739
Dive into the research topics of 'Increased nucleic acid receptor expression in chronic periodontitis'. Together they form a unique fingerprint.
Toll-Like Receptor 9 Medicine & Life Sciences 100%
Chronic Periodontitis Medicine & Life Sciences 87%
Nucleic Acids Medicine & Life Sciences 70%
Melanoma Medicine & Life Sciences 47%
Immunohistochemistry Medicine & Life Sciences 19%
Bacterial DNA Medicine & Life Sciences 18%
Inflammation Medicine & Life Sciences 16%
Sahingur, S. E., Xia, X. J., Voth, S. C., Yeudall, W. A., & Gunsolley, J. C. (2013). Increased nucleic acid receptor expression in chronic periodontitis. Journal of periodontology, 84(10), e48-e57. https://doi.org/10.1902/jop.2013.120739
Increased nucleic acid receptor expression in chronic periodontitis. / Sahingur, S. Esra; Xia, Xia Juan; Voth, Stephanie C.; Yeudall, W. Andrew; Gunsolley, John C.
In: Journal of periodontology, Vol. 84, No. 10, 10.2013, p. e48-e57.
Sahingur, SE, Xia, XJ, Voth, SC, Yeudall, WA & Gunsolley, JC 2013, 'Increased nucleic acid receptor expression in chronic periodontitis', Journal of periodontology, vol. 84, no. 10, pp. e48-e57. https://doi.org/10.1902/jop.2013.120739
Sahingur SE, Xia XJ, Voth SC, Yeudall WA, Gunsolley JC. Increased nucleic acid receptor expression in chronic periodontitis. Journal of periodontology. 2013 Oct;84(10):e48-e57. https://doi.org/10.1902/jop.2013.120739
Sahingur, S. Esra ; Xia, Xia Juan ; Voth, Stephanie C. ; Yeudall, W. Andrew ; Gunsolley, John C. / Increased nucleic acid receptor expression in chronic periodontitis. In: Journal of periodontology. 2013 ; Vol. 84, No. 10. pp. e48-e57.
@article{79ac256efea446c7a2d9b36153a8f92f,
title = "Increased nucleic acid receptor expression in chronic periodontitis",
abstract = "Background: Nucleic acid sensing has emerged as one of the important components of the immune system triggering inflammation. The aim of this study is to determine the expression of bacterial DNA sensors, including Toll-like receptor 9 (TLR-9), DNA-dependent activator of interferonregulatory factors (DAI), and absent in melanoma 2 (AIM2) in chronic periodontitis (CP versus healthy) (H) tissues. Methods: Thirty-five CP and 27 H gingival biopsies were included. Real-time quantitative polymerase chain reaction was performed to determine mRNA levels of AIM2, DAI, and TLRs (TLR-1 through TLR-9). The difference in gene expression for each sensor between CP and H tissues was calculated using analysis of covariance. The Spearman test was used to determine correlations among innate receptors. The expression of TLR-9, AIM2, and DAI in gingival tissues was further confirmed using immunohistochemistry. Results: The present results reveal statistically significant upregulation of TLR-9 (P <0.006), DAI (P <0.001), and TLR-8 (P <0.01) in CP tissues compared to H sites. Although mRNA expression was not changed significantly between groups for other receptors, the present results reveal significant correlations between receptors (P <0.05), suggesting that cooperation between multiple components of the host immune system may influence the overall response. Immunohistochemistry further confirmed expression of TLR-9, AIM2, and DAI in gingival tissues. Conclusions: This study highlights a possible role for nucleic acid receptors in periodontal inflammation. Future investigations will determine whether cytoplasmic receptors and their ligands can be targeted to improve clinical outcomes in periodontitis. J Periodontol 2013;84:e48-e57.",
keywords = "AIM2, Bacterial, DNA, Inflammation, Periodontal disease, Periodontitis, Toll-like receptors",
author = "Sahingur, {S. Esra} and Xia, {Xia Juan} and Voth, {Stephanie C.} and Yeudall, {W. Andrew} and Gunsolley, {John C.}",
doi = "10.1902/jop.2013.120739",
journal = "Journal of Periodontology",
publisher = "American Academy of Periodontology",
T1 - Increased nucleic acid receptor expression in chronic periodontitis
AU - Sahingur, S. Esra
AU - Xia, Xia Juan
AU - Voth, Stephanie C.
AU - Yeudall, W. Andrew
AU - Gunsolley, John C.
N2 - Background: Nucleic acid sensing has emerged as one of the important components of the immune system triggering inflammation. The aim of this study is to determine the expression of bacterial DNA sensors, including Toll-like receptor 9 (TLR-9), DNA-dependent activator of interferonregulatory factors (DAI), and absent in melanoma 2 (AIM2) in chronic periodontitis (CP versus healthy) (H) tissues. Methods: Thirty-five CP and 27 H gingival biopsies were included. Real-time quantitative polymerase chain reaction was performed to determine mRNA levels of AIM2, DAI, and TLRs (TLR-1 through TLR-9). The difference in gene expression for each sensor between CP and H tissues was calculated using analysis of covariance. The Spearman test was used to determine correlations among innate receptors. The expression of TLR-9, AIM2, and DAI in gingival tissues was further confirmed using immunohistochemistry. Results: The present results reveal statistically significant upregulation of TLR-9 (P <0.006), DAI (P <0.001), and TLR-8 (P <0.01) in CP tissues compared to H sites. Although mRNA expression was not changed significantly between groups for other receptors, the present results reveal significant correlations between receptors (P <0.05), suggesting that cooperation between multiple components of the host immune system may influence the overall response. Immunohistochemistry further confirmed expression of TLR-9, AIM2, and DAI in gingival tissues. Conclusions: This study highlights a possible role for nucleic acid receptors in periodontal inflammation. Future investigations will determine whether cytoplasmic receptors and their ligands can be targeted to improve clinical outcomes in periodontitis. J Periodontol 2013;84:e48-e57.
AB - Background: Nucleic acid sensing has emerged as one of the important components of the immune system triggering inflammation. The aim of this study is to determine the expression of bacterial DNA sensors, including Toll-like receptor 9 (TLR-9), DNA-dependent activator of interferonregulatory factors (DAI), and absent in melanoma 2 (AIM2) in chronic periodontitis (CP versus healthy) (H) tissues. Methods: Thirty-five CP and 27 H gingival biopsies were included. Real-time quantitative polymerase chain reaction was performed to determine mRNA levels of AIM2, DAI, and TLRs (TLR-1 through TLR-9). The difference in gene expression for each sensor between CP and H tissues was calculated using analysis of covariance. The Spearman test was used to determine correlations among innate receptors. The expression of TLR-9, AIM2, and DAI in gingival tissues was further confirmed using immunohistochemistry. Results: The present results reveal statistically significant upregulation of TLR-9 (P <0.006), DAI (P <0.001), and TLR-8 (P <0.01) in CP tissues compared to H sites. Although mRNA expression was not changed significantly between groups for other receptors, the present results reveal significant correlations between receptors (P <0.05), suggesting that cooperation between multiple components of the host immune system may influence the overall response. Immunohistochemistry further confirmed expression of TLR-9, AIM2, and DAI in gingival tissues. Conclusions: This study highlights a possible role for nucleic acid receptors in periodontal inflammation. Future investigations will determine whether cytoplasmic receptors and their ligands can be targeted to improve clinical outcomes in periodontitis. J Periodontol 2013;84:e48-e57.
KW - AIM2
KW - Bacterial
KW - Inflammation
KW - Periodontal disease
KW - Periodontitis
KW - Toll-like receptors
U2 - 10.1902/jop.2013.120739
DO - 10.1902/jop.2013.120739
JO - Journal of Periodontology
JF - Journal of Periodontology | {
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Duran Duran 'excited' to be in Malta
July 25, 2008 Press
The ever popular band Duran Duran arrived in Malta this afternoon for their first ever concert in Malta on Saturday.
The members of the group - the original line-up of Simon Le Bon, John Taylor, Nick Rhodes and Roger Taylor - were given an enthusiastic welcome by a small roup of fans and paused to sign autographs.
The 'boys' said they were excited to be performing in Malta for the first time, seeing it as a fitting follow-up to their successful concerts in Italy. Their last concert, in Calabria, drew 50,000. While in Malta they also intend to take a short break.
The concert, at the Luxol Grounds, forms part of the band's successful Red Carpet Massacre tour
Ira Losco and Tony Moore will provide the backing acts.
Duran Duran have been brought to Malta by NnG Promotions Ltd.
Courtesy Times of Malta
The Wild Boys are here
A Blog From John | {
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} | 8,224 |
{"url":"https:\/\/www.quizover.com\/physics1\/course\/11-2-angular-momentum-by-openstax?page=4","text":"# 11.2 Angular momentum \u00a0(Page 5\/8)\n\n Page 5 \/ 8\n\n## Angular momentum of a robot arm\n\nA robot arm on a Mars rover like Curiosity shown in [link] is 1.0 m long and has forceps at the free end to pick up rocks. The mass of the arm is 2.0 kg and the mass of the forceps is 1.0 kg. See [link] . The robot arm and forceps move from rest to $\\omega =0.1\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{rad}\\text{\/}\\text{s}$ in 0.1 s. It rotates down and picks up a Mars rock that has mass 1.5 kg. The axis of rotation is the point where the robot arm connects to the rover. (a) What is the angular momentum of the robot arm by itself about the axis of rotation after 0.1 s when the arm has stopped accelerating? (b) What is the angular momentum of the robot arm when it has the Mars rock in its forceps and is rotating upwards? (c) When the arm does not have a rock in the forceps, what is the torque about the point where the arm connects to the rover when it is accelerating from rest to its final angular velocity?\n\n## Strategy\n\nWe use [link] to find angular momentum in the various configurations. When the arm is rotating downward, the right-hand rule gives the angular momentum vector directed out of the page, which we will call the positive z -direction. When the arm is rotating upward, the right-hand rule gives the direction of the angular momentum vector into the page or in the negative z- direction. The moment of inertia is the sum of the individual moments of inertia. The arm can be approximated with a solid rod, and the forceps and Mars rock can be approximated as point masses located at a distance of 1 m from the origin. For part (c), we use Newton\u2019s second law of motion for rotation to find the torque on the robot arm.\n\n## Solution\n\n1. Writing down the individual moments of inertia, we have\nRobot arm: ${I}_{\\text{R}}=\\frac{1}{3}{m}_{\\text{R}}{r}^{2}=\\frac{1}{3}\\left(2.00\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\\right){\\left(1.00\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)}^{2}=\\frac{2}{3}\\phantom{\\rule{0.1em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}.$\nForceps: ${I}_{\\text{F}}={m}_{\\text{F}}{r}^{2}=\\left(1.0\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\\right){\\left(1.0\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)}^{2}=1.0\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}.$\nMars rock: ${I}_{\\text{MR}}={m}_{\\text{MR}}{r}^{2}=\\left(1.5\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\\right){\\left(1.0\\phantom{\\rule{0.2em}{0ex}}\\text{m}\\right)}^{2}=1.5\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}.$\nTherefore, without the Mars rock, the total moment of inertia is\n${I}_{\\text{Total}}={I}_{\\text{R}}+{I}_{\\text{F}}=1.67\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}$\nand the magnitude of the angular momentum is\n$L=I\\omega =1.67\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}\\left(0.1\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{rad}\\text{\/}\\text{s}\\right)=0.17\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}\\text{\/}\\text{s}.$\n\nThe angular momentum vector is directed out of the page in the $\\stackrel{^}{k}$ direction since the robot arm is rotating counterclockwise.\n2. We must include the Mars rock in the calculation of the moment of inertia, so we have\n${I}_{\\text{Total}}={I}_{\\text{R}}+{I}_{\\text{F}}+{I}_{\\text{MR}}=3.17\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}$\n\nand\n$L=I\\omega =3.17\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}\\left(0.1\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{rad}\\text{\/}\\text{s}\\right)=0.32\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}\\text{\/}\\text{s}\\text{.}$\n\nNow the angular momentum vector is directed into the page in the $\\text{\u2212}\\stackrel{^}{k}$ direction, by the right-hand rule, since the robot arm is now rotating clockwise.\n3. We find the torque when the arm does not have the rock by taking the derivative of the angular momentum using [link] $\\frac{d\\stackrel{\\to }{L}}{dt}=\\sum \\stackrel{\\to }{\\tau }.$ But since $L=I\\omega$ , and understanding that the direction of the angular momentum and torque vectors are along the axis of rotation, we can suppress the vector notation and find\n$\\frac{dL}{dt}=\\frac{d\\left(I\\omega \\right)}{dt}=I\\frac{d\\omega }{dt}=I\\alpha =\\sum \\tau ,$\n\nwhich is Newton\u2019s second law for rotation. Since $\\alpha =\\frac{0.1\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{rad}\\text{\/}\\text{s}}{0.1\\phantom{\\rule{0.2em}{0ex}}\\text{s}}=\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{rad}\\text{\/}{\\text{s}}^{2}$ , we can calculate the net torque:\n$\\sum \\tau =I\\alpha =1.67\\phantom{\\rule{0.2em}{0ex}}\\text{kg}\u00b7{\\text{m}}^{2}\\left(\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{rad}\\text{\/}{\\text{s}}^{2}\\right)=1.67\\pi \\phantom{\\rule{0.2em}{0ex}}\\text{N}\u00b7\\text{m}.$\n\n## Significance\n\nThe angular momentum in (a) is less than that of (b) due to the fact that the moment of inertia in (b) is greater than (a), while the angular velocity is the same.\n\nV=E\u00bd-P-\u00bd where v; velocity, P; density and E; constant. Find dimension and it's units of E (constant)\nML-3\nLAWAL\nderivation of simple harmonic equation\nif an equation is dimensionally correct does this mean that equation must be true?\nhow do I calculate angular velocity\nw=vr where w, angular velocity. v; velocity and r; radius of a circle\nmichael\nsorry I meant Maximum positive angular velocity of\nPriscilla\nCan any one give me the definition for Bending moment plz...\nI need a question for moment\nwhat is charge\nAn attribution of particle that we have thought about to explain certain things like Electomagnetism\nNikunj\nplease what is the formula instantaneous velocity in projectile motion\nA computer is reading from a CD-ROM that rotates at 780 revolutions per minute.What is the centripetal acceleration at a point that is 0.030m from the center of the disc?\nchange revolution per minute by multiplying from 2pie and devide by 60.and take r=.030 and use formula centripital acceleration =omega sqare r.\nKumar\nOK thank you\nRapqueen\nobservation of body boulded\na gas is compressed to 1\/10 0f its original volume.calculate the rise temperature if the original volume is 400k. gamma =1.4\nthe specific heat of hydrogen at constant pressure and temperature is 14.16kj|k.if 0.8kg of hydrogen is heated from 55 degree Celsius to 80 degree Celsius of a constant pressure. find the external work done .\nCeline\nhi\nshaik\nhy\nPrasanna\ng\nwhat is imaginary mass and how we express is\nwhat is imaginary mass how we express it\nYash\ncentre of mass is also called as imaginary mass\nLokmani\nl'm from Algeria and fell these can help me\nMany amusement parks have rides that make vertical loops like the one shown below. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if: (a) The car goes over the top at faster than this speed? (b) The car goes over the top at slower than this speed?\nhow can I convert mile to meter per hour\n1 mile * 1609m\nBoon\nhey can someone show me how to solve the - \"Hanging from the ceiling over a baby bed ....\" question\ni wanted to know the steps\nShrushti\nsorry shrushti..\nRashid\nwhich question please write it briefly\nAsutosh","date":"2018-11-15 20:59:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 15, \"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\": 0.7867985963821411, \"perplexity\": 662.2539442876965}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-47\/segments\/1542039742937.37\/warc\/CC-MAIN-20181115203132-20181115225132-00106.warc.gz\"}"} | null | null |
Where the Mountain Meets the Moon (Paperback)
By Grace Lin
1 on hand, as of Jan 28 12:07am
(JUV-)
Autumn 2009 Kids' Indie Next List
"In the poor village of Fruitless Mountain -- where there is always lots of work to do and little food or time to rest -- a young girl named Minli decides to set off on a journey to find a way to improve her family's plight. Lin's story takes you on a magical adventure with a vivid setting and wondrous characters."
— Lisa Fabiano, Wellesley Booksmith, Wellesley, MA
This stunning fantasy inspired by Chinese folklore is a companion novel to Starry River of the Sky and the New York Times bestselling and National Book Award finalist When the Sea Turned to Silver
In the valley of Fruitless mountain, a young girl named Minli lives in a ramshackle hut with her parents. In the evenings, her father regales her with old folktales of the Jade Dragon and the Old Man on the Moon, who knows the answers to all of life's questions. Inspired by these stories, Minli sets off on an extraordinary journey to find the Old Man on the Moon to ask him how she can change her family's fortune. She encounters an assorted cast of characters and magical creatures along the way, including a dragon who accompanies her on her quest for the ultimate answer.
Grace Lin, author of the beloved Year of the Dog and Year of the Rat returns with a wondrous story of adventure, faith, and friendship. A fantasy crossed with Chinese folklore, Where the Mountain Meets the Moon is a timeless story reminiscent of The Wizard of Oz and Kelly Barnhill's The Girl Who Drank the Moon. Her beautiful illustrations, printed in full-color, accompany the text throughout. Once again, she has created a charming, engaging book for young readers.
Grace Lin is the award-winning and bestselling author and illustrator of Starry River of the Sky, Where the Mountain Meets the Moon, The Year of the Dog, The Year of the Rat, Dumpling Days, and Ling & Ting, as well as pictures books such as The Ugly Vegetables and Caldecott Honor book A Big Mooncake for Little Star. Grace lives with her husband and daughter in Western MA, where they get plenty of winter snow. Her website is www.gracelin.com.
Juvenile Fiction / Fantasy & Magic
Juvenile Fiction / Legends, Myths, Fables / Asian
Juvenile Fiction / Historical / Asia
Juvenile Fiction / Girls & Women
Hardcover (July 1st, 2009): $18.99
Paperback (October 8th, 2019): $36.00
Library Binding, Large Print (February 19th, 2020): $27.59 | {
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To expand the air connectivity and hold the rising air passenger numbers, the government plans to construct about 100 more airports in the country in next 15 years. Once constructed, the number of airports in the country would double. These airports will be built at an estimated investment of Rs 4 lakh crore. Of the 100 airports, 70 will be constructed in cities which do not have such a facility and the remaining 30 will be second airports or the expansion of existing airfields to handle commercial flights.
The Supreme Court has refused to allow Jaypee Associates to deposit Rs 400 crore with its registry as against Rs 2,000 crore directed by it. The apex court has asked the developer to submit a substantial amount to prove its bonafide. On September 11, the developer was asked to deposit Rs 2,000 crore with the apex court registry by October 27.
The trial runs on the Delhi Metro's Mundka-Bahadurgarh stretch is set to begin in December. The 11-km-long line from west Delhi will make Bahadurgarh the third Haryana township in the Metro's fold, after Gurgaon and Faridabad. The corridor, which is an extension of the Inderlok- Mundka Green Line (Line 5), is scheduled to be commissioned in June 2018.
The Greater Mohali Area Development Authority (GMADA) has sent the final notices for land acquisition for the development of the third phase of the residential scheme Eco City. For the project, GMADA will acquire 322 acres of land from six villages in New Chandigarh. | {
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How to download IFS Admit Card 2018?
Indian Forest Services, IFS Admit Card for mains examination can be downloaded now. Union Public Service Commission released the official IFS Mains Admit Card on 12th November 2018. All the candidates who are eligible to write mains examination can download their official IFS Mains Hall Ticket through the direct link provided in this post. It is advised that candidates should download their IFS Mains Call Letter as soon as possible to avoid any technical glitch.
Select the option, Registration ID/Roll No.
Now enter your Roll No,/ Registration ID along with Date of Birth.
Confirm random image and click on submit.
A new page will appear with downloaded IFS Admit card for Mains 2018.
Save and take a print out of the official IFS 2018 Mains Hall Ticket for future reference.
Do not forget to carry the IFS Admit Card to the examination hall as any candidate without the official Admit Card will not be allowed to write the exam.
Carry a govt. valid identity proof along with the IFS Hall Ticket 2018.
Hope this post on IFS Admit Card was useful for you. If you have any query or suggestion, Do write us in the comments below. | {
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Three new titles make their debut on Blu-Ray and DVD this week, highlighted by the Academy Award nominated War Horse. Also landing is Cameron Crowe's We Bought a Zoo and the documentary Being Elmo.
We Bought a Zoo Quotes: This Place is Perfect!
We Bought a Zoo is an utter delight and stars Matt Damon and Scarlett Johansson in a Cameron Crowe film. Like so many Crowe films of the past, it has a ton of great movie quotes.
Cameron Crowe is back to his best with We Bought a Zoo. Matt Damon and Scarlett Johansson star in a true story that will warm the heart and is perfect for the holiday season.
More advance screenings have been added for the fan favorite We Bought a Zoo. The Cameron Crowe-directed film scored huge with advance preview audiences when it screened over Thanksgiving.
We Bought a Zoo, starring Matt Damon and Scarlett Johansson, had a sneak peek over the weekend that produced astounding positive numbers from audiences. We Bought a Zoo lands in theaters December 23 and thus far, seems to be a must see.
We Bought a Zoo has debuted a new trailer and poster. Cameron Crowe directs We Bought a Zoo, that stars Matt Damon and Scarlett Johansson.
'We Bought a Zoo' takes place in Southern California. A family moves to the country side because the father wants to try to renovate and re-open a struggling zoo.
Benjamin: This place is perfect! Why didn't you mention it earlier?
Mr. Stevens: It's a bit complicated.
Benjamin: Complicated's okay. What's so complicated about this place?
Mr. Stevens: Well, you see, it's uhhh... (Lion roars, loudly) It's a zoo. | {
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\section{Foreword}
The original author of the manuscript, Professor Seyyed Mahmoud Hessaby
\cite{1}, devised the initial theory of Continuous Particles
while he attended a visiting position at the Princeton's Institute of
Advanced Study by invitation of Albert Einstein. He published the initial
line of thoughts in 1947 in the Proceedings of National Academy of Sciences
\cite{2}. He also published a second paper in 1957 in French language
entitled Mod\`ele de particule infinie \cite{3} (Model of Infinite Particles).
About twenty years later in 1977, he managed to formulate a fully extended
and unified version of his theory but for unclear reasons it remained as
unpublished; he only reproduced it in very few numbers at Tehran University
Press \cite{4}, where he had founded himself. This TeX article has been typeset
based on a rare remaining copy obtained from within the US market.
This paper presents the only existing theory which unifies the three forces
and also successfully estimates the mass ratios of various elementary particles.
At present, there is simply no other theory being capable of estimation of
particle mass ratios, at least within the framework of Standard Model.
Aside from a couple of minor calculation errors and typos which have been
removed during the reproduction, Professor Hessaby's derivations seem to be
completely flawless.
Professor Mahmoud Hessaby's contributions to the infrastructure of
science in the Iranian society was just overwhelming. He founded the
Tehran University, Iranian Physical Society, Atomic Energy Organization
of Iran, Institute for Geophysics, the first Radio Broadcasting Center,
the first Private Hospital, and much more.
Professor Seyyed Mahmoud Hessaby passed away in 1992. He is correctly recognized
as the \textit{Father of Modern Physics in Iran}.
\section{The Generalized Field}
\subsection{The Four-Potential}
Our postulate is that the gravitational, electric, and nuclear fields are special cases of a more general field. Designating the four potential by $\Phi_\mu$, the field in the general case is given by the expression
\begin{equation}
\label{i1}
F_{\mu\nu}=\frac{\partial \Phi_{\mu}}{\partial x_\nu}-\frac{\partial \Phi_\nu}{\partial x_\mu}
\end{equation}
\noindent
The charge-current vector is
\begin{equation}
\label{i2}
J^{\mu}=\frac{1}{2\pi}(F^{\mu\nu})_{,\nu}
\end{equation}
\noindent
and the energy tensor is
\begin{equation}
\label{i3}
U^{\nu}_{\mu}=\frac{1}{2\pi}(-F^{\nu\alpha}F_{\mu\alpha}+\frac{1}{4}g^{\nu}_{\mu}F^{\alpha\beta}F_{\alpha\beta})
\end{equation}
\noindent
By contraction of (\ref{i3}) we have also
\begin{equation}
\label{i4}
U=0
\end{equation}
We assume first the existence of only two components of $\Phi_\mu$ viz, $\Phi_3$ and $\Phi_4$, $\Phi_4$ depending only on $r$, while $\Phi_3$ may depend on both $r$ and $\theta$. The components of the field are as given in the table~\ref{tab1.1}.
\begin{table}
\caption{\label{tab1.1}Components of the field tensor $F_{\mu\nu}$.}
\begin{ruledtabular}
\begin{tabular}{cccc}
0 & 0 & $-\frac{\partial \Phi_3}{\partial x_1}$ & $-\frac{\partial \Phi_4}{\partial x_1}$ \\
0 & 0 & $-\frac{\partial \Phi_3}{\partial x_2}$ & 0 \\
$\frac{\partial \Phi_3}{\partial x_1}$ & $\frac{\partial \Phi_3}{\partial x_2}$ & 0 & 0 \\
$\frac{\partial \Phi_4}{\partial x_1}$ & 0 & 0 & 0 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{The Potential $\Phi_4$}
We take first the case in which the component $\Phi_3=0$. We write the line element in the form
\begin{equation}
\label{ii1}
ds^2=-e^\alpha dr^2-e^\beta r^2 d\theta^2-e^\gamma r^2 \sin^2\theta d\phi^2+e^\delta dt^2
\end{equation}
\noindent
where we assume that $\alpha$, $\beta$, $\gamma$, and $\delta$ depend only on $r$.
In order to write down the components of $U^\nu_\mu$, we calculate first $F^{\alpha\beta}F_{\alpha\beta}$. Since we assume that $\Phi_4$ depends only on $r$, we have $\frac{\partial \Phi_4}{\partial \theta}=\frac{\partial \Phi_4}{\partial \phi}=0$, so that
\begin{equation}
\label{ii2}
F^{\alpha\beta}F_{\alpha\beta}=F^{14}F_{14}+F^{41}F_{41}=2g^{11}g^{44}(F_{14})^2=-2e^{-(\alpha+\delta)}(F_{14})^2
\end{equation}
\noindent
Also
\begin{equation}
\label{ii3}
F^{1\alpha}F_{1\alpha}=F^{4\alpha}F_{4\alpha}=g^{11}g^{44}(F_{14})^2=-e^{-(\alpha+\delta)}(F_{14})^2
\end{equation}
\noindent
so that
\begin{equation}
\label{ii4}
U^1_1=-U^2_2=-U^3_3=U^4_4=\frac{1}{4\pi}e^{-(\alpha+\delta)}(F_{14})^2
\end{equation}
\noindent
The contracted tensor is
\begin{equation}
\label{ii5}
U=\sum U^\nu_\mu=0
\end{equation}
\noindent
Designating the Ricci tensor by $R^\nu_\mu$, the expression for the energy tensor is
\begin{equation}
\label{ii6}
T^\nu_\mu=-\frac{a}{4\pi}(R^\nu_\mu-\frac{1}{2}g^\nu_\mu R)
\end{equation}
\noindent
where $a$ is a dimensional constant.
Since we have identically $U=0$, the identification of $T^\nu_\mu$ with $U^\nu_\mu$, gives by contraction
\begin{equation}
\label{ii7}
U=T=\frac{a}{4\pi}R=0
\end{equation}
\noindent
so that we have identically $R=0$.
Taking account of (\ref{ii7}), relation (\ref{ii6}) becomes
\begin{equation}
\label{ii8}
U^\nu_\mu=T^\nu_\mu=-\frac{a}{4\pi}R^\nu_\mu
\end{equation}
\noindent
relations (\ref{ii4}) and (\ref{ii8}) give
\begin{equation}
\label{ii9}
R^1_1=-R^2_2=-R^3_3=R^4_4=-\frac{1}{a}e^{-(\alpha+\delta)}(F_{14})^2
\end{equation}
\noindent
We write $R^\nu_\mu$ in terms of $\alpha$, $\beta$, $\gamma$, $\delta$. The only Christoffel symbols which do not vanish are enlisted in table~\ref{tab2.1}.
\begin{table}
\caption{\label{tab2.1}Non-vanishing Christoffel symbols.}
\begin{ruledtabular}
\begin{tabular}{ll}
$\{11,1\}=\frac{1}{2}\alpha^\prime$ & $\{21,2\}=\frac{1}{2}\beta^\prime+\frac{1}{r}$\\
$\{12,2\}=\frac{1}{2}\beta^\prime+\frac{1}{r}$ & $\{22,1\}=-e^{\beta-\alpha}(\frac{1}{2}r^{2\beta^\prime}+r)$\\
$\{13,3\}=\frac{1}{2}\gamma^\prime+\frac{1}{r}$ & $\{23,3\}=\frac{\cos\theta}{\sin\theta}$\\
$\{14,4\}=\frac{1}{2}\delta^\prime$ & \\
$\{31,3\}=\frac{1}{2}\gamma^\prime+\frac{1}{r}$ & $\{41,4\}=\frac{1}{2}\delta^\prime$\\
$\{32,3\}=\frac{\cos\theta}{\sin\theta}$ & $\{44,1\}=\frac{1}{2}\delta^\prime e^{\delta-\alpha}$\\
$\{33,1\}=-e^{\gamma-\alpha} \sin^2\theta (\frac{1}{2}r^2\gamma^\prime+r)$ & \\
$\{33,2\}=-e^{\gamma-\beta} \sin\theta\cos\theta$ & \\
\end{tabular}
\end{ruledtabular}
\end{table}
\noindent
The expressions that we obtain for the $R^\nu_\mu$ are
\begin{eqnarray}
\label{ii10}
R^1_1=e^{-\alpha} &&(-\frac{1}{2}\beta^{\prime\prime}-\frac{1}{2}\gamma^{\prime\prime}-\frac{1}{2}\delta^{\prime\prime}-\frac{1}{4}\beta^{\prime 2}-
\frac{1}{4}\gamma^{\prime 2}-\frac{1}{4}\delta^{\prime 2}+\frac{1}{4}\alpha^\prime\beta^\prime\\&&
+\frac{1}{4}\alpha^\prime\gamma^\prime+\frac{1}{4}\alpha^\prime\delta^\prime+\frac{\alpha^\prime}{r}-\frac{\beta^\prime}{r}-\frac{\delta^\prime}{r})
\nonumber
\end{eqnarray}
\begin{eqnarray}
\label{ii11}
R^2_2=e^{-\alpha}&&(-\frac{1}{2}\beta^{\prime\prime}-\frac{1}{4}\beta^{\prime 2}+\frac{1}{4}\alpha^\prime\beta^\prime-
\frac{1}{4}\beta^\prime\gamma^\prime-\frac{1}{4}\beta^\prime\delta^\prime\\&&
+\frac{1}{2}\frac{\alpha^\prime}{r}-\frac{3}{2}\frac{\beta^\prime}{r}-\frac{1}{2}\frac{\gamma^\prime}{r}-\frac{1}{2}\frac{\delta^\prime}{r}-
\frac{1}{r^2})+\frac{e^{-\beta}}{r^2}\nonumber
\end{eqnarray}
\begin{eqnarray}
\label{ii12}
R^3_3=e^{-\alpha}&&(-\frac{1}{2}\gamma^{\prime\prime}-\frac{1}{4}\gamma^{\prime 2}+\frac{1}{4}\alpha^\prime\gamma^\prime-\frac{1}{4}\beta^\prime\gamma^\prime-\frac{1}{4}\gamma^\prime\delta^\prime\\&&
+\frac{1}{2}\frac{\alpha^\prime}{r}-\frac{1}{2}\frac{\beta^\prime}{r}-\frac{3}{2}\frac{\gamma^\prime}{r}-
\frac{1}{2}\frac{\delta^\prime}{r}-\frac{1}{r^2})+\frac{e^{-\beta}}{r^2}\nonumber
\end{eqnarray}
\begin{equation}
\label{ii13}
R^4_4=e^{-\alpha}(-\frac{1}{2}\delta^{\prime\prime}-\frac{1}{4}\delta^{\prime 2}+\frac{1}{4}\alpha^\prime\delta^\prime-\frac{1}{4}\beta^\prime\delta^\prime-
\frac{1}{4}\gamma^\prime\delta^\prime-\frac{\delta^\prime}{r})
\end{equation}
By relation (\ref{ii7}) the contracted tensor $R$ is zero
\begin{eqnarray}
\label{ii14}
R=\sum R^\mu_\mu=e^{-\alpha}&&(-\beta^{\prime\prime}-\gamma^{\prime\prime}-\delta^{\prime\prime}-\frac{1}{2}\beta^{\prime 2}
-\frac{1}{2}\gamma^{\prime 2}-\frac{1}{2}\delta^{\prime 2}\\&&
+\frac{1}{2}\alpha^\prime\beta^\prime+\frac{1}{2}\alpha^\prime\gamma^\prime+\frac{1}{2}\alpha^\prime\delta^\prime
-\frac{1}{2}\beta^\prime\delta^\prime-\frac{1}{2}\gamma^\prime\delta^\prime-\frac{1}{2}\beta^\prime\delta^\prime \nonumber\\&&
+\frac{2\alpha^\prime}{r}-\frac{3\beta^\prime}{r}-\frac{3\gamma^\prime}{r}-\frac{2\delta^\prime}{r}-\frac{2}{r^2})+\frac{2e^{-\beta}}{r^2}
\nonumber\\&&=0\nonumber
\end{eqnarray}
We remark that expressions (\ref{ii11}) and (\ref{ii12}) are identical in form except for the interchange of $\beta^\prime$ and $\gamma^\prime$. This fact, together with the equality $R^2_2=R^3_3$, suggests that we assume the equality of $\beta^\prime$ and $\gamma^\prime$. Replacing $\gamma^\prime$ by $\beta^\prime$ in relations (\ref{ii10}) to (\ref{ii14}), and making use of the equality $R^1_1=R^4_4$, we obtain
\begin{equation}
\label{ii15}
e^{-\alpha}(\beta^{\prime\prime}+\frac{1}{2}\beta^{\prime 2}+2\frac{\beta^\prime}{r}-\frac{1}{2}\alpha^\prime\beta^\prime-\frac{1}{2}\beta^\prime\delta^\prime-\frac{\alpha^\prime}{r}-\frac{\delta^\prime}{r})=0
\end{equation}
\subsubsection{Case I}
We consider first the case where $\beta=\gamma=0$. Relation (\ref{ii15}) gives then $\delta^\prime=-\alpha^\prime$, and relation (\ref{ii14}), with $\delta=-\alpha$ becomes
\begin{equation}
\label{ii16}
e^\delta(-\delta^{\prime\prime}-\delta^{\prime 2}-4\frac{\delta^\prime}{r}-\frac{2}{r^2})+\frac{2}{r^2}=0
\end{equation}
\noindent
The solution
\begin{equation}
\label{ii17}
e^\delta=(1+\frac{K}{r})^2
\end{equation}
\noindent
where $K$ is a constant depending on the mass of the particle and having the dimensions of a length, satisfies equation (\ref{ii16}). In this special case, we have therefore
\begin{equation}
\label{ii18}
e^\alpha=e^{-\delta}=(1+\frac{K}{r})^{-2}
\end{equation}
\noindent
and since we have assumed $\beta=\gamma=0$, the line element becomes
\begin{equation}
\label{ii19}
ds^2=-\frac{1}{(1+\frac{K}{r})^2}dr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2+(1+\frac{K}{r})^2 dt^2
\end{equation}
\subsubsection{Case II}
If we extend the validity of the equality $\delta^\prime=-\alpha^\prime$ to the general case where $\beta^\prime\neq 0$, the differential equation (\ref{ii15}) reduces to
\begin{equation}
\label{ii20}
\beta^{\prime\prime}+\frac{1}{2}\beta^{\prime 2}+2\frac{\beta^\prime}{r}=0
\end{equation}
\noindent
which admits the solution
\begin{equation}
\label{ii21}
\beta^\prime=-\frac{2A}{r(r+A)}
\end{equation}
\noindent
giving
\begin{equation}
\label{ii22}
\beta=2\log \frac{r+A}{r}=\log(1+\frac{A}{r})^2
\end{equation}
\noindent
so that
\begin{equation}
\label{ii23}
e^\beta=(1+\frac{A}{r})^2
\end{equation}
\noindent
To find the expression for $\alpha$, we make use of the relation $R=0$ in equation (\ref{ii14}), which when set $\gamma=\beta$ becomes
\begin{equation}
\label{ii24}
e^{-\alpha}(-2\beta^{\prime\prime}-\frac{3}{2}\beta^{\prime 2}-6\frac{\beta^\prime}{r}+\alpha^{\prime\prime}-\alpha^{\prime 2}+4\frac{\alpha^\prime}{r}+2\alpha^\prime\beta^\prime-\frac{1}{r^2})+\frac{e^{-\beta}}{r^2}=0
\end{equation}
\noindent
The solution
\begin{equation}
\label{ii25}
\alpha^\prime=\beta^\prime=-\frac{2A}{r(r+A)}
\end{equation}
\noindent
satisfies relation (\ref{ii24}). We have thus, with $\delta=-\alpha$
\begin{eqnarray}
\label{ii26}
&&e^\alpha=e^\beta=e^\gamma=(1+\frac{A}{r})^2\\
&&e^\delta=(1+\frac{A}{r})^{-2}\nonumber
\end{eqnarray}
\noindent
and the line element assumes the form
\begin{equation}
\label{ii27}
ds^2=-(1+\frac{A}{r})^2(dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2)+\frac{1}{(1+\frac{A}{r})^2}dt^2
\end{equation}
\section{The Gravitational Field}
\subsection{First Solutions of the Field Equation for $\Phi_4$}
We consider the form of the line element as found in (\ref{ii19})
\begin{equation}
\label{iii1}
ds^2=-\frac{1}{(1+\frac{K}{r})^2}dr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2+(1+\frac{K}{r})^2 dt^2
\end{equation}
\noindent
The field $F_{14}$ is given by relation (\ref{ii4}), which $\delta=-\alpha$
\begin{equation}
\label{iii2}
U^4_4=\frac{1}{4\pi}e^{-(\alpha+\delta)}(F_{14})^2=\frac{1}{4\pi}(F_{14})^2
\end{equation}
\noindent
Relation (\ref{ii13}) becomes, since we have here $\delta=-\alpha$ and $\beta=\gamma=0$
\begin{equation}
\label{iii3}
R^4_4=e^\delta(-\frac{1}{2}\delta^{\prime\prime}-\frac{1}{2}\delta^{\prime 2}-\frac{\delta^\prime}{r})
\end{equation}
\noindent
By (\ref{ii17}), we have $e^\delta=(1+\frac{K}{r})^2$, so that
\begin{eqnarray}
\nonumber
&&\delta^\prime=-\frac{2K}{r(r+K)}\\&&
\delta^{\prime\prime}=2K [\frac{1}{r^2(r+K)}+\frac{1}{r(r+K)^2} ] \nonumber
\end{eqnarray}
\noindent
and we get
\begin{equation}
\label{iii4}
R^4_4=-\frac{K}{r^4}
\end{equation}
\noindent
Relation (\ref{iii2}), together with relation (\ref{ii8}) gives then
\begin{equation}
\label{iii5}
F_{14}=\frac{\sqrt{a}K}{r^2}
\end{equation}
\noindent
The energy density (\ref{ii8}), becomes
\begin{equation}
\label{iii6}
U^{44}=-\frac{a}{4\pi}R^{44}=-\frac{a}{4\pi}g^{44}R^4_4=-\frac{a}{4\pi}e^{-\delta}R^4_4=\frac{a}{4\pi}\frac{K^2}{r^2(r+K)^2}
\end{equation}
\noindent
The integral of energy density over the whole space is
\begin{equation}
\label{iii7}
W=\frac{a}{4\pi}K^2\int{\frac{1}{r^2(r+K)^2}r^2\sin^2\theta d\theta d\phi dr}=aK^2[-\frac{1}{r+K}]^\infty_0=aK
\end{equation}
\noindent
The mass of the particle being $m$, expression (\ref{iii7}) must be equal to $mc^2$
\begin{equation}
\label{iii8}
aK=mc^2
\end{equation}
The dimension of $K$ being that of a length, the dimensions of $a$ must be that of a force. We remark that the combination of $\frac{Gm}{c^2}$, where $G$ is the gravitational constant, has the dimension of a length, and that the combination $\frac{c^4}{G}$ has the dimension of a force. We set
\begin{eqnarray}
\label{iii9}
&&a=\frac{c^4}{G}\\
&&K=\frac{Gm}{c^2}\nonumber
\end{eqnarray}
We see that in the integral of the energy density (\ref{iii7}) we encounter no infinities, and that the mass of the particle consists of the integral of its energy density over all space. The particle is seen to have infinite extension, the greatest part of its mass being nevertheless concentrated near the center of spherical pattern constituting the particle.
The line element (\ref{ii19}) becomes now
\begin{equation}
\label{iii10}
ds^2=-\frac{1}{(1+\frac{Gm}{c^2 r})^2} dr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2+(1+\frac{Gm}{c^2 r})^2 dt^2
\end{equation}
\noindent
The difference between this solution and Einstein's solution
\begin{equation}
\label{iii11}
ds^2=-\frac{1}{(1-\frac{2m}{r})}dr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2+(1-\frac{2m}{r})dt^2
\end{equation}
\noindent
is that in the line element (\ref{iii10}) the $g_{\mu\nu}$ are perfect squares, and also that they do not have any singularities apart from the origin, giving withal a finite value for the energy of the particle.
The contravariant charge current-density vector is
\begin{eqnarray}
\label{iii12}
J^4&&=\frac{1}{2\pi}(F^{\mu\nu})_{,\nu}\\&&\nonumber
=\frac{1}{2\pi\sqrt{-g}}\frac{\partial}{\partial r}(F^{4\nu}\sqrt{-g})\\&& \nonumber
=\frac{1}{2\pi r^2}\frac{\partial}{\partial r}(g^{44}g^{11}F_{41}r^2)\\&& \nonumber
=\frac{1}{2\pi r^2}\frac{\partial}{\partial r}(F_{41}r^2)\\&& \nonumber
=\frac{1}{r^2}\frac{\partial}{\partial r}(\frac{\sqrt{G}m}{r^2}r^2)\\&& \nonumber
=0
\end{eqnarray}
\noindent
The vanishing of $J^4$ is of course to be expected in the case of the gravitational field.
The interpretation of $g_{11}=-(1+\frac{Gm}{c^2 r})^{-2}$ is that of a strain in the direction of the radius vector, reducing the length of the unit mesh in that direction by the factor $(1+\frac{Gm}{c^2 r})^{-2}$. The $R_{\mu\nu}$ are not interpreted in terms of a curvature, but rather in terms of strains in space. The identity $R\equiv0$ means here that the total strain at any point of space is zero.
\section{The Electric Field and Leptons}
\subsection{Second Solution of the Field Equations for $\Phi_4$}
We now consider the line element as given in (\ref{ii27})
\begin{equation}
\label{iv1}
ds^2=-(1+\frac{A}{r})^2(dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2)+\frac{1}{(1+\frac{A}{r})^2}dt^2
\end{equation}
The two quantities $R^\nu_\mu$ and $U^\nu_\mu$ being proportional to each other, we set as in (\ref{ii8}) and (\ref{ii4})
\begin{equation}
\label{iv2}
U^4_4=-\frac{b}{4\pi}R^4_4=\frac{1}{4\pi}e^{-(\alpha+\delta)}(F_{14})^2
\end{equation}
\noindent
By (\ref{ii13}) we have
\begin{equation}
\label{iv3}
R^4_4=e^{-\alpha}(-\frac{1}{2}\delta^{\prime\prime}-\frac{1}{4}\delta^{\prime 2}+\frac{1}{4}\alpha^\prime\delta^\prime-\frac{1}{4}\beta^\prime\delta^\prime-\frac{1}{4}\gamma^\prime\delta^\prime-\frac{\delta^\prime}{r})
\end{equation}
\noindent
Here, since we have $\alpha=\beta=\gamma=-\delta$ we get
\begin{equation}
\nonumber
R^4_4=e^\delta(-\frac{1}{2}\delta^{\prime\prime}-\frac{\delta^\prime}{r})
\end{equation}
\noindent
Replacing $e^\delta$ by $(1+\frac{A}{r})^{-2}$ we find
\begin{equation}
\label{iv4}
R^4_4=-\frac{A^2}{(r+A)^4}
\end{equation}
\noindent
so that
\begin{eqnarray}
\label{iv5}
&&R^4_4=-\frac{1}{b}(F_{14})^2=-\frac{A^2}{(r+A)^4}\\ && \nonumber
F_{14}=\sqrt{b}\frac{A}{(r+A)^2}\\ && \nonumber
U^4_4=-\frac{bA^2}{4\pi (r+A)^4}
\end{eqnarray}
The integral over all space is
\begin{equation}
\label{iv6}
\int U^4_4 dv=\frac{bA}{3}
\end{equation}
\noindent
Setting this equal to the mass of the particle, we get
\begin{equation}
\label{iv7}
\frac{bA}{3}=mc^2
\end{equation}
\noindent
This expression has the dimensions of an energy. As $A$ has the dimension of length, $b$ must have the dimensions of a force. We remark that the combination $\frac{e^2}{mc^2}$ has the dimension of a length, and the combination $\frac{m^2c^4}{e^2}$ has the dimensions of a force. We set
\begin{eqnarray}
\label{iv8}
&& b=\frac{9m^2c^4}{e^2}\\&& \nonumber
A=\frac{1}{3}\frac{e^2}{mc^2}
\end{eqnarray}
\noindent
so that
\begin{equation}
\label{iv9}
bA^2=e^2
\end{equation}
The electric field is by (\ref{iv4}) and (\ref{iv9})
\begin{equation}
\label{iv10}
E=F_{14}=\frac{\sqrt{b}A}{(r+\frac{e^2}{3mc^2})^2}=\frac{e}{(r+\frac{e^2}{3mc^2})^2}
\end{equation}
The distance $A$ is equal to
\begin{equation}
\label{iv11}
A=\frac{e^2}{3mc^2}=\frac{1}{3}\frac{e^2}{\hbar c}\frac{\hbar}{mc}=\frac{1}{3}\alpha\lambdabar=0.9393\times 10^{-13} \textrm {cm}
\end{equation}
The covariant charge current-density vector gives
\begin{equation}
\label{iv12}
J_4=\frac{1}{2\pi}(F^\nu_4)_\nu=\frac{1}{2\pi\sqrt{-g}}\frac{\partial}{\partial x_\nu}(F^\nu_4\sqrt{-g})-\frac{1}{2}\frac{\partial g^{\alpha\beta}}{\partial x_4}F_{\alpha\beta}
\end{equation}
In the static case the last term is zero, and we have, taking account of (\ref{iv1})
\begin{equation}
\label{iv13}
J_4=\frac{1}{2\pi(r+A)^2}\frac{\partial}{\partial r}[F^1_4(r+A)^2]
\end{equation}
\noindent
Since $F^1_4=g^{11}F_{41}=-(1+\frac{A}{r})^{-2}e(r+A)^2$ we get
\begin{equation}
\label{iv14}
J_4=-\frac{e}{2\pi(r+A)^2}\frac{\partial}{\partial r}\frac{1}{(1+\frac{A}{r})^2}=\frac{eAr}{\pi(r+A)^5}
\end{equation}
The integral of this expression over all space is
\begin{equation}
\label{iv15}
\int J_4 dv=4\pi\frac{eA}{\pi}[-\frac{1}{r+A}+\frac{3}{2}\frac{A}{(r+A)^3}+\frac{A^2}{(r+A)^3}+\frac{A^3}{4(r+A)^4}]_0^\infty=e
\end{equation}
\noindent
that is, the integral of the charge density over all space is equal to the charge of the particle.
We note that we do not encounter any infinities in our calculations. The mass of the particle is spread out over all space, and so is its charge. We interpret the $g_{\mu\nu}$ not in terms of a curvature, but in terms of constraints in the unit lengths at every point of space. The equality of $\alpha$, $\beta$, and $\gamma$ in the line element (\ref{iv1}), is interpreted as an isotropic constraint in all three directions at every point of space. We note that in the gravitational case the vanishing of $\gamma$ and $\beta$ in the line element (\ref{ii1}) denotes that in that case the only constraint is in the direction of the radius vector.
\subsection{The Muon}
The relativistic wave equations for a particle in a central potential are
\begin{eqnarray}
\label{v1}
&&(E+mc^2-V)F-2\pi hc\frac{dG}{dr}-\frac{2\pi hck}{r}G=0 \\ && \nonumber
(E-mc^2-V)G+2\pi hc\frac{dF}{dr}-\frac{2\pi hck}{r}F=0
\end{eqnarray}
\noindent
where $F$ and $G$ are two wave functions and $r$ is the distance to the origin. We substitute the quantities $V=-(r+\frac{e^2}{3mc^2})^{-2}e^2$, $z=\frac{E}{mc^2}$, $\beta_1=1+z$, $\beta_2=1-z$, $\beta=\sqrt{\beta_1\beta_2}=\sqrt{1-z^2}$, and the fine-structure constant $\alpha=\frac{e}{\hbar c}=0.00729735$ in equations (\ref{v1}), and introduce two new wave functions $f$ and $g$ defined by
\begin{eqnarray}
\nonumber && F(\rho)=e^{-\rho}f(\rho) \\
\nonumber && G(\rho)=e^{-\rho}g(\rho)
\end{eqnarray}
\noindent
and we obtain the following relations
\begin{eqnarray}
\label{v2}
&& g^\prime-g+\frac{kg}{\rho}-(\frac{\beta_1}{\beta}+\frac{\alpha}{\rho+\frac{\alpha\beta}{3}})f=0\\ \nonumber
&& f^\prime-f-\frac{kf}{\rho}-(\frac{\beta_2}{\beta}-\frac{\alpha}{\rho+\frac{\alpha\beta}{3}})g=0
\end{eqnarray}
We introduce two new wave functions $\phi$ and $\psi$ defined by
\begin{eqnarray}
\nonumber && f=\rho^s (\rho+\frac{\alpha\beta}{3})^t \phi \\
\nonumber && g=\rho^s (\rho+\frac{\alpha\beta}{3})^t \psi
\end{eqnarray}
\noindent
and we obtain the equations
\begin{eqnarray}
\label{v3}
\rho (\rho+\frac{\alpha\beta}{3})\psi^\prime+&&[(\rho+\frac{\alpha\beta}{3})(s+k-\rho)+t\rho]\psi\\ \nonumber &&
-[\frac{\beta_1}{\beta}(\rho+\frac{\alpha\beta}{3})+\alpha]\rho\phi=0\\ \nonumber
\rho (\rho+\frac{\alpha\beta}{3})\psi^\prime+&&[(\rho+\frac{\alpha\beta}{3})(s-k-\rho)+t\rho]\phi\\ \nonumber &&
-[\frac{\beta_2}{\beta}(\rho+\frac{\alpha\beta}{3})-\alpha]\rho\psi=0
\end{eqnarray}
\noindent
We expand $\phi$ and $\psi$ in terms of $\rho$
\begin{eqnarray}
\label{v4}
\phi=\sum a_n \rho^n \\ \nonumber
\psi=\sum b_n \rho^n
\end{eqnarray}
\noindent
and substitute these expansions in the equations (\ref{v3}); we equate to zero the constant terms
\begin{eqnarray}
\nonumber (s+k)\frac{\alpha\beta}{3}b_0=0\\
\nonumber (s-k)\frac{\alpha\beta}{3}a_0=0
\end{eqnarray}
\noindent
We must therefore have either $s=+k$ and $b_0=0$, or $s=-k$ and $a_0$=0. We choose $s=+k$ and therefore $b_0=0$.
We introduce a new variable
\begin{equation}
\nonumber
x=\rho+\frac{\alpha\beta}{3}
\end{equation}
\noindent
and substitute in the relations (\ref{v3}), and we obtain the two equations
\begin{eqnarray}
\label{v5}
x(x-\frac{\alpha\beta}{3})\psi^\prime+&&[2kx+(t-x)(x-\frac{\alpha\beta}{3})]\psi\\ \nonumber &&
-(\frac{\beta_1}{\beta}x+\alpha)(x-\frac{\alpha\beta}{3})\phi=0 \\ \nonumber
x(x-\frac{\alpha\beta}{3})\phi^\prime+&&[(t-x)(x-\frac{\alpha\beta}{3})]\phi\\ \nonumber &&
-(\frac{\beta_2}{\beta}x-\alpha)(x-\frac{\alpha\beta}{3})\psi=0
\end{eqnarray}
\noindent
We expand $\phi$ and $\psi$ in terms of $x$
\begin{eqnarray}
\nonumber \phi=\sum c_n x^n\\
\nonumber \psi=\sum d_n x^n
\end{eqnarray}
\noindent
and substitute in the equations (\ref{v5}); we equate to zero the constant terms
\begin{eqnarray}
\nonumber -t\frac{\alpha\beta}{3}d_0+\frac{\alpha^2\beta}{3}c_0=0\\
\nonumber -t\frac{\alpha\beta}{3}c_0-\frac{\alpha^2\beta}{3}d_0=0
\end{eqnarray}
\noindent
which give $t^2=-\alpha^2$, that is $t=\pm i \alpha$. We choose $t=+i\alpha$. Replacing $t$ by this value in equation (\ref{v3}), we obtain the two equations
\begin{eqnarray}
\label{v6}
\rho(\rho+\frac{\alpha\beta}{3})\psi^\prime&&+[-\rho^2+(2k+i\alpha-\frac{\alpha\beta}{3})\rho+2k \frac{\alpha\beta}{3}]\psi\\
\nonumber && -[\frac{\beta_1}{\beta}(\rho+\frac{\alpha\beta}{3})+\alpha]\rho\phi=0\\ \nonumber
(\rho+\frac{\alpha\beta}{3})\phi^\prime&&+[-\rho+i\alpha-\frac{\alpha\beta}{3}]\phi\\ \nonumber &&
-[\frac{\beta_2}{\beta}(\rho+\frac{\alpha\beta}{3})-\alpha]\psi=0
\end{eqnarray}
We substitute the expansions (\ref{v4}) into equations (\ref{v6}) and equate to zero the coefficients of successive powers of $\rho$. We obtain thus an infinite sequence of equations connecting the different coefficients $a_n$, $b_n$ of expansions (\ref{v4})
\begin{eqnarray}
\label{v7}
&&\frac{\alpha\beta}{3}a_1+(i\alpha-\frac{\alpha\beta}{3})a_0=0\\ \nonumber
&&\frac{\alpha\beta}{3}(1+2k)b_1-(\frac{\alpha\beta_1}{3}+\alpha)a_0=0\\ \nonumber
&&\frac{2\alpha\beta}{3}a_2+(1+i\alpha-\frac{\alpha\beta}{3})a_1-(\frac{\alpha\beta_2}{3}-\alpha)b_1-a_0=0\\ \nonumber
&&(2+2k)\frac{\alpha\beta}{3}b_2+(1+2k+i\alpha-\frac{\alpha\beta}{3})b_1-(\frac{\alpha\beta_1}{3}+\alpha)a_1-\frac{\beta_1}{\beta}a_0=0\\ \nonumber
&&3\frac{\alpha\beta}{3}a_3+(2+i\alpha-\frac{\alpha\beta}{3})a_2-(\frac{\alpha\beta_2}{3}-\alpha)b_2-a_1-\frac{\beta_2}{\beta}b_1=0\\ \nonumber
&&(3+2k)\frac{\alpha\beta}{3}b_3+(2+2k+i\alpha-\frac{\alpha\beta}{3})b_2-(\frac{\alpha\beta_1}{3}+\alpha)a_2-b_1-\frac{\beta_1}{\beta}a_1=0\\ \nonumber
&&\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\\ \nonumber
&&n\frac{\alpha\beta}{3}a_n+(n-1+i\alpha-\frac{\alpha\beta}{3})a_{n-1}-(\frac{\alpha\beta_2}{3}-\alpha)b_{n-1}-a_{n-2}-\frac{\beta_2}{\beta}b_{n-2}=0\\ \nonumber
&&(n+2k)\frac{\alpha\beta}{3}b_n+(n-1+2k+i\alpha-\frac{\alpha\beta}{3})b_{n-1}-(\frac{\alpha\beta_1}{3}+\alpha)a_{n-1}-b_{n-2}-\frac{\beta_1}{\beta}a_{n-2}=0
\end{eqnarray}
\noindent
The expansions (\ref{v4}) must terminate at some power of $\rho$. If the expansions terminate at the first power of $\rho$, then all coefficients after $a_1$, $b_1$ being zero, the third pair of relations (\ref{v7}) give
\begin{equation}
\label{v8}
\frac{b_1}{a_1}=-\frac{\beta_1}{\beta}
\end{equation}
\noindent
We substitute this value of $b_1$ in terms of $a_1$ in the second pair of equations (\ref{v7}), we multiply the first of these by $\beta_1$ and the second by $\beta$ and subtract the one from the other. We obtain the relation
\begin{equation}
\label{v9}
\beta_1(1+i\alpha)+\beta\alpha+\beta_1(1+2k+i\alpha)-\frac{\beta_1}{\beta}\alpha=0
\end{equation}
\noindent
which gives the value of $k$
\begin{equation}
\label{v10}
k=-1-i\alpha+\frac{\alpha z}{\beta}
\end{equation}
\noindent
In general, if expansion (\ref{v4}) terminates in $\rho^n$ we have
\begin{equation}
\label{v11}
k=-n-i\alpha+\frac{\alpha z}{\beta}
\end{equation}
\noindent
we substitute (\ref{v8}) and (\ref{v10}) in the first pair of relations (\ref{v7}), we eliminate the coefficients $a_1$ and $b_1$, and obtain
\begin{equation}
\label{v12}
-\frac{\beta_2}{\beta}=\frac{\beta_1+3}{(\beta-3i)(-1-2i\alpha+\frac{2\alpha z}{\beta})}
\end{equation}
\noindent
which yields the complex equation of the second degree
\begin{equation}
\label{v13}
(2\alpha^2+6i\alpha)z^2+(-9-18i\alpha)z+9-20\alpha^2+12i\alpha=0
\end{equation}
The moduli of the two roots of this equation give the masses of two charged particles in units of the electron mass. These particles
have no nuclear interaction. The roots are
\begin{eqnarray}
\label{v14}
&&z_1=0.999892+i 5.18192\times10^{-7}\\ \nonumber
&&z_2=2.50009-i 205.546
\end{eqnarray}
\noindent
and their moduli are
\begin{eqnarray}
\label{v15}
&&|z_1|=0.999893\\ \nonumber
&&|z_2|=205.561=105.042 \textrm{MeV}
\end{eqnarray}
\noindent
The first root gives the mass of the electron in the field of a positive charge; the second root gives the mass of the muon in the field of a positive charge. The experimental value of the muon mass is 206.768 (105.659MeV).
Further values of masses of heavy leptons may be obtained by terminating expansions (\ref{v4}) at higher powers of $\rho$.
\section{The Nuclear Field and The Hadrons}
\subsection{The Potential $\Phi_3$}
Referring to table~\ref{tab1.1}, we see that the components of the field due to $\Phi_3$, are $F_{13}=-\frac{\partial \Phi_3}{\partial x_1}$ and $F_{23}=-\frac{\partial \Phi_3}{\partial x_2}$. The energy tensor (\ref{i3}) is
\begin{equation}
\label{vi1}
U^\nu_\mu=\frac{1}{2\pi}(-F^{\nu\alpha}F_{\mu\alpha}+\frac{1}{4}g^\nu_\mu F^{\alpha\beta}F_{\alpha\beta})
\end{equation}
\noindent
we calculate first $F^{\alpha\beta}F_{\alpha\beta}$. The only components of $F_{\mu\nu}$ being $F_{13}=-F_{31}$ and $F_{23}=-F_{32}$ , we have
\begin{eqnarray}
\label{vi2}
F^{\alpha\beta}F_{\alpha\beta}&&=2F^{13}F_{13}+2F^{23}F_{23}\\ \nonumber &&
=2[g^{11}g^{33}(F_{13})^2+g^{22}g^{33}(F_{23})^2]\\ \nonumber &&
=2g^{33}[g^{11}(F_{13})^2+g^{22}(F_{23})^2]\\ \nonumber &&
=2e^{-\gamma}r^{-2}\sin^{-2}\theta [e^{-\alpha}(F_{13})^2+e^{-\beta}r^{-2}(F_{23})^2]
\end{eqnarray}
\noindent
and
\begin{eqnarray}
\label{vi3}
&&F^{1\alpha}F_{2\alpha}=F^{13}F_{23}=g^{11}g^{33}F_{13}F_{23}=e^{-(\alpha+\gamma)}r^{-2}F_{13}F_{23}\\ \nonumber
&&F^{2\alpha}F_{1\alpha}=F^{23}F_{13}=g^{22}g^{33}F_{23}F_{13}=e^{-(\beta+\gamma)}r^{-2}F_{13}F_{23}\\ \nonumber
&&F^{13}F^{13}=g^{11}g^{33}(F_{13})^2=e^{-(\alpha+\gamma)}r^{-2}(F_{13})^2\\ \nonumber
&&F^{23}F^{23}=g^{22}g^{33}(F_{23})^2=e^{-(\beta+\gamma)}r^{-2}(F_{23})^2
\end{eqnarray}
\noindent
Also
\begin{eqnarray}
\label{vi4}
&&U^1_1=\frac{1}{4\pi}e^{-\gamma}r^{-2}\sin^{-2}\theta [+e^{-\alpha}(F_{13})^2+e^{-\beta}r^{-2}(F_{23})^2]\\ \nonumber
&&U^2_2=\frac{1}{4\pi}e^{-\gamma}r^{-2}\sin^{-2}\theta [-e^{-\alpha}(F_{13})^2-e^{-\beta}r^{-2}(F_{23})^2]\\ \nonumber
&&U^3_3=\frac{1}{4\pi}e^{-\gamma}r^{-2}\sin^{-2}\theta [-e^{-\alpha}(F_{13})^2-e^{-\beta}r^{-2}(F_{23})^2]\\ \nonumber
&&U^4_4=\frac{1}{4\pi}e^{-\gamma}r^{-2}\sin^{-2}\theta [+e^{-\alpha}(F_{13})^2+e^{-\beta}r^{-2}(F_{23})^2]\\ \nonumber
&&U^2_1=\frac{1}{2\pi}e^{-(\beta+\gamma)}r^{-4}\sin^{-2}\theta (F_{13}F_{23})\\ \nonumber
&&U^1_2=\frac{1}{2\pi}e^{-(\alpha+\gamma)}r^{-4}\sin^{-2}\theta (F_{13}F_{23})\\ \nonumber
&&U=U^1_1+U^2_2+U^3_3+U^4_4=0
\end{eqnarray}
\noindent
We remark that since $e^\alpha$, $e^\beta$, $e^\gamma$, $e^\delta$ are pure numbers, the expressions in square brackets show that $F_{13}$ and $\frac{1}{r}F_{23}$ have the same dimensions. This fact suggests an analogy with the field of a dipole, which is of the form
\begin{eqnarray}
\label{vi5}
H_r=\frac{2\mu \cos\theta}{r^3} \\ \nonumber
H_\theta=\frac{\mu \sin\theta}{r^2}
\end{eqnarray}
\noindent
deriving from a potential
\begin{equation}
\label{vi6}
\Phi=\frac{\mu \cos\theta}{r^2}
\end{equation}
We assume therefore the following expression for the potential $\Phi_3$
\begin{equation}
\label{vi7}
\Phi_3=\frac{\sigma \cos\theta}{(r+D)^2}
\end{equation}
\noindent
where $\sigma$ is the strength of the source, $D$ is a basic length introduced in analogy with formula (\ref{ii17}), and $\theta$ is the angle between the axis of the dipole and the radius vector to the point of observation. The field due to the potential $\Phi_3$ is
\begin{eqnarray}
\label{vi8}
F_{13}=-\frac{\partial \Phi_3}{\partial r}=\frac{-2\sigma \cos\theta}{(r+D)^3}\\
\label{vi9}
F_{23}=-\frac{\partial \Phi_3}{\partial \theta}=\frac{\sigma \sin\theta}{(r+D)^2}
\end{eqnarray}
The energy density, taken in covariant form, is
\begin{equation}
\label{vi10}
U_{33}=g_{33}U^3_3=-\frac{1}{4\pi}[-e^{-\alpha}(F_{13})^2-e^{-\beta}r^{-2}(F_{23})^2]
\end{equation}
\noindent
As a first approximation we assume Galilean coordinates, so that $U_{33}$ is equal to
\begin{equation}
\label{vi11}
U_{33}=\frac{1}{4\pi}[(F_{13})^2+r^{-2}(F_{23})^2]
\end{equation}
\noindent
Substituting (\ref{vi8}) and (\ref{vi9}) in (\ref{vi11}) we get
\begin{equation}
\label{vi12}
U_{33}=\frac{1}{4\pi}\sigma^2[\frac{4\cos^2\theta}{(r+D)^6}+\frac{\sin^2\theta}{r^2(r+D)^4}]
\end{equation}
\noindent
Integrating over all space we obtain the total energy
\begin{equation}
\label{vi13}
W=\frac{4}{15}\frac{\sigma^2}{D^3}
\end{equation}
To find the values of $\sigma$ and $D$, we first seek to find an order of magnitude for $\sigma$. We make the assumption that the field due to the potential $\Phi_3$ is a constituent of the nuclear field. We consider then the force between two protons in a nucleus. The field of a proton for $\theta=0$ being $2\sigma(r+D)^{-3}$, the energy of the second proton in the presence of the first is
\begin{equation}
\label{vi14}
V=-\frac{2\sigma^2}{(r+D)^3}
\end{equation}
\noindent
The nuclear interaction between the two protons will be
\begin{equation}
\label{vi15}
\frac{\partial V}{\partial r}=\frac{6\sigma^2}{(r+D)^4}
\end{equation}
The electric repulsion between the two protons is $e^2(r+A)^{-2}$, where $A$ is the basic length for the electric field $A=0.9393\times 10^{-13}\rm cm$, see (\ref{iv11}). We equate the two forces
\begin{equation}
\nonumber
\frac{6\sigma^2}{(r+D)^4}=\frac{e^2}{(r+A)^2}
\end{equation}
\noindent
If we adopt for $D$ the Compton wavelength for the proton $\lambdabar=0.21\times 10^{-13}\rm cm$, we obtain
\begin{equation}
\nonumber
\frac{\sigma}{e}=\frac{(r+0.21\times 10^{-13})^2\textrm{cm}}{\sqrt{6} (r+0.94\times 10^{-13})\textrm{cm}}
\end{equation}
\noindent
The distance $r$ between protons being of the order of a fermi, $\frac{\sigma}{e}$ will also be of the order of a fermi, and $\sigma$ will be of the order
\begin{equation}
\label{vi16}
\sigma\simeq 10^{-13} e\simeq 10^{-23}
\end{equation}
We can find the order of magnitude of the energy stored by the field in the whole space by inserting the values of $\sigma$ and $D$ in (\ref{vi13})
\begin{equation}
\label{vi17}
W\sim \frac{4\times 10^{-46}}{15\times(0.21\times 10^{-13})^3}\sim 10^{-5} \textrm{erg}
\end{equation}
The mass of proton being $M=1.6\times 10^{-13} \rm erg$, the ratio of the energy of the nuclear field to the mass of the proton will be of the order of $\frac{W}{M}\sim\frac{10^{-5}}{10^{-3}}\sim 10^{-2}\sim\alpha\simeq\frac{1}{137}$, that is, the fine structure constant. We equate therefore the expression (\ref{vi13}) to $\alpha Mc^2$
\begin{equation}
\nonumber
\frac{4}{15}\frac{\sigma^2}{\lambdabar^3}=\alpha Mc^2=\frac{e^2}{\hbar c} Mc^2=\frac{e^2}{\lambdabar}
\end{equation}
\noindent
so that $\sigma^2=3.75e^2\lambdabar^2$, or
\begin{equation}
\label{vi18}
\sigma=1.936e\lambdabar
\end{equation}
\noindent
For the value of $M$ which we shall use, we deduct from the mass $M_p=938.26\rm MeV$ of the proton the mass $m$ of its positive charge, which is of electromagnetic nature. We thus get
\begin{eqnarray}
\label{vi19}
&&M=938.26-0.51=937.75 \textrm{MeV} \\
\label{vi20}
&&\lambdabar=0.2104\times 10^{-13}\textrm{cm} \\
\label{vi21}
&& \sigma=1.9569\times 10^{-23}\textrm{cgs}
\end{eqnarray}
\subsection{The Baryons}
If we substitute the potential energy of the nuclear field (\ref{vi14}) in the wave equation (\ref{v1}), we must obtain the masses of baryons, following the same method that was used to obtain the leptons. The wave equations are
\begin{eqnarray}
\label{vii1}
&& (E-Mc^2-W)G+\hbar c \frac{dF}{dr}-\frac{\hbar c k}{r}F=0 \\ \nonumber
&& (E+Mc^2-W)F-\hbar c \frac{dG}{dr}-\frac{\hbar c k}{r}G=0
\end{eqnarray}
\noindent
We make the following substitutions in these equations
\begin{eqnarray}
\label{vii2}
&& z=\frac{E}{Mc^2} \\ \nonumber
&& \beta_1=1+z \\ \nonumber
&& \beta_2=1-z \\ \nonumber
&& \beta=\sqrt{\beta_1 \beta_2}=\sqrt{1-z^2} \\ \nonumber
&& \alpha=\frac{e^2}{\hbar c} \\ \nonumber
&& \rho=\frac{\beta r}{\lambdabar} \\ \nonumber
&& W=-\frac{2\sigma^2}{(r+\lambdabar)^3} \\ \nonumber
\end{eqnarray}
\noindent
where $W$ has been taken for $\theta=0$.
We take two new wave functions defined by
\begin{eqnarray}
\label{vii3}
f(\rho)=e^\rho F \\ \nonumber
g(\rho)=e^\rho G
\end{eqnarray}
\noindent
and obtain the two equations
\begin{eqnarray}
\label{vii4}
&& f^\prime-f-\frac{kf}{\rho}-(\frac{\beta_2}{\beta}-\frac{W}{\beta Mc^2})g=0 \\ \nonumber
&& g^\prime-g+\frac{kg}{\rho}-(\frac{\beta_1}{\beta}+\frac{W}{\beta Mc^2})f=0
\end{eqnarray}
\noindent
We transform the expression $\frac{W}{\beta Mc^2}$ using relations (\ref{vi14}) and (\ref{vi19})
\begin{eqnarray}
\nonumber
\frac{W}{\beta Mc^2}&&=-\frac{2\sigma^2}{(r+\lambdabar)^3}\frac{1}{\beta Mc^2} \\ \nonumber
&& =-\frac{7.5e^2\lambdabar^2}{(\frac{\lambdabar}{\beta})^3(\rho+\beta)^3\beta Mc^2} \\ \nonumber
&& =-\frac{7.5e^2 \beta^2}{\lambdabar (\rho+\beta)^3 Mc^2}
\end{eqnarray}
\noindent
Substituting $\frac{\hbar}{Mc}=\lambdabar$ and $\frac{e^2}{\hbar c}=\alpha$ we get
\begin{equation}
\label{vii5}
\frac{W}{\beta Mc^2}=-\frac{7.5\alpha\beta^2}{(\rho+\beta)^3}=-0.05473\frac{\beta^2}{(\rho+\beta)^3}
\end{equation}
\noindent
We denote the number $7.5\alpha$ by $p$.
\begin{equation}
\label{vii6}
p=7.5\alpha=0.05473
\end{equation}
\noindent
We take two new wave functions $\phi$ and $\psi$ defined by
\begin{eqnarray}
\label{vii7}
&& f=\rho^u \exp[\frac{s}{2(\rho+\beta)^2}] \phi \\ \nonumber
&& g=\rho^u \exp[\frac{s}{2(\rho+\beta)^2}] \psi
\end{eqnarray}
\noindent
Equations (\ref{vii4}) become
\begin{eqnarray}
\label{vii8}
&& \rho \phi^\prime+[u+\rho\frac{s}{(\rho+\beta)^3}-\rho-k]\phi-[\frac{\beta_2}{\beta}-\frac{p\beta^2}{(\rho+\beta)^3}]\rho\psi=0 \\ \nonumber
&& \rho \psi^\prime+[u+\rho\frac{s}{(\rho+\beta)^3}-\rho+k]\psi-[\frac{\beta_2}{\beta}+\frac{p\beta^2}{(\rho+\beta)^3}]\rho\phi=0
\end{eqnarray}
\noindent
We expand $\phi$ and $\psi$ in terms of $\rho$
\begin{eqnarray}
\label{vii9}
&& \phi=A_0+A_1\rho+A_2\rho^2+A_3\rho^3+\cdots \\ \nonumber
&& \psi=B_0+B_1\rho+B_2\rho^2+B_3\rho^3+\cdots
\end{eqnarray}
\noindent
Substituting these expansions in (\ref{vii8}), and equating to zero the constant terms, we get $u=k$, $B_0=0$, or $u=-k$, $A_0=0$. We choose
\begin{eqnarray}
\label{vii10}
&&u=-k \\ \nonumber
&&A_0=0
\end{eqnarray}
\noindent
Writing $\xi=\rho+\beta$ equations (\ref{vii8}) become
\begin{eqnarray}
\label{vii11}
&&(\xi-\beta)\phi^\prime+[-2k+\frac{(\xi-\beta)s}{\xi^3}-(\xi-\beta)]\phi-[\frac{\beta_2}{\beta}-\frac{p\beta^2}{\xi^3}](\xi-\beta)\psi=0 \\ \nonumber
&&(\xi-\beta)\psi^\prime+[\frac{(\xi-\beta)s}{\xi^3}-(\xi-\beta)]\psi-[\frac{\beta_1}{\beta}+\frac{p\beta^2}{\xi^3}](\xi-\beta)\phi=0
\end{eqnarray}
\noindent
We expand $\phi$ and $\psi$ in powers of $\xi$
\begin{eqnarray}
\label{vii12}
&& \phi=a_0+a_1\xi+a_2\xi^2+a_3\xi^3+\cdots \\ \nonumber
&& \psi=b_0+b_1\xi+b_2\xi^2+b_3\xi^3+\cdots
\end{eqnarray}
\noindent
We substitute these expressions in equations (\ref{vii11}) and equate constant terms to zero, and we obtain
\begin{eqnarray}
\nonumber
&&-sa_0-p\beta^2 b_0=0\\ \nonumber
&&-sb_0+p\beta^2 a_0=0
\end{eqnarray}
\noindent
which gives $s^2=-p^2\beta^4$, or $s=\pm ip\beta^2$.
We choose $s=ip\beta^2$ which gives
\begin{equation}
\label{vii13}
b_0=-i a_0
\end{equation}
\noindent
Equations (\ref{vii11}) become, after dividing the first by $(\xi-\beta)$ and multiplying both by $\xi^3$
\begin{eqnarray}
\label{vii14}
&&\xi^3(\xi-\beta)\phi^\prime+[-2k+ip\beta^2(\xi-\beta)-\xi^3(\xi-\beta)]\phi-[\frac{\beta_2}{\beta}\xi^3-p\beta^2](\xi-\beta)\psi=0\\ \nonumber
&&\xi^3\psi^\prime+[ip\beta^2-\xi^3]\psi-[\frac{\beta_1}{\beta}\xi^3+p\beta^2]\phi=0
\end{eqnarray}
\noindent
Equating the coefficients of $\xi$ to zero we get
\begin{equation}
\label{vii15}
b_1=-ia_1
\end{equation}
\noindent
Equating the coefficients of $\xi^2$ to zero we get
\begin{equation}
\label{vii16}
b_2=-ia_2
\end{equation}
We continue by equating the coefficients of $\xi^3$, $\xi^4$, $\xi^5$ and successive powers of $\xi$ to zero and obtain the following sequence of equations
\begin{eqnarray}
\label{vii17}
&& ip\beta^3 a_3+p\beta^3 b_3+\beta a_1+(2k-\beta+i\beta_2)a_0=0 \\ \nonumber
&& ip\beta^2 b_3-p\beta^2 a_3-i a_1+(i-\frac{\beta_1}{\beta})a_0=0 \\ \nonumber
&& ip\beta^3 a_4+p\beta^3 b_4-i p\beta^2 a_3-p\beta^2 b_3+2\beta a_2+(2k-1-\beta+i\beta_2)a_1+(1-i\frac{\beta_2}{\beta})a_0=0\\ \nonumber
&& ip\beta^2 b_4-p\beta^2 a_4-2 i a_2+(i-\frac{\beta_1}{\beta})a_1=0\\ \nonumber
&& ip\beta^3 a_5+p\beta^3 b_5-i p\beta^2 a_4-p\beta^2 b_4+3\beta a_3+(2k-2-\beta+i \beta_2)a_2+(1-i\frac{\beta_2}{\beta})a_1=0\\ \nonumber
&& ip\beta^2 b_5-p\beta^2 a_5+3 b_3+(i-\frac{\beta_1}{\beta})a_2=0\\ \nonumber
&& ip\beta^3 a_6+p\beta^3 b_6-i p\beta^2 a_5-p\beta^2 b_5+4\beta a_4+(2k-3-\beta)a_3+\beta_2 b_3+(1-i\frac{\beta_2}{\beta})a_2=0 \\ \nonumber
&& ip\beta^2 b_6-p\beta^2 a_6+4 b_4+b_3-\frac{\beta1}{\beta}a_3=0\\ \nonumber
&& ip\beta^3 a_7+p\beta^3 b_7-i p\beta^2 a_6-p\beta^2 b_6+5\beta a_5+(2k-4-\beta)a_4+\beta_2 b_4+\frac{\beta_2}{\beta}b_3+a_3=0\\ \nonumber
&& ip\beta^2 b_7-p\beta^2 a_7+5 b_5-b_4-\frac{\beta_1}{\beta}a_4=0\\ \nonumber
&& \cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\\ \nonumber
&& \cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\\ \nonumber
&& ip\beta^3 a_n+p\beta^3 b_n-ip\beta^2 a_{n-1}-p\beta^2 b_{n-1}+(n-2)\beta a_{n-2}+(2k-n+3-\beta)a_{n-3}\\ \nonumber
&& \quad\quad\quad\quad -\beta_2 b_{n-3}+\frac{\beta_2}{\beta}b_{n-4}+a_{n-4}=0 \\ \nonumber
&& ip\beta^2 b_n-p\beta^2 a_n+(n-2)b_{n-2}-b_{n-3}-\frac{\beta_1}{\beta}a_{n-3}=0
\end{eqnarray}
\subsubsection{Expansions (\ref{vii12}) terminate with $\xi^3$}
If the expansions (\ref{vii12}) terminate at $\xi^3$ so that $a_4$, $b_4$ and all coefficients after $a_4$ and $b_4$ vanish, the second equation of the fourth pair in (\ref{vii17}) will give
\begin{equation}
\label{vii18}
b_3=-\frac{\beta_1}{\beta}a_3
\end{equation}
Multiplying the first equation of the fourth pair in (\ref{vii17}) by $\frac{\beta_1}{\beta}$ and subtracting it from the second equation of the third pair in (\ref{vii17}), we find
\begin{equation}
\label{vii19}
\frac{\beta_1}{\beta}(6-2k)a_3=0
\end{equation}
\noindent
which has the two solutions
\begin{eqnarray}
\label{vii20}
\beta_1=0\\
k=3 \nonumber
\end{eqnarray}
The solution $\beta_1=0$, that is $z=-1$, represents a particle with a negative mass equal to the neutron mass. The second solution $k=3$ substituted in the first equation of the third pair in (\ref{vii17}) gives
\begin{equation}
\label{vii21}
-3\beta a_3+(-4+\beta-i \beta_2)a_2+(i\frac{\beta_2}{\beta}-1)a_1=0
\end{equation}
We eliminate $a_1$ between (\ref{vii21}) and the second equation of the second pair in (\ref{vii17}). Multiplying (\ref{vii21}) by $\frac{\beta_1}{\beta}$ and subtracting from second equation of the second pair in (\ref{vii17}) we obtain the equation
\begin{equation}
\label{vii22}
3\beta_1 a_3+(4\frac{\beta_1}{\beta}-\beta_1+i\beta-2i)a_2=0
\end{equation}
We have now to eliminate $a_2$ between (\ref{vii22}) and the second equation of the third pair in (\ref{vii17}). Multiplying this last equation by $\beta$ and taking account of relation (\ref{vii18}) and adding to (\ref{vii22}) we get
\begin{equation}
\label{vii23}
\beta(\frac{\beta_1}{\beta}-i)+4\frac{\beta_1}{\beta}-\beta_1+i\beta-2i=0
\end{equation}
\noindent
which may be written as
\begin{equation}
\label{vii24}
\beta(i+\beta_1)=\beta_1(2+i\beta_2)
\end{equation}
\noindent
Squaring, we get
\begin{equation}
\label{vii25}
(1-z^2)[-1+2i(1+z)+(1+z)^2]=(1+z)^2[4+2i(1-z)-(1-z)^2]
\end{equation}
\noindent
which reduces to the equation of the second order degree in $z$
\begin{equation}
\label{vii26}
(2-2i)z^2+3z+3+2i=0
\end{equation}
\noindent
Writing $z=x+iy$ and separating real and imaginary parts, we obtain the two simultaneous equations
\begin{eqnarray}
\label{vii27}
&&2x^2+4xy-2y^2+3x+3=0 \\ \nonumber
&&2x^2-4xy-2y^2-3y-2=0
\end{eqnarray}
\noindent
whose solutions are
\begin{eqnarray}
\label{vii28}
&&(x_1,y_1)=(0.23757,-1.16572)\\ \nonumber
&&(x_2,y_2)=(-0.98757,0.41572)
\end{eqnarray}
We consider only the first root whose real part is positive. Its modulus is $|z|=1.18968$. Multiplying this by the reduced mass of the nucleon $937.75 \rm MeV$ from (\ref{vi19}) we get
\begin{equation}
\nonumber
M=1115.62 \textrm{MeV}
\end{equation}
\noindent
which is to be compared with the mass of the baryon
\begin{equation}
\label{vii29}
\Lambda=1115.6\pm0.05 \textrm{MeV}
\end{equation}
\subsubsection{Expansions (\ref{vii12}) terminate with $\xi^4$}
In equations (\ref{vii17}), $a_5$, $b_5$ and all the following coefficients vanish. So the second equation of the fifth pair in (\ref{vii17}) gives
\begin{equation}
\label{vii30}
b_4=-\frac{\beta_1}{\beta}a_4
\end{equation}
Multiplying the second equation of the fourth pair in (\ref{vii17}) by $\frac{\beta_2}{\beta}$ and subtracting from the first equation of the fifth pair in (\ref{vii17}) we get
\begin{equation}
\label{vii31}
4-2k+4=0
\end{equation}
\noindent
which gives $k=4$.
Replacing $b_4$ from (\ref{vii30}) we get
\begin{equation}
\label{vii32}
a_4=-\frac{1}{4}(\frac{\beta}{\beta_1}b_3+a_3)
\end{equation}
Multiplying the second equation of the third pair in (\ref{vii17}) by $\frac{\beta_2}{\beta}$ and subtracting from the first equation of the fourth pair in (\ref{vii17}) we get
\begin{equation}
\label{vii33}
-4\beta a_4+(-5+\beta)a_3+(\beta_2-3\frac{\beta_2}{\beta})b_3=0
\end{equation}
\noindent
Replacing $a_4$ from (\ref{vii32}) into the second equation of the fourth pair in (\ref{vii17}) we get
\begin{equation}
\label{vii34}
b_3=-\frac{\beta}{\beta_2}\frac{5-2\beta}{3-2\beta}a_3
\end{equation}
\noindent
We replace in the second equation of the fourth pair in (\ref{vii17}) $b_3$ by its value from (\ref{vii34}) and we get
\begin{equation}
\label{vii35}
a_4=\frac{1}{2(3-2\beta)}a_3
\end{equation}
\noindent
We multiply the second equation of the second pair (\ref{vii17}) by $\frac{\beta_2}{\beta}$ and subtract from the first equation of the third pair in (\ref{vii17})
\begin{equation}
\label{vii36}
2p\beta(i\beta-z)a_4-3\beta a_3+(-6+\beta-i\beta_2+2i\frac{\beta_2}{\beta})a_2=0
\end{equation}
\noindent
We eliminate $a_2$ between (\ref{vii36}) and the second equation of the third pair in (\ref{vii6}) and get
\begin{equation}
\label{vii37}
2p\beta(i\beta-z)(i\beta-\beta_1)a_4-3(i\beta-\beta_1)a_3+3(i\beta_2-\beta+4)b_3=0
\end{equation}
\noindent
Replacing $a_4$ and $b_3$ by their values (\ref{vii35}) and (\ref{vii34}) we obtain the equation
\begin{equation}
\label{vii38}
\beta[p(-1+2z)+12i \beta_2+48]=\beta_2[ip(1+2z)+24i+12\beta_1]+60
\end{equation}
\noindent
Squaring and dividing by $12$, we obtain the equation of the third degree
\begin{eqnarray}
\label{vii39}
&&(4p+2ip+12-24i)z^3+(2p-12ip+24i)z^2\\ \nonumber
&&+(\frac{p^2}{12}-8p+3ip+36-96i)z-\frac{p^2}{12}+2p+7ip+102+96i=0
\end{eqnarray}
In a first approximation we neglect $p$ as found in (\ref{vii6}) $p=0.05473$. We have then after dividing by $6$, the equation to be solved
\begin{equation}
\label{vii40}
(2-4i)z^3+4iz^2+(6-16i)z+17+16i=0
\end{equation}
\noindent
We write $z=x+iy$ and obtain the two simultaneous equations
\begin{eqnarray}
\label{vii41}
&&2x^3-4y^3-12x^2y-3xy^2-8xy+6x+16y+17=0\\ \nonumber
&&2x^3+y^3-3x^2y-6xy^2-2x^2+2y^2-3y+8x-8=0
\end{eqnarray}
\noindent
It is possible to directly solve for the roots of (\ref{vii41}), however, numerical solutions to (\ref{vii39}) can be obtained exactly as
\begin{eqnarray}
\label{vii42}
&& (x_1,y_1)=(0.258,2.22)\\ \nonumber
&& (x_2,y_2)=(0.994,-1.15)\\ \nonumber
&& (x_3,y_3)=(-0.466,-1.46)
\end{eqnarray}
We consider only the first two solutions whose real parts are positive. The modulus of the first solution is $|z_1|=2.2332$ which when multiplied by the reduced mass of the nucleon $937.75\rm MeV$ as assessed in (\ref{vi19}), gives the mass
\begin{equation}
\nonumber
M_1=2094\textrm{MeV}
\end{equation}
\noindent
which is to be compared to the mass of the baryon
\begin{equation}
\label{vii43}
\Lambda=2100(-10,+20)\textrm{MeV}
\end{equation}
\noindent
The modulus of the second solution is $|z|=1.5225$ which gives the mass
\begin{equation}
\label{vii44}
M_2=1428\textrm{MeV}
\end{equation}
\noindent
to be compared with the mass of the baryon $N=1450\pm 32$.
\subsubsection{Expansions (\ref{vii12}) terminate with $\xi^5$}
Following the same procedure as above, and beginning with the second equation of the fifth pair in (\ref{vii17}), we find first $b_5=-\frac{\beta_1}{\beta}a_5$ and ending up with the second equation (\ref{vii11}), we arrive at the equation
\begin{eqnarray}
\label{vii45}
&&p\beta\{ [\beta(5-5z-10z^2-18i-25iz)+\beta_1(-13\beta_2+12z+5\beta_2 i+10\beta_2 iz)]\\ \nonumber
&&-2(\beta^2+i\beta z)a_4+2(i\beta^2-\beta z)\}+6\beta_1[\beta(i\beta_2+4)+\beta_2(-\beta_1-1)]a_4\\ \nonumber
&&+6[\beta(6\beta_2 i+\beta^2+28)+\beta_2(-9\beta_1-i\beta^2-7i)]b_4=0
\end{eqnarray}
In a first approximation, if we neglect $p$, we have the equation
\begin{eqnarray}
\label{vii46}
&&\beta_1[\beta(\beta_2 i+4)+\beta_2(-\beta_1-i)]a_4+\\ \nonumber
&&[\beta(6\beta_2 i+\beta^2+28)+\beta_2(-9\beta_1-i\beta^2-7i)]b_4=0
\end{eqnarray}
\noindent
Replacing $b_4$ in terms of $a_4$ derived from the second of the fourth equation and the fifth pair of equations in (\ref{vii17}), that is
\begin{equation}
\label{vii47}
b_4=-\frac{\beta}{\beta_1}\frac{3-\beta}{2-\beta}a_4
\end{equation}
\noindent
we arrive at the equation
\begin{eqnarray}
\nonumber
\beta&&[102-18z^2+i(29-29z-2z^2+2z^3)]=\\ \nonumber
&&\beta_2[65+65z-2z^2-2z^3+i(33-12z^2)]
\end{eqnarray}
\noindent
Squaring, we obtain the equation of the sixth degree
\begin{eqnarray}
\label{vii48}
(8-120i)z^6+36z^5&&+(84-1524i)z^4+648z^3+(978-1386i)z^2\\ \nonumber
&&+5931z+6427+1586i=0
\end{eqnarray}
The roots which have positive real parts are
\begin{eqnarray}
\nonumber
&&(x_1,y_1)=(1.502,-0.9444)\\ \nonumber
&&(x_2,y_2)=(0.1888,1.950)\\ \nonumber
&&(x_3,y_3)=(0.059,3.278)
\end{eqnarray}
The modulus of the first root is $|z_1|=1.77457$ giving, when multiplied by the mass of the nucleon $937.75$ (\ref{vi19})
\begin{equation}
\nonumber
M_1=1664\textrm{MeV}
\end{equation}
\noindent
to be compared with the mass of the baryon $\Delta=1650(-35,+45)\rm MeV$. The second root gives $|z_2|=1.95939$, that is
\begin{equation}
\nonumber
M_2=1837.4\textrm{MeV}
\end{equation}
\noindent
to be compared with the mass of the baryon $\Sigma=1840\pm 10 \rm MeV$. The third root gives $|z_3|=3.27879$
\begin{equation}
\nonumber
M_3=3074.7\textrm{MeV}
\end{equation}
\noindent
to be compared with the mass of the baryon $N=3030\textrm{MeV}$.
In similar fashion, by terminating the expressions (\ref{vii12}) at higher powers of $\xi$, we shall obtain equations of increasing degree in $z$. The roots of these equations will give successive values of the masses of baryons.
\subsection{The Mesons: The Klein-Gordon Equation}
We insert the potential energy $V=-\frac{\sigma^2}{(\rho+\lambdabar)^3}$ found in (\ref{vi14}), into the Klein-Gordon radial equation
\begin{equation}
\label{viii1}
[-\frac{1}{r^2}\frac{d}{dr}(r^2\frac{d}{dr})+\frac{l(l+1)}{r^2}]\phi=\frac{(E-V)^2-m^2 c^4}{\hbar^2 c^2}\phi
\end{equation}
\noindent
and make the change of variable $\rho=\frac{z}{\lambdabar}r$ where $z=\frac{E}{mc^2}$ and $\lambdabar=\frac{\hbar}{mc}$. We get the equation
\begin{equation}
\label{viii2}
\frac{z^2}{\lambdabar^2}[-\frac{1}{\rho^2}\frac{d}{d\rho}(\rho^2\frac{d}{d\rho})+\frac{l(l+1)}{\rho^2}]\phi=
\frac{(mc^2z+\frac{z^3}{\lambdabar^3}\frac{2\sigma^2}{(\rho+z)^3})^2 m^2 c^4}{\hbar^2 c^2}\phi
\end{equation}
\noindent
In (\ref{vi18}) we have found that $\sigma^2=3.75e^2\lambdabar^2$. Equation (\ref{viii2}) becomes therefore
\begin{equation}
\label{viii3}
\{z^2[\frac{d^2}{dz^2}+\frac{2}{\rho}\frac{d}{d\rho}-\frac{l(l+1)}{\rho^2}+(1+\frac{7.5\alpha z^2}{(\rho+z)^2})^2]-1\}\phi=0
\end{equation}
\noindent
where $\alpha=\frac{e^2}{\hbar c}$ is the fine-structure constant.
We write $\phi=e^{-\frac{1}{2}\rho}F$ and get
\begin{equation}
\label{viii4}
z^2\{F^{\prime\prime}+(\frac{2}{\rho}-1)F^\prime+[\frac{1}{4}-\frac{1}{\rho}-\frac{l(l+1)}{\rho^2}+(1+\frac{7.5\alpha z^2}{(\rho+z)^3})^2]F\}-F=0
\end{equation}
\noindent
We write next $F=\rho^s f$ and get
\begin{eqnarray}
\label{viii5}
&&z^2\{\rho^2 F^{\prime\prime}+[2(s+1)\rho-\rho^2]f^\prime+[s(s+1)-(s+1)\rho+\frac{\rho^2}{4}-l(l+1)\\ \nonumber
&&\quad\quad\quad\quad +\rho^2(1+\frac{7.5 \alpha z^2}{(\rho+z)^3})^2]f\}-\rho^2 f=0
\end{eqnarray}
\noindent
Equating constant terms to zero we have
\begin{equation}
\label{viii6}
s(s+1)-l(l+1)=0
\end{equation}
\noindent
which gives $s=l$ and $s=-l-1$. We here take $s=l$ for regular solutions.
Equation (\ref{viii5}) becomes therefore after dividing by $\rho$
\begin{eqnarray}
\label{viii7}
&& z^2\{\rho f^{\prime\prime}+[2(l+1)-\rho]f^\prime+[-(l+1)+\frac{\rho}{4} \\ \nonumber
&& \quad\quad\quad\quad +\rho(1+\frac{7.5\alpha z^2}{(\rho+z)^3})^2]f\}-\rho f=0
\end{eqnarray}
\noindent
We multiply throughout by $(\rho+z)^6$
\begin{eqnarray}
\label{viii8}
&& z^2(\rho+z)^6\{\rho f^{\prime\prime}+[2(l+1)-\rho]f^\prime+[-(l+1)+\frac{5}{4}\rho]f\} \\ \nonumber
&& \quad\quad\quad\quad +[15\alpha z^4(\rho+z)^3\rho+56.25\alpha^2 z^6 \rho-(\rho+z)^6\rho]f=0
\end{eqnarray}
\noindent
We expand $f$ in powers of $\rho$
\begin{equation}
\label{viii9}
f=a_0+a_1\rho+a_2\rho^2+a_3\rho^3+\cdots
\end{equation}
\noindent
We equate constant terms in equation (\ref{viii8}) to zero
\begin{equation}
\label{viii10}
z^8[2(l+1)a_1-(l+1)a_0]=0
\end{equation}
\noindent
which gives $a_1=\frac{1}{2}a_0$.
We equate to zero the coefficients of $\rho$ in equation (\ref{viii8})
\begin{equation}
\label{viii11}
z^8[2(2l+3)a_2-(l+2)a_1+\frac{5}{4}a_0]+(15\alpha z^7+56.25 \alpha^2 z^6-z^6)a_0=0
\end{equation}
\noindent
Taking the account of (\ref{viii10}) this equation becomes
\begin{equation}
\label{viii12}
z^2\{2(2l+3)a_2 z^2+[-(l-\frac{1}{2})z^2+30\alpha z+2(56.25\alpha^2-1)]a_1\}=0
\end{equation}
\noindent
which gives
\begin{equation}
\label{viii13}
a_2=\frac{1}{2(2l+3)}[l-\frac{1}{2}-\frac{30\alpha}{z}+\frac{2(1-56.25\alpha^2)}{z^2}]a_1
\end{equation}
If expansion (\ref{viii9}) terminates so that $a_2$ and following coefficients vanish, we have the equation
\begin{equation}
\label{viii14}
(l-\frac{1}{2})z^2-30\alpha z+2(1-56.25 \alpha^2)=0
\end{equation}
\noindent
that is
\begin{equation}
\label{viii15}
(l-\frac{1}{2})z^2-0.21892 z+1.994=0
\end{equation}
For $l=0$ from (\ref{viii14}) we have the equation of the second degree
\begin{equation}
\label{viii16}
z^2+0.43784z-3.988=0
\end{equation}
\noindent
The positive root is $z=1.79004$. This value, multiplied by the reduced mass of the nucleon, $937.75\rm MeV$ as assessed in (\ref{vi19}) gives the mass
\begin{equation}
\label{viii17}
M=1679 \textrm{MeV}
\end{equation}
\noindent
to be compared with the meson $g=1680\pm 20 \rm MeV$.
We equate next to zero the coefficient of $\rho^2$ in (\ref{viii8})
\begin{eqnarray}
\label{viii18}
&&6(l+2)a_3z^8+[-(l+3)a_2+\frac{5}{4}a_1]z^3 \\ \nonumber
&&\quad\quad +[12(2l+3)a_2-6(l-\frac{1}{2})a_1+15\alpha a_1]z^7 \\ \nonumber
&&\quad\quad +[90\alpha+56.25\alpha^2-1]a_1 z^6-12a_1 z^5=0
\end{eqnarray}
\noindent
We substitute the expression for $a_2$ found in (\ref{viii13})
\begin{equation}
\label{viii19}
a_2=\frac{1}{2(2l+3)}[l-\frac{1}{2}-\frac{30\alpha}{z}+\frac{2(1-56.25\alpha^2)}{z^2}]a_1
\end{equation}
\noindent
Dividing by $z^5$ we find
\begin{eqnarray}
\label{viii20}
&& 6(l+2)a_3 z^3+\{[-\frac{(l+3)(l+\frac{1}{2})}{2(2l+3)}+\frac{5}{4}]z^3 \\ \nonumber
&& \quad\quad +\frac{l+2}{2l+3}45\alpha z^2-[\frac{3(l+2)}{2l+3}(1-56.25\alpha)^2-90\alpha]z-675\alpha^2\}a_1=0
\end{eqnarray}
If in expansion (\ref{viii9}), $a_3$ and following coefficients vanish, we have the equation
\begin{eqnarray}
\label{viii21}
&&\frac{(2-l)(2l+9)}{4(2l+3)}z^3+0.32838\frac{l+2}{2l+3}z^2 \\ \nonumber
&&\quad\quad\quad\quad -(2.991\frac{l+2}{2l+3}+0.65676)z-0.036=0
\end{eqnarray}
For $l=0$, equation (\ref{viii21}) becomes
\begin{equation}
\label{viii22}
z^3+0.14595z^2-1.76714z-0.024=0
\end{equation}
\noindent
which has the positive root $z=1.2641$, which multiplied by the reduced mass of the nucleon $937.75\rm MeV$, gives the mass
\begin{equation}
\label{viii23}
M=1185\textrm{MeV}
\end{equation}
\noindent
to be compared with the meson $\epsilon=1200\pm 100 \rm MeV$.
We equate next to zero the coefficients of $\rho^3$ in (\ref{viii8})
\begin{eqnarray}
\label{viii24}
&&4(2l+5)z^4 a_4+[-(l+4)a_3+\frac{5}{4}a_2]z^4 \\ \nonumber
&&\quad\quad +[36(l+2)a_3+(15\alpha-6(l+3))a_2+\frac{15}{2}a_1]z^3 \\ \nonumber
&&\quad\quad +[(30(2l+3)56.25\alpha^2-1)a_2+(45\alpha-15(l+2)+37.5)a_1]z^2 \\ \nonumber
&&\quad\quad +(90\alpha-6)a_1 z-30a_1=0
\end{eqnarray}
\noindent
We substitute the expressions for $a_2$ and $a_3$ from (\ref{viii13}) and (\ref{viii20}), and obtain the relation
\begin{equation}
\label{viii25}
50.52631z^4a_4+[z^4+0.04605z^3-0.70615z^2-16.48146z-1.06517]a_1=0
\end{equation}
If $a_4$ and following coefficients in expansion (\ref{viii9}) vanish, relation (\ref{viii25}) gives an equation whose positive root is $z=2.641$, which multiplied by the reduced mass of the nucleon as assessed in (\ref{vi19}), gives the mass
\begin{equation}
\nonumber
M=2476\textrm{MeV}
\end{equation}
\noindent
to be compared with the meson $X=2500\pm32 \rm MeV$.
Further values of meson masses may be obtained by terminating expansion (\ref{viii9}) at higher powers of $\rho$.
\section{Conclusions}
In this monograph, the premise is set forth, that the difficulties which to day beset particle physics, are due to our adherence to the concept of point-like particles. A departure from that concept is proposed, namely, that we base our quest on the postulate that particles are infinitely extended in space.
An interpretation of general relativity that differs in some respects from that generally accepted, results in the formulation of expressions for the gravitational, electric, and nuclear potentials, which include each a basic length, and which are free from the infinities which occur in the integrations for the self energy. The particle is seen to be spread over all space, and its potentials are not interpreted in terms of a curvature of space, but as local stresses in the unit mesh at a point.
When the potentials are inserted in the wave equations they yield in a simple manner the masses of the muon, of baryons, and of mesons, as summarized in Table III.
\begin{table}
\caption{List of Particles (mass in MeV).}
\begin{ruledtabular}
\begin{tabular}{lll}
& Calculated & Compared With \\
\hline
Leptons & 105.04 & Muon 105.66 \\
\hline
& 1115.62 & $\Lambda\quad 1115.6\pm 0.05$ \\
& 2094 & $\Lambda\quad 2100(-10,+20)$\\
Baryons & 1428 & $N\quad 1450\pm 32$ \\
& 1664 & $\Delta\quad 1650(-35,+45)$\\
& 1837 & $\Sigma\quad 1840\pm 10$ \\
& 3074 & $N\quad 3030$\\
\hline
& 1679 & $g\quad 1680\pm20$ \\
Mesons & 1185 & $\epsilon\quad 1200\pm100$\\
& 2476 & $X\quad 2500\pm32$
\end{tabular}
\end{ruledtabular}
\end{table}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,310 |
Q: mystery Error #1063: Argument count mismatch So I'm getting a "Error #1063: Argument count mismatch" error. The weird thing is that it isn't keeping the game from running, but I would like to know why I'm even getting an error in the first place. The full error is:
ArgumentError: Error #1063: Argument count mismatch on Hock(). Expected 3, got 0.
at flash.display::Sprite/constructChildren()
at flash.display::Sprite()
at flash.display::MovieClip()
at PlayScreen()[Z:\PROJECTS\Silversound\Occulus Squish\Oculus Squish\Classes\PlayScreen.as:30]
at Main/addPlayscreen()[Z:\PROJECTS\Silversound\Occulus Squish\Oculus Squish\Classes\Main.as:17]
at Main()[Z:\PROJECTS\Silversound\Occulus Squish\Oculus Squish\Classes\Main.as:13]
at runtime::ContentPlayer/loadInitialContent()
at runtime::ContentPlayer/playRawContent()
at runtime::ContentPlayer/playContent()
at runtime::AppRunner/run()
at ADLAppEntry/run()
at global/runtime::ADLEntry()
The Code for PlayScreen is:
import flashx.textLayout.formats.BackgroundColor;
import flash.display.SimpleButton;
import flash.ui.Mouse;
import flash.text.TextField;
import flash.display.MovieClip;
import flash.events.Event;
import flash.utils.Timer;
import flash.events.TimerEvent;
import flash.events.MouseEvent;
import flash.events.KeyboardEvent;
import flash.ui.Keyboard;
public class PlayScreen extends MovieClip
{
public var batArmy:Array;
public var hockArmy:Array;
public var shadow:Shadow;
public var crossHairs:CrossHairs;
var Layer01:MovieClip;
var Layer02:MovieClip;
var Layer03:MovieClip;
var Layer04:MovieClip;
var Layer05:MovieClip;
var randomX:Number = 300 + (660 - 300) * Math.random();
public function PlayScreen()
{
//Mouse.hide();
addBatButton.addEventListener(MouseEvent.CLICK, addBat);
addHockButton.addEventListener(MouseEvent.CLICK, addHock);
batArmy = new Array();
hockArmy = new Array();
//addEventListener(Event.ENTER_FRAME, crossHairsMove);
//stage.addEventListener( KeyboardEvent.KEY_DOWN, onKeyPress );
Layer01 = new MovieClip;
this.addChild(Layer01);
Layer02 = new MovieClip;
this.addChild(Layer02);
Layer03 = new MovieClip;
this.addChild(Layer03);
Layer04 = new MovieClip;
this.addChild(Layer04);
Layer05 = new MovieClip;
this.addChild(Layer05);
//add crossHair
/*crossHairs = new CrossHairs(mouseX,mouseY,this);
Layer05.addChild (crossHairs);
addEventListener(Event.ENTER_FRAME, crossHairsMove);*/
}
/*public function onKeyPress( keyboardEvent:KeyboardEvent ):void
{
if ( keyboardEvent.keyCode == Keyboard.DOWN )
{
trace("yar");
addBat;
}
}*/
public function addBat( mouseEvent:MouseEvent ):void
{
var randomX:Number = 300 + (660 - 300) * Math.random();
var newBat = new Bat( randomX, -50, this);
batArmy.push ( newBat );
Layer02.addChild (newBat);
}
public function addHock( mouseEvent:MouseEvent ):void
{
var newHock = new Hock(-72, 170, this);
hockArmy.push ( newHock );
Layer02.addChild (newHock);
}
/*public function crossHairsMove ( e:Event ):void
{
crossHairs.x = mouseX;
crossHairs.y = mouseY;
}*/
}
and from the looks of it the error has something to do with the Hock class, so here's the code for that:
import flash.display.MovieClip;
import flash.utils.Timer;
import flash.events.TimerEvent;
import flash.ui.Mouse;
import flash.events.KeyboardEvent;
import flash.ui.Keyboard;
import flash.events.Event;
public class Hock extends MovieClip
{
private var _screen: PlayScreen;
public var xSpeed:Number;
public function Hock( startX:Number, startY:Number, screen:PlayScreen )
{
_screen = screen;
x = startX;
y = startY;
width = 100;
scaleY = scaleX;
addEventListener(Event.ENTER_FRAME, moveRightFar);
addEventListener(Event.ENTER_FRAME, moveSpeed)
}
public function moveSpeed( e:Event ):void
{
x += xSpeed;
}
public function moveRightFar ( e:Event): void
{
if (x < 0)
{
gotoAndStop("rollRight");
xSpeed = 13;
}
else if (x >= 240)
{
gotoAndStop("still")
xSpeed = 0;
}
}
}
Now I could be wrong but I think it's having a problem with var newHock = new Hock(-72, 170, this); in the "addHock" function, but I have 3 arguments there, not 0. Right? Anyway, like I said, it's not keeping the game from running but it is kind of annoying, so any insight is welcome. I'm sure it's something obvious. Thanks!
A: I have a guess, but I'll explain how i got there first ...
the first line of the stacktrace pointing at your source code is
at PlayScreen()[Z:\PROJECTS\Silversound\Occulus Squish\Oculus Squish\Classes\PlayScreen.as:30]
which points at the first line of the constructor of PlayScreen: addBatButton.addEventListener(MouseEvent.CLICK, addBat);
but obviously the problem is not there ...
But PlayScreen extends MovieClip and since you didn't specifically included a super() statement, the compiler will put that at as the first command. In fact, the previous lines of the stack point to the constructor of MovieClip and then to a mysterious constructChildren() method of Sprite
That happens to be an internal method used to create the childs of the Sprite that you might have setup on your clip's stage directly from Animate.
So my guess is, the player is trying to instantiate a symbol that extends Hock and that you positioned on the stage somewhere, and of course is doing it by passing zero arguments because that's what a normal Sprite would expect.
Check your library to see what extends Hock and then see which one of those is placed in some other symbol's stage. Your options then will be to remove that and create it from code or to rework the class signature to take zero arguments.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,199 |
\section{Proofs and extended discussion for regret upper-bounds}
\label{app:upper_bounds}
\subsection{Further discussion on Opportunity~\ref{enum_prob_3}}
The example in \pref{fig:summary} does not illustrate \ref{enum_prob_3} to its fullest extent. We now expand this example and elaborate why it is important to address Opportunity~\ref{enum_prob_3}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{imgs/gap_example_tikz3.png}
\caption{Example for Opportunity~\ref{enum_prob_3}}
\label{fig:fail_gap2}
\end{figure}
Our example can be found in \pref{fig:fail_gap2}. The MDP is an extension of the one presented in Figure~\ref{fig:summary} with the new addition of actions $a_5$ and $a_6$ in state $s_3$ and the new state following action $a_6$. Again there is only a single action available at all other states than $s_1,s_2, s_3$. The reward of the state following action $a_6$ is set as $r=c+\epsilon/2$. This defines a new sub-optimal policy $\pi_3$ and the gap $\gap(s_3,a_6) = \frac{\epsilon}{2}$. Information theoretically it is impossible to distinguish $\pi_3$ as sub-optimal in less than $\Omega(\log(K)/\epsilon^2)$ rounds and so any uniformly good algorithm would have to pay at least $O(\log(K)/\epsilon)$ regret. However, what we observed previously still holds true, i.e., we should not have to play more than $\log(K)/c^2$ rounds to eliminate both $\pi_1$ and $\pi_2$ as sub-optimal policies. Prior work now suffers Opportunity~\ref{enum_prob_3} as it would pay $\log(K)/\epsilon$ regret for all zero gap state-action pairs belonging to either $\pi_1$ or $\pi_2$, essentially evaluating to $SA\log(K)/\epsilon$. On the other hand our bounds will only pay $\log(K)/\epsilon$ regret for zero gap state-action pairs belonging to $\pi_3$.
\subsection{Useful decomposition lemmas}
We start by providing the following lemma that establishes that the instantaneous regret can be decomposed into gaps defined w.r.t. any optimal (and not necessarily Bellman optimal) policy.
\begin{lemma}[General policy gap decomposition]
\label{lem:gap_decomp_pi}
Let $\gap^{\hat \pi}(s,a) = V^{\hat \pi}(s) - Q^{\hat \pi}(s,a)$ for any optimal policy $\hat \pi \in \Pi^*$. Then the difference in values of $\hat \pi$ and any policy $\pi \in \Pi$ is
\begin{align}
V^{\hat \pi}(s) - V^{\pi}(s)
= \EE_{\pi}\left[\sum_{h=\kappa(s)}^H \gap^{\hat \pi}(S_h, A_h) ~ \bigg| ~ S_{\kappa(s)} = s\right]
\end{align}
and, further, the instantaneous regret of $\pi$ is
\begin{align}
\label{eq:gap_decomp_pi}
\return{*} - \return{\pi} = \sum_{s,a} w^{\pi}(s,a) \gap^{\hat \pi}(s,a).
\end{align}
\end{lemma}
\begin{proof}
We start by establishing a recursive bound for the value difference of $\pi$ and $\hat \pi$ for any $s$
\begin{align*}
V^{\hat \pi}(s) - V^{\pi}(s)
&= V^{\hat \pi}(s)
- Q^{\hat \pi}(s, \pi(s))
+ Q^{\hat \pi}(s, \pi(s))
- V^{\pi}(s)\\
&= \gap^{\hat \pi}(s, \pi(s)) + Q^{\hat \pi}(s, \pi(s))
- Q^{\pi}(s, \pi(s))\\
&= \gap^{\hat \pi}(s, \pi(s))
+ \sum_{s'} P_\theta(s' | s, \pi(s))
[V^{\hat \pi}(s') - V^{\pi}(s')].
\end{align*}
Unrolling this recursion for all layers gives
\begin{align*}
V^{\hat \pi}(s) - V^{\pi}(s)
= \EE_{\pi}\left[\sum_{h=\kappa(s)}^H \gap^{\hat \pi}(S_h, A_h) ~ \bigg| ~ S_{\kappa(s)} = s\right].
\end{align*}
To show the second identity, consider $s=s_1$ and note that $\return{\pi} = V^\pi(s_1)$ and $\return{*} = \return{\hat \pi} = V^{\hat \pi}(s_1)$ because $\hat \pi$ is an optimal policy.
\end{proof}
For the rest of the paper we are going to focus only on the Bellman optimal policy from each state and hence only consider $\gap^{\hat\pi}(s,a) = \gap(s,a)$. All of our analysis will also go through for arbitrary $\gap^{\hat\pi},\hat\pi\in \Pi^*$, however, this did not provide us with improved regret bounds.
We now show the following technical lemma which generalizes the decomposition of value function differences and will be useful in the surplus clipping analysis.
\begin{lemma}
\label{lem:rec_rel}
Let $\Psi:\mathcal{S} \rightarrow \RR$, $\Delta: \mathcal{S} \times \mathcal{A} \rightarrow \RR$ be functions satisfying $\Psi(s) = 0$ for any $s$ with $\kappa(s) = H+1$ and $\pi \colon \mathcal{S} \rightarrow \mathcal{A}$ a deterministic policy. Further, assume that the following relation holds
\begin{align*}
\Psi(s) = \Delta(s, \pi(s)) + \langle P(\cdot|s,\pi(s)), \Psi \rangle,
\end{align*}
and let $\mathcal{A}$ be any event that is $\mathcal{H}_h$-measurable where $\mathcal{H}_h = \sigma(S_1, A_1, R_1, \dots, S_h)$ is the sigma-field induced by the episode up to the state at time $h$.
Then, for any $h \in [H]$ and $h' \in \NN$ with $h \leq h' \leq H+1$, it holds that
\begin{align*}
\EE_{\pi}[\indicator{\mathcal{A}}\Psi(S_h))] = \EE_{\pi}\left[\indicator{\mathcal{A}}
\left(\sum_{t=h}^{h' - 1} \Delta(S_{t}, A_t) + \Psi(S_{h'+1})\right)\right]
= \EE_{\pi}\left[\indicator{\mathcal{A}}
\sum_{t=h}^{H} \Delta(S_{t}, A_t)\right]
.
\end{align*}
\end{lemma}
\begin{proof}
First apply the assumption of $\Psi$ recursively to get
\begin{align*}
\Psi(s) = \EE_\pi \left[ \sum_{t=\kappa(s)}^{h' - 1} \Delta(S_t, A_t) + \Psi(S_{h'}) ~\Bigg|~ S_{\kappa(s)} = s\right].
\end{align*}
Plugging this identity into $\EE_{\pi}[\indicator{\mathcal{A}}\Psi(S_h))]$ yields
\begin{align*}
\EE_{\pi}[\indicator{\mathcal{A}}\Psi(S_h))]
&= \EE_\pi \left[
\indicator{\mathcal{A}}
\EE_\pi \left[ \sum_{t=h}^{h' - 1} \Delta(S_t, A_t) + \Psi(S_{h'})~\Bigg|~ S_{h}\right]\right]\\
&\overset{(i)}{=} \EE_\pi \left[
\indicator{\mathcal{A}}
\EE_\pi \left[ \sum_{t=h}^{h' - 1} \Delta(S_t, A_t) + \Psi(S_{h'}) ~\Bigg|~ \mathcal{H}_h \right]\right]\\
&\overset{(ii)}{=} \EE_\pi \left[
\EE_\pi \left[ \indicator{\mathcal{A}}\left( \sum_{t=h}^{h' - 1} \Delta(S_t, A_t) + \Psi(S_{h'}) \right) ~\Bigg|~ \mathcal{H}_h \right]\right]
\\
&\overset{(iii)}{=}\EE_\pi \left[\indicator{\mathcal{A}}\left( \sum_{t=h}^{h' - 1} \Delta(S_t, A_t) + \Psi(S_{h'}) \right) \right]
\end{align*}
where $\mathcal{H}_h = \sigma(S_1, A_1, R_1, \dots, S_h)$ is the sigma-field induced by the episode up to the state at time $h$. Identity $(i)$ holds because of the Markov-property and $(ii)$ holds because $\mathcal{A}$ is $\mathcal{H}_h$-measurable. The final identity $(iii)$ uses the tower-property of conditional expectations.
\end{proof}
\subsection{General surplus clipping for optimistic algorithms}
\paragraph{Clipped operators.}
One of the main arguments to derive instance dependent bounds is to write the instantaneous regret in terms of the surpluses which are clipped to the minimum positive gap. We now define the clipping threshold $\epsilon_{k} : \mathcal{S} \times \mathcal{A} \rightarrow \RR^+_0$ and associated clipped surpluses
\begin{align}
\label{eq:clipped_surp_def}
\ddot E_{k}(s,a) = \clip\left[E_k(s,a) \mid \epsilon_{k}(s,a)\right] = \indicator{ E_{k}(s,a) \geq \epsilon_k(s,a)} E_{k}(s,a).
\end{align}
Next, define the clipped $Q$- and value-function as
\begin{equation}
\label{eq:clipped_value_def}
\begin{aligned}
\ddot Q_{k}(s,a) &= \ddot E_{k}(s,a) + r(s,a) + \langle P(\cdot|s,a), \ddot V_{k}, \rangle \quad\textrm{and}
&
\ddot V_{k}(s) &= \ddot Q_{k}(s,\pi_k(s)).
\end{aligned}
\end{equation}
The random variable which is the state visited by $\pi_k$ at time $h$ throughout episode $k$ is denoted by $S_h$ and $A_h$ is the action at time $h$.
\paragraph{Events about encountered gaps} Define the event
$\mathcal{E}_h = \{\gap(S_h, A_h) > 0\}$ that at time $h$ an action with a positive gap played,
the $\mathcal{P}_{1:h} = \bigcap_{h' =1 }^{h-1} \mathcal{E}_{h'}^{c}$ that only actions with zero gap have been played until $h$
and the event
$\mathcal{A}_h = \mathcal{E}_h \cap \mathcal{P}_{1:h}$ that the first positive gap was encountered at time $h$. Let $\mathcal{A}_{H+1}= \mathcal{P}_{1:H}$ be the event that only zero gaps were encountered.
Further, let
\begin{align*}
B = \min\{ h \in [H+1] \colon \gap(S_h, A_h) > 0 \}
\end{align*}
be the first time a non-zero gap is encountered. Note that $B$ is a stopping time w.r.t. the filtration $\mathcal{F}_h = \sigma(S_1, A_1, \dots, S_h, A_h)$.
The proof of \citet{simchowitz2019non} consists of two main steps. First show that for their definition of clipped value functions one can bound $\ddot V_k(s_1) - V^{\pi_k}(s_1) \geq \frac{1}{2}(\bar V_k(s_1) - V^{\pi_k}(s_1))$. Next, using optimism together with the fact that $\pi_k$ has highest value function at episode $k$ it follows that $\bar V_k(s_1) - V^{\pi_k}(s_1) \geq V^*(s_1) - V^{\pi_k}(s_1)$. The second main step is to use a high-probability bound on the clipped surpluses to relate them to the probability to visit the respective state-action pair and the proof is finished via an integration lemma. We now show that the first step can be carried out in greater generality by defining a less restrictive clipping operator. This operator is independent of the details in the definition of gap at each state-action pair but rather only uses a certain property which allows us to decompose the episodic regret as a sum over gaps. We will also further show that one does not need to use an integration lemma for the second step but can rather reformulate the regret bound as an optimization problem. This will allow us to clip surpluses at state-action pairs with zero gaps beyond the $\gap_{\min}$ rate.
\paragraph{Clipping with an arbitrary threshold.} Recall the definition of the clipped surpluses and clipped value function in Equation~\ref{eq:clipped_surp_def} and Equation~\ref{eq:clipped_value_def}. We begin by showing a general relation between the clipped value function difference and the non-clipped surpluses for any clipping threshold $\epsilon_{k} : \mathcal{S} \rightarrow \RR$. This will help in establishing $\ddot V_{k}(s_1) - V^{\pi_k}(s_1) \geq \frac{1}{2}(\bar V_k(s_1) - V^{\pi_k}(s_1))$.
\begin{lemma}
\label{lem:Vdd_lb1}
Let $\epsilon_k : \mathcal{S} \times \mathcal{A} \rightarrow \RR^+_0$ be arbitrary. Then for any optimistic algorithm it holds that
\begin{equation}
\label{eq:clipped_ineq_tight}
\ddot V_k(s_1) - V^{\pi_k}(s_1) \geq
\mathbb{E}_{\pi_k}\left[
\sum_{h=B}^H\left( \gap(S_{h},A_h) - \epsilon_{k}(S_{h}, A_h)\right)
\right].
\end{equation}
\end{lemma}
\begin{proof}
We use $W_k(s) = \ddot V_k(s) - V^{\pi_k}(s)$ in the following and first show that $W(s_1) \geq \EE_{\pi_k}[ W_k(S_B)]$.
As a precursor, we prove
\begin{align}
\EE_{\pi_k}\left[ \indicator{\mathcal{P}_{1:h} } W_k(S_h)\right]
\geq
\EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h+1} } W_k(S_{h+1})\right]
+
\EE_{\pi_k}\left[ \indicator{ \mathcal{P}_{1:h+1} } W_k(S_{h+1})\right].
\label{eq:first_rec}
\end{align}
To see this, plug the definitions into $W_k(s)$ which gives $W_k(s) = \ddot V_k(s) - V^{\pi_k}(s) = \ddot E_k(s, \pi_k(s)) + \langle P(\cdot | s, \pi_k(s)), W_{k} \rangle $ and use this in the LHS of \pref{eq:first_rec} as
\begin{align*}
\EE_{\pi_k}\left[ \indicator{\mathcal{P}_{1:h} } W_k(S_h)\right]
&= \EE_{\pi_k}\big[ \indicator{\mathcal{P}_{1:h} }
\underset{\geq 0}{\underbrace{ \ddot E_k(S_h, A_h)}\big]}
+
\EE_{\pi_k}\left[ \indicator{\mathcal{P}_{1:h}} \EE[ W_k(S_{h+1}) \mid S_h]\right]
\\
& \overset{(i)}{\geq}
\EE_{\pi_k}\left[ \indicator{\mathcal{P}_{1:h}} \EE_{\pi_k}[ W_k(S_{h+1}) \mid \mathcal{H}_{h}]\right]
\\
& \overset{(ii)}{=}
\EE_{\pi_k}\left[ \EE_{\pi_k}[ \indicator{\mathcal{P}_{1:h} } W_k(S_{h+1}) \mid \mathcal{H}_{h}]\right] =
\EE_{\pi_k}\left[ \indicator{\mathcal{P}_{1:h}} W_k(S_{h+1}) \right]
\end{align*}
where $\mathcal{H}_h = \sigma(S_1, A_1, R_1, \dots, S_h)$ is the sigma-field induced by the episode up to the state at time $h$. Step $(i)$ follows from $\clip[\cdot | c] \geq 0$ for any $c \geq 0$ and the Markov property and $(ii)$ holds because $\mathcal{P}_{1:h}$ is $\mathcal{H}_h$-measurable.
We now rewrite the RHS by splitting the expectation based on whether event $\mathcal{E}_{h+1}$ occurred as
\begin{align*}
\EE_{\pi_k}\left[ \indicator{\mathcal{P}_{1:h} } W_k(S_{h+1}) \right]
=
\EE_{\pi_k}\left[ \indicator{\mathcal{P}_{1:h+1} } W_k(S_{h+1}) \right]
+ \EE_{\pi_k}\left[ \indicator{\mathcal{A}_{h+1}} W_k(S_{h+1}) \right].
\end{align*}
We have now shown \pref{eq:first_rec}, which we will now use to lower-bound $W_k(s_1)$ as
\begin{align*}
W_k(s_1)
&= \EE_{\pi_k}[\indicator{\mathcal{E}_1} W_1(S_1)] + \EE_{\pi_k}[\indicator{\mathcal{E}_1^c} W_1(S_1)]\\
&= \EE_{\pi_k}[\indicator{\mathcal{A}_1} W_1(S_1)] + \EE_{\pi_k}[\indicator{\mathcal{P}_{1:1}} W_1(S_1)]\\
& \geq \EE_{\pi_k}[\indicator{\mathcal{A}_1} W_1(S_1)] + \sum_{h=2}^H \EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } W_k(S_{h})\right]\\
&= \sum_{h=1}^H \EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } W_k(S_{h})\right] =
\EE_{\pi_k}[ W_k(S_B)].
\end{align*}
Applying \pref{lem:rec_rel} with $\mathcal{A} = \mathcal{A}_h$, $\Psi = W_k$ and $\Delta = \ddot E_k$ yields
\begin{align*}
W_k(s_1)
&\geq \sum_{h=1}^H \EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h}} \sum_{h' = h}^H \ddot E_k(S_{h'}, A_{h'})\right]\\
&\geq \sum_{h=1}^H \EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H E_k(S_{h'}, A_{h'})\right] - \sum_{h=1}^H \EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H \epsilon_{k}(S_{h'}, A_{h'})\right],
\end{align*}
where we applied the definition clipped surpluses which gives $\ddot E_k(s,a) = \clip[ E_k(s,a) \mid \epsilon_{k}(s,a)] \geq E_k(s,a) - \epsilon_{k}(s,a)$. It only remains to show that
\begin{align*}
\EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H E_k(S_{h'}, A_{h'})\right] \geq
\EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H \gap(S_{h'}, A_{h'})\right].
\end{align*}
To do so, we apply \pref{lem:rec_rel} twice, first with $\mathcal{A} = \mathcal{A}_h$, $\Psi = \bar V_k - V^{\pi_k}$ and $\Delta = E_k$ and then again with $\mathcal{A} = \mathcal{A}_h$, $\Psi = V^{*} - V^{\pi_k}$ and $\Delta = \gap$ which gives
\begin{align*}
\EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H E_k(S_{h'}, A_{h'})\right]
&=
\EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } (\bar V_k(S_h) - V^{\pi_k}(S_{h}))\right] \\
&\geq
\EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } (V^*(S_h) - V^{\pi_k}(S_{h}))\right]\\
&=
\EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H \gap(S_{h'}, A_{h'})\right].
\end{align*}
Thus, we have shown that
\begin{align*}
& \ddot V_k(s_1) - V^{\pi_k}(s_1) = W_k(s_1) \\
&\geq \sum_{h=1}^H \EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H \gap(S_{h'}, A_{h'})\right] - \sum_{h=1}^H \EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H \epsilon_{k}(S_{h'}, A_{h'})\right]\\
&= \sum_{h=1}^H \EE_{\pi_k}\left[ \indicator{ \mathcal{A}_{h} } \sum_{h' = h}^H \left( \gap(S_{h'}, A_{h'}) - \epsilon_{k}(S_{h'}, A_{h'}) \right)\right]\\
&= \EE_{\pi_k}\left[ \sum_{h = B}^H \left( \gap(S_{h}, A_{h}) - \epsilon_{k}(S_{h}, A_{h}) \right)\right]
\end{align*}
where the last equality uses the definition of $B$, the first time step at which a non-zero gap was encountered.
\end{proof}
\begin{lemma}[Optimism of clipped value function]
\label{lem:opt_clipped_weak_stopping_time}
Let the clipping thresholds $\epsilon_k \colon \mathcal{S} \times \mathcal{A} \rightarrow \RR^+_0$ used in the definition of $\ddot V_k$ satisfy
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^H \epsilon_k(S_h, A_h) \right]
\leq \frac{1}{2} \EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h) \right]
\end{align*}
for some optimal policy $\hat \pi$. Then scaled optimism holds for the clipped value function, i.e.,
\begin{align*}
\ddot V_k(s_1) - V^{\pi_k}(s_1)
\geq \frac{1}{2} (V^*(s_1) - V^{\pi_k}(s_1)).
\end{align*}
\end{lemma}
\begin{proof}
The proof works by establishing the following chain of inequalities:
\begin{align*}
\frac{V^*(s_1) - V^{\pi_k}(s_1)}{2}
&\overset{(a)}{=}
\frac{1}{2} \EE_{\pi_k}\left[\sum_{h=1}^H\gap(S_{h},A_{h})\right]
\overset{(b)}{=}
\frac{1}{2} \EE_{\pi_k}\left[\sum_{h=B}^H\gap(S_{h},A_{h}))\right]
\\
&\overset{(c)}{=}
\EE_{\pi_k}\left[\sum_{h=B}^H\left( \gap(S_{h},A_{h})) - \frac{1}{2}\gap(S_{h},A_{h}))\right)\right]\\
&\overset{(d)}{\leq}
\EE_{\pi_k}\left[\sum_{h=B}^H\left( \gap(S_{h},A_{h})) - \epsilon_k(S_{h},A_{h}))\right)\right]\\
&\overset{(e)}{\leq} \ddot V_k(s_1) - V^{\pi_k}(s_1).
\end{align*}
Here, $(a)$ uses Lemma~\ref{lem:gap_decomp_pi} and $(b)$ uses the definition of $B$. Step $(c)$ is just algebra and
step $(d)$ uses the assumption on the threshold function.
The last step $(e)$ follows from Lemma~\ref{lem:Vdd_lb1}.
\end{proof}
\surplusclippingbound*
\begin{proof}
Applying \pref{lem:opt_clipped_weak_stopping_time} which ensures scaled optimism of the clipped value function gives
\begin{align*}
V^*(s_1) - V^{\pi_k}(s_1) \leq 2(\ddot V_k(s_1) - V^{\pi_k}(s_1))
= 2\sum_{s,a} w^{\pi_k}(s,a) \ddot E_k(s,a),
\end{align*}
where the equality follows from the definition of $\ddot V_k(s_1)$ and \pref{lem:rec_rel}. Subtracting $\frac 1 2 (V^*(s_1) - V^{\pi_k}(s_1))$ from both sides gives
\begin{align*}
\frac{1}{2}( V^*(s_1) - V^{\pi_k}(s_1) )\leq 2\sum_{s,a} w^{\pi_k}(s,a) \left( \ddot E_k(s,a) - \frac{\gap(s,a)}{4} \right)
\end{align*}
because \pref{lem:gap_decomp_pi} ensures that $\frac{1}{2}(V^*(s_1) - V^{\pi_k}(s_1)) = \frac{1}{2} \sum_{s,a} w^{\pi_k}(s,a) \gap(s,a)$. Reordering terms yields
\begin{align*}
V^*(s_1) - V^{\pi_k}(s_1)
& \leq 4 \sum_{s,a} w^{\pi_k}(s,a) \left( \ddot E_k(s,a) - \frac{\gap(s,a)}{4} \right)\\
& = 4 \sum_{s,a} w^{\pi_k}(s,a) \left( \clip\left[ E_k(s,a) ~\bigg| ~ \epsilon_{k}(s, a) \right] - \frac{\gap(s,a)}{4} \right)\\
& \leq 4 \sum_{s,a} w^{\pi_k}(s,a) \clip\left[ E_k(s,a) ~\bigg| ~ \epsilon_{k}(s, a) \vee \frac{\gap(s,a)}{4} \right],
\end{align*}
where the final inequality follows from the general properties of the clipping operator,
which satisfies
\begin{align*}
\clip[a | b ] - c =
\begin{cases}
a - c \leq a & \textrm{for } a \geq b \vee c\\
0 - c \leq 0 & \textrm{for } a \leq b\\
a - c \leq 0 & \textrm{for } a \leq c
\end{cases}
\leq \clip[a | b \vee c].
\end{align*}
\end{proof}
\subsection{Definition of valid clipping thresholds $\epsilon_{k}$}
\pref{prop:surplus_clipping_bound} establishes a sufficient condition on the clipping thresholds $\epsilon_k$ that ensures that the penalized surplus clipping bounds holds.
We now discuss several choices for this threshold that satisfy this condition.
\paragraph{Minimum positive gap $\gap_{\min}$:}
We now make the quick observation that taking $\epsilon_{k} \equiv \frac{\gap_{\min}}{2H}$ will satisfy the condition of Proposition~\ref{prop:surplus_clipping_bound}, because on the event $\mathcal{B} \equiv \mathcal{A}_{H+1}^c$ there exists at least one positive gap in the sum $\sum_{h=1}^H\gap(S_h,A_h)$, which, by definition, is at least $\gap_{\min}$. This shows that our results already can recover the bounds in prior work, with significantly less effort.
\paragraph{Average gaps:}Instead of the minimum gap which was used in existing analyses, we now show that we can also use the marginalized average gap which we will define now.
Recall that $B = \min\{ h \in [H+1] \colon \gap(S_h, A_h) > 0 \}$ is the first time a non-zero gap is encountered. Note that $B$ is a stopping time w.r.t. the filtration $\mathcal{F}_h = \sigma(S_1, A_1, \dots, S_h, A_h)$. Further let
\begin{align}
\mathcal{B}(s,a) \equiv \{B \leq \kappa(s), S_{\kappa(s)} = s, A_{\kappa(s)} = a\}
\end{align}
be the event that $(s,a)$ was visited after a non-zero gap in the episode.
We now define this clipping threshold
\begin{align}
\label{eqn:new_avgclip_clean}
\epsilon_k(s,a) \equiv
\begin{cases}
\frac{1}{2H}
\EE_{\pi_k}\left[ \sum_{h=1}^H \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a) \right]
& \textrm{if }\PP_{\pi_k}(\mathcal{B}(s,a)) > 0\\
\infty & \textrm{otherwise}
\end{cases}
\end{align}
As the following lemma shows, this is a valid choice which satisfies the condition of \pref{prop:surplus_clipping_bound}.
\begin{lemma}
\label{lem:clipping_gaps_rel}
The expected sum of clipping thresholds in Equation~\eqref{eqn:new_avgclip_clean} over all state-action pairs encountered after a positive gap is at most half the expected total gaps per episode. That is,
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^H \epsilon_k(S_h, A_h) \right]
\leq \frac{1}{2} \EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h) \right].
\end{align*}
\end{lemma}
\begin{proof}
We rewrite the LHS of the inequality to show as
$\EE_{\pi_k}\left[\sum_{h=1}^H \indicator{B \leq h}\epsilon_k(S_h, A_h) \right]$ and from now on consider the random variable $f_h(B, S_h, A_h) = \indicator{B \leq h}\epsilon_k(S_h, A_h)$ where $f_h(b, s, a) = \indicator{b \leq h}\epsilon_k(s,a)$ is a deterministic function\footnote{It may still depend on the current policy $\pi_k$ which is determined by observations in episodes $1$ to $k-1$. But, crucially, $f_h$ does not depend on any realization in the $k$-th episode}.
We will show below that $\EE_{\pi_k}\left[ f_h(B, S_h, A_h)\right] \linebreak[1]\leq \linebreak[1]\frac{1}{2H}\EE_{\pi_k}\left[\sum_{h=B}^H \gap(S_{h}, A_{h}) \right]$. This is sufficient to prove the statement, because
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^H \epsilon_k(S_h, A_h) \right]
&= \sum_{h=1}^H \EE_{\pi_k}\left[ f_h(B, S_h, A_h)\right]\\
& \leq \frac{1}{2H}\sum_{h=1}^H\EE_{\pi_k}\left[
\sum_{h'=B}^H \gap(S_{h'}, A_{h'})
\right]\\
&= \frac{1}{2}\EE_{\pi_k}\left[
\sum_{h=B}^H \gap(S_{h}, A_{h})
\right]
= \frac{1}{2}\EE_{\pi_k}\left[
\sum_{h=1}^H \gap(S_{h}, A_{h})
\right].
\end{align*}
To bound the expected value of $f_h(B, S_h, A_h)$, we first write $f_h$ for all triples $b, s, a$ such that $\PP_{\pi_k}(B = b, A_h = a, S_h = s) > 0$ as
\begin{align*}
f_h(b, s, a) &\overset{(i)}{=}
\indicator{b \leq h}
\frac{1}{2H}
\EE_{\pi_k}\left[ \sum_{h'=1}^H \gap(S_{h'}, A_{h'}) ~ \bigg| ~B \leq h, ~S_h = s, A_h = a \right]\\
&\overset{(ii)}{=}
\indicator{b \leq h}
\frac{1}{2H}
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) ~ \bigg| ~B \leq h, ~S_h = s, A_h = a \right]\\
&\quad + \indicator{b \leq h}
\frac{1}{2H}
\EE_{\pi_k}\left[ \sum_{h'=h+1}^H \gap(S_{h'}, A_{h'}) ~ \bigg| S_h = s, A_h = a \right],
\end{align*}
where $(i)$ expands the definition of $\epsilon_k$ and $(ii)$ decomposes the sum inside the conditional expectation and uses the Markov-property to simplify the conditioning for terms after $h$.
Before taking the expectation of $f_h(B,S_h, A_h)$, we first rewrite the conditional expectation in the first term above, which will be useful later.
\begin{align*}
&\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) ~ \bigg| ~B \leq h, ~S_h = s, A_h = a \right]\\
&\overset{(i)}{=} \frac{
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) \indicator{A_h = a, S_h = s} \indicator{B \leq h}\right]
}{ \PP_{\pi_k}\left[ B \leq h, ~S_h = s, A_h = a \right]}\\
&\overset{(ii)}{=} \frac{
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) \indicator{A_h = a, S_h = s} \right]
}{ \PP_{\pi_k}\left[ B \leq h, ~S_h = s, A_h = a \right]}
\\
&= \frac{
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) ~ \bigg| ~ S_h = s, A_h = a \right]
}{ \PP_{\pi_k}\left[ B \leq h ~ \mid ~S_h = s, A_h = a \right]}.
\end{align*}
Here, step $(i)$ uses the property of conditional expectations with respect to an event with nonzero probability and $(ii)$ follows from the definition of $B$: When $B > h$, the sum of gaps until $h$ is zero.
Consider now the expectation of $f_h(B, S_h, A_h)$
\begin{align}
&\EE_{\pi_k}\left[ f_h(B, S_h, A_h)\right]\nonumber\\
&=
\frac{1}{2H}\EE_{\pi_k}\left[\indicator{B \leq h}
\frac{
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) ~ \bigg| ~ S_h, A_h \right]
}{ \PP_{\pi_k}\left[ B \leq h ~ \mid ~S_h, A_h \right]}
\right]
\label{eqn:term1_fexp}
\\
&\quad +
\frac{1}{2H} \EE_{\pi_k}\left[\indicator{B \leq h}
\EE_{\pi_k}\left[ \sum_{h'=h+1}^H \gap(S_{h'}, A_{h'}) ~ \bigg| S_h , A_h \right]\right]
\label{eqn:term2_fexp}
\end{align}
The term in \eqref{eqn:term2_fexp} can be bounded using the tower-property of expectations as
\begin{align*}
&\frac{1}{2H} \EE_{\pi_k}\left[\indicator{B \leq h}
\EE_{\pi_k}\left[ \sum_{h'=h+1}^H \gap(S_{h'}, A_{h'}) ~ \bigg| S_h , A_h \right]\right]\\
&\leq
\frac{1}{2H} \EE_{\pi_k}\left[
\EE_{\pi_k}\left[ \sum_{h'=h+1}^H \gap(S_{h'}, A_{h'}) ~ \bigg| S_h , A_h \right]\right]
= \frac{1}{2H} \EE_{\pi_k}\left[\sum_{h'=h+1}^H \gap(S_{h'}, A_{h'}) \right].
\end{align*}
For the term in \eqref{eqn:term1_fexp}, we also use the tower-property to rewrite it as
\begin{align*}
&\frac{1}{2H}\EE_{\pi_k}\left[\indicator{B \leq h}
\frac{
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) ~ \bigg| ~ S_h, A_h \right]
}{ \PP_{\pi_k}\left[ B \leq h ~ \mid ~S_h, A_h \right]}
\right]
\\
&=
\frac{1}{2H}\EE_{\pi_k}\left[
\EE_{\pi_k}\left[
\indicator{B \leq h}
\frac{
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) ~ \bigg| ~ S_h, A_h \right]
}{ \PP_{\pi_k}\left[ B \leq h ~ \mid ~S_h, A_h \right]}
~ \bigg| ~ S_h, A_h\right]
\right]\\
&=
\frac{1}{2H}\EE_{\pi_k}\left[
\EE_{\pi_k}\left[
\indicator{B \leq h} ~ \bigg| ~ S_h, A_h\right]
\frac{
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) ~ \bigg| ~ S_h, A_h \right]
}{ \PP_{\pi_k}\left[ B \leq h ~ \mid ~S_h, A_h \right]}
\right]\\
&=
\frac{1}{2H}\EE_{\pi_k}\left[
\EE_{\pi_k}\left[ \sum_{h'=B}^h \gap(S_{h'}, A_{h'}) ~ \bigg| ~ S_h, A_h \right]
\right]\\
&=\frac{1}{2H}\EE_{\pi_k}\left[
\sum_{h'=B}^h \gap(S_{h'}, A_{h'})
\right].
\end{align*}
Summing both terms yields the required upper-bound $\frac{1}{2H}\EE_{\pi_k}\left[\sum_{h=B}^H \gap(S_{h}, A_{h}) \right]$ on the expectation $\EE_{\pi_k}\left[ f_h(B, S_h, A_h)\right]$.
\end{proof}
\subsection{Policy-dependent regret bound for \textsc{StrongEuler}}
We now show how to derive a regret bound for \textsc{StrongEuler} algorithm in \citet{simchowitz2019non} that depends on the gaps of the played policies throughout the $K$ episodes.
To build on parts of the analysis in \citet{simchowitz2019non}, we first define some useful notation analogous to \citet{simchowitz2019non} but adapted to our setting:
\begin{align*}
\bar n_k(s,a) &= \sum_{j=1}^k w^{\pi_k}(s,a),\\
M &= (SAH)^3,\\
\mathcal{V}^{\pi}(s,a) &= \VV[R(s,a)] + \VV_{s'\sim P(\cdot|s,a)}[V^{\pi}(s')],\\
\mathcal{V}_k(s,a) &= \mathcal{V}^{\pi_k}(s,a) \wedge \mathcal{V}^*(s,a)
\end{align*}
We will use their following results:
\begin{proposition}[Proposition~F.1, F.9 and B.4 in \citet{simchowitz2019non}]
\label{prop:strongeuler_surplus_bound}
There is a good event $\mathcal{A}^{\mathrm{conc}}$ that holds with probability $1 - \delta / 2$. In this event,
\textsc{StrongEuler} is strongly optimistic (as well as optimistic). Further, there is a universal constant $c \geq 1$ so that for all $k \geq 1$, $s \in \mathcal{S}$, $a \in \mathcal{A}$, the surpluses are bounded as
\begin{align*}
0 \leq \frac{1}{c} E_{k}(s,a) \leq B^{\mathrm{lead}}_{k}(s,a) + \sum_{h=\kappa(s)}^H \EE_{\pi_k}\left[B^{\mathrm{fut}}_{k}(S_h ,A_h) \mid (S_{\kappa(s)},A_{\kappa(s)}) = (s,a) \right],
\end{align*}
where $B^{\mathrm{lead}}, B^{\mathrm{fut}}$ are defined as
\begin{align*}
B^{\mathrm{lead}}_{k}(s,a) &= H \wedge \sqrt{\frac{\mathcal{V}_{k}(s,a)\log(M n_k(s,a)/\delta) }{ n_k(s,a)}},\\
B^{\mathrm{fut}}_{k}(s,a) &= H^3 \wedge H^3\left(\sqrt{\frac{S\log(M n_k(s,a)/\delta) }{n_k(s,a)}} + \frac{S\log(M n_k(s,a)/\delta) }{n_k(s,a)}\right)^2.
\end{align*}
\end{proposition}
\begin{lemma}[Lemma~B.3 in \citet{simchowitz2019non}]
\label{lem:clip_dist}
Let $m \geq 2$, $a_1, \dots, a_m \geq 0$ and $\epsilon \geq 0$. Then $\clip\left[\sum_{i=1}^m a_i \big| \epsilon \right] \leq 2 \sum_{i=1}^m \clip\left[ a_i | \frac{\epsilon}{2m} \right]$.
\end{lemma}
Equipped with these results and our improved surplus clipping proposition in \pref{prop:strongeuler_surplus_bound}, we can now derive the following bound on the regret of \textsc{StrongEuler}
\begin{lemma}
\label{lem:strongeuler_regB_bound}
In event $\mathcal{A}^{\mathrm{conc}}$, the regret of \textsc{StrongEuler} is bounded for all $k\geq 1$ as
\begin{align*}
\mathfrak{R}(K) \leq &
8 \sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{lead}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{4}\right]\\
& +
16 \sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{fut}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{8SA} \right],
\end{align*}
with a universal constant $c \geq 1$ and $\breve \gap_k(s,a) = \frac{\gap(s,a)}{4} \lor \epsilon_{k}(s,a)$.
\end{lemma}
\begin{proof}
We now use our improved surplus clipping result from \pref{prop:surplus_clipping_bound} as a starting point to bound the instantaneous regret of \textsc{StrongEuler} in the $k$th episode as
\begin{align}
V^*(s_1) - V^{\pi_k}(s_1)
\leq
4\sum_{s,a} w^{\pi_k}(s,a) \clip\left[E_{k}(s,a)~ \bigg| \breve \gap_k(s,a) ~\right].
\label{eq:surplus_clip1}
\end{align}
Next, we write the bound on the surpluses from \pref{prop:strongeuler_surplus_bound} as
\begin{align*}
E_{k}(s,a) \leq &~ cB^{\mathrm{lead}}_{k}(s,a) \\
&+ c \sum_{s', a'} \indicator{\kappa(s') \geq \kappa(s)}\PP^{\pi_k}\left[ S_{\kappa(s')} = s', A_{\kappa(s')} = a' \mid (S_{\kappa(s)},A_{\kappa(s)}) = (s,a) \right] B^{\mathrm{fut}}_{k}(s' ,a')
\end{align*}
and plugging it in \pref{eq:surplus_clip1} and applying \pref{lem:clip_dist} gives
\begin{align*}
V^*(s_1) - V^{\pi_k}(s_1)
\leq & ~
8 \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{lead}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{4}\right]\\
& +
16 \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{fut}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{8SA} \right].
\end{align*}
The statement to show follows now by summing over $k \in [K]$. The form of the second term in the previous display follows from the inequality
\begin{align*}
&\sum_{s,a}w^{\pi_k}(s,a) \indicator{\kappa(s') \geq \kappa(s)}\PP^{\pi_k}\left[ S_{\kappa(s')} = s', A_{\kappa(s')} = a' \mid (S_{\kappa(s)},A_{\kappa(s)})
= (s,a) \right] \\
&\leq \sum_{s,a}w^{\pi_k}(s,a) \PP^{\pi_k}\left[ S_{\kappa(s')} = s', A_{\kappa(s')}
= a' \mid (S_{\kappa(s)},A_{\kappa(s)}) = (s,a) \right] = w^{\pi_k}(s', a').
\end{align*}
\end{proof}
We note that if $\pi_k \equiv \hat\pi$ for any $\hat\pi \in \Pi^*$ then $V^*(s_1) - V^{\pi_k}(s_1) = 0$, and WLOG we can disregard such terms in the total regret.
The next step is to relate $\bar n_k(s,a)$ to $n_k(s,a)$ via the following lemma.
\begin{lemma}[Lemma~B.7 in \citet{simchowitz2019non}]
\label{lem:b7_simchowitz}
Define the event $\mathcal{A}^{\mathrm{samp}}$
\begin{align*}
\mathcal{A}^{\mathrm{samp}} = \left\{\forall (s,a) \in \mathcal{S}\times\mathcal{A}, \forall k \geq \tau(s,a) \colon n_k(s,a) \geq \frac{\bar n_k(s,a)}{4}\right\},
\end{align*}
where $\tau(s,a) = \inf\{k : \bar n_k(s,a) \geq H_{\mathrm{samp}}\}$ and $H_{\mathrm{samp}} = c' \log(M/\delta)$ for a universal constant $c'$. Then event $\mathcal{A}^{\mathrm{samp}}$ holds with probability $1 - \delta/2$.
\end{lemma}
\begin{proof}
This can be proved analogously to Lemma~B.7 in \citet{simchowitz2019non} and Lemma~6 in \citet{dann2019policy} with the difference that in our case, there can only be at most one observation of $(s,a)$ per episode for each $(s,a)$ due to our layered assumption. Thus, there is no need to sum over observations accumulated for each $h \in [H]$ and our $H_{\mathrm{samp}} = O(\log(H))$ as opposed to $O(H \log(H))$.
\end{proof}
\begin{lemma}
\label{lem:fntofbarn}
Let $f_{s,a} \colon \NN \rightarrow \RR$ be non-increasing with $\sup_{u} f_{s,a}(u) \leq \hat f < \infty$ for all $s,a \in \mathcal{S} \times \mathcal{A}$. Then on event $\mathcal{A}^{\mathrm{samp}}$ in \pref{lem:b7_simchowitz}, we have
\begin{align*}
\sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) f_{s,a}(n_k(s,a)) \leq S A \hat f H_{\mathrm{samp}} +
\sum_{s,a} \sum_{k=\tau(s,a)}^K w^{\pi_k}(s,a) f_{s,a}(\bar n_k(s,a) / 4).
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
&\sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) f_{s,a}(n_{k}(s,a)) \\
=&~\sum_{s,a}\sum_{k=1}^{\tau(s,a)-1} w^{\pi_k}(s,a)
f_{s,a}(n_{k}(s,a))
+\sum_{s,a}\sum_{k=\tau(s,a)}^{K} w^{\pi_k}(s,a)
f_{s,a}(n_{k}(s,a))\\
\leq&~\sum_{s,a}\left(\sum_{k=1}^{\tau(s,a)-1} w^{\pi_k}(s,a)\right) \hat f
+\sum_{s,a}\sum_{k=\tau(s,a)}^{K} w^{\pi_k}(s,a)
f_{s,a}(\bar n_{k}(s,a) / 4)\\
=& ~
\sum_{s,a}n_{\tau(s,a)}(s,a) \hat f
+\sum_{s,a}\sum_{k=\tau(s,a)}^{K} w^{\pi_k}(s,a)
f_{s,a}(\bar n_{k}(s,a) / 4)\\
\leq&~ SA H_{\mathrm{samp}} \hat f
+\sum_{s,a}\sum_{k=\tau(s,a)}^{K} w^{\pi_k}(s,a)
f_{s,a}(\bar n_{k}(s,a) / 4).
\end{align*}
\end{proof}
\begin{theorem}[Regret Bound for \textsc{StrongEuler}]
\label{thm:reg_bound_gen_se}
With probability at least $1 - \delta$, the regret of \textsc{StrongEuler} is bounded for all number of episodes $K \in \NN$ as
\begin{align*}
\mathfrak{R}(K) \lesssim &~
\sum_{s,a} \min_{t \in [K_{(s,a)}]}\Bigg\{\frac{\mathcal{V}^*(s,a)\mathcal{LOG}(M/\delta,t,\breve\gap_t(s,a))}{\breve\gap_t(s,a)}\\
&+ \sqrt{(K_{(s,a)}-t)\mathcal{LOG}(M/\delta,K_{(s,a)},\breve\gap_{K_{(s,a)}}(s,a))}\Bigg\}
\\&
+ \sum_{s,a} SH^3 \log\frac{MK}{\delta} \min\left\{ \log \frac{MK}{\delta}, \log \frac{MH}{\breve \gap_{\min}(s,a)}\right\}
\\& + SAH^3 (S \vee H)\log\frac{M}{\delta}.
\end{align*}
Here, $K_{(s,a)}$ is the last round during which a policy $\pi$ was played such that $w^{\pi}(s,a)>0$, $\breve\gap_{t}(s,a) = \gap(s,a) \lor \epsilon_{t}(s,a)$, $\breve \gap_{\min}(s,a) = \min_{k \in [K] \colon \breve\gap_k(s,a) > 0} \breve\gap_k(s,a)$ is the smallest gap encountered for each $(s,a)$, and $\mathcal{LOG}(M/\delta,t,\breve\gap_t(s,a)) = \log\left(\frac{M}{\delta}\right)\log\left(t\land 1 + \frac{16\mathcal{V}^*(s,a)\log(M/\delta)}{\breve\gap_t(s,a)^2}\right)$.
\end{theorem}
\begin{proof}
We here consider the event $\mathcal{A}^{\mathrm{conc}} \cap \mathcal{A}^{\mathrm{samp}}$ which has probability at least $1 - \delta$ by \pref{prop:strongeuler_surplus_bound} and \pref{lem:b7_simchowitz}. We now start with the regret bound in \pref{lem:strongeuler_regB_bound} and bound the two terms individually in the following:
\paragraph{Bounding the $B^{\mathrm{lead}}$ term}
We have
\begin{align}
& \sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{lead}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{4}\right]\nonumber\\
&\overset{(i)}{\leq}
SAH H_{\mathrm{samp}} + \sum_{s,a}\sum_{k=\tau(s,a)}^{K} w^{\pi_k}(s,a)
\clip\left[c\sqrt{\frac{4\mathcal{V}_{k}(s,a)\log(M \bar n_k(s,a)/4\delta) }{ \bar n_k(s,a)}} ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{4}\right]
\nonumber\\
&\overset{(ii)}{\leq} \!
SAH H_{\mathrm{samp}} +\! \sum_{s,a}\sum_{k=\tau(s,a)}^{K_{(s,a)}} \!\!\! w^{\pi_k}(s,a)\!
\clip\left[2c\sqrt{\mathcal{V}^*(s,a)\log\frac{M}{\delta}}\sqrt{\frac{\log(\bar n_k(s,a)) }{ \bar n_k(s,a)}} ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{4}\right],
\label{eq:optapp1}
\end{align}
where step $(i)$ applies \pref{lem:fntofbarn} and $(ii)$ follows from the definition of $\mathcal{V}_k(s,a)$, the definition of $K_{(s,a)}$ and
\begin{align*}
&\log\left(\frac{M \bar n_k(s,a)}{4\delta}\right) =
\log\left(\frac{M}{4\delta}\right)
+\log\left(\bar n_k(s,a)\right)\\
&\leq \left(\log\left(\frac{M}{4\delta}\right) + 1\right)\log(\bar n_k(s,a))
= \log\left(\frac{Me}{4\delta}\right) \log(\bar n_k(s,a)) \leq \log(M/ \delta) \log(\bar n_k(s,a)).
\end{align*}
We now apply our optimization lemma (\pref{lem:sa_regret_bound}) with $x_k = w^{\pi_k}(s,a)$, $v_k = 2c\sqrt{\mathcal{V}^*(s,a)\log(M /\delta)}$, and $\epsilon_{k} = \frac{\breve \gap_{k}(s,a)}{4 v_k}$ to bound each $(s,a)$-term in \pref{eq:optapp1} for any $t \in [K]$ as
\begin{align*}
&4\frac{v_t}{\epsilon_t}\log\left(t\land 1 + \frac{1}{\epsilon_t^2}\right) + 4v_t\sqrt{\log\left(K\land 1+\frac{1}{\epsilon_K^2}(K-t)\right)}\\
=&\frac{32c^2\mathcal{V}^*(s,a)\log\left(\frac{M}{\delta}\right)\log\left(t\land 1 + \frac{16\mathcal{V}^*(s,a)\log(M/\delta)}{\breve\gap_t(s,a)^2}\right)}{\breve\gap_t(s,a)}\\
+&8c\sqrt{(K-t)\mathcal{V}^*(s,a)\log\left(\frac{M}{\delta}\right)\log\left(K\land 1 + \frac{16\mathcal{V}^*(s,a)\log(M/\delta)}{\breve\gap_K(s,a)^2}\right)}.
\end{align*}
Let $\mathcal{LOG}(M/\delta,t,\breve\gap_t(s,a)) = \log\left(\frac{M}{\delta}\right)\log\left(t\land 1 + \frac{16\mathcal{V}^*(s,a)\log(M/\delta)}{\breve\gap_t(s,a)^2}\right)$. We have
\begin{align*}
& \sum_{k=\tau(s,a)}^{K} w^{\pi_k}(s,a)
\clip\left[2c\sqrt{\mathcal{V}^*(s,a)\log(M /4\delta)}\sqrt{\frac{\log(\bar n_k(s,a)) }{ \bar n_k(s,a)}} ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{4}\right]\\
\leq &\frac{32c^2\mathcal{V}^*(s,a)\mathcal{LOG}(M/\delta,t,\breve\gap_t(s,a))}{\breve\gap_t(s,a)} + 8c\sqrt{(K-t)\mathcal{LOG}(M/\delta,K,\breve\gap_K(s,a))}.
\end{align*}
Plugging this bound back in \pref{eq:optapp1} gives
\begin{align*}
&\sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{lead}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{4}\right]\\
&\lesssim SAH \log\frac{M}{\delta}\\
&+ \sum_{s,a} \min_{t \in [K_{(s,a)}]}\Bigg\{\frac{\mathcal{V}^*(s,a)\mathcal{LOG}(M/\delta,t,\breve\gap_t(s,a))}{\breve\gap_t(s,a)} + \sqrt{(K_{(s,a)}-t)\mathcal{LOG}(M/\delta,K,\breve\gap_{K_{(s,a)}}(s,a))}
\Bigg\}
\end{align*}
where $\lesssim$ only ignores absolute constant factors.
\paragraph{Bounding the $B^{\mathrm{fut}}$ term}
Consider the second term in \pref{lem:strongeuler_regB_bound} and event $\mathcal{A}^{\mathrm{conc}} \cap \mathcal{A}^{\mathrm{samp}}$. Then by \pref{lem:fntofbarn}
\begin{align*}
&\sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{fut}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{8SA} \right]\\
&\leq SAH^3 H_{\mathrm{samp}} + \sum_{s,a} \sum_{k=\tau(s,a)}^{K} w^{\pi_k}(s,a) f_{s,a}(\bar n_k(s,a))
\end{align*}
where $f_{s,a}$ is
\begin{align*}
f_{s,a}(\bar n_k(s,a)) = \clip\left[2c H^3 \wedge 2cH^3 \left(\sqrt{\frac{S\log(M \bar n_k(s,a)/\delta) }{\bar n_k(s,a)}} + \frac{S\log(M \bar n_k(s,a)/\delta) }{\bar n_k(s,a)}\right)^2 ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{4}\right].
\end{align*}
We now apply Lemma~C.1 by \citet{simchowitz2019non} which gives
\begin{align*}
&\sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{fut}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{8SA} \right]\\
&\leq SAH^3 H_{\mathrm{samp}} + \sum_{s,a} H f_{s,a}(H) + \sum_{s,a}\int_{H}^{\bar n_K(s,a)} f_{s,a}(u) du\\
&\leq SAH^4 c'\log(M/\delta) + \sum_{s,a}\int_{H}^{\bar n_K(s,a)} f_{s,a}(u) du.
\end{align*}
The remaining integral term is bounded with Lemma~B.9 (b) by \citet{simchowitz2019non} with $C' = S, C= H^3$ and $\epsilon = \breve \gap_{\min}(s,a) = \min_{k \in [K_{(s,a)}] \colon \breve\gap_k(s,a) > 0} \breve\gap_k(s,a)$ as follows.
\begin{align*}
&\sum_{k=1}^K \sum_{s,a} w^{\pi_k}(s,a) \clip\left[cB^{\mathrm{fut}}_{k}(s,a) ~ \bigg| ~ \frac{\breve\gap_k(s,a)}{8SA} \right]\\
&\lesssim SAH^4 \log\frac{M}{\delta} + \sum_{s,a}\left(SH^3 \log\frac{M}{\delta}
+ SH^3 \log\frac{MK}{\delta} \min\left\{ \log \frac{MK}{\delta}, \log \frac{MH}{\breve \gap_{\min}(s,a)}\right\}\right)\\
&\lesssim SAH^3 (S \vee H)\log\frac{M}{\delta} + \sum_{s,a} SH^3 \log\frac{MK}{\delta} \min\left\{ \log \frac{MK}{\delta}, \log \frac{MH}{\breve \gap_{\min}(s,a)}\right\}
\end{align*}
\end{proof}
\paragraph{Comparing with the bound in \citet{simchowitz2019non}.} We now proceed to compare our bound directly to the one stated in Corollary B.1~\citep{simchowitz2019non}. We will ignore the factors with only poly-logarithmic dependence on gaps as they are are common between both bounds. We now recall the regret bound presented in Corollary B.1, modulo said factors:
\begin{align*}
\mathfrak{R}(K) \leq O\Bigg(\sum_{(s,a) \in \mathcal{Z}_{sub}} \frac{\alpha H\mathcal{V}^*(s,a)}{\gap(s,a)}\mathcal{LOG}(M/\delta,K,\gap(s,a)) + |\mathcal{Z}_{opt}|\frac{H\mathcal{V}^*}{\gap_{\min}}\mathcal{LOG}(M/\delta,K,\gap_{\min})\Bigg),
\end{align*}
where $\mathcal{V}^* = \max_{(s,a)}\mathcal{V}(s,a)$, $\mathcal{Z}_{opt}$ is the set on which $\gap(s,\pi^*(s)) = 0$, i.e., the set of state-action pairs assigned to $\pi^*$ according to the Bellman optimality condition, and $\mathcal{Z}_{sub}$ is the complement of $\mathcal{Z}_{opt}$. If we take $t = K$ in Theorem~\ref{thm:reg_bound_gen_se}, we have the following upper bound:
\begin{align*}
\mathfrak{R}(K) \leq O\Bigg(\sum_{(s,a) \in \mathcal{Z}_{sub}} \frac{\mathcal{V}^*(s,a)\mathcal{LOG}(M/\delta,K,\gap(s,a))}{\gap(s,a)} + \frac{H\mathcal{V}^*|\mathcal{S}_{opt}|\mathcal{LOG}(M/\delta,K,\gap_{\min})}{\min_{k,s,a}\epsilon_k(s,a)}\Bigg),
\end{align*}
where $\mathcal{S}_{opt}$ is the set of all states for $s\in\mathcal{S}$ for which $\gap(s,\pi^*(s)) = 0$ and there exists at least one state $s'$ with $\kappa(s')<s$ for which $\gap(s',\pi^*(s))>0$. We note that this set is no larger than the set $\mathcal{Z}_{opt}$ and further that even the smallest $\epsilon_k(s,a)$ can still be much larger than $\gap_{\min}$, as it is the conditional average of the gaps. In particular, this leads to an arbitrary improvement in our example in Figure~\ref{fig:summary} and an improvement of $SA$ in the example in Figure~\ref{fig:fail_gap2}.
\subsection{Nearly tight bounds for deterministic transition MDPs}
We recall that for deterministic MDPs, $\epsilon_{k}(s,a) = \frac{V^*(s_1) - V^{\pi_k}(s_1)}{2H},\forall a$ and the definition of the set $\Pi_{s,a}$:
$$
\Pi_{s,a} \equiv \{\pi \in \Pi ~\colon s^\pi_{\kappa(s)} = s ,a^\pi_{\kappa(s)} = a, \exists~ h \leq \kappa(s), \gap(s^\pi_{h},a^\pi_{h})>0\}.
$$
We note that $\mathcal{V}(s,a) \leq 1$ as this is just the variance of the reward at $(s,a)$.
Theorem~\ref{thm:reg_bound_gen_se} immediately yields the following regret bound by taking $t=K$.
\begin{corollary}[Explicit bound from \pref{eq:det_trans_reg}]
\label{cor:det_trans_formal}
Suppose the transition kernel of the MDP consists only of point-masses. Then with probability $1-\delta$, \texttt{StrongEuler}'s regret is bounded as
\begin{align*}
\mathfrak{R}(K) &\leq O\Bigg(\sum_{(s,a) : \Pi_{s,a}\neq \emptyset} \frac{H\mathcal{LOG}\left(M/\delta,K,\overline{\gap}(s,a)\right)}{\return{*}-\return{*}_{s,a}}\\
\\&+ \sum_{s,a} SH^3 \log\frac{MK}{\delta} \min\left\{ \log \frac{MK}{\delta}, \log \frac{MH}{\overline{\gap}(s,a)}\right\}
\\& + SAH^3 (S \vee H)\log\frac{M}{\delta}\Bigg),
\end{align*}
where $\return{*}_{s,a} = \max_{\pi\in\Pi_{s,a}}\return{\pi}$.
\end{corollary}
We now compare the above bound with the one in \citep{simchowitz2019non} again. For simplicity we are going to take $K$ to be the smaller of the two quantities in the logarithm. To compare the bounds, we compare $\sum_{(s,a) : \Pi_{s,a}\neq \emptyset} \frac{H(\log(KM/\delta)))}{\return{*}- \return{*}_{(s,a)}}$ to $\sum_{(s,a) \in \mathcal{Z}_{sub}} \frac{\alpha H \log(KM/\delta)}{\gap(s,a)} + \frac{|\mathcal{Z}_{opt}|H}{\gap_{\min}}$. Recall that $\alpha \in [0,1]$ is defined as the smallest value such that for all $(s,a,s') \in \mathcal{S}\times\mathcal{A}\times\mathcal{S}$ it holds that
\begin{align*}
P(s'|s,a) - P(s'|s,\pi^*(s)) \leq \alpha P(s'|s,a).
\end{align*}
For any deterministic transition MDP with more than one layer and one sub-optimal action it holds that $\alpha = 1$. We will compare $V^*(s_1) - V^{\pi^*_{(s,a)}}(s_1)$ to $\gap(s,a) = Q^*(s,\pi^*(s)) - Q^*(s,a)$. This comparison is easy as by Lemma~\ref{lem:gap_decomp_pi} we can write
\begin{align*}
V^*(s_1) - V^{\pi^*_{(s,a)}}(s_1) = \sum_{(s',a') \in \pi^*_{(s,a)}} w_{\pi^*_{(s,a)}(s',a')} \gap(s',a') = \sum_{(s',a') \in \pi^*_{(s,a)}}\gap(s',a') \geq \gap(s,a).
\end{align*}
Hence, our bound in the worst case matches the one in \citet{simchowitz2019non} and can actually be significantly better. We would further like to remark that we have essentially solved all of the issues presented in the example MDP in Figure~\ref{fig:summary}. In particular we do not pay any gap-dependent factors for states which are only visited by $\pi^*$, we do not pay a $\gap_{\min}$ factor for any state and we never pay any factors for distinguishing between two suboptimal policies. Finally, we compare this bound to the lower bound derived Theorem~\ref{thm:lower_bound_deterministic} only with respect to number of episodes and gaps. Let $\mathcal{S}^*$ be the set of all states in the support of an optimal policy
\begin{align*}
\sum_{(s,a) \in \mathcal{S}\setminus\mathcal{S}^*\times\mathcal{A}} \frac{\log(K)}{H(\return{*} - \return{\pi^*_{(s,a)}}(s_1))} \leq \mathfrak{R}(K) \leq \sum_{(s,a) : \Pi_{s,a}\neq \emptyset}\frac{H\log(K)}{\return{*} - \return{*}_{s,a}}.
\end{align*}
The difference between the two bounds, outside of an extra $H^2$ factor, is in the sets $\mathcal{S}^*$ and the set $\{s,a : \Pi_{s,a}=\emptyset\}$. We note that $\{s,a : \Pi_{s,a}=\emptyset\} \subseteq \mathcal{S}^*$. Unfortunately there are examples in which $\{s,a : \Pi_{s,a}=\emptyset\}$ is $O(1)$ and $\mathcal{S}^* = \Omega(S)$ leading to a discrepancy between the upper and lower bounds of the order $\Omega(S)$. As we show in \pref{thm:det_lower_bound} this discrepancy can not really be avoided by optimistic algorithms.
\subsection{Tighter bounds for unique optimal policy.}
\label{app:unique_opt_pol}
If we further assume that the optimal policy is unique on its support, then we can show \textsc{StrongEuler} will only incur regret on sub-optimal state-action pairs. This matches the information theoretic lower bound up to horizon factors. We begin by showing a different type of upper bound on the expected gaps by the surpluses.
Define the set $\beta_k = range(B)$ where $B$ is the r.v. which is the stopping time with respect to $\pi_k$. For any $\pi^*$, define the set
\begin{align*}
\mathcal{O}_k(\pi^*) = \bigcup_{s_b \in \beta_k} \{(s,a)\in \mathcal{S}\times\mathcal{A} : \PP_{\pi^*}((S_h,A_h) = (s,a)|S_{\kappa(s_b)}= s_b) \geq \PP_{\pi_k}((S_h,A_h) = (s,a)|S_{\kappa(s_b)}= s_b)\}.
\end{align*}
This set has the following intuitive definition -- whenever $\mathcal{A}_B$ occurs we restrict our attention to the MDP with initial state $S_B$. On this restricted MDP, $\mathcal{O}_k$ is the set of state-action pairs which have greater probability to be visited by the optimal $\pi^*$ than by $\pi_k$.
\begin{lemma}
\label{lem:gap_surp_bound}
Assume strong optimism and greedy $\bar V_k,$ i.e., $\bar V_k(s) \geq \max_a \bar Q_k(s,a)$ for all $s \in \mathcal{S}$. Then there exists an optimal $\pi^*$ for which
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^H\gap(S_h,A_h)\right] \leq \EE_{\pi_k}\left[\sum_{h=B}^H \chi(S_h,A_h \not \in \mathcal{O}_k(\pi^*))E_k(S_h,A_h)\right].
\end{align*}
\end{lemma}
\begin{proof}
One can write the optimistic value function for any $s$ and $\pi$ as follows
\begin{align*}
\bar V^{\pi}(s) &= \EE_{\pi}\left[\sum_{h=\kappa(s)}^H E_k(S_h,A_h) + r(S_h,A_h)\big\vert S_{\kappa(s)} = s\right]\\
&= E_k(s,\pi(s)) + r(s,\pi(s)) + \langle P(\cdot | s,\pi(s)), \bar V^{\pi} \rangle.
\end{align*}
By backwards induction on $H$ we show that for any $s$, $\kappa(s) \leq H$ $\bar V^{\pi} \leq \bar V_k$. The base case holds from the fact that on all $s: \kappa(s)=H$, $\bar V_k(s)$ is just the largest optimistic reward over all actions at $s$. For the induction step it holds that
\begin{align*}
\bar V^{\pi}(s) &= E_k(s,\pi(s)) + r(s,\pi(s)) + \langle P(\cdot|s,\pi(s)),\bar V^{\pi} \rangle\\
&\leq E_k(s,\pi(s)) + r(s,\pi(s)) + \langle P(\cdot|s,\pi(s)),\bar V_k \rangle\\
& = \bar Q_k(s,\pi(s)) \leq \bar V_k(s),
\end{align*}
where the first inequality holds from the induction hypothesis and the second inequality holds by definition of the value function.
We now have
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^H \gap(S_h,A_h)\right] &= \EE_{\pi_k}\left[V^*(S_B) - V_k(S_B)\right]\\
&\leq \EE_{\pi_k}\left[\bar V_k(S_B) - V_k(S_B)\right] - \EE_{\pi_k}\left[\bar V^*(S_B) - V^*(S_B)\right].
\end{align*}
Let us focus on the term $\EE_{\pi_k}\left[\bar V^*(S_B) - V^*(S_B)\right]$
\begin{align*}
\EE_{\pi_k}\left[\bar V^*(S_B) - V^*(S_B)\right] &= \EE_{\pi_k}\left[\EE_{\pi_k}\left[\bar V^*(S_B) - V^*(S_B)|S_B\right]\right]\\
&=\EE_{\pi_k}\left[ \sum_s \frac{\bar V^*(s) - V^*(s)}{\PP_{\pi_k}(S_B=s)}\chi(S_B=s)\right]\\
&=\EE_{\pi_k}\left[ \sum_s \frac{ \EE_{\pi^*}\left[ \sum_{h=\kappa(s)}^H E_k(S_h,A_h)| S_{\kappa(s)}=s\right] }{\PP_{\pi_k}(S_B=s)}\chi(S_B=s)\right].
\end{align*}
We can similarly expand the term $\EE_{\pi_k}\left[\bar V_k(S_B) - V_k(S_B)\right]$. By the definition of $\mathcal{O}_k(\pi^*)$ it holds that for any $h\geq\kappa(s)$
\begin{align*}
\EE_{\pi_k}\left[E_k(S_h,A_h)|S_{\kappa(s)} = s\right] &- \EE_{\pi^*}\left[E_k(S_h,A_h)|S_{\kappa(s)} = s\right]\\
&\leq \EE_{\pi_k}\left[\chi(S_h,A_h \not \in \mathcal{O}_k(\pi^*))E_k(S_h,A_h)|S_{\kappa(s)} = s\right].
\end{align*}
This implies
\begin{align*}
\EE_{\pi_k}\left[\bar V^*(S_B) - V^*(S_B)\right] &\leq \EE_{\pi_k}\left[ \sum_s \frac{ \EE_{\pi_k}\left[ \sum_{h=\kappa(s)}^H \chi(S_h,A_h \not \in \mathcal{O}_k(\pi^*))E_k(S_h,A_h)| S_{\kappa(s)}=s\right] }{\PP_{\pi_k}(S_B=s)}\chi(S_B=s)\right]\\
&= \EE_{\pi_k}\left[\sum_{h=B}^H \chi(S_h,A_h \not \in \mathcal{O}_k(\pi^*))E_k(S_h,A_h)\right].
\end{align*}
\end{proof}
We next show a version of Lemma~\ref{lem:Vdd_lb1} which takes into account the set $\mathcal{O}_k(\pi^*)$.
\begin{lemma}
\label{lem:clipped_val_lower}
With the same assumptions as in Lemma~\ref{lem:gap_surp_bound}, there exists an optimal $\pi^*$ for which
\begin{align*}
\ddot V_k(s_1) - V_k(s_1) \geq \EE_{\pi_k}\left[\sum_{h=B}^H \gap(S_h,A_h) - \sum_{h=B}^H\chi(S_h,A_h \not \in \mathcal{O}_k(\pi^*))\epsilon_k(S_h,A_h)\right],
\end{align*}
where $\epsilon_k$ is arbitrary.
\end{lemma}
\begin{proof}
Since $\ddot E_k$ is non-negative on all state-action pairs we have
\begin{align*}
\ddot V_k(s_1) - V^{\pi_k}(s_1) &= \EE_{\pi_k}\left[\sum_{h=1}^{H} \ddot E_k(S_h,A_h)\right]
\geq \EE_{\pi_k}\left[\sum_{h=B}^{H} \ddot E_k(S_h,A_h)\right]\\
& \geq \EE_{\pi_k}\left[\sum_{h=B}^{H}
\indicator{(S_h,A_h) \not\in \mathcal{O}_k}
\ddot E_k(S_h,A_h)\right]\\
&\geq \EE_{\pi_k}\left[\sum_{h=B}^{H}\chi((S_h,A_h) \not\in \mathcal{O}_k) E_k(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=B}^{H}\chi((S_h,A_h) \not\in \mathcal{O}_k) \epsilon_k(S_h,A_h)\right]\\
&\geq \EE_{\pi_k}\left[\sum_{h=B}^H \gap(S_h,A_H)\right] - \EE_{\pi_k}\left[\sum_{h=B}^{H}\chi((S_h,A_h) \not\in \mathcal{O}_k) \epsilon_k(S_h,A_h)\right],
\end{align*}
where the second to last inequality follows from the definition of $\ddot E_k$ and the last inequality follows from Lemma~\ref{lem:gap_surp_bound}.
\end{proof}
Next, we define $\bar\epsilon_k$ in the following way. Let
\begin{align}
\bar\epsilon_k(s,a) &\equiv
\begin{cases}
\epsilon_k(s,a) &\textrm{if } (s,a) \not\in \mathcal{O}_k(\pi^*)\\
\infty & \textrm{otherwise},
\end{cases}
\end{align}
where $\epsilon_k$ is the clipping function defined in \pref{eqn:new_avgclip_clean}.
Lemma~\ref{lem:clipped_val_lower} now implies that
\begin{align*}
\ddot V_k(s_1) - V_k(s_1) \geq \EE_{\pi_k}\left[\sum_{h=B}^H \gap(S_h,A_h) - \sum_{h=B}^H \bar \epsilon_k(S_h,A_h)\right].
\end{align*}
This is sufficient to argue Lemma~\ref{lem:strongeuler_regB_bound} with $\breve\gap_k(s,a) = \frac{\gap(s,a)}{4}\lor \bar\epsilon_k(s,a)$ and hence arrive at a version of Corollary~\ref{cor:det_trans_formal} which uses $\bar\epsilon_k$ as the clipping thresholds. Let us now argue that $\bar\epsilon_k(s,a) = \infty$ for all $(s,a)\in\pi^*$ whenever $\pi^*$ is the unique optimal policy for the deterministic MDP. To do so consider $(s,a)\in\pi^*$ and $\pi_k \neq \pi^*$. Since the MDP is deterministic, $\beta_k$ is a singleton and is the the first state $s_b$ at which $\pi_k$ differs from $\pi^*$. We now observe that if $\kappa(s) < \kappa(s_b)$, this implies $\epsilon_k(s,a) = \infty$ as $\mathcal{B}(s,a)$ does not occur. Further, the conditional probabilities $\PP_{\pi^*}((S_h,A_h) = (s,a)|S_{\kappa(s_b)}= s_b)$ and $\PP_{\pi_k}((S_h,A_h) = (s,a)|S_{\kappa(s_b)}= s_b)$ are both equal to $1$ if $\kappa(s) > \kappa(s_b)$ and so $(s,a) \in \mathcal{O}_k(\pi^*)$ which implies $\bar\epsilon_k(s,a) = \infty$. Thus we can clip all gaps at $(s,a) \in \pi^*$ to infinity and they will never appear in the regret bound. With the notation from Corollary~\ref{cor:det_trans_formal} we have the following tighter bound.
\begin{corollary}
\label{cor:det_upper_tight}
Suppose the transition kernel of the MDP consists only of point-masses and there exists a unique optimal $\pi^*$. Then with probability $1-\delta$, \texttt{StrongEuler}'s regret is bounded as
\begin{align*}
\mathfrak{R}(K) &\leq O\Bigg(\sum_{(s,a) \not\in \pi^*} \frac{\mathcal{LOG}\left(M/\delta,K,\overline{\gap}(s,a)\right)}{\overline{\gap}(s,a)}\\
\\&+ \sum_{(s,a)\not\in\pi^*} SH^3 \log\frac{MK}{\delta} \min\left\{ \log \frac{MK}{\delta}, \log \frac{MH}{\overline{\gap}(s,a)}\right\}
\\& + SAH^3 (S \vee H)\log\frac{M}{\delta}\Bigg).
\end{align*}
\end{corollary}
Comparing terms which depend polynomially on $1/\overline{\gap}$ to the information theoretic lower bound in \pref{thm:lower_bound_deterministic} we observe only a multiplicative difference of $H^2$.
\subsection{Alternative to integration lemmas}
The following lemma is an alternative to the integration lemmas when bounding the sum of the clipped surpluses and in some cases allows us to save additional factors of $H$.
\begin{restatable}[]{lemma}{optimlemma}
Consider the following optimization problem
\begin{equation}
\begin{aligned}
\label{eq:opt_prob_regret_upper}
\maximize{x_1,\ldots,x_K}{\sum_{k=1}^K \frac{v_k x_k\sqrt{\log(\sum_{j=1}^k x_j)}}{\sqrt{\sum_{j=1}^k x_j}}}
{
1 \leq x_1,\quad
0\leq x_k\leq 1, \quad
\frac{\sqrt{\log(\sum_{j=1}^k x_j)}}{\sqrt{\sum_{j=1}^k x_j}} \geq \epsilon_k\qquad\forall~ k\in[K]},
\end{aligned}
\end{equation}
with $(v_i)_{i \in [K]} \in \RR_{+}^K$ and $(\epsilon_i)_{i \in [K]} \in \RR_+^{K}$. Then the optimal value of Problem~\ref{eq:opt_prob_regret_upper} is bounded for any $t \in [K]$ as
\begin{align}
4\frac{\bar v_{t}}{\epsilon_t}\log\left(t \land 1+\frac{1}{\epsilon_{t}^2}\right) + 4v^*_t\sqrt{\log\left(K\land 1+ \frac{1}{\epsilon_{K}^2}\right)(K-t)},
\label{eq:opt_prob_bound}
\end{align}
where $\bar v_t = \max_{k \in [t]} v_k$ and $v^*_t = \max_{K\geq k\geq t} v_k$.
\label{lem:sa_regret_bound}
\end{restatable}
\begin{proof}Denote by $X_k = \sum_{t=1}^k x_t$ the cumulative sum of $x_t$.
The proof consists of splitting the objective of \prettyref{eq:opt_prob_regret_upper} into
two terms:
\begin{align}
\sum_{k=1}^t \frac{v_k x_k\sqrt{\log( X_k)}}{\sqrt{X_k}} + \sum_{k=t+1}^K \frac{v_k x_k\sqrt{\log(X_k)}}{\sqrt{X_k}}
\label{eq:obj_split}
\end{align}
and bounding each by the corresponding one in \prettyref{eq:opt_prob_bound} respectively.
Before doing so, we derive the following bound on the sum of $\frac{x_k}{\sqrt{X_k}}$ terms:
\begin{align}
\label{eq:sqrt_X_bound}
\sum_{k=m+1}^M\frac{x_k}{\sqrt{X_k}}=\sum_{k=m+1}^M\frac{X_k-X_{k-1}}{\sqrt{X_k}}\leq \int_{X_m}^{X_M}\frac{1}{\sqrt{x}}\,dx = 2(\sqrt{X_M}-\sqrt{X_m})\,,
\end{align}
where the inequality is due to $X_k$ being non-decreasing.
Consider now each term in the objective in \prettyref{eq:obj_split} separately.
\paragraph{Summands up to $t$:}
Since $X_k$ is non-decreasing, we can bound
\begin{align*}
\sum_{k=1}^t \frac{v_k x_k \sqrt{\log(X_k)}}{\sqrt{X_k}}
\leq
\bar v_t \sqrt{\log(X_t)}
\sum_{k=1}^t \frac{x_k}{\sqrt{X_k}}
& \overset{(i)}{\leq} 2\bar v_t \sqrt{\log(X_t)}\sqrt{ X_t}\\
& \overset{(ii)}{\leq}
2\frac{\bar v_t}{\epsilon_t} \log(X_t),
\end{align*}
where $(i)$ follows from \prettyref{eq:sqrt_X_bound} using the convention $X_0=0$ and $(ii)$ from the optimization constraint $\sqrt{\log(X_t)} \geq \epsilon_t \sqrt{X_t}$. It remains to bound $\log(X_t)$ by $2\log\left(t \land 1+\frac{1}{\epsilon_{t}^2}\right)$. Since all increments $x_j$ are at most $1$, the bound $\log(X_t) \leq \log(t)$ holds.
We claim the following:
\begin{claim}
\label{claim:log_ineq}
For any $x$ s.t. $\log(x) \leq \log(\log(x)/a)$ it holds that $\log(x) \leq 2\log(1+1/a)$.
\end{claim}
\begin{proof}
First, we note that if $0<x\leq e$, then $\log(\log(x)) < 0$ and thus the assumption of the claim implies $\log(x) \leq \log(1/a)$. Next, assume that $x > e$. Then we have $\frac{\log(\log(x))}{\log(x)} \leq 1/e$, which together with the assumption of the claim implies $\log(x) \leq 1/e\log(x) + \log(1/a)$ or equivalently $\log(x) \leq \frac{e}{e-1}\log(1/a)$. Noting that $e/(e-1) \leq 2$ completes the proof.
\end{proof}
The constraints of the problem enforce $\sqrt{X_k} \leq \frac{\sqrt{\log(X_k)}}{\epsilon_k}$,
which implies after squaring and taking the $\log$:
$\log(X_k) \leq \log(\log(X_k)/\epsilon_k^2)$. Thus, using Claim~\ref{claim:log_ineq} yields:
\begin{align}
\log(X_k) \leq 2\log(k\land 1+1/\epsilon_k^2).
\label{eq:logX_bound}
\end{align}
\paragraph{Summands larger than $t$:}
Let $v^*_t = \max_{k \colon t < k \leq K} v_k$. For this term, we have
\begin{align*}
\sum_{k=t+1}^K \frac{v_k x_k \sqrt{\log(X_k)}}{\sqrt{X_k}}
&\overset{\prettyref{eq:logX_bound}}{\leq} 2v^*_t\sqrt{\log(K\land 1+1/\epsilon^2_K)}\sum_{k=t+1}^K \frac{x_k}{\sqrt{X_k}}\\
&\overset{\prettyref{eq:sqrt_X_bound}}{\leq} 4v^*_t\sqrt{\log(K\land 1+1/\epsilon^2_K)}(\sqrt{X_K}-\sqrt{X_t})\\
&\leq 4v^*_t\sqrt{\log(K\land 1+1/\epsilon^2_K)}(\sqrt{X_K-X_t})\\
&\leq 4v^*_t\sqrt{\log(K\land 1+1/\epsilon^2_K)}(\sqrt{K-t}),
\end{align*}
where we first bounded $\log(X_k) \leq \log(X_K)$, because $X_k$ is non-decreasing, and used the upper bound on $\log(X_K)$.
Then we applied \prettyref{eq:sqrt_X_bound} and finally used $0\leq x_k\leq 1$.
\end{proof}
\section{Model-based optimistic algorithms for tabular RL}
\label{app:opt_algs}
This section is a general discussion of optimistic algorithms for the tabular setting. Our regret upper bounds can be extended to other model based optimistic algorithms or in general any optimistic algorithm for which we can show a meaningful bound on the surpluses in terms of the number of times a state-action pair has been visited throughout the $K$ episodes.
\begin{algorithm}[t]
\caption{Generic Model-Based Optimistic Algorithm for Tabular RL}
\label{alg:model_based_alg}
\begin{algorithmic}[1]
\REQUIRE{Number of episodes $K$, horizon $H$, number of states $S$, number of actions $A$, probability of failure $\delta$.}
\ENSURE{A sequence of policies $(\pi_k)_{k=1}^K$ with low regret.}
\STATE Initialize empirical transition kernel $\hat P \in [0,1]^{S\times A\times S}$, empirical reward kernel $\hat r \in [0,1]^{S\times A}$, bonuses $b \in [0,1]^{S\times A}$.
\FOR{$k\in [K]$}
\STATE $h=H$, $Q_k(s_{H+1},a_{H+1}) = 0,\forall (s,a) \in \mathcal{S}\times\mathcal{A}$.
\WHILE{$h>0$}
\STATE $Q_k(s,a) = \hat r(s,a) + \langle \hat P(\cdot|s,a), V_k \rangle + b(s,a)$.
\STATE $\pi_k(s) := \argmax_{a} Q_k(s,a)$.
\STATE $h-=1$
\ENDWHILE
\STATE Play $\pi_k$, collect observations from transition kernel $P$ and reward kernel $r$ and update $\hat P$, $\hat r$, $b$.
\ENDFOR
\end{algorithmic}
\end{algorithm}
Pseudo-code for a generic algorithm can be found in Algorithm~\ref{alg:model_based_alg}. The algorithm begins by initializing an empirical transition kernel $\hat P \in [0,1]^{S\times A\times S}$, empirical reward kernel $\hat r \in [0,1]^{S\times A}$, and bonuses $b \in [0,1]^{S\times A}$. If we let $n_k(s,a)$ be the number of times we have observed state-action pair $(s,a)$ up to episode $k$ and $n_k(s',s,a)$ the number of times we have observed state $s'$ after visiting $(s,a)$ then one standard way to define the empirical kernels at episode $k$ are as follows:
\begin{equation}
\hat r(s,a) = \frac{1}{n_k(s,a)}\sum_{j=1}^k R_j(s,a), \qquad
\hat P(s'|s,a) =
\begin{cases*}
\frac{n_k(s',s,a)}{n_k(s,a)} &\text{if} $n_k(s,a)>0$ \\
0 &otherwise
\end{cases*}\\
\end{equation}
where $R_j(s,a)$ is a sample from $r(s,a)$ at episode $j$ if $(s,a)$ was visited and $0$ otherwise.
At every episode the generic algorithm constructs an policy $\pi_k$ using the empirical model together with bonus terms $b(s,a),\forall (s,a)\in\mathcal{S}\times\mathcal{A}.$ Bonuses are constructed by using concentration of measure results relating $\hat r(s,a)$ to $r(s,a)$ and $\hat P(\cdot|s,a)$ to $P(\cdot|s,a)$. These bonuses usually scale inversely with the empirical visitations $n_k(s,a), \forall (s,a) \in \mathcal{S}\times \mathcal{A}$, as $O(1/\sqrt{n_k(s,a)})$. Further, depending on the type of concentration of measure result, the bonuses could either have a direct dependence on $K,H,S,A,\delta$ (following from Azuma-Hoeffding style concentration bounds) or replace $H$ with the empirical estimator (following Freedman style concentration bounds).
The bonus terms ensure that optimism is satisfied for $\pi_k$, that is $Q_k(s,a) \geq Q^{\pi_k}(s,a)$ for all $(s,a)\in \mathcal{S}\times\mathcal{A}$ and all episodes $k \in [K]$ with probability at least $1-\delta$. Algorithms such as \textsc{UCBVI}~\citep{azar2017minimax}, \textsc{Euler}~\citep{zanette2019tighter} and \textsc{StrongEuler}~\citep{simchowitz2019non} are all versions of Algorithm~\ref{alg:model_based_alg} with different instantiations of the bonus terms.
The greedy choice of $\pi_k$ together with optimism also ensures that $V_k(s) \geq V^*(s)$. This has been key in prior work as it is what allows to bound the instantaneous regret by the sum of surpluses and ultimately relate the regret upper bound back to the bonus terms and the number of visits of each state-action pair respectively.
Our regret upper bounds are also based on this decomposition and as such are not really tied to the \textsc{StrongEuler} algorithm but would work with any model-based optimistic algorithm for the tabular setting. The main novelty in this work is a way to control the surpluses by clipping them to a gap-like quantity which better captures the sub-optimality of $\pi_k$ compared to $\pi^*$. We remark that our analysis can be extended to any algorithm which follows Algorithm~\ref{alg:model_based_alg} so as long as we can control the bonus terms sufficiently well.
\section{Related work}
We now discuss related work carefully. Instance dependent regret lower bounds for the MAB were first introduced in \citet{lai1985asymptotically}. Later \citet{graves1997asymptotically} extend such instance dependent lower bounds to the setting of controlled Markov chains, while assuming infinite horizon and certain properties of the stationary distribution of each policy. Building on their work, more recently \citet{combes2017minimal} establish instance dependent lower bounds for the Structured Stochastic Bandit problem. Very recently, in the stochastic MAB, \citet{garivier2019explore} generalize and simplify the techniques of \citet{lai1985asymptotically} to completely characterize the behavior of uniformly good algorithms. The work of \citet{ok2018exploration} builds on these ideas to provide an instance dependent lower bound for infinite horizon MDPs, again under assumptions of how the stationary distributions of each policy will behave and irreducibility of the Markov chain. The idea behind deriving the above bounds is to use the uniform goodness of the studied algorithm to argue that the algorithm must select a certain policy or action at least a fixed number of times. This number is governed by a change of environment under which said policy/action is now the best overall. The reasoning now is that unless the algorithm is able to distinguish between these two environments it will have to incur linear regret asymptotically. Since the algorithm is uniformly good this can not happen.
For infinite horizon MDPs with additional assumptions the works of \citet{auer2007logarithmic,tewari2008optimistic,auer2009near,filippi2010optimism,ok2018exploration} establish logarithmic in horizon regret bounds of the form $O(D^2S^2A\log(T)/\delta)$, where $\delta$ is a gap-like quantity and $D$ is a diameter measure. We now discuss the works of \citep{tewari2008optimistic,ok2018exploration}, which should give more intuition about how the infinite horizon setting differs from our setting. Both works consider the non-episodic problem and therefore make some assumptions about the MDP $\mathcal{M}$. The main assumption, which allows for computationally tractable algorithms is that of irreducibility. Formally both works require that under any policy the induced Markov chain is irreducible. Intuitively, the notion of irreducibility allows for coming up with exploration strategies, which are close to min-max optimal and are easy to compute. In \citep{ok2018exploration} this is done by considering the same semi-infinite LP~\ref{eq:opt_prob} as in our work. Unlike our work, however, assuming that the Markov chain induced by the optimal policy $\pi^*$ is irreducible allows for a nice characterization of the set $\Lambda(\theta)$ of "confusing" environments. In particular the authors manage to show that at every state $s$ it is enough to consider the change of environment which makes the reward of any action $a : (s,a) \not\in \pi^*$ equal to the reward of $a' : (s,a') \in \pi^*$. Because of the irreducability assumption we know that the support of $P(\cdot|s,a)$ is the same as the support of $P(\cdot |s,a')$ and this implies that the above change of environment makes the policy $\pi$ which plays $(s,a)$ and then coincides with $\pi^*$ optimal. Some more work shows that considering only such changes of environment is sufficient for an equivalent formulation to the LP\ref{eq:opt_prob}. Since this is an LP with at most $S\times A$ constraints it is solvable in polynomial time and hence a version of the algorithm in \citep{combes2017minimal} results in asymptotic min-max rates for the problem. The exploration in \citep{tewari2008optimistic} is also based on a similar LP, however, slightly more sophisticated.
Very recently there has been a renewed interest in proposing instance dependent regret bounds for finite horizon tabular MDPs~\citep{simchowitz2019non,lykouris2019corruption,jin2020simultaneously}. The works of \citep{simchowitz2019non,lykouris2019corruption} are based on the OFU principle and the proposed regret bounds scale as $O(\sum_{(s,a) \not\in \pi^*} H\log(T)/\gap(s,a) + SH\log(T)/\gap_{\min})$, disregarding variance terms and terms depending only poli-logarithmically on the gaps. The setting in \citep{lykouris2019corruption} also considers adversarial corruptions to the MDP, unknown to the algorithm, and their bound scales with the amount of corruption. \citet{jin2020simultaneously} derive similar upper bounds, however, the authors assume a known transition kernel and take the approach of modelling the problem as an instance of Online Linear Optimization, through using occupancy measures~\citep{zimin2013online}. For the problem of $Q$-learning, \citet{yang2020q,du2020agnostic}, also propose algorithms with regret scaling as $O(SAH^6\log(T)/\gap_{\min})$.
All of these bounds scale at least as $\Omega(SH\log(T)/\gap_{\min})$.
\citet{simchowitz2019non} show an MDP instance on which no optimistic algorithm can hope to do better.
\section{Proofs and extended discussion for regret lower-bounds}
\label{app:lower_bounds}
Let $N_{\psi,\pi}(k)$ be the random variable denoting the number of times policy $\pi$ has been chosen by the strategy $\psi$. Let $N_{\psi, (s,a)}(k)$ be the number of times the state-action pair has been visited up to time $k$ by the strategy $\psi$.
\subsection{Lower bound as an optimization problem}
\label{sec:opt_lower_bound}
We begin by formulating an LP characterizing the minimum regret incurred by any uniformly good algorithm $\psi.$
\begin{theorem}
\label{thm:lower_bound_gen1}
Let $\psi$ be a uniformly good RL algorithm for $\Theta$, that is, for all problem instances $\theta \in \Theta$ and exponents $\alpha > 0$, the regret of $\psi$ is bounded as $\EE[\mathfrak{R}_\theta(K)] \leq o(K^{\alpha})$. Then, for any $\theta \in \Theta$, the regret of $\psi$ satisfies
\begin{align*}
\liminf_{K \to \infty} \frac{\EE[\mathfrak{R}_\theta(K)]}{\log{K}} \geq C(\theta),
\end{align*}
where $C(\theta)$ is the optimal value of the following optimization problem
\begin{equation}
\label{eq:opt_prob1}
\begin{aligned}
\minimize{\eta(\pi)\geq 0}{\sum_{\pi \in \Pi} \eta(\pi)\left(\return{*}_{\theta} - \return{\pi}_{\theta}\right)}
{
\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1 \qquad \textrm{for all } \,\,\lambda \in \Lambda(\theta)
},
\end{aligned}
\end{equation}
where $\Lambda'(\theta) = \{ \lambda \in \Theta \colon \Pi^{*}_{\lambda} \cap \Pi^{*}_{\theta} = \varnothing, KL(\PP_\theta^{\pi^*_\theta}, \PP_\lambda^{\pi^*_\theta}) = 0\}$ are all environments that share no optimal policy with $\theta$ and do not change the rewards or transition kernel on $\pi^*$.
\end{theorem}
\begin{proof}
We can write the expected regret as $\EE[\mathfrak{R}_\theta(K)] = \sum_{\pi \in \Pi} \mathbb{E}_\theta[N_{\psi,\pi}(K)](\return{*}_\theta - \return{\pi}_\theta)$. We will show that $\eta(\pi) = \mathbb{E}_\theta[N_{\psi,\pi}(K)]/\log{K}$ is feasible for the optimization problem in \pref{eq:opt_prob}. This is sufficient to prove the theorem. To do so we follow the techniques of~\cite{garivier2019explore}.
With slight abuse of notation, let $\mathbb{P}_\theta^{I_k}$ be the law of all trajectories up to episode $k$, where $I_k$ is the history up to and including time $k$. Let $Y_k$ be the random variable which is the value function of the policy, $\psi(I_k)$, selected at episode $k$. We have
\begin{equation}
\label{eq:kl-expand}
\begin{aligned}
KL(\mathbb{P}_\theta^{I_{k+1}},\mathbb{P}_\lambda^{I_{k+1}}) &= KL(\mathbb{P}_\theta^{Y_{k+1},I_{k}},\mathbb{P}_\lambda^{Y_{k+1},I_{k}})\\
&=
KL(\mathbb{P}_\theta^{I_{k}},\mathbb{P}_\lambda^{I_{k}})
+\mathbb{E}\left[ \mathbb{E}_{\PP^{\psi(I_k)}_\theta}\left[\log{\frac{\PP^{\psi(I_k)}_\theta(Y_{k+1})}{\PP^{\psi(I_k)}_\lambda(Y_{k+1})}} ~\bigg|~I_k\right]\right]\\
&=KL(\mathbb{P}_\theta^{I_{k}},\mathbb{P}_\lambda^{I_{k}})
+ \mathbb{E}\left[\sum_{\pi \in \Pi}\chi(\psi(I_k) = \pi)KL(\PP_\theta^{\pi},\PP_\lambda^\pi)\right].
\end{aligned}
\end{equation}
Iterating the argument we arrive at $\sum_{\pi\in\Pi}\mathbb{E}_\theta[N_{\psi,\pi}(K)]
KL(\PP_\theta^\pi,\PP_\lambda^\pi) = KL(\PP_{\theta}^{I_K},\PP_{\lambda}^{I_K})$ where $\EE_\theta$ denotes expectation in problem instance $\theta$.
Next one shows that for any measurable $Z \in [0,1]$, with respect to the natural sigma-algebra induced by $I_K$, it holds that $KL(\PP_{\theta}^{I_K},\PP_{\lambda}^{I_K}) \geq kl(\mathbb{E}_\theta[Z],\mathbb{E}_\lambda[Z])$ where $kl(p,q) = p\log{p/q} + (1-p)\log{(1-p)/(1-q)}$ denotes the KL-divergence between two Bernoulli random variables $p$ and $q$. This follows directly from Lemma~1 by \citet{garivier2019explore}.
Finally we choose $Z = N_{\psi,\Pi^*_\lambda}(K) / K$ as the fraction of episodes where an optimal policy for $\lambda$ was played (here we use the short-hand notation $N_{\psi,\Pi^*_\lambda}(K) = \sum_{\pi \in \Pi^*_\lambda} N_{\psi,\pi}(K)$). Evaluating the $kl$-term we have
\begin{align*}
&kl\left(\frac{\mathbb{E}_\theta[ N_{\psi,\Pi^*_\lambda}(K)]}{K},\frac{\mathbb{E}_\lambda[ N_{\psi,\Pi^*_\lambda}(K)]}{K}\right) \geq \left(1 - \frac{\mathbb{E}_\theta[ N_{\psi,\Pi^*_\lambda}(K)]}{K}\right)\log{\frac{K}{K - \mathbb{E}_\lambda[N_{\psi,\Pi^*_\lambda}(K)]}}
-\log{2}.
\end{align*}
Since $\psi$ is a uniformly good algorithm it follows that for any $\alpha>0$, $K - \mathbb{E}_\lambda[N_{\psi,\Pi^*_\lambda}(K)] = o(K^\alpha)$. By assuming that $\Pi^*_\theta \cap \Pi^*_\lambda = \varnothing$, we get $\mathbb{E}_\theta[N_{\psi,\Pi^*_\lambda}(K)] = o(K)$. This implies that for $K$ sufficiently large and all $1\geq\alpha>0$
\begin{align*}
kl\left(\frac{\mathbb{E}_\theta[N_{\psi,\Pi^*_\lambda}(K)]}{K},\frac{\mathbb{E}_\lambda[N_{\psi,\Pi^*_\lambda}(K)]}{K}\right) \geq \log{K} - \log{K^\alpha} = (1-\alpha)\log{K} \xrightarrow{\alpha \rightarrow 0} \log{K}.
\end{align*}
\end{proof}
The set $\Lambda'(\theta)$ is uncountably infinite for any reasonable $\Theta$ we consider. What is worse the constraints of LP~\ref{eq:opt_prob} will not form a closed set and thus the value of the optimization problem will actually be obtained on the boundary of the constraints. To deal with this issue it is possible to show the following.
\generallb*
\begin{proof}
For the rest of this proof we identify $\Lambda'(\theta) = \{\lambda \in \Theta : \Pi^*_\lambda \cap \Pi^*_\theta = \emptyset, KL(\PP^{\pi^*_\theta}_\theta,\PP^{\pi^*_\theta}_\lambda) = 0,\forall \pi^*_\theta \in \Pi^*_\theta\}$ as the set from Theorem~\ref{thm:lower_bound_gen1} and $\tilde \Lambda(\theta) = \{\lambda \in \Theta : \return{\pi^*_\lambda}_\lambda \geq \return{\pi^*_\theta}_\theta, \pi^*_\lambda \not\in \Pi^*_\theta, KL(\PP^{\pi^*_\theta}_\theta,\PP^{\pi^*_\theta}_\lambda) = 0\}$. From the proof of Theorem~\ref{thm:lower_bound_gen1} it is clear that we can rewrite $\Lambda'(\theta)$ as the union $\bigcup_{\pi\in\Pi} \Lambda_{\pi}(\theta)$, where $\Lambda_{\pi}(\theta) = \{\lambda \in \Theta: KL(\PP_\theta^{\pi^*_\theta},\PP_\lambda^{\pi^*_\theta}) = 0, \return{\pi^*_\lambda} > \return{\pi^*_\theta}_\theta, \pi^*_\lambda = \pi\}$ is the set of all environments which make $\pi$ the optimal policy. This implies that we can equivalently write LP~\ref{eq:opt_prob} as
\begin{equation}
\label{eq:opt_prob_equiv}
\begin{aligned}
\minimize{\eta(\pi)\geq 0}{\sum_{\pi \in \Pi} \eta(\pi)\left(\return{*}_{\theta} - \return{\pi}_{\theta}\right)}
{
\inf_{\lambda \in \Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1 \qquad \textrm{for all } \,\,\pi' \in \Pi
}.
\end{aligned}
\end{equation}
The above formulation now minimizes a linear function over a finite intersection of sets, however, these sets are still slightly inconvenient to work with. We are now going to try to make these sets more amenable to the proof techniques we would like to use for deriving specific lower bounds. We begin by noting that $\Lambda_{\pi}(\theta)$ is bounded in the following sense. We identify each $\lambda$ with a vector in $[0,1]^{S^2A}\times[0,1]^{SA}$ where the first $S^2A$ coordinates are transition probabilities and the last $SA$ coordinates are the expected rewards. From now on we work with the natural topology on $[0,1]^{S^2A}\times[0,1]^{SA}$, induced by the $\ell_1$ norm. Further, we claim that we can assume that $KL(\PP_\theta^\pi,\PP_\lambda^\pi)$ is a continuous function over $\Lambda_{\pi'}(\theta)$. The only points of discontinuity are at $\lambda$ for which the support of the transition kernel induced by $\lambda$ does not match the support of the transition kernel induced by $\theta$. At such points the $KL(\PP_\theta^\pi,\PP_\lambda^\pi) = \infty$. This implies that such $\lambda$ does not achieve the infimum in the set of constraints so we can just restrict $\Lambda_{\pi'}(\theta)$ to contain only $\lambda$ for which $KL(\PP_\theta^\pi,\PP_\lambda^\pi) < \infty$. With this restriction in hand the KL-divergence is continuous in $\lambda$.
Fix a $\pi'$ and consider the set $\{ \eta:\inf_{\lambda \in \Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1\}$ corresponding to one of the constraints in LP~\ref{eq:opt_prob_equiv}. Denote $\tilde \Lambda_{\pi'}(\theta) = \{\lambda \in \Theta: KL(\PP_\theta^{\pi^*_\theta},\PP_\lambda^{\pi^*_\theta}) = 0, \return{\pi^*_\lambda}_\lambda \geq \return{\pi^*_\theta}_\theta, \pi^*_\lambda \not\in \Pi^*_\theta, \pi^*_\lambda = \pi'\}$. $\tilde \Lambda_{\pi'}(\theta)$ is closed as $KL(\PP_\theta^{\pi^*_\theta},\PP_\lambda^{\pi^*_\theta})$ and $\return{\pi^*_\lambda}_\lambda - \return{\pi^*_\theta}_\theta$ are both continuous in $\lambda$. To see the statement for $\return{\pi^*_\lambda}_\lambda$, notice that this is the maximum over the continuous functions $\return{\pi}_\lambda$ over $\pi\in\Pi$. Take any $\eta \in \Lambda_{\pi'}(\theta)$ and let $\{\lambda_{j}\}_{j=1}^\infty, \lambda_j \in \Lambda_{\pi'}(\theta)$ be a sequence of environments such that $\sum_{\pi\in \Pi}\eta(\pi)KL(\PP_\theta^{\pi}, \PP_{\lambda_j}^{\pi}) \geq 1+ 2^{-j}$. If there is no convergent subsequence of $\{\lambda_{j}\}_{j=1}^\infty$ in $\Lambda_{\pi'}(\theta)$ we claim it is because of the constraint $\return{\pi^*_\lambda}_\lambda > \return{\pi^*_\theta}_\theta$. Take the limit $\lambda$ of any convergent subsequence of $\{\lambda_{j}\}_{j=1}^\infty$ in the closure of $\Lambda_{\pi'}(\theta)$. Then by continuity of the divergence we have $0=\lim_{j\rightarrow\infty}KL(\PP_\theta^{\pi^*_\theta},\PP_{\lambda_j}^{\pi^*_\theta}) = KL(\PP_\theta^{\pi^*_\theta},\PP_\lambda^{\pi^*_\theta})$, thus it must be the case that $\return{\pi^*_\lambda}_\lambda \leq \return{\pi^*_\theta}_\theta$. This shows that $\tilde \Lambda_{\pi'}(\theta)$ is a subset of the closure of $\Lambda_{\pi'}(\theta)$ which implies it is the closure of $\Lambda_{\pi'}(\theta)$, i.e., $\bar \Lambda_{\pi'}(\theta) = \tilde \Lambda_{\pi'}(\theta)$.
Next, take $\eta \in \{ \eta:\min_{\lambda \in \bar\Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1\}$ and let $\lambda_{\pi',\eta}$ be the environment on which the minimum is achieved. Such $\lambda_{\pi',\eta}$ exists because we just showed that $\bar \Lambda_{\pi'}(\theta)$ is closed and bounded and hence compact and the sum consists of a finite number of continuous functions. If $\lambda_{\pi',\eta} \in \Lambda_{\pi'}(\theta)$ then $\eta \in \{ \eta:\inf_{\lambda \in \Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1\}$. If $\lambda_{\pi',\eta} \not\in \Lambda_{\pi'}(\theta)$ then $\lambda_{\pi',\eta}$ must be a limit point of $\Lambda_{\pi'}(\theta)$. By definition we can construct a convergent sequence of $\{\lambda_j\}_{j=1}^\infty,\lambda_j \in \Lambda_{\pi'}(\theta)$ to $\lambda_{\pi',\eta}$ such that $\sum_{\pi\in\Pi} \eta(\pi) KL(\PP_\theta^{\pi}, \PP_{\lambda_{j}}^\pi) \geq 1$. This implies
$\sum_{\pi\in\Pi} \eta(\pi) KL(\PP_\theta^{\pi}, \PP_{\lambda_{j}}^\pi) \geq \inf_{\lambda \in \Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi)$. Using the continuity of the KL term and taking limits, the above implies that the minimum upper bounds the infimum.
Since we argued that $\Lambda_{\pi'}(\theta)$ is bounded and $\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^{\pi},\PP_{\lambda_j}^{\pi})$ is also bounded from below this implies $\bar \Lambda_{\pi'}(\theta)$ contains the infimum $\inf_{\lambda \in \Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi)$. This implies $\inf_{\lambda \in \Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq \min_{\lambda \in \bar \Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi)$
, and so the infimum over $\Lambda_{\pi}(\theta)$ equals the minimum over $\bar\Lambda_{\pi}(\theta)$. Which finally implies that $\eta \in \{ \eta:\inf_{\lambda \in \Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1\}$. This shows that LP~\ref{eq:opt_prob_equiv} is equivalent to
\begin{align*}
\minimize{\eta(\pi)\geq 0}{\sum_{\pi \in \Pi} \eta(\pi)\left(\return{*}_{\theta} - \return{\pi}_{\theta}\right)}
{
\min_{\lambda \in \bar\Lambda_{\pi'}(\theta)}\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1 \qquad \textrm{for all } \,\,\pi' \in \Pi
},
\end{align*}
or equivalently that we can consider the closure of $\Lambda(\theta)$ in LP~\ref{eq:opt_prob}, $\bar\Lambda(\theta) = \{ \lambda \in \Theta \colon \return{\pi^*_\lambda}_\lambda \geq \return{\pi^*_\theta}_\theta,\pi^*_\lambda \not\in \Pi^*_\theta, KL(\PP_\theta^{\pi^*_\theta}, \PP_\lambda^{\pi^*_\theta}) = 0\}$ i.e. the set of environments which makes any $\pi$ optimal without changing the environment on state-action pairs in $\pi^*_\theta$.
\end{proof}
\subsection{Lower bounds for full support optimal policy}
\label{app:lower_bounds_full_supp}
\gaplemmafullsupp*
\begin{proof}Let
$\lambda$ be the environment that is identical to $\theta$ except for the immediate reward for state-action pair for $(s,a)$. Specifically, let $R_{\lambda}(s,a)$ so that $r_{\lambda}(s,a) = r_\theta(s,a) + \Delta$ with $\Delta = \gap_\theta(s,a)$ . Since we assume that rewards are Gaussian, it follows that
\begin{align*}
KL(\PP_\theta^{\pi}, \PP_\lambda^{\pi})
&= w^{\pi}_\lambda(s, a) KL(R_\theta(s,a), R_\lambda(s,a))
\leq KL(R_\theta(s,a), R_\lambda(s,a))\\
&\leq \gap_\theta(s,a)^2
\end{align*}
for any policy $\pi \in \Pi$.
We now show that the optimal value function (and thus return) of $\lambda$ is uniformly upper-bounded by the optimal value function of $\theta$. To that end, consider their difference in any state $s'$, which we will upper-bound by their difference in $s$ as
\begin{align*}
V_\lambda^*(s') - V_\theta^*(s')
&\leq
\chi(\kappa(s) > \kappa(s')) \PP_\theta^{\pi^*_\lambda}(s_{\kappa(s)} = s | s_{\kappa(s')} = s')[V_\lambda^*(s) - V_\theta^*(s)] \\
&\leq V_\lambda^*(s) - V_\theta^*(s).
\end{align*}
Further, the difference in $s$ is exactly
\begin{align*}
V_\lambda^*(s) - V_\theta^*(s)
&= r_{\lambda}(s,a) + \langle P_\theta(\cdot | s,a), V_\theta^*\rangle - V_\theta^*(s)\\
&= r_{\theta}(s,a) + \langle P_\theta(\cdot | s,a), V_\theta^*\rangle +\gap_\theta(s,a) - V_\theta^*(s)
= 0.
\end{align*}
Hence, $V^*_\lambda = V^*_\theta \leq 1$ and thus $\lambda \in \Theta$.
We will now show that there is a policy that is optimal in $\lambda$ but not in $\theta$.
Let $\pi^* \in \Pi^*_\theta$ be any optimal policy for $\theta$ that has non-zero probability of visiting $s$ and consider the policy
\begin{align*}
\tilde \pi(\tilde s) =
\begin{cases}
\pi^*(\tilde s) & \textrm{if } s \neq \tilde s
\\
a & \textrm{if } s = \tilde s
\end{cases}
\end{align*}
that matches $\pi^*$ on all states except $s$. We will now show that $\tilde \pi$ achieves the same return as $\pi^*$ in $\lambda$. Consider their difference
\begin{align*}
\return{\tilde \pi}_\lambda -
\return{\pi^*}_\lambda
\overset{(i)}{=} &~
w_\lambda^{\tilde \pi}(s, \tilde \pi(s)) [r_\lambda(s, \tilde \pi(s)) + \langle P_\lambda(\cdot | s, \tilde \pi(s)), V_\lambda^{\tilde \pi} \rangle ]\\
&~-
w_\lambda^{\pi^*}(s, \pi^*(s)) [r_\lambda(s, \pi^*(s)) + \langle P_\lambda(\cdot | s, \pi^*(s)), V_\lambda^{\pi^*} \rangle ]\\
\overset{(ii)}{=} &~
w_\lambda^{\pi^*}(s, \pi^*(s)) [r_\lambda(s, \tilde \pi(s)) - r_\lambda(s, \pi^*(s)) + \langle P_\lambda(\cdot | s, \tilde \pi(s)) - P_\lambda(\cdot | s, \pi^*(s)), V_\lambda^{\pi^*} \rangle ]\\
\overset{(iii)}{=} &~
w_\theta^{\pi^*}(s, \pi^*(s)) [\Delta + r_\theta(s, \tilde \pi(s)) - r_\theta(s, \pi^*(s)) + \langle P_\theta(\cdot | s, \tilde \pi(s)) - P_\theta(\cdot | s, \pi^*(s)), V_\theta^* \rangle ]\\
\overset{(iv)}{=} &~
w_\theta^{\pi^*}(s, \pi^*(s)) [\Delta - \gap_\theta(s, \tilde \pi(s)) ]
\end{align*}
where $(i)$ and $(ii)$ follow from the fact that $\tilde \pi$ and $\pi^*$ only differ on $s$ and hence, their probability at arriving at $s$ and their value for any successor state of $s$ is identical.
Step $(iii)$ follows from the fact that $\lambda$ and $\theta$ only differ on $(s,a)$ which is not visited by $\pi^*$. Finally, step $(iv)$ applies the definition of optimal value functions and value-function gaps.
Since $\Delta = \gap_\theta(s, \tilde \pi(s))$, it follows that $\return{\tilde \pi}_\lambda = \return{\pi^*}_\lambda = \return{\pi^*}_\theta = \return{*}_\theta$. As we have seen above, the optimal value function (and return) is identical in $\theta$ and $\lambda$ and, hence, $\tilde \pi$ is optimal in $\lambda$.
Note that the we can apply the chain of equalities above in the same manner to $\return{\tilde \pi}_\theta -
\return{\pi^*}_\theta$ if we consider $\Delta = 0$. This yields
\begin{align*}
\return{\tilde \pi}_\theta -
\return{\pi^*}_\theta = - w_\theta^{\pi^*}(s, \pi^*(s)) \gap_\theta(s, a) < 0
\end{align*}
because $w_\theta^{\pi^*}(s, \pi^*(s)) > 0$ and $\gap_\theta(s, a) < 0$ by assumption. Hence $\tilde \pi$ is not optimal in $\theta$, which completes the proof.
\end{proof}
\begin{lemma}[Optimization problem over $\mathcal{S} \times \mathcal{A}$ instead of $\Pi$]
\label{lem:LP_(s,a)_relaxed}
Let optimal value $C(\theta)$ of the optimization problem \pref{eq:opt_prob} in \pref{thm:lower_bound_gen} is lower-bound by the optimal value of the problem
\begin{equation}
\label{eq:LP_(s,a)_relaxed}
\begin{aligned}
\underset{\eta(s,a) \geq 0}{\operatorname{minimize}} \quad
&
\sum_{s,a} \eta(s,a)\gap_{\theta}(s,a)
\\
\textrm{s.t.} \quad &
\sum_{s,a}
\eta(s,a)
KL(R_\theta(s,a), R_\lambda(s,a)) \\
& + \sum_{s,a}
\eta(s,a) KL(P_\theta(\cdot|s,a), P_\lambda(\cdot|s,a))
\geq 1\qquad \textrm{for all } \,\,\lambda \in \Lambda(\theta)
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
First, we rewrite the objective of \pref{eq:opt_prob} as
\begin{align*}
\sum_{\pi \in \Pi} \eta(\pi) (\return{*}_\theta - \return{\pi}_\theta)
\overset{(i)}{=} \sum_{\pi \in \Pi} \eta(\pi) \sum_{s,a} w^\pi_\theta(s,a) \gap_\theta(s,a)
= \sum_{s,a} \left(\sum_{\pi \in \Pi} \eta(\pi) w^\pi_\theta(s,a)\right) \gap_\theta(s,a)
\end{align*} where step $(i)$ applies \pref{lem:gap_decomp_pi} proved in Appendix~\ref{app:upper_bounds}. Here, $w^\pi_\theta(s,a)$ is the probability of reaching $s$ and taking $a$ when playing policy $\pi$ in MDP $\theta$.
Similarly, the LHS of the constraints of \pref{eq:opt_prob} can be decomposed as
\begin{align*}
&\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi, \PP_\lambda^\pi)\\
&= \sum_{\pi \in \Pi} \eta(\pi) \sum_{s,a} w_\theta^\pi(s,a)
\left( KL(R_\theta(s,a), R_\lambda(s,a)) + KL(P_\theta(\cdot|s,a), P_\lambda(\cdot|s,a))
\right) \\
&= \sum_{s,a} \left[\sum_{\pi \in \Pi} \eta(\pi) w_\theta^\pi(s,a) \right]
\left( KL(R_\theta(s,a), R_\lambda(s,a)) + KL(P_\theta(\cdot|s,a), P_\lambda(\cdot|s,a))
\right)
\end{align*}
where the first equality follows from writing out the definition of the KL divergence.
Let now $\eta(\pi)$ be a feasible solution to the original problem \pref{eq:opt_prob}. Then the two equalities we just proved show that $\eta(s,a) = \sum_{\pi \in \Pi} \eta(\pi) w_\theta^\pi(s,a)$ is a feasible solution for the problem in \pref{eq:LP_(s,a)_relaxed} with the same value. Hence, since \pref{eq:LP_(s,a)_relaxed} is a minimization problem, its optimal value cannot be larger than $C(\theta)$, the optimal value of \pref{eq:opt_prob}.
\end{proof}
\fullsupportlb*
\begin{proof}
Let $\bar \Lambda(\theta)$ be a set of all confusing MDPs from \pref{lem:non-empty_change_env}, that is, for every suboptimal $(s,a)$, $\bar \Lambda(\theta)$ contains exactly one confusing MDP that differs with $\theta$ only in the immediate reward at $(s,a)$.
Consider now the relaxation of \pref{thm:lower_bound_gen} from \pref{lem:LP_(s,a)_relaxed} and further relax it by reducing the set of constraints induced by $\Lambda(\theta)$ to only the set of constraints induced by $\bar \Lambda(\theta)$:
\begin{equation*}
\begin{aligned}
\underset{\eta(s,a) \geq 0}{\operatorname{minimize}} \quad
&
\sum_{s,a} \eta(s,a)\gap_{\theta}(s,a)
\\
\textrm{s.t.} \quad &
\sum_{s,a}
\eta(s,a)
KL(R_\theta(s,a), R_\lambda(s,a))
\geq 1\qquad \textrm{for all } \,\,\lambda \in \bar \Lambda(\theta)
\end{aligned}
\end{equation*}
Since all confusing MDPs only differ in rewards, we dropped the KL-term for the transition probabilities. We can simplify the constraints by noting that for each $\lambda$, only one KL-term is non-zero and it has value $\gap_{\theta}(s,a)^2$. Hence, we can write the problem above equivalently as
\begin{equation*}
\begin{aligned}
\underset{\eta(s,a) \geq 0}{\operatorname{minimize}} \quad
&
\sum_{s,a} \eta(s,a)\gap_{\theta}(s,a)
\\
\textrm{s.t.} \quad &
\eta(s,a)\gap_{\theta}(s,a)^2 \geq 1 \qquad \textrm{for all } \,\,(s,a) \in \mathcal{S} \times \mathcal{A} \textrm{ with }\,\, \gap_{\theta}(s,a) > 0
\end{aligned}
\end{equation*}
Rearranging the constraint as $\eta(s,a) \geq 1 / \gap_{\theta}(s,a)^2$, we see that the value is lower-bounded by \begin{align*}
\sum_{s,a} \eta(s,a)\gap_{\theta}(s,a) \geq
\sum_{s,a \colon \gap_\theta(s,a) > 0} \eta(s,a)\gap_{\theta}(s,a) \geq
\sum_{s,a \colon \gap_\theta(s,a) > 0} \frac{1}{\gap_\theta(s,a)},
\end{align*}
which completes the proof.
\end{proof}
We note that because the relaxation in \pref{lem:LP_(s,a)_relaxed} essentially allows the algorithm to choose which state-action pairs to play instead of just policies, the final lower bound in \pref{thm:lower_bound_all_states_supp} may be loose, especially in factors of $H$. However, it is unlikely that the $\gap_{\min}$ term arising in the upper bound of \citet{simchowitz2019non} can be recovered. We conjecture that such a term can be avoided by algorithms, which do not construct optimistic estimators for the $Q$-function at each state-action pair but rather just work with a class of policies and construct only optimistic estimators of the return.
\subsection{Lower bounds for deterministic MDPs}
\label{app:lower_bounds_det}
We will show that we can derive lower bounds in two cases:
\begin{enumerate}
\item We show that if the graph induced by the MDP is a tree, then we can formulate a finite LP which has value at most a polynomial factor of $H$ away from the value of LP~\ref{eq:opt_prob}.
\item We show that if we assume that the value function for any policy is at most $1$ and the rewards of each state-action pair are at most $1$, then we can derive a closed form lower bound. This lower bound is also at most a polynomial factor of $H$ away from the solution to LP~\ref{eq:opt_prob}.
\end{enumerate}
We begin by stating a helpful lemma, which upper and lower bounds the $KL$-divergence between two environments on any policy $\pi$. Since we consider Gaussian rewards with $\sigma =1/\sqrt{2}$ it holds that $KL(R_\theta(s,a),R_{\lambda}(s,a)) = (r_\theta(s,a) - r_\lambda(s,a))^2$. Further for any $\pi$ and $\lambda$ it holds that $KL(\theta(\pi),\lambda(\pi)) = \sum_{(s,a)\in\pi} KL(R_\theta(s,a),R_\lambda(s,a)) = \sum_{(s,a)\in\pi}(r_\theta(s,a) - r_\lambda(s,a))^2$. We can now show the following lower bound on $KL(\theta(\pi),\lambda(\pi))$.
\begin{lemma}
\label{lem:kl_lower_bound}
Fix $\pi$ and suppose $\lambda$ is such that $\pi^*_\lambda = \pi$. Then $(\return{*}-\return{\pi})^2 \geq KL(\theta(\pi),\lambda(\pi)) \geq \frac{(\return{*} - \return{\pi})^2}{H}$.
\end{lemma}
\begin{proof}
The second inequality follows from the fact that the optimization problem
\begin{align*}
\minimize{\theta,\lambda \in \Lambda(\theta) : \pi^*_\lambda = \pi}{\sum_{(s,a)\in\pi}(r_\theta(s,a) - r_\lambda(s,a))^2}{\sum_{(s,a)\in\pi} r_{\lambda}(s,a) - r_\theta(s,a) \geq \return{*} - \return{\pi}},
\end{align*}
admits a solution at $\theta,\lambda$ for which $r_\lambda(s,a) - r_\theta(s,a) = \frac{\return{*} - \return{\pi}}{H}, \forall (s,a) \in \pi$. The first inequality follows from considering the optimization problem
\begin{align*}
\maximize{\theta,\lambda \in \Lambda(\theta) : \pi^*_\lambda = \pi}{\sum_{(s,a)\in\pi}(r_\theta(s,a) - r_\lambda(s,a))^2}{\sum_{(s,a)\in\pi} r_{\lambda}(s,a) - r_\theta(s,a) \geq \return{*} - \return{\pi}},
\end{align*}
and the fact that it admits a solution at $\theta,\lambda$ for which there exists a single state-action pair $(s,a) \in \pi$ such that $r_\theta(s,a)-r_\lambda(s,a) = \return{*} - \return{\pi}$ and for all other $(s,a)$ it holds that $r_\lambda(s,a) = r_\theta(s,a)$.
\end{proof}
Using the above Lemma~\ref{lem:kl_lower_bound} we now show that we can restrict our attention only to environments $\lambda \in \Lambda(\theta)$ which make one of $\pi^*_{(s,a)}$ optimal and derive an upper bound on $C(\theta)$ which we will try to match, up to factors of $H$, later. Define the set $\tilde\Lambda(\theta) = \{\lambda \in \Lambda(\theta) : \exists (s,a)\in \mathcal{S}\times\mathcal{A}, \pi^*_{\lambda} = \pi^*_{(s,a)}\}$ and $\Pi^* = \{\pi \in \Pi,\pi\neq\pi^*_\theta: \exists (s,a) \in \mathcal{S}\times\mathcal{A}, \pi = \pi^*_{(s,a)}\}$. We have
\begin{lemma
\label{lem:primal_opt_upper_bound}
Let $\tilde C(\theta)$ be the value of the optimization problem
\begin{equation}
\label{eq:opt_primal_relaxed}
\begin{aligned}
\minimize{\eta(\pi)\geq 0}{\sum_{\pi\in \Pi^*} \eta(\pi)(\return{*} - \return{\pi})}{ \sum_{\pi \in \Pi^*} \eta(\pi)KL(\theta(\pi),\lambda(\pi)) \geq 1,\forall \lambda \in \tilde\Lambda(\theta)}.
\end{aligned}
\end{equation}
Then $\sum_{\pi \in \Pi^*} \frac{H}{\return{*} - \return{\pi}} \geq C(\theta) \geq \frac{\tilde C(\theta)}{H}$.
\end{lemma}
\begin{proof}
We begin by showing $C(\theta) \geq \frac{\tilde C(\theta)}{H}$ holds. Fix a $\pi \not\in\Pi^*$ s.t. the solution of LP~\ref{eq:opt_prob} implies $\eta(\pi)>0$. Let $\lambda \in \tilde\Lambda(\theta)$ be a change of environment for which $KL(\theta(\pi),\lambda(\pi))>0$. We can now shift all of the weight of $\eta(\pi)$ to $\eta(\pi^*_\lambda)$ while still preserving the validity of the constraint. Further doing so to all $\pi^*_{(s,a)}$ for which $\pi^*_{(s,a)} \cap \pi \neq \emptyset$ will not increase the objective by more than a factor of $H$ as $\return{*} - \return{\pi} \geq \frac{1}{H}\sum_{(s,a) \in \pi} \return{*} - \return{\pi^*_{(s,a)}}$. Thus, we have converted the solution to LP~\ref{eq:opt_prob} to a feasible solution to LP~\ref{eq:opt_primal_relaxed} which is only a factor of $H$ larger.
Next we show that $\sum_{\pi \in \Pi^*}\frac{H}{\return{*} - \return{\pi}} \geq C(\theta)$. Set $\eta(\pi) = 0,\forall \pi \in \Pi\setminus \Pi^*$ and set $\eta(\pi) = \frac{H}{(\return{*} - \return{\pi})^2}, \forall \pi \in \Pi^*$.
If $\pi$ is s.t. $\eta(\pi) > 0$ then for any $\lambda$ which makes $\pi$ optimal it holds that
\begin{align*}
1 &\leq \frac{H}{(\return{*} - \return{\pi^*_\lambda})^2} \times \frac{(\return{*} - \return{\pi^*_\lambda})^2}{H} \leq \frac{H}{(\return{*} - \return{\pi^*_\lambda})^2} KL(\theta(\pi^*_\lambda),\lambda(\pi^*_\lambda))\\
&= \eta(\pi^*_\lambda) KL(\theta(\pi^*_\lambda),\lambda(\pi^*_\lambda)) \leq \sum_{\pi' \in \Pi} \eta(\pi')KL(\theta(\pi'),\lambda(\pi')),
\end{align*}
where the second inequality follows from Lemma~\ref{lem:kl_lower_bound}.
Next, if $\pi$ is s.t. $\eta(\pi) = 0$ then for any $\lambda$ which makes $\pi$ optimal it holds that
\begin{align*}
\sum_{\pi' \in \Pi} \eta(\pi')KL(\theta(\pi'),\lambda(\pi')) &\geq \sum_{(s,a) \in \pi^*_\lambda} \eta(\pi^*_{(s,a)}) KL(\theta(\pi^*_{(s,a)}),\lambda(\pi^*_{(s,a)}))\\
&=\sum_{(s,a) \in \pi^*_\lambda} \frac{H}{(\return{*} - \return{\pi^*_{(s,a)}})^2}KL(\theta(\pi^*_{(s,a)}),\lambda(\pi^*_{(s,a)}))\\
&\geq \frac{H}{(\return{*} - \return{\pi^*_{\lambda}})^2}\sum_{(s,a) \in \pi^*_\lambda} KL(\theta(\pi^*_{(s,a)}),\lambda(\pi^*_{(s,a)}))\\
&\geq \frac{H}{(\return{*} - \return{\pi^*_{\lambda}})^2}\sum_{(s,a) \in \pi^*_\lambda} KL(R_\theta(s,a),R_\lambda(s,a))\\
&= \frac{H}{(\return{*} - \return{\pi^*_{\lambda}})^2} KL(\theta(\pi^*_{\lambda}),\lambda(\pi^*_\lambda))\geq 1,
\end{align*}
where the second inequality follows from the fact that $\return{\pi^*_\lambda} \leq \return{\pi^*_{(s,a)}},\forall (s,a) \in \pi^*_{\lambda}$.
\end{proof}
\subsubsection{Lower bound for Markov decision processes with bounded value function}
\begin{lemma}
\label{lem:confusing_mdps_det}
Let $\Theta$ be the set of all episodic MDPs with Gaussian immediate rewards and optimal value function uniformly bounded by $1$. Consider an MDP $\theta \in \Theta$ with deterministic transitions. Then, for any reachable state-action pair $(s,a)$ that is not visited by any optimal policy, there exists a confusing MDP $\lambda \in \Lambda(\theta)$ with
\begin{itemize}
\item $\lambda$ and $\theta$ only differ in the immediate reward at $(s,a)$
\item $KL(\PP_\theta^\pi, \PP_\lambda^\pi) = w_\theta^\pi(s,a) (\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta)^2$ for all $\pi \in \Pi$ where $\return{\pi_{(s,a)}^*}_\theta = \max_{\pi \colon w^\pi(s,a) > 0} \return{\pi}_\theta$.
\end{itemize}
\end{lemma}
\begin{proof}
Let $(s,a) \in \mathcal{S} \times \mathcal{A}$ be any state-action pair that is not visited by any optimal policy. Then $\return{\pi_{(s,a)}^*}_\theta = \max_{\pi \colon w^\pi(s,a) > 0} \return{\pi}_\theta \leq \return{*}_\theta$ is strictly suboptimal in $\theta$. Let $\tilde \pi$ be any policy that visits $(s,a)$ and achieves the highest return $\return{\pi_{(s,a)}^*}_\theta$ in $\theta$ possible among such policies.
Define $\lambda$ to be the MDP that matches $\theta$ except in the immediate reward at $(s,a)$, which we set as $R_\lambda(s,a) = \mathcal{N}(r_\theta(s,a) + \Delta, 1/2)$ with $\Delta = \return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta$. That is, the expected reward of $\lambda$ in $(s,a)$ is raised by $\Delta$.
For any policy $\pi$, it then holds
\begin{align*}
KL(\PP_\theta^\pi, \PP_\lambda^\pi) &= w_\theta^\pi(s,a) KL(R_\theta(s,a), R_\lambda(s,a))\\
\return{\pi}_\lambda &= w_\theta^\pi(s,a) \Delta + \return{\pi}_\theta
\end{align*}
due to the deterministic transitions. Hence, while $\return{*}_\lambda = \return{*}_\theta$ and all optimal policies of $\theta$ are still optimal in $\lambda$, now policy $\tilde \pi$, which is not optimal in $\theta$ is optimal in $\lambda$.
By the choice of Gaussian rewards with variance $1/2$, we have $KL(R_\theta(s,a), R_\lambda(s,a)) = (\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta)^2$ and thus $KL(\PP_\theta^\pi, \PP_\lambda^\pi) = w_\theta^\pi(s,a) (\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta)^2$ for all $\pi \in \Pi$.
It only remains to show that $\lambda \in \Theta$, i.e., that all immediate rewards and optimal value function is bounded by $1$. For rewards, we have
\begin{align*}
r_\lambda(s,a) = r_\theta(s,a) + \Delta = r_\theta(s,a) + \return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta
= \return{*}_\theta - \underset{\geq 0}{\underbrace{(\return{\pi_{(s,a)}^*}_\theta - r_\theta(s,a))}} \leq \return{*}_\theta \leq 1
\end{align*}
for $(s,a)$ and for all other $(s', a')$, $r_\lambda(s', a') = r_\theta(s', a') \leq 1$. Finally, the value function at any reachable state is bounded by the optimal return $\return{*}_\lambda = \return{*}_\theta \leq 1$ and for any unreachable state, the optimal value function of $\lambda$ is identical to the optimal value function of $\theta$. Hence, $\lambda \in \Theta$.
\end{proof}
\lowerbounddeterministic*
\begin{proof}
The proof
works by first relaxing the general LP~\ref{eq:opt_prob} and then considering its dual. We now define the set $\check\Lambda(\theta)$ which consists of all changes of environment which make $\pi^*_{(s,a)}$ optimal by only changing the distribution of the reward at $(s,a)$ by making it $\return{*}_\theta - \return{\pi^*_{(s,a)}}_\theta$ larger.
Formally, the set is defined as
\begin{align*}
\check\Lambda(\theta) = \big\{\lambda_{(s,a)}\colon \lambda \in \Lambda(\theta), KL(R_\theta(s,a), R_{\lambda}(s,a)) = (\return{*}_\lambda - \return{\pi^*_{(s,a)}})^2,\\ KL(R_\theta(s',a'), R_{\lambda}(s',a')) = 0,KL(P_\theta(s',a'), P_{\lambda}(s',a')) = 0,\forall (s',a')\neq (s,a)\big\}.
\end{align*}
This set is guaranteed to be non-empty (for any reasonable MDP) by Lemma~\ref{lem:confusing_mdps_det}.
The relaxed LP is now give by
\begin{equation}
\begin{aligned}
\minimize
{\eta(\pi)\geq 0}
{
\sum_{\pi \in \Pi} \eta(\pi)(\return{*}_\theta - \return{\pi}_\lambda)
}
{
\sum_{\pi \in \Pi} \eta(\pi)KL(\PP^\pi_\theta,\PP^\pi_\lambda) \geq 1
\qquad \textrm{for all } \lambda \in \check\Lambda(\theta)
}.
\end{aligned}
\end{equation}
The dual of the above LP is given by
\begin{equation}
\label{eq:opt_dual_gen}
\begin{aligned}
\maximize
{\mu(\lambda)\geq 0}
{
\sum_{\lambda \in \check\Lambda(\theta)} \mu(\lambda)
}
{
\sum_{\lambda \in \check \Lambda(\theta)} \mu(\lambda)KL(\PP^\pi_\theta,\PP^\pi_\lambda) \leq \return{*}_\theta - \return{\pi}_\theta
\qquad \textrm{for all } \pi \in \Pi
}.
\end{aligned}
\end{equation}
By weak duality, the value of any feasible solution to \pref{eq:opt_dual_gen} produces a lower bound on $C(\theta)$ in \pref{thm:lower_bound_gen}.
Let
\begin{align*}
\mathcal{X} = \{ (s,a) \in \mathcal{S} \times \mathcal{A} \colon w^\pi_\theta(s,a) = 0 \textrm{ for all }\pi \in \Pi^*_\theta \textrm{ and } w^\pi_\theta(s,a) > 0 \textrm{ for some } \pi \in \Pi \setminus \Pi^*_\theta \}
\end{align*}
be the set of state-action pairs that are reachable in $\theta$ but no optimal policy visits.
Then consider a dual solution $\mu$ that puts $0$ on all confusing MDPs except on the $|\mathcal{X}|$ many MDPs from \pref{lem:confusing_mdps_det}. Since each such confusing MDP is associated with an $(s,a) \in \mathcal{X}$, we can rewrite $\mu$ as a mapping from $\mathcal{X}$ to $\RR$ sending $(s,a) \rightarrow \lambda_{(s,a)}$. Specifically, we set
\begin{align*}
\mu(s,a) &= \frac{1}{H} \left(\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta\right)^{-1} & \textrm{for all } & (s,a) \in \mathcal{X}.
\end{align*}
To show that this $\mu$ is feasible, consider the LHS of the constraints in \pref{eq:opt_dual_gen}
\begin{align*}
\sum_{\lambda \in \check \Lambda(\theta)} \mu(\lambda)KL(\PP^\pi_\theta,\PP^\pi_\lambda)
&= \sum_{(s,a) \in \mathcal{X}} \frac{1}{H} \left(\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta\right)^{-1} KL(\PP^\pi_\theta,\PP^\pi_{(s,a)})\\
&= \sum_{(s,a) \in \mathcal{X}} \frac{1}{H} \left(\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta\right)^{-1} w_\theta^\pi(s,a) (\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta)^2\\
&= \sum_{(s,a) \in \mathcal{X}} \frac{1}{H} w_\theta^\pi(s,a) (\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta)
\end{align*}
where the first equality applies our definition of $\mu$ and the second uses the expression for the KL-divergence from \pref{lem:confusing_mdps_det}. By definition of $\return{\pi_{(s,a)}^*}_\theta$, we have $\return{\pi_{(s,a)}^*}_\theta \geq \return{\pi}_\theta$ for all policies $\pi$ with $w_\theta^\pi(s,a) > 0$. Thus,
\begin{align*}
\sum_{(s,a) \in \mathcal{X}} \frac{1}{H} w_\theta^\pi(s,a) (\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta)
&\leq
\sum_{(s,a) \in \mathcal{X}} \frac{1}{H} w_\theta^\pi(s,a) (\return{*}_\theta - \return{\pi}_\theta)
\\
& \leq \return{*}_\theta - \return{\pi}_\theta
\end{align*}
where the second inequality holds because each policy visits at most $H$ states. Thus proves that $\mu$ defined above is indeed feasible. Hence, its objective value
\begin{align*}
\sum_{\lambda \in \Lambda(\theta)} \mu(\lambda)
= \sum_{(s,a) \in \mathcal{X}} \frac{1}{H} \left(\return{*}_\theta - \return{\pi_{(s,a)}^*}_\theta\right)
\end{align*}
is a lower-bound for $C(\theta)$ from \pref{thm:lower_bound_gen} which finishes the proof.
\end{proof}
\subsubsection{Tree-structured MDPs}
Even though Lemma~\ref{lem:primal_opt_upper_bound} restricts the set of confusing environments from $\Lambda(\theta)$ to $\tilde\Lambda(\theta)$, this set could still have exponential or even infinite cardinality. In this section we show that for a type of special MDPs we can restrict ourselves to a finite subset of $\tilde\Lambda(\theta)$ of size at most $SA$.
Arrange $\pi^*_{(s,a)},(s,a)\in\mathcal{S}\times\mathcal{A}$ according to the value functions $\return{\pi^*_{(s,a)}}$. Under this arrangement let $\pi_1 \succeq \pi_2 \succeq,\ldots,\succeq \pi_m$. Let $\pi_0 = \pi_\theta^*$. We will now construct $m$ environments $\lambda_1,\ldots,\lambda_m$, which will constitute the finite subset. We begin by constructing $\lambda_1$ as follows. Let $\mathcal{B}_1$ be the set of all $(s_h,a_h) \in \pi_1$ and $(s_h,a_h)\not\in \pi_0$. Arrange the elements in $\mathcal{B}_1$ in inverse dependence on horizon $(s_{h_1},a_{h_1}) \preceq (s_{h_2},a_{h_2}) \preceq \ldots \preceq (s_{h_{H_1}},a_{h_{H_1}})$, where $H_1 = |\mathcal{B}_1|$, so that $h_1 > h_2 >,\ldots, h_{H_1}$. Let $\lambda_1$ be the environment which sets
\begin{align*}
R_{\lambda_1}(s_{h_1},a_{h_1}) &= \min(1,\return{\pi_0} - \return{\pi_1})\\
R_{\lambda_1}(s_{h_2},a_{h_2}) &= \min(1,\max(R_{\theta}(s_{h_2},a_{h_2}),R_{\theta}(s_{h_2},a_{h_2})+\return{\pi_0} - (\return{\pi_1} - R_{\theta}(s_{h_1},a_{h_1})) - 1)))\\
&\vdots\\
R_{\lambda_1}(s_{h_i},a_{h_i}) &= \min(1,\max(R_{\theta}(s_{h_i},a_{h_i}),R_{\theta}(s_{h_i},a_{h_i})+\return{\pi_0} - (\return{\pi_1} - \sum_{\ell=1}^i R_{\theta}(s_{h_\ell},a_{h_\ell})) - i))\\
&\vdots
\end{align*}
Clearly $\lambda_1$ makes $\pi_1$ optimal and also does not change the value of any state-action pair which belongs to $\pi_0$ so it agrees with $\theta$ on $\pi_0$. Further $\pi_2,\pi_3,\ldots,\pi_m$ are still suboptimal policies under $\lambda_1$. This follows from the fact that for any $i>1$, $\return{\pi_1} > \return{\pi_i}$ and there exists $(s,a)$ such that $(s,a)\in \pi_i$ but $(s,a)\not \in \pi_1$ so $R_{\lambda_1}(s,a) = R_{\theta}(s,a)$. Further $\lambda_1$ only increases the rewards for state-action pairs in $\pi_1$ and hence $\return{\pi_1}_{\lambda_1} > \return{\pi_i}_{\lambda_1}$. Notice that there exists an index $\tilde H_1$ at which $R_{\lambda_1}(s_{h_{\tilde H_1}}, a_{h_{\tilde H_1}}) = \return{\pi_0} - (\return{\pi_1} - \sum_{\ell=1}^{\tilde H_1} R_{\theta}(s_{h_\ell},a_{h_\ell})) - \tilde H_1) \geq R_{\theta}(a_{\tilde H_1},s_{\tilde H_1})$. For this index it holds that for $h < \tilde H_1$, $R_{\lambda_1}(s_h,a_h) = 1$ and for $h> \tilde H_1$, $R_{\lambda_1}(s_h,a_h) = R_{\theta}(s_h,a_h)$.
Let
\begin{align*}
\mathcal{B}_i &= \{(s,a) \in \pi_i : (s,a)\not\in \bigcup_{\ell < i} \pi_\ell\}\\
\tilde\mathcal{B}_i &= \{(s,a) \in \pi_i : (s,a)\in \bigcup_{\ell < i} \pi_\ell\}.
\end{align*}
We first define an environment $\tilde\lambda_i$ on $(s,a)\in \tilde\mathcal{B}_i$ as follows. $R_{\lambda_{i}}(s,a) = R_{\lambda_{\ell}}(s,a)$, where $\ell < i$ is such that $(s,a) \in \mathcal{B}_\ell$.
Let $\return{\pi_i}_{\tilde\lambda_i}$ be the value function of $\pi_i$ with respect to $\tilde\lambda_i$.
\begin{lemma}
\label{lem:val_func_comp}
It holds that $\return{\pi_i}_{\tilde\lambda_i} \leq \return{\pi_0}$.
\end{lemma}
\begin{proof}
Let $\tilde H_i$ be the index for which it holds that for $\ell \leq \tilde H_i$, $(s_{h_\ell},a_{h_\ell}) \in \pi_i \iff (s_{h_\ell},a_{h_\ell}) \in \mathcal{B}_i$. Such a $\tilde H_i$ exists as there is a unique sub-tree $\mathcal{M}_i$, of maximal depth, for which it holds that if $\pi_j \bigcap \mathcal{M}_i \neq \emptyset \iff \pi_i \succeq \pi_j$. The root of this subtree is exactly at depth $H - h_{\tilde H_i}$. Let $\pi_j$ be any policy such that $\pi_j \succeq \pi_i$ and $\exists (s_{h_{\tilde H_i}},a_{h_{\tilde H_i}}) \in \pi_j$. By the maximality of $\mathcal{M}_i$ such a $\pi_j$ exists. Because of the tree structure it holds that for any $h' > h_{\tilde H_i}$ if $(s_{h'},a_{h'}) \in \pi_{i} \implies (s_{h'},a_{h'}) \in \pi_{j}$ and hence $\tilde\lambda_i = \lambda_j$ up to depth $h_{\tilde H_i}$. Since $\pi_i$ and $\pi_j$ match up to depth $H - h_{\tilde H_i}$ and $\pi_j \succeq \pi_i$ it also holds that
\begin{align*}
\sum_{\ell \leq \tilde H_i} R_{\lambda_j}(s_{h_\ell}^{\pi_j},a_{h_\ell}^{\pi_j}) \geq \sum_{\ell \leq \tilde H_i} R_{\theta}(s_{h_\ell}^{\pi_j},a_{h_\ell}^{\pi_j}) \geq \sum_{\ell \leq \tilde H_i} R_{\theta}(s_{h_\ell}^{\pi_i},a_{h_\ell}^{\pi_i}) = \sum_{\ell \leq \tilde H_i} R_{\tilde\lambda_i}(s_{h_\ell}^{\pi_i},a_{h_\ell}^{\pi_i}).
\end{align*}
Since $\pi_j$ is optimal under $\lambda_j$ the claim holds.
\end{proof}
For all $(s_{h_j},a_{h_j}) \in \mathcal{B}_i$ we now set
\begin{align}
\label{eq:ith_env_constr}
R_{\lambda_i}(s_{h_j},a_{h_j}) = \min(1,\max(R_{\theta}(s_{h_j},a_{h_j}), R_{\theta}(s_{h_j},a_{h_j}) + \return{\pi_0} - (\return{\pi_i}_{\tilde\lambda_i} - \sum_{\ell=1}^j R_{\tilde\lambda_i}(s_{h_\ell},a_{h_\ell})) -j )),
\end{align}
and for all $(s_h,a_h) \in \tilde\mathcal{B}_i$ we set $R_{\lambda_i}(s_h,a_h) = R_{\tilde\lambda_i}(s_h,a_h)$. From the definition of $\tilde\mathcal{B}_i$ it follows that $\lambda_i$ agrees with all $\lambda_j$ for $j\leq i$ on state-action pairs in $\pi_i$. Finally we need to show that the construction in Equation~\ref{eq:ith_env_constr} yields an environment $\lambda_i$ for which $\pi_i$ is optimal.
\begin{lemma}
\label{lem:opt_of_lambdai}
Under $\lambda_i$ it holds that $\pi_i$ is optimal.
\end{lemma}
\begin{proof}
Let $\tilde H_i$ and $\pi_j$ be as in the proof of Lemma~\ref{lem:val_func_comp}. We now show that $\sum_{\ell \leq \tilde H_i} R_{\lambda_j}(s_{h_\ell}^{\pi_j},a_{h_\ell}^{\pi_j}) \leq \sum_{\ell \leq \tilde H_i} R_{\lambda_i}(s_{h_\ell}^{\pi_i},a_{h_\ell}^{\pi_i})$.
We only need to show that $\sum_{\ell \leq \tilde H_i} R_{\lambda_i}(s_{h_\ell}^{\pi_i},a_{h_\ell}^{\pi_i}) \geq \return{\pi_0} - \return{\pi_i}_{\tilde\lambda_i}$. From Equation~\ref{eq:ith_env_constr} we have $R_{\lambda_i}(s_{h_1},a_{h_1}) = \min(1,\return{\pi_0} - \return{\pi_i}_{\tilde\lambda_i})$. If $R_{\lambda_i}(s_{h_1},a_{h_1}) = \return{\pi_0} - \return{\pi_i}_{\tilde\lambda_i}$ then the claim is complete. Suppose $R_{\lambda_i}(s_{h_1},a_{h_1}) = 1$. This implies $\return{\pi_0} - \return{\pi_i}_{\tilde\lambda_i} \geq 1 - R_{\theta}(s_{h_1},a_{h_1})$. Next the construction adds the remaining gap of $\return{\pi_0} - \return{\pi_i}_{\tilde\lambda_i} + R_{\theta}(s_{h_1},a_{h_1}) - 1$ to $R_\theta(s_{h_2},a_{h_2})$ and clips $R_{\lambda_i}(s_{h_2},a_{h_2})$ to $1$ if necessary. Continuing in this way we see that if ever $R_{\lambda_i}(s_{h_j},a_{h_j}) = R_{\theta}(s_{h_j},a_{h_j}) + \return{\pi_0} - (\return{\pi_i}_{\tilde\lambda_i} - \sum_{\ell=1}^j R_{\tilde\lambda_i}(s_{h_\ell},a_{h_\ell})) -j$ then $\return{\pi_0} - V_{\tilde\lambda_i}^{\pi_i} \leq \sum_{\ell \leq \tilde H_i} R_{\lambda_i}(s_{h_\ell}^{\pi_i},a_{h_\ell}^{\pi_i})$. On the other hand if this never occurs, we must have $R_{\lambda_i}(s_{h_\ell}^{\pi_i},a_{h_\ell}^{\pi_i}) = 1 \geq R_{\lambda_j}(s_{h_\ell}^{\pi_j},a_{h_\ell}^{\pi_j})$ which concludes the claim.
\end{proof}
Let $\hat\Lambda(\theta) = \{\lambda_1,\ldots,\lambda_m\}$ be the set of the environments constructed above. We now show that the value of the optimization problem is not too much smaller than the value of Problem~\ref{eq:opt_prob}.
\begin{theorem}
\label{thm:tree_mdp_bound}
The value $\hat C(\theta)$ of the LP
\begin{align*}
\minimize{\eta(\pi)\geq 0}{\sum_{\pi\in \Pi^*} \eta(\pi)(\return{*} - \return{\pi})}{ \sum_{\pi \in \Pi^*} \eta(\pi)KL(\theta(\pi),\lambda(\pi)) \geq 1,\forall \lambda \in \hat\Lambda(\theta)},
\end{align*}
satisfies $\hat C(\theta) \geq \frac{C(\theta)}{H^2}$ and $C(\theta) \geq \frac{\hat C(\theta)}{H}$.
\end{theorem}
\begin{proof}
The inequality $C(\theta) \geq \frac{\hat C(\theta)}{H}$ follows from Lemma~\ref{lem:primal_opt_upper_bound} and the fact that the above optimization problem is a relaxation to LP~\ref{eq:opt_primal_relaxed}.
To show the first inequality we consider the following relaxed LP
\begin{align*}
\minimize{\eta(\pi)\geq 0}{\sum_{\pi\in \Pi} \eta(\pi)(\return* - \return{\pi})}{ \sum_{\pi \in \Pi} \eta(\pi)KL(\theta(\pi),\lambda(\pi)) \geq 1,\forall \lambda \in \hat\Lambda(\theta)}.
\end{align*}
Any solution to the LP in the statement of the theorem is feasible for the above LP and thus the value of the above LP is no larger. We now show that the value of the above LP is greater than or equal to $\frac{C(\theta)}{H^2}$. Fix $\lambda \in \hat \Lambda(\theta)$. We show that for any $\lambda' \in \Lambda(\theta)$ such that $\pi^*_{\lambda} = \pi^*_{\lambda'}$ it holds that $KL(\theta(\pi),\lambda(\pi)) \leq H^2 KL(\theta(\pi),\lambda'(\pi)),\forall \pi \in \Pi$. This would imply that if $\eta$ is a solution to the above LP, then $H^2 \eta$ is feasible for LP~\ref{eq:opt_prob} and therefore $\hat C(\theta) \geq \frac{C(\theta)}{H^2}$.
Arrange $\pi \in \Pi : KL(\theta(\pi),\lambda(\pi)) > 0$ according to $KL(\theta(\pi),\lambda(\pi))$ so that
\begin{align*}
\pi_i \preceq \pi_j \iff KL(\theta(\pi_i),\lambda(\pi_i)) \geq KL(\theta(\pi_j),\lambda(\pi_j)).
\end{align*}
Consider the optimization problem
\begin{align*}
\minimize{\lambda' \in \Lambda(\theta)}{KL(\theta(\pi_i),\lambda'(\pi_i))}{\pi^*_{\lambda'} = \pi^*_\lambda}.
\end{align*}
If we let $\Delta_{\lambda'}(s_h,a_h),(s_h,a_h) \in \pi^*_\lambda$ denote the change of reward for $(s_h,a_h)$ under environment $\lambda'$, then the above optimization problem can be equivalently written as
\begin{align*}
\minimize{\lambda' \in \Lambda(\theta)}{\sum_{h=1}^{h_{\tilde H_i}} \Delta_{\lambda'}(s_h,a_h)^2}{\sum_{h=1}^H r(s_h,a_h) + \Delta_{\lambda'}(s_h,a_h) \geq \return*}.
\end{align*}
It is easy to see that the solution to the above optimization problem is to set $r(s_h,a_h) + \Delta_{\lambda'}(s_h,a_h) = 1$ for all $h \in [h_{\tilde H_i}+1,H]$ and spread the remaining mass of $\return* - \tilde H_i - (\return{\pi^*_\lambda} - \sum_{\ell=1}^{\tilde H_i}) R_\theta(s_{h_\ell},a_{h_\ell})$ as uniformly as possible on $\Delta_{\lambda'}(s_h,a_h)$, $h \in [1, h_{\tilde H_i}]$. Notice that under this construction the solution to the above optimization problem and $\lambda$ match for $h \in [h_{\tilde H_i}+1,H]$. Since the remaining mass is now the same it now holds that for any $\lambda'$, $\sum_{h=1}^{h_{\tilde H_i}} \Delta_{\lambda'}(s_h,a_h)^2 \geq \frac{1}{h_{\tilde H_i}^2} \sum_{h=1}^{h_{\tilde H_i}} \Delta_{\lambda}(s_h,a_h)^2$. This implies $KL(\theta(\pi_i),\lambda'(\pi_i)) \geq \frac{1}{\tilde H_i ^2} KL(\theta(\pi),\lambda(\pi))$ and the result follows as $\tilde H_i \leq H,\forall i \in [H]$.
\end{proof}
\subsubsection{Issue with deriving a general bound}
\label{app:lower_bounds_issues}
We now try to give some intuition regarding why we could not derive a generic lower bound for deterministic transition MDPs. We have already outlined our general approach of restricting the set $\Pi$ and $\Lambda(\theta)$ to finite subsets of manageable size and then showing that the value of the LP on these restricted sets is not much smaller than the value of the original LP. One natural restriction of $\Pi$ is the set $\Pi^*$ from Theorem~\ref{thm:lower_bound_deterministic}. Suppose we restrict ourselves to the same set and consider only environments making policies in $\Pi^*$ optimal as the restriction for $\Lambda(\theta)$. We now give an example of an MDP for which such a restriction will lead to an $\Omega(SA)$ multiplicative discrepancy between the value of the original semi-infinite LP and the restricted LP.
\begin{figure}
\centering
\includegraphics{imgs/lower_bound_counterexamp.pdf}
\caption{Issue with restricting LP to $\Pi^*$}
\label{fig:lower_bound_counterexample}
\end{figure}
The MDP can be found in Figure~\ref{fig:lower_bound_counterexample}. The rewards for each action for a fixed state $s$ are equal and are shown in the vertices corresponding to the states. The number of states in the second and last layer of the MDP are equal to $(SA-3)/2$. The optimal policy takes the red path and has value $V^{\pi^*} = 3$. The set $\Pi^*$ consists of all policies $\pi_{j,i}$ which visit one of the states in green. The policies $\pi_{1,i}$, in blue, visit the green state in the second layer of the MDP and one of the states in the final layer, following the paths in blue. Similarly the policies $\pi_{2,i}$, in orange, visit one of the state in the second layer and the green state in the last layer, following the orange paths. The value function of $\pi_{j,i}$ is $V^{\pi_{j,i}} = 3 - \frac{3}{SA} - i\epsilon$, where $0\leq i \leq (SA-4)/2$. We claim that playing each $\pi_{j,i}$ $\eta(\pi_{j,i}) = \Omega(SA)$ times is a feasible solution to the LP restricted to $\Pi^*$. Fix $i$, the $\lambda_{\pi_{1,i}}$ must put weight at least $1/SA$ on the green state in layer 2. Coupling with the fact that for all $i'$ the rewards $\pi_{1,i'}$ are also changed under this environment we know that the constraint of the restricted LP with respect to $\lambda_{\pi_{1,i}}$ is lower bounded by
$\sum_{i'} \eta(\pi_{1,i'})/(SA)^2$. Since there are $\Omega(SA)$ policies $\{\pi_{1,i'}\}_{i'}$, this implies that $\eta(\pi_{1,i}) = \Omega(SA)$ is feasible. A similar argument holds for any $\pi_{2,i}$. Thus the value of the restricted LP is at most $O(SA)$, for any $\epsilon \ll SA$.
However, we claim that the value of the semi-infinite LP which actually characterizes the regret is at least $\Omega(S^2A^2)$. First, to see that the above assignment of $\eta$ is not feasible for the semi-infinite LP, consider any policy $\pi \not \in \Pi^*$, e.g. take the policy which visits the state in layer $2$ with reward $1-1/SA - \epsilon$ and the state in layer $4$ with reward $1-2/SA - \epsilon$. Each of these states have been visited $O(SA)$ times and $\eta(\pi) = 0$ hence the constraint for the environment $\lambda_{\pi}$ is upper bounded by $SA\left(\left(\frac{1}{SA} + \epsilon\right)^2 + \left(\left(\frac{2}{SA} + \epsilon\right)^2\right)\right) \approx 1/SA$. In general each of the states in black in the second layer and the fourth layer have been visited $1/SA$ times less than what is necessary to distinguish any $\pi \not \in \Pi^*$ as sub-optimal. If we define the $i$-th column of the MDP as the pair consisting of the states with rewards $1-1/SA - i\epsilon$ and $1-2/SA - i\epsilon$ then to distinguish the policy visiting both of these states as sub-optimal we need to visit at least one of these $\Omega(S^2A^2)$ times. This implies we need to visit each column of the MDP $\Omega(S^2A^2)$ times and thus any strategy must incur regret at least $\Omega\left(\sum_{i} S^2A^2 \frac{1}{SA}\right) = \Omega(S^2A^2)$, leading to the promised multiplicative gap of $\Omega(SA)$ between the values of the two LPs.
Why does such a gap arise and how can we hope to fix it this issue? Any feasible solution to the LP restricted to $\Pi^*$ essentially needs to visit the states in green $\Theta(S^2A^2)$ times. This is sufficient to distinguish the green states as sub-optimal to visit and hence any strategy visiting these states would be also deemed sub-optimal. This is achievable by playing each strategy in $\Pi^*$ in the order of $\Theta(SA)$ times as already discussed. Now, even though $\Pi^*$ covers all other states, from our argument above we see that we need to play each $\pi \in \Pi^*$ in the order of $\Theta(S^2A^2)$ times to be able to determine all sub-optimal states. To solve this issue, we either have to increase the size of $\Pi^*$ to include for example all policies visiting each column of the MDP or at the very least include changes of environments in the constraint set which make such policies optimal. This is clearly computationally feasible for the MDP in Figure~\ref{fig:lower_bound_counterexample}, however, it is not clear how to proceed for general MDPs, without having to include exponentially many constraints. This begs the question about the computational hardness of achieving both upper and lower regret bounds in a factor of $o(SA)$ from what is optimal.
\subsection{Lower bounds for optimistic algorithms in MDPs with deterministic transitions}
In this section we prove a lower bound on the regret of optimistic algorithms, demonstrating that optimistic algorithms can not hope to achieve the information-theoretic lower bounds even if the MDPs have deterministic transitions. While the result might seem similar to the one proposed by \citet{simchowitz2019non} (Theorem 2.3) we would like to emphasize that the construction of \citet{simchowitz2019non} does not apply to MDPs with deterministic transitions, and that the idea behind our construction is significantly different.
\begin{figure}
\centering
\includegraphics[scale=0.5]{imgs/Det_mdplower.png}
\caption{Deterministic MDP instance for optimistic lower bound}
\label{fig:mdp_det_lower}
\end{figure}
Consider the MDP in Figure~\ref{fig:mdp_det_lower}. This MDP has $2n+9$ states and $4n+8$ actions. The rewards for each action are either $1/12$ or $1/12+\epsilon/2$ and can be found next to the transitions from the respective states. We are going to label the states according to their layer and their position in the layer so that the first state is $s_{1,1}$ the state which is to the left of $s_{1,1}$ in layer 2 is $s_{2,1}$ and to the right $s_{2,2}$. In general the $i$-th state in layer $h$ is denoted as $s_{h,i}$. The rewards in all states are deterministic, with a single exception of a Bernoulli reward from state $s_{4,1}$ to $s_{5,2}$ with mean $1/12$. From the construction it is clear that $V^*(s_{1,1}) = 1/2+\epsilon$. Further there are two sets of optimal policies with the above value function -- the $n$ optimal policies which visit state $s_{2,2}$ and the $n$ optimal policies which visit $s_{5,1}$. Notice that the information-theoretic lower bound for this MDP is in $O(\log(K)/\epsilon)$ as only the transition from state $s_{4,1}$ to $s_{5,2}$ does not belong to an optimal policy. In particular, there is no dependence on $n$. Next we try to show that the class of optimistic algorithms will incur regret at least $\Omega(n\log(\delta^{-1})/\epsilon)$.
\paragraph{Class of algorithms.}
We adopt the class of algorithms from Section G.2 in \citep{simchowitz2019non} with an additional assumption which we clarify momentarily. Recall that the class of algorithms assumes access to an optimistic value function $\bar V_k(s) \geq V^*(s)$ and optimistic Q-functions.
In particular the algorithms construct optimistic Q and value functions as
\begin{align*}
\bar V_k(s) &= \max_{a\in\mathcal{A}} \bar Q_k(s,a)\\
Q_k(s,a) &= \hat r_k(s,a) + b_k^{rw}(s,a) + \hat p_k(s,a)^\top \bar V_k + b_k(s,a).
\end{align*}
We assume that there exists a $c\geq 1$ such that
\begin{align*}
\frac{c}{2}\sqrt{\frac{\log(M(1\lor n_k(s,a)))/\delta}{(1\lor n_k(s,a))}}\leq b_k^{rw}(s,a) \leq c\sqrt{\frac{\log(M(1\lor n_k(s,a)))/\delta}{(1\lor n_k(s,a))}}\,,
\end{align*}
where $M = \theta(n)$ and $b_k(s,a) \sim \sqrt{S}f_k(s,a)b_k^{rw}(s,a)$, where $f_k$ is a decreasing function in the number of visits to $(s,a)$ given by $n_k(s,a)$.
For $n_k(s,a) = \Omega(n\log(n))$, we assume $b_k(s,a) \leq b_k^{rw}(s,a)$.
One can verify that this is true for the the Q and value functions of StrongEuler.
\paragraph{Lower bound.}
Let $\epsilon>0$ be sufficiently small to be specified later and let $N$ be such that
\begin{align*}
N = \lfloor\frac{c^2n\log(MN/(n\delta))}{16\epsilon^2}\rfloor\,.
\end{align*}
\begin{lemma}
\label{lem:det_low_bound1}
There exists $n_0,\epsilon_0$ such that for any pair of $n\geq n_0$ and $\epsilon\leq\epsilon_0$ and any $k\leq N$, with probability at least $1-\delta$, it holds that either $n_{k}(s_{5,1}) < N/4$, or $\bar Q_k(s_{4,1},1)<\bar Q_k(s_{4,1},2)$.
\end{lemma}
\begin{proof}
Assume $n_{k}(s_{5,1}) \geq N/4$, then we have
\begin{align*}
\bar Q_k(s_{4,1},1) &= \frac{1}{4}+\epsilon +\sum_{i=4}^6 b_k^{rw}(s_{i,1},1)+b_k(s_{i,1})\\
&\leq \frac{1}{4}+\epsilon + 6c\sqrt{\frac{\log(MN/(4\delta))}{N/4}}\leq \frac{1}{4}+\epsilon+\frac{48\epsilon}{\sqrt{n}}\,,
\end{align*}
where we assume $\epsilon$ is sufficiently small such that $b_k(s,a) \leq b_k^{rw}(s,a)$ for $n_k(s,a)\geq N/4$.
On the other hand, we have have with probability at least 1-$\delta$, that
$\forall k:\hat r_k(s_{4,1},2)+b_k^{rw}(s_{4,1},2)\geq 1/12$. Hence conditioned under that event, we have
\begin{align*}
\bar Q_k(s_{4,1},2) &= \frac{1}{4} +b_k^{rw}(s_{4,1},2)+b_k(s_{4,1},2)+\max_{j\in\{2,\dots n+1}\sum_{i=5}^6 b_k^{rw}(s_{i,j},1)+b_k(s_{i,j},1)\\
&\geq \frac{1}{4}+ c\sqrt{\frac{\log(MN/(n\delta))}{N/n}}
\geq \frac{1}{4} + 4 \epsilon\,.
\end{align*}
The proof is completed for $n_0 = 48^2$.
\end{proof}
We can show the same for the upper part of the MDP.
\begin{lemma}
\label{lem:det_low_bound2}
There exists $n_0,\epsilon_0$ such that for any pair of $n\geq n_0$ and $\epsilon\leq\epsilon_0$ and any $k\leq N$, with probability at least $1-\delta$, it holds that either $n_{k}(s_{1,2}) < N/4$, or $\bar Q_k(s_{1,1},2)<\bar Q_k(s_{1,1},1)$.
\end{lemma}
\begin{proof}
First we split $\bar Q_k(s_{1,1},2)$ into the observed sum of mean rewards and bonuses from $s_{1,1}$ to $s_{5,2}$ and the value $\bar V_k(s_{5,2})$.
Then we upper bound $\bar Q_k(s_{1,1},1)$ by $\bar V_k(s_{5,2})$ and the maximum observed sum of mean rewards and bonuses along the paths passing by $s_{3,j}$ for $j\in[n]$.
Finally analogous to the proof of Lemma~\ref{lem:det_low_bound1}, it is straightforward show that the latter is always larger as long as the visitation count for $s_{2,2}$ exceeds $N/4$.
\end{proof}
\begin{theorem}
\label{thm:det_lower_bound}
There exists an MDP instance with deterministic transitions on which any optimistic algorithm with confidence parameter $\delta$ will incur expected regret of at least $\Omega(S\log(\delta^{-1})/\epsilon))$ while it is asymptotically possible to achieve $\Omega(\log(K)/\epsilon)$ regret.
\end{theorem}
\begin{proof}
Taking the MDP from Figure~\ref{fig:mdp_det_lower}.
Applying Lemma~\ref{lem:det_low_bound1} and \ref{lem:det_low_bound2} shows that after $N$ episodes with probability at least $1-2\delta$, the visitation count of $s_{2,2}$ and $s_{5,1}$ each do not exceed $N/4$.
Hence there are at least $N/2$ episodes in which neither of them is visited, which means an $\epsilon$-suboptimal policy is taken. Hence the expected regret after $N$ episodes is at least
\begin{align*}
(1-2\delta)\epsilon N/2 = \Omega\left(\frac{S\log(\delta^{-1})}{\epsilon}\right)\,.
\end{align*}
\end{proof}
Theorem~\ref{thm:det_lower_bound} has two implications for optimistic algorithms in MDPs with deterministic transitions.
\begin{itemize}
\item It is impossible to be asymptotically optimal if the confidence parameter $\delta$ is tuned to the time horizon $K$.
\item It is impossible to have an anytime bound matching the information-theoretic lower bound.
\end{itemize}
\section{Some thoughts}
\subsection{Results for Recent Action Elimination Algorithm?}
Here are some random sketches on whether we could say anything about the action elimination algorithm by \citet{xu2021fine}.
Their algorithm uses upper- and lower-confidence bounds $\bar Q, \underline{Q}$ on $Q^\star$ in each state-action pair which are computed by a mix of (the usual) optimistic bootstrap and Monte-Carlo estimate.
\begin{definition}[Range functions]
For all $s \notin G_k$ and $a \in A_k(s)$, we define
\begin{align*}
\Delta Q_k(s, a) &= \alpha^0_{n_k} H
+ 4 b_{n_{k}}(s, a)
+ \sum_{t=1}^{n_{k}} \alpha^t_{n_k} \Delta V_{k[t]}(x'_{k[t]})\\
\Delta \ddot Q_k(s, a) &= \alpha^0_{n_k} H
+ \clip\left[4 b_{n_{k}}(s, a) \mid \epsilon_k(s,a)\right]
+ \sum_{t=1}^{n_{k}} \alpha^t_{n_k} \Delta \ddot V_{k[t]}(x'_{k[t]})\\
\Delta V_k(s) &= \Delta Q_k(s, a_k)
\qquad \qquad \Delta \ddot V_k(s) = \Delta \ddot Q_k(s, a_k)
\end{align*}
where $a_k = \argmax_{a \in A_k(s)} \bar Q_k(s, a) - \underline Q_k(s,a)$.
\end{definition}
Their Lemma~B.3 shows that the difference of lower and upper-bounds are bounded by this range function:
\begin{align*}
\Delta Q_k(s, a) &\geq \bar Q_k(s, a) - \underline Q_k(s, a)\\
\Delta V_k(s) &\geq \bar V_k(s) - \underline V_k(s)
\end{align*}
for all $s,a$ (there seems to be some ambiguity here on what these range function should be for states in $G_k$.
The proof of Lemma~B.6 gives
\begin{align*}
\EE_k[V^*(S_1) - V^{\pi_k}(S_1)]
= \EE_k\left[ \sum_{h=B}^H \gap(S_h, A_h)\right]
\leq 2\EE_{\pi_k}\left[ \sum_{h=B}^H \Delta Q_k(S_h, A_h)\indicator{S_h \notin G_k}\right]~.
\end{align*}
In order to get a similar result for the half-clipped range function, we need to show the following for our chosen clipping thresholds
\begin{align}
\sum_{k=1}^T \EE_{\pi_k}\left[\sum_{h=B}^H \indicator{S_h \notin G_k}
(\Delta Q_k(S_h, A_h) - \Delta \ddot Q_k(S_h, A_h))\right] \leq
\sum_{k=1}^T \EE_{\pi_k}\left[ \sum_{h=1}^H \gap(S_h, A_h)\right]
\end{align}
\begin{align*}
\Delta Q_k(s, a) - \Delta \ddot Q_k(s, a)
\leq \epsilon_k(s,a) + \sum_{t=1}^{n_{k}} \alpha^t_{n_k} (\Delta V_{k}(x'_{k[t]}) - \Delta \ddot V_{k}(x'_{k[t]}))
\end{align*}
\tm{More attempts}
Lemma 4.4 in \cite{xu2021fine} seems to imply
\begin{align*}
\EE_{k}[V^*(S_1) - V_k(S_1)|\mathcal{E}_{k-1}] \leq 2\EE_{k}\left[\sum_{h=B}^H (\bar Q_k(S_h,A_h) - \underline Q_k(S_h,A_h))\chi(S_h\not\in G_k)\vert \mathcal{E}_{k-1}\right],
\end{align*}
where $\mathcal{E}_{k-1}$ is some nice event which \cite{xu2021fine} define. Our \pref{lem:clipping_gaps_rel} together with \pref{lem:gap_decomp_pi} imply that
\begin{align*}
\frac{1}{2}\EE_{k}[V^*(S_1) - V_k(S_1)|\mathcal{E}_{k-1}] \geq \EE_k\left[\sum_{h=B}^H \epsilon_k(S_h,A_h)\right]
\end{align*}
We can now bound the half-clipped range function from below as \cd{this doesn't work}
\begin{align*}
&\EE_{\pi_k} \left[\sum_{h=1}^H \Delta \ddot Q_k(S_h, A_h) \indicator{S_h \notin G_k}\right]\\
& \geq
\EE_{\pi_k} \left[\sum_{h=B}^H \Delta \ddot Q_k(S_h, A_h) \indicator{S_h \notin G_k}\right]\\
& \geq
\EE_{\pi_k} \left[\sum_{h=B}^H (\Delta Q_k(S_h, A_h) - H \epsilon_k(S_h, A_h))\indicator{S_h \notin G_k}\right]\\
& =
\EE_{\pi_k} \left[\sum_{h=B}^H \Delta Q_k(S_h, A_h)\indicator{S_h \notin G_k} \right]
- H \EE_{\pi_k} \left[\sum_{h=B}^H \epsilon_k(S_h, A_h))\right],
\end{align*}
where the first inequality is true because $\Delta \ddot Q_k(S_h, A_h) \indicator{S_h \notin G_k} \geq 0$ for any $h \in [H]$. The second inequality follows from
The additional $H$ factor in the second term is likely gonna hurt us but maybe this is a start. Using \eqref{eqn:range_gap_bound}, we can lower-bound the first term as
\begin{align*}
\EE_{\pi_k}[V^*(S_1) - V^{\pi_k}(S_1)]
= \EE_{\pi_k}\left[ \sum_{h=B}^H \gap(S_h, A_h)\right]
\leq 2\EE_{\pi_k}\left[ \sum_{h=B}^H \Delta Q_k(S_h, A_h)\indicator{S_h \notin G_k}\right]~.
\end{align*}
Combining them gives
\begin{align*}
&2\EE_{\pi_k} \left[\sum_{h=1}^H \Delta \ddot Q_k(S_h, A_h) \indicator{S_h \notin G_k}\right]
\geq \EE_{\pi_k} \left[\sum_{h=B}^H (\gap(S_h, A_h) - 2H \epsilon_k(S_h, A_h))\right]~.
\end{align*}
Using the self-normalizing trick we get for any thresholds that satisfy
$ \EE_{\pi_k} \left[\sum_{h=B}^H \epsilon_k(S_h, A_h))\right]
\leq \frac{1}{4H}\EE_{\pi_k} \left[\sum_{h=B}^H \gap(S_h, A_h) \right]$
\begin{align*}
\EE_{\pi_k}[V^*(S_1) - V^{\pi_k}(S_1)]
&=\EE_{\pi_k} \left[\sum_{h=B}^H \gap(S_h, A_h) \right] - \frac{1}{2}\EE_{\pi_k} \left[\sum_{h=B}^H \gap(S_h, A_h) \right]
\\
&\leq \EE_{\pi_k} \left[\sum_{h=B}^H \gap(S_h, A_h) \right] - 2H \EE_{\pi_k} \left[\sum_{h=B}^H \epsilon_k(S_h, A_h) \right]
\\
& \leq
2\EE_{\pi_k} \left[\sum_{h=1}^H \Delta \ddot Q_k(S_h, A_h) \indicator{S_h \notin G_k}\right]
\end{align*}
\subsection{Correctness things to be double-checked}
\begin{enumerate}
\item Corollary 8 needs to be stated differently or proved differently. We do pay a gap in state-action pairs that can be visited by some suboptimal policy even if there is an optimal policy passing through it.
See also the discussion around $\widetilde \gap$ vs. $\overline{\gap}$ below.
\item Figure out $V^* \leq 1$ vs $V^* < 1$ issue in general lower-bound
\item Avoid uniqueness assumptions and notation of optimal policies in the upper-bounds
Double-check the bounds stated in the table in Figure 1
\item Clean up and ensure consistency in Appendix C.3 ("value function" vs. "return")
Double-check the simplified optimization lemma in the main text matches the one in the appendix.
\end{enumerate}
\subsection{Suggestions}
\paragraph{Why policy-dependent gap?} What is the benefit for using a policy-dependent gap? We argue that this helps in Figure 1 but the improved clipping is also sufficient to give the desired rate in Figure 1 (and in the example in D.2). I currently do not know any concrete example where optimizing over $\pi^*_k$ is beneficial. If there is one, we should include it in the paper and otherwise remove this optimization.
\paragraph{State full-support lower bound more generally} State the full-support bound in the more general way and have the condition in the sum -- similar to what we have in the table in Figure 1?
\paragraph{Some generalization / equivalent to notion of return gap in stochastic MDPs?}
Let $B = \min\{ h \in [H+1] \colon \gap(S_h, A_h) > 0 \}$ be the first time a non-zero gap is encountered. Note that $B$ is a stopping time w.r.t. the filtration $\mathcal{F}_h = \sigma(S_1, A_1, \dots, S_h, A_h)$. Further let
\begin{align}
\mathcal{B}(s,a) \equiv \{B \leq \kappa(s), S_{\kappa(s)} = s, A_{\kappa(s)} = a\}
\end{align}
be the event that $(s,a)$ was visited after a non-zero gap in the episode.
We now define this notion of average gap:
\begin{align}
\widetilde{\gap}_k(s,a) \equiv
\begin{cases}
\frac{1}{H}
\EE_{\pi_k}\left[ \sum_{h=1}^H \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a) \right]
& \textrm{if }\PP_{\pi_k}(\mathcal{B}(s,a)) > 0\\
\infty & \textrm{otherwise}
\end{cases}
\end{align}
The condition for the first case ensures that the conditional expectation is unique.
For any suboptimal state-action pair $(s,a)$ (i.e. that satisfies $\gap(s,a) >0$), the expression in the first case simplifies to just $\frac{1}{H}
\EE_{\pi_k}\left[ \sum_{h=1}^H \gap(S_h, A_h) ~ \bigg| ~S_{\kappa(s)} = s, A_{\kappa(s)} = a \right]$, the average gap conditioned on the policy visiting $(s,a)$.
Using this definition, we can state our main result as
\begin{theorem}[Informal]
The regret $\mathfrak{R}(K)$ of \texttt{StrongEuler} is bounded as
\begin{align*}
\mathfrak{R}(K) \lessapprox&~
\sum_{s,a} \min_{k \in [K]} \left\{
\frac{\mathcal{V}^*(s,a)}{\gap(s,a) + \widetilde{\gap}_k(s,a)}
+ \sqrt{\mathcal{V}^*(s,a) (K - k)}\right\} \log(K)
\\&
+ \sum_{s,a} SH^3 \log\frac{MK}{\delta} \min\left\{ \log \frac{MK}{\delta}, \log \frac{MH}{\gap(s,a) + \widetilde \gap(s,a)}\right\}
\\& + SAH^3 (S \vee H)\log\frac{M}{\delta}.
\end{align*}
\end{theorem}
We can also remove the dependency on the random policy $\pi_k$ by taking the minimum over all policies in the gap definition as
\begin{align}
\widetilde{\gap}(s,a) \equiv \min_{\substack{\pi \in \Pi \colon \\\PP_{\pi}(\mathcal{B}(s,a)) > 0}}&~
\frac{1}{H}
\EE_{\pi}\left[ \sum_{h=1}^H \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a) \right]\\
= \min_{\substack{\pi \in \Pi \colon \\\PP_{\pi}(\mathcal{B}(s,a)) > 0}}&~
\frac{1}{H}
\EE_{\pi}\left[ \sum_{h=1}^{\kappa(s)} \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a) \right]
\end{align}
Then the main regret upper-bound becomes
\begin{align*}
\mathfrak{R}(K) \lessapprox&~
\sum_{s,a}
\frac{\mathcal{V}^*(s,a)}{\gap(s,a) + \widetilde{\gap}(s,a)} \log(K)
\\&
+ \sum_{s,a} SH^3 \log\frac{MK}{\delta} \min\left\{ \log \frac{MK}{\delta}, \log \frac{MH}{\gap(s,a) + \widetilde \gap(s,a)}\right\}
\\& + SAH^3 (S \vee H)\log\frac{M}{\delta}.
\end{align*}
Comparison to other notions of gap:
To see how much we might have lost when removing the policy-dependency, we compare this gap to other notions:
\begin{align}
\widetilde \gap(s,a) & > 0\\
\widetilde \gap(s,a) &\geq \frac{\gap(s,a)}{H}\\
\widetilde \gap(s,a) &\geq \frac{\gap_{\min}}{H}\\
\widetilde \gap(s,a) & = \frac{\overline{\gap}(s,a)}{H} \quad \textrm{when } \overline{\gap}(s,a) > 0\\
\widetilde \gap(s,a) & \geq \frac{\overline{\gap}(s,a)}{H}
\end{align}
\paragraph{Tightness of upper-bound in MDPs with deterministic transitions}
There is a discrepancy between $\widetilde \gap(s,a)$ and $\overline{\gap}(s,a)$ because $\widetilde \gap(s,a)$ takes the minimum over all policies that visit $(s,a)$ and have made a mistake before but $\overline{\gap}(s,a)$ only takes the minimum over all policies that visit $(s,a)$. When $(s,a)$ can be visited through an optimal and a suboptimal policy, then $\overline{\gap}(s,a) = 0$ but $0 < \widetilde \gap(s,a) < \infty$. Those pairs thus appear in the upper-bound but not in the regret lower-bound. Note that this is not an issue of writing things as $\widetilde \gap$ but also occurs in our results as currently written.
Essentially there are 3 cases in MDPs with deterministic transitions:
\begin{itemize}
\item The state-action pair can only be visited by a suboptimal policy: We are tight with the lower-bound up to a $H^2$ factor. \text{\ding{51}}
\item The state-action pair cannot be visited by any suboptimal policy: Neither our lower-bound nor our upper bound pays for such pairs. \text{\ding{51}}
\item The state-action pair can be visited by an optimal and a suboptimal policy: These pairs do not appear in the lower-bound but we pay for them with average gap across the episode in the upper-bound.
\text{\ding{55}}
\end{itemize}
I think we might be able to tighten our upper-bound regret analysis to show that we can also clip state-action pairs of the 3rd case to $\infty$ in deterministic MDPs.
A key property we need to prove and use for this is
\begin{align}
\EE_{\pi_k}\left[ \sum_{h=B}^{h'} E_k(S_h, A_h) \right] \geq
\EE_{\pi_k}\left[ \sum_{h=B}^{h'} \gap(S_h, A_h) \right] \end{align}
for all $h' \in [H]$ (or maybe only as long as $\pi_k$ is optimal from $h'$ on?). Recall the surplus definition $E_k(s,a) = \bar Q_k(s,a) - r(s,a) - \langle P(\cdot | s,a) , \bar V_k \rangle$ and the form $\bar Q_k$ which is
\begin{align*}
\bar Q_k(s,a) = H \wedge \widehat r(s,a) + \langle \widehat P(\cdot | s,a), \bar V_k \rangle + \textrm{bonus}(s,a)
\end{align*}
The usual notion of optimism gives this inequality for $h' = H$.
\paragraph{Deriving the key property for deterministic MDPs:} Let the bonus added at episode $k$ for state-action pair $(s,a)$ be $b_k(s,a)$.
For a deterministic policy $\pi$, denote the (deterministic) trajectory of state-action pairs be $s_1^\pi, a_1^\pi, \dots, s_H^\pi, a_H^\pi$.
We assume that there is a layer $h'$ such that there is a $\hat \pi \in \Pi$ with $V^{\hat \pi} = V^*$ and $(s_h^{\hat \pi}, a_h^{\hat \pi}) = (s_h^{\pi_k}, a_h^{\pi_k})$ for all $h \geq h'$.
Because we are considering deterministic transitions we can write
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^{h'} \gap(S_h,A_h)\right] &= \sum_{h=B}^{h'} \gap(s_h^{\pi_k},a_h^{\pi_k})\\
\EE_{\pi_k}\left[ \sum_{h=B}^{h'} E_k(S_h, A_h) \right] &= \sum_{h=B}^{h'} E_k(s_h^{\pi_k}, a_h^{\pi_k})
\end{align*} where where $s_h^{\pi}$ denotes the state visited by $\pi$ at time $h$. Note that $B$ is also fixed given $\pi_k$ in deterministic MDPs.
Euler computes the optimistic Q-function as $\bar Q_k(s,a) = H \wedge \hat r_k(s,a) + \langle \hat P_k(\cdot|s,a), \bar V_k \rangle + b_k(s,a)$.
There are two cases which we treat separately:
\paragraph{Case 1: $(s,a)$ has been visited often enough so that $\bar Q_k(s,a) = \hat r_k(s,a) + \langle \hat P_k(\cdot|s,a), \bar V_k \rangle + b_k(s,a)$:}
Since the MDP is deterministic, we have $\hat P_k(\cdot|s,a) = P(\cdot|s,a)$. This implies that $E_k(s,a) = \hat r_k(s,a) - r(s,a) + b_k(s,a)$. Further, we can write the sum of the gaps as
\begin{align*}
\sum_{h=1}^{h'} \gap(s^{\pi_k}_h,a^{\pi_k}_h) &= V^{*}(s^{\pi_k}_1) - V^{\pi_k}(s^{\pi_k}_1) - \left(V^{*}(s^{\pi_k}_{h'+1}) - V^{\pi_k}(s^{\pi_k}_{h'+1})\right)\\
&= V^{*}(s^{\pi_k}_1) - V^{\pi_k}(s^{\pi_k}_1)\\
&= \sum_{h=B}^{H} (r(s_h^{\hat \pi},a_h^{\hat \pi}) - r(s_h^{\pi_k},a_h^{\pi_k}))
= \sum_{h=B}^{h'} (r(s_h^{\hat \pi},a_h^{\hat \pi}) - r(s_h^{\pi_k},a_h^{\pi_k})).
\end{align*}
Because we have picked $\pi_k$ at episode $k$ it further must hold that $\bar V^{\pi_k}(s_1) \geq \bar V^{\hat \pi}(s_1)$ but what is more it must hold that $\sum_{h=1}^{h'} \hat r(s_h^{\pi_k},a_h^{\pi_k}) + b_{k}(s_h^{\pi_k},a_h^{\pi_k}) \geq \sum_{h=1}^{h'} \hat r(s_h^{\hat \pi},a_h^{\hat \pi}) + b_{k}(s_h^{\hat \pi},a_h^{\hat \pi})$ as otherwise the policy which follows $\hat \pi$ up to $h'$ and $\pi_k$ from $h'+1$ to $H$ would have higher value function in the empirical MDP \cd{this assumes that all sa along the paths of both policies have been visited often enough so that the optimistic Q-function does not get clipped}. Further, because the bonuses include optimism of the rewards it holds that $\sum_{h=1}^{h'} \hat r(s_h^{\hat \pi},a_h^{\hat \pi}) + b_{k}(s_h^{\hat \pi},a_h^{\hat \pi}) \geq \sum_{h=1}^{h'} r(s_h^{\hat \pi},a_h^{\hat \pi})$ with high probability. This implies
\begin{align*}
\sum_{h=1}^{h'}\gap(s_h^{\pi_k},a_h^{\pi_k}) &= \sum_{h=1}^{h'} r(s_h^{\hat \pi},a_h^{\hat \pi}) - r(s_h^{\pi_k},a_h^{\pi_k}) \leq \sum_{h=1}^{h'} \hat r(s_h^{\pi_k},a_h^{\pi_k}) - r(s_h^{\pi_k},a_h^{\pi_k}) + b_{k}(s_h^{\pi_k},a_h^{\pi_k})\\
&=\sum_{h=1}^{h'} E_k(s_h^{\pi_k},a_h^{\pi_k}).
\end{align*}
\paragraph{Case 2: $(s,a)$ has not been visited often enough so that $\bar Q_k(s,a) = H < \hat r(s,a) + \langle \hat P(\cdot|s,a), \bar V_k \rangle + b_k(s,a)$:}
\cd{We need to handle this case as well. Clipping to $H$ is somewhat non-optional in Euler and ideally we can handle clipping here directly. If this turns out to be difficult, we may bound the number of episodes where a clipped optimistic Q-function could occur. This could be made easier by clipping to 2H instead (which should also be sufficient) and bound the number of times until $b_k(s,a) \leq H$.}
\paragraph{Both cases at the same time?} Let $\bar \pi_k$ be the policy which follows $\hat\pi$ up to and including layer $h'$, and from $h'+1$ to $H$ it follows $\pi_k$. Then we know that it must hold $\bar V^{\bar\pi}(s_{h'+1}) \geq \bar V^{\hat\pi}(s_{h'+1})$ as otherwise the policy which follows $\pi_k$ until $h'$ and then follows $\hat\pi$ would have higher value function than $\pi_k$. Suppose we can show that $\sum_{h=1}^{h'} r(s_h^{\hat \pi},a_h^{\hat \pi}) \leq \bar V^{\bar \pi}(s_1) - \bar V^{\bar\pi}(s_{h'+1}^{\pi_k})$. Then we could write
\begin{align*}
\sum_{h=1}^{h'} r(s_h^{\hat \pi},a_h^{\hat \pi}) - r(s_h^{\pi_k},a_h^{\pi_k})
& \leq
\bar V^{\bar \pi}(s_1) - \bar V^{\bar \pi}(s_{h'+1}^{\pi_k}) - \sum_{h=1}^{h'} r(s_h^{\pi_k},a_h^{\pi_k})\\
&= \bar V^{\bar \pi}(s_1) - \bar V_k(s_{h'+1}^{\pi_k}) - \sum_{h=1}^{h'} r(s_h^{\pi_k},a_h^{\pi_k})\\
&\leq \bar V_k(s_1) - \bar V_k(s_{h'+1}^{\pi_k}) - \sum_{h=1}^{h'} r(s_h^{\pi_k},a_h^{\pi_k}).
\end{align*}
Further, it holds
\begin{align*}
\sum_{h=1}^{h'} E_k(s_h^{\pi_k},a_h^{\pi_k}) &= \sum_{h=1}^{h'} \bar V_k(s_h^{\pi_k}) - r(s_h^{\pi_k},a_h^{\pi_k}) - \langle P(\cdot|s_h^{\pi_k},a_h^{\pi_k}), \bar V_k \rangle\\
&=\sum_{h=1}^{h'} \bar V_k(s_h^{\pi_k}) - \sum_{h=2}^{h'+1} \bar V_k(s_h^{\pi_k}) - \sum_{h=1}^{h'} r(s_h^{\pi_k},a_h^{\pi_k})\\
&= \bar V_k(s_1) - \bar V_k(s_{h'+1}^{\pi_k})- \sum_{h=1}^{h'} r(s_h^{\pi_k},a_h^{\pi_k}).
\end{align*}
Combining with the inequality on the sum of rewards we arrive at the desired result. To show $\sum_{h=1}^{h'} r(s_h^{\hat \pi},a_h^{\hat \pi}) \leq \bar V^{\bar \pi}(s_1) - \bar V^{\bar\pi}(s_{h'+1}^{\pi_k})$ argue as follows. Let $\tilde h$ be the smallest layer for which $\bar V^{\bar \pi}(s_{\tilde h}^{\hat\pi}) = H - \tilde h + 1$. Because $\bar V^{\bar\pi}(s_{h'+1}^{\pi_k}) \leq H-h'$ we have
\begin{align*}
\bar V^{\bar \pi}(s_1) - \bar V^{\bar\pi}(s_{h'+1}^{\pi_k}) &= \bar V^{\bar \pi}(s_1) - \bar V^{\bar \pi}(s_{\tilde h}^{\hat\pi}) + \bar V^{\bar \pi}(s_{\tilde h}^{\hat\pi}) - \bar V^{\bar \pi}(s_{h'+1}^{\hat\pi})\\
&\geq \bar V^{\bar \pi}(s_1) - \bar V^{\bar \pi}(s_{\tilde h}^{\hat\pi}) + h' - \tilde h + 1 \geq \bar V^{\bar \pi}(s_1) - \bar V^{\bar \pi}(s_{\tilde h}^{\hat\pi}) + \sum_{h=\tilde h}^{h'} r(s_h^{\hat\pi},a_h^{\hat\pi}).
\end{align*}
Finally by our definition of $\tilde h$ it must hold that for all $h < \tilde h$, $\hat r(s_h^{\hat\pi},a_h^{\hat\pi}) + b_k(s_h^{\hat\pi},a_h^{\hat\pi}) \leq 1$ and hence we can write $\bar V^{\bar\pi}(s_1) = \sum_{h=1}^{\tilde h-1} \hat r(s_h^{\hat\pi},a_h^{\hat\pi}) + b_k(s_h^{\hat\pi},a_h^{\hat\pi}) + \bar V^{\bar \pi}(s_{\tilde h}^{\hat\pi})$, which implies
\begin{align*}
\bar V^{\bar \pi}(s_1) - \bar V^{\bar \pi}(s_{\tilde h}^{\hat\pi}) \geq \sum_{h=1}^{\tilde h-1} \hat r(s_h^{\hat\pi},a_h^{\hat\pi}) + b_k(s_h^{\hat\pi},a_h^{\hat\pi}) \geq \sum_{h=1}^{\tilde h-1} r(s_h^{\hat\pi},a_h^{\hat\pi}),
\end{align*}
where the last inequality holds with probability $1-\delta$ because $\hat r(s_h^{\hat\pi},a_h^{\hat\pi}) + b_k(s_h^{\hat\pi},a_h^{\hat\pi}) \geq r(s_h^{\hat\pi},a_h^{\hat\pi})$ for all $h \in [H], (s,a) \in \mathcal{S}\times\mathcal{A}$ by the construction of the bonuses.
It would be interesting if could show something similar for non-deterministic transitions. In particular if there exists a layer $h'$ such that with probability $1$ $\pi_k$ agrees with $\Pi^*$ on every state in the layer then does the inequality between surpluses and gaps still hold up to $h'$?
\paragraph{The argument for non-deterministic transitions:} Assume that at time $h'$ it holds that there exists an optimal policy $\hat \pi$ (not necessarily deterministic), such that $\pi_k$ and $\hat\pi$ have the same distribution over state-action pairs $(s,a)$ at $h'$. Define $\bar\pi$ as in the deterministic transitions MDP case, i.e., $\bar\pi$ matches $\pi_k$ from layer $h'+1$ to $H$ and otherwise follows $\hat\pi$ from layer $1$ to layer $h'$. Suppose we can show $\EE_{\hat\pi}[\sum_{h=1}^{h'}r(S_h,A_h)] \leq \bar V^{\bar\pi}(s_1) - \EE_{\pi_k}[\bar V_k(S_{h'+1})]$. We can write
\begin{align*}
\EE_{\hat\pi}\left[\sum_{h=1}^{h'} r(S_h,A_h)\right] \leq V^{\bar\pi}(s_1) - \EE_{\pi_k}[\bar V_k(S_{h'+1})] \leq \bar V_k(s_1) - \EE_{\pi_k}[\bar V_k(S_{h'+1})].
\end{align*}
Further we have
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=1}^{h'} E_k(S_h,A_h)\right] &= \EE_{\pi_k}\left[\sum_{h=1}^{h'} \bar V_k(S_h) - r(S_h,A_h) - \langle P(\cdot|S_h,A_h), \bar V_k \rangle\right]\\
&= \EE_{\pi_k}\left[\sum_{h=1}^{h'} \bar V_k(S_h) - \EE_{\pi_k}\left[\bar V_k(S_{h+1})|S_h\right]\right] - \sum_{h=1}^{h'}\EE_{\pi_k}[r(S_h,A_h)]\\
&=\bar V_k(s_1) - \EE_{\pi_k}[\bar V_k(S_{h'+1})] - \sum_{h=1}^{h'}\EE_{\pi_k}[r(S_h,A_h)].
\end{align*}
Next we decompose the sum of the gaps as a difference of rewards by using Lemma~\ref{lem:gap_decomp_pi}
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=1}^{h'} \gap(S_h,A_h)\right] &= \EE_{\pi_k}\left[\sum_{h=1}^{H} \gap(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=h'+1}^{H} \gap(S_h,A_h)\right]\\
&= V^{\hat\pi}(s_1) - V_k(s_1) - \EE_{\pi_k}\left[\EE_{\pi_k}\left[\sum_{h=h'+1}^{H} \gap(S_h,A_h)|S_{h'+1}=s_{h'+1}\right]\right]\\
&= \EE_{\hat\pi}\left[\sum_{h=1}^H r(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=1}^H r(S_h,A_h)\right]\\
&- \EE_{\pi_k}\left[\EE_{\pi_k}\left[V^{\hat\pi}(s_{h'+1})|S_{h'+1}=s_{h'+1}\right] -\EE_{\pi_k}\left[ V_k(s_{h'+1})|S_{h'+1}=s_{h'+1}\right]\right]\\
&\overset{(i)}{=} \EE_{\hat\pi}\left[\sum_{h=1}^H r(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=1}^H r(S_h,A_h)\right]\\
&-\EE_{\hat\pi}\left[V^{\hat\pi}(S_{h'+1})\right] + \EE_{\pi_k}\left[V_k(S_{h'+1})\right]\\
&=\EE_{\hat\pi}\left[\sum_{h=1}^H r(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=1}^H r(S_h,A_h)\right]\\
&-\EE_{\hat\pi}[\sum_{h=h'+1}^H r(S_h,A_h)] + \EE_{\pi_k}[\sum_{h=h'+1}^H r(S_h,A_h)]\\
&= \EE_{\hat\pi}\left[\sum_{h=1}^{h'} r(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=1}^{h'}r(S_h,A_h)\right],
\end{align*}
where $(i)$ follows from the fact that $\pi_k$ and $\hat\pi$ have the same distribution over state-action pairs in layer $h'$ and hence $\PP_{\hat\pi}(S_{h'+1} = s_{h'+1}) = \PP_{\pi_k}(S_{h'+1} = s_{h'+1})$.
Combining the above with the inequality on difference of rewards and the equality for surpluses we have
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=1}^{h'} \gap(S_h,A_h)\right] &= \EE_{\hat\pi}\left[\sum_{h=1}^{h'} r(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=1}^{h'}r(S_h,A_h)\right]\\
&\leq \bar V_k(s_1) - \EE_{\pi_k}[\bar V_k(S_{h'+1})] - \sum_{h=1}^{h'}\EE_{\pi_k}[r(S_h,A_h)]\\
&=\EE_{\pi_k}\left[\sum_{h=1}^{h'} E_k(S_h,A_h)\right].
\end{align*}
Next we try to show $\EE_{\hat\pi}[\sum_{h=1}^{h'}r(S_h,A_h)] \leq \bar V^{\bar\pi}(s_1) - \EE_{\pi_k}[\bar V_k(S_{h'+1})]$ or equivalently $\EE_{\bar\pi}[\sum_{h=1}^{h'}r(S_h,A_h)] + \EE_{\bar\pi}[\bar V^{\bar\pi}(S_{h'+1})]\leq \bar V^{\bar\pi}(s_1)$. Fix $(s_h,a_h) \in \bar\pi$ and consider
\begin{align*}
r(s_h,a_h) + \mathbb{E}_{\bar\pi}[\bar V^{\bar\pi}(S_{h+1}) ~|~ S_h = a_h, A_h = a_h] = r(s_h,a_h) + \langle P(\cdot|s_h,a_h),\bar V^{\bar\pi} \rangle.
\end{align*}
It either holds that $\bar V^{\bar\pi}(s_h) = H-h+1$ or $\bar V^{\bar\pi}(s_h) = \hat r(s_h,a_h) + b_k^{rw}(s_h,a_h) + b_k^{prob}(s_h,a_h)+ b_k^{str}(s_h,a_h) + \langle \hat P(\cdot|s_h,a_h), \bar V^{\bar\pi} \rangle$.
Suppose that $\bar V^{\bar\pi}(s_h) = H-h+1$, then we must have $r(s_h,a_h) \leq 1 \leq \bar V^{\bar\pi}(s_h) - \langle P(\cdot|s_h,a_h),\bar V^{\bar\pi} \rangle$. Otherwise we have
\begin{align*}
r(s_h,a_h) &\leq \hat r(s_h,a_h) + b_k^{rw}(s_h,a_h) = \bar V^{\bar\pi}(s_h) - b_k^{prob}(s_h,a_h) - b_k^{str}(s_h,a_h) - \langle \hat P(\cdot|s_h,a_h), \bar V^{\bar\pi} \rangle\\
&=\bar V^{\bar\pi}(s_h) - \langle P(\cdot|s_h,a_h), \bar V^{\bar\pi} \rangle\\
&+ \langle P(\cdot|s_h,a_h) - \hat P(\cdot|s_h,a_h), \bar V^{\bar\pi} \rangle - b_k^{prob}(s_h,a_h) - b_k^{str}(s_h,a_h).
\end{align*}
By definition of $\bar V^{\bar\pi}$ it holds that $\bar V^{\hat\pi}(s_h) \leq \bar V^{\bar\pi}(s_h) \leq \bar V_k(s_h)$ on all $s_h$, because $\pi_k$ is the Bellman optimal policy with respect to the empirical MDP. This implies we can use Lemma F.2 in \citep{simchowitz2019non} to show that $|\langle P(\cdot|s_h,a_h) - \hat P(\cdot|s_h,a_h), \bar V^{\bar\pi} \rangle| \leq b_k^{prob}(s_h,a_h)$ and thus $r(s_h,a_h) \leq \bar V^{\bar\pi}(s_h) - \langle P(\cdot|s_h,a_h), \bar V^{\bar\pi} \rangle$. Combining everything, we have
\begin{align*}
\EE_{\bar\pi}\left[\sum_{h=1}^{h'}r(S_h,A_h)\right] + \EE_{\bar\pi}[\bar V^{\bar\pi}(S_{h'+1})]&= \EE_{\bar\pi}\left[\sum_{h=1}^{h'-1}r(S_h,A_h)\right] + \EE_{\bar\pi}[r(S_{h'},A_{h'})+ \langle P(\cdot|S_{h'},A_{h'}),\bar V^{\bar\pi}\rangle]\\
&\leq \EE_{\bar\pi}\left[\sum_{h=1}^{h'-1}r(S_h,A_h)\right] + \EE_{\bar\pi}[\bar V^{\bar\pi}(S_{h'})] \leq \ldots \leq \bar V^{\bar\pi}(s_1).
\end{align*}
\subsection{Extension to linear MDPs?}
I do not think the clipped surplus bound we derive relies on finite state-spaces. We could also think about what a good definition of clipping threshold $\epsilon_k$ would be in linear MDPs \citep{jin2020provably}.
From a quick look, I think Lemma~B.4 by \citet{jin2020provably} ensures that the surpluses of their LSVI-UCB algorithm satisfy
\begin{align}
0 \leq E_k(s,a) \leq 2 \beta \sqrt{\phi(s,a)^\top \Lambda_k^{-1} \phi(s,a)}.
\end{align}
This means their algorithm is strongly optimistic and our surplus clipping bound from \pref{prop:surplus_clipping_bound} gives
\begin{align}
&V^\star(s_1) - V^{\pi_k}(s_1)\\
&\leq 4 \EE_{\pi_k} \left[ \sum_{h=1}^H \clip \left[ E_k(S_h,A_h) ~ \bigg| ~ \frac{1}{4}\gap(S_h,A_h) \vee \epsilon_k(S_h,A_h) \right] \right]\\
& \leq
4\EE_{\pi_k} \left[ \sum_{h=1}^H \clip \left[ 2 \beta \sqrt{\phi(S_h,A_h)^\top \Lambda_k^{-1} \phi(S_h,A_h)} ~ \bigg| ~ \frac{1}{4}\gap(S_h,A_h) \vee \epsilon_k(S_h,A_h) \right] \right].
\end{align}
Assume for now, we settle for $\epsilon(s,a) = \inf_k (\gap(s,a) \vee \epsilon_k(s,a) )$, a policy independent clipping threshold. Then we have to bound an expression of the form
\begin{align*}
\mathfrak{R}(T)
\lesssim \sum_{h=1}^H \sum_{k=1}^T \EE_{\pi_k} \left[ \clip\left[\sqrt{\phi(S_h,A_h)^\top \Lambda_k^{-1} \phi(S_h,A_h)} ~ \bigg| ~ \epsilon(S_h,A_h) \right] \right]
\end{align*}
where $\Lambda_k$ is the regularized covariance matrix with all features up to episode $k-1$.
Handling this expression is not trivial but having a good gap-dependent bound for linear MDPs would certainly be a nice contribution.
\paragraph{Idea how to proceed:}
One could possibly get a $\gap_{\min}$ bound by noticing that
\begin{align*}
\EE_{\pi_k} \left[ \clip\left[\sqrt{\phi(S_h,A_h)^\top \Lambda_k^{-1} \phi(S_h,A_h)} ~ \bigg| ~ \epsilon(S_h,A_h) \right] \right] &\leq \EE_{\pi_k} \left[\frac{\phi(S_h,A_h)^\top \Lambda_k^{-1} \phi(S_h,A_h)}{\epsilon(S_h,A_h)} \right]\\
&\leq \EE_{\pi_k} \left[\frac{\phi(S_h,A_h)^\top \Lambda_k^{-1} \phi(S_h,A_h)}{\gap_{\min}} \right],
\end{align*}
and then argue similarly to how the log-det lemmas are proved for the contextual linear bandits problems.
Taking this idea a little further -- instead of replacing all $\epsilon(S_h,A_h)$ by $\gap_{\min}$ we can partition the gaps into $O(\log(T))$ ranges as follows. Define a grid on the interval $[1,T]$ (could also be $[1,\sqrt{T}]$) such that there are $O(\log(T))$ partitions. The first partition includes all gaps in $[1/2,1]$, the second all gaps in $[1/4,1/2]$, etc., where the last partition includes gaps in $[2/T, T]$. Now we can replace $\epsilon(S_h,A_h)$ by the lower limit of the partition in which it falls e.g., if $\epsilon(S_h,A_h) \in [1/2^{\ell}, 1/2^{\ell+1}]$, then bound
\begin{align*}
\clip\left[\sqrt{\phi(S_h,A_h)^\top \Lambda_k^{-1} \phi(S_h,A_h)} ~ \bigg| ~ \epsilon(S_h,A_h) \right] \leq
\frac{\phi(S_h,A_h)^\top \Lambda_k^{-1} \phi(S_h,A_h)}{1/2^{\ell+1}}.
\end{align*}
Now, we could try to come up with a better version of the log-det lemma which uses the fact that $\epsilon(S_h,A_h) > 1/2^{\ell+1}$ to shave off factors of dimensionality from the sum over terms grouped in the $\ell$-th interval.
\paragraph{The formal argument:} We first note that \pref{lem:Vdd_lb1} only requires that the algorithm be optimistic i.e. $\bar Q_k(s,a) \geq Q^*(s,a), \forall (s,a) \in \mathcal{S}\times\mathcal{A}$. Lemma B.5 in \cite{jin2020provably} guarantees that optimism holds for their algorithm $\textsc{LSVI-UCB}$, where the Q-functions are defined as $\hat Q_k(s,a) := \min(\langle \phi(s,a), \mathbf{w}^{k}_{\kappa(s)} \rangle + \beta\sqrt{\phi(s,a)^\top (\Lambda^k)_{\kappa(s)}^{-1}\phi(s,a)} H)$, where $\mathbf{w}^k_{\kappa(s)}$ are the weights computed by the algorithm at episode $k$ for layer $\kappa(s)$ and $\Lambda^k_{\kappa(s)}$ is the design matrix at episode $k$ for layer $\kappa(s)$. We can now apply \pref{prop:surplus_clipping_bound} with $\epsilon_k(s,a) = \gap_{\min}/H$ to bound the regret at episode $k$ as
\begin{align*}
V^*(s_1) - V^{\pi_k}(s_1) \leq 4\EE_{\pi_k}\left[\sum_{h=B}^H\clip[E_k(S_h,A_h)|\epsilon_k(S_h,A_h)]\right].
\end{align*}
Further we have that
\begin{align*}
E_k(s,a) = \hat Q_k(s,a) - r(s,a) - \langle P(\cdot|s,a),\hat V_k \rangle &\leq \langle \phi(s,a), \mathbf{w}^k_{\kappa(s)} \rangle + \beta\sqrt{\phi(s,a)^\top (\Lambda^k_{\kappa(s)})^{-1}\phi(s,a)}\\
&- r(s,a) - \langle P(\cdot|s,a),\hat V_k\rangle\\
&= \langle \phi(s,a), \mathbf{w}^k_{\kappa(s)} \rangle - Q^*(s,a) - \langle P(\cdot|s,a),\hat V_k - V^*\rangle\\
&+\beta\sqrt{\phi(s,a)^\top (\Lambda^k_{\kappa(s)})^{-1}\phi(s,a)}\\
&\leq 2\beta\sqrt{\phi(s,a)^\top (\Lambda^k_{\kappa(s)})^{-1}\phi(s,a)},
\end{align*}
where the last inequality follows from Lemma B.4 in \citet{jin2020provably}. And so we can bound the regret as
\begin{align*}
\mathfrak{R}(T) &\leq 4\sum_{k=1}^K\sum_{h=1}^H \EE_{\pi_k}\left[\clip\left[2\beta\sqrt{\phi(S_h,A_h)^\top (\Lambda^k_h)^{-1}\phi(S_h,A_h)} \vert \epsilon_k(S,A)\right]\right]\\
&\leq 4\sum_{k=1}^K\sum_{h=1}^H \EE_{\pi_k}\left[\frac{4\beta^2\phi(S_h,A_h)^\top (\Lambda^k_{h})^{-1}\phi(S_h,A_h)}{\epsilon_k(S_h,A_h)}\right]\\
&\leq16H\beta^2\sum_{h=1}^H\sum_{k=1}^K \EE_{\pi_k}\left[\frac{\phi(S_h,A_h)^\top (\Lambda^k_{h})^{-1}\phi(S_h,A_h)}{\gap_{\min}}\right]\\
&\leq \frac{32H^2\beta^2d\log(2dT/\delta)}{\gap_{\min}} = O\left(\frac{d^3H^4\log^2(dT/\delta)}{\gap_{\min}}\right),
\end{align*}
where the last inequality follows from Lemma D.2 in \citet{jin2020provably}, in the following way. Since $\Lambda_h^k$ is a random variable depending on $\pi_{1:k-1}$ we can further take the expectation for a fixed $h$ as
\begin{align*}
\sum_{k=1}^K \EE_{\pi_{1:k}}[\phi(S_h,A_h)^\top (\Lambda^k_{h})^{-1}\phi(S_h,A_h)] = \EE_{\pi_{1:K}}\left[\sum_{k=1}^K\phi(S_h,A_h)^\top (\Lambda^k_{h})^{-1}\phi(S_h,A_h)\right],
\end{align*}
and now apply Lemma D.2 in \citet{jin2020provably} for every realization of state-action pairs at layer $h$ played throughout the $K$ episodes.
\section{Tighter Clipping for Optimal Pairs Reachable Similarly by an Optimal Policy}
\begin{lemma}\label{lem:Oclipping_perfect}
Let $\mathcal{O} \subseteq [H]$ be the set of time steps where for every $h \in \mathcal{O}$ there is an optimal policy $\pi^\star$ such that $\PP_{\pi^\star}(S_h, A_h) = \PP_{\pi_k}(S_h, A_h)$. Then
\begin{align}
\EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h)\right] \leq
\EE_{\pi_k} \left[ \sum_{h = 1}^{H} \indicator{h \notin \mathcal{O}} E_k(S_h, A_h)\right].
\end{align}
\end{lemma}
\cd{I think this may be too strong to be true? I will first show a weaker version:}
\begin{lemma}Assume strong optimism and greedy $\bar V_k$, i.e., $\bar V_k(s) \geq \max_{a} \bar Q_k(s,a)$ for all $s \in \mathcal{S}$.
Let $\mathcal{O} \subseteq [H]$ be the set of time steps such that there is an optimal policy $\pi^\star$ which satisfies $\PP_{\pi^\star}(S_h, A_h) = \PP_{\pi_k}(S_h, A_h)$ for every $h \in \mathcal{O}$. Then
\begin{align}
\EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h)\right] \leq
\EE_{\pi_k} \left[ \sum_{h = 1}^{H} \indicator{h \notin \mathcal{O}} E_k(S_h, A_h)\right].
\end{align}
\end{lemma}
\begin{proof}
We show this by proving that for any $\underline{h}, \bar h \in \mathcal{O}$ with $\underline{h} \leq \bar h$, we have
\begin{align*}
\EE_{\pi_k} \left[\sum_{h=\underline h + 1}^{\bar h - 1} \gap(S_h, A_h)\right] \leq
\EE_{\pi_k} \left[ \sum_{h=\underline h + 1}^{\bar h - 1} E_k(S_h, A_h)\right].
\end{align*}
Since $\gap(S_h ,A_h) = 0$ for $h \in \mathcal{O}$ and we can assume that there is a dummy $H+1 \in \mathcal{O}$, this is sufficient to prove the statement to show.
We now rewrite both sides of the inequality as
\begin{align*}
\EE_{\pi_k} \left[ \sum_{h=\underline h + 1}^{\bar h - 1} E_k(S_h, A_h)\right]
&=
\EE_{\pi_k} \left[
\bar V_k(S_{\underline h + 1}) - \bar V_k(S_{\bar h})
+ \sum_{h=\underline h + 1}^{\bar h - 1} r(S_h, A_h)
\right]\\
\EE_{\pi_k} \left[\sum_{h=\underline h + 1}^{\bar h - 1} \gap(S_h, A_H)\right]
&=
\EE_{\pi_k} \left[ V^\star(S_{\underline h + 1}) - V^{\pi_k}(S_{\underline h + 1}) -
V^\star(S_{\bar h}) + V^{\pi_k}(S_{\bar h})
\right]\\
&=\EE_{\pi_k} \left[ V^\star(S_{\underline h + 1}) -
V^\star(S_{\bar h}) + \sum_{h=\underline h + 1}^{\bar h - 1} r(S_h, A_h)
\right].
\end{align*}
\textcolor{olive}{JZ: isn't there a sign error in front of the rewards in the first line? If I am correct this breaks the whole proof.}
Thus, it suffices to show
$\EE_{\pi_k} \left[ V^\star(S_{\underline h + 1}) -
V^\star(S_{\bar h}) \right] \leq
\EE_{\pi_k} \left[
\bar V_k(S_{\underline h + 1}) - \bar V_k(S_{\bar h})\right]$.
Note that since $P_{\pi_k}(S_{\underline h}, A_{\underline h}) = P_{\pi^\star}(S_{\underline h}, A_{\underline h})$, this also implies that
$P_{\pi_k}(S_{\underline h + 1}) = P_{\pi^\star}(S_{\underline h + 1})$ by the Markov property of the MDP. Hence
\begin{align*}
\EE_{\pi_k} \left[ V^\star(S_{\underline h + 1}) -
V^\star(S_{\bar h}) \right]
=
\EE_{\pi^\star} \left[ V^\star(S_{\underline h + 1}) -
V^\star(S_{\bar h}) \right]
= \EE_{\pi^\star} \left[
\sum_{h=\underline h + 1}^{\bar h - 1} r(S_h, A_h)
\right]
\end{align*}
and all that is left to show is
\begin{align*}
\EE_{\pi^\star}\left[
\bar V_k(S_{\underline h + 1})
\right] \geq
\EE_{\pi^\star}\left[
\sum_{h=\underline h + 1}^{\bar h - 1} r(S_h, A_h)
+
\bar V_k(S_{\bar h })
\right].
\end{align*}
This holds due to strong optimism and the greedy nature of $\bar V_k$ by \pref{lem:multi_strong_optimism}.
\end{proof}
\begin{lemma}[Multi-step Strong Optimism]
\label{lem:multi_strong_optimism}
Assume an algorithm is strongly optimistic and it computes its optimistic V-function greedily based on its optimistic Q-function, i.e., $\bar V_k(s) \geq \bar Q_{k}(s,a)$ holds always. Then it satisfies in all rounds $k$, stopping times $N, N'$ with $N \leq N'$ a.s. and policies $\pi$ that
\begin{align*}
\EE_{\pi}\left[
\bar V_k(S_{N})
-
\sum_{t=N}^{N' - 1} r(S_t, A_t)
+
\bar V_k(S_{N'})
\right] \geq 0.
\end{align*}
\end{lemma}
This property can be seen as a multi-step generalization of strong optimism. While strong optimism only considers the surplus in one time-step, we here consider the surplus over multiple time steps under any policy.
\cd{This lemma definitely holds when $N$ and $N'$ are regular indices. The stopping time version should be double-checked carefully!}
\begin{proof} We lower-bound the LHS in the condition as
\begin{align*}
&\EE_{\pi}\left[
\bar V_k(S_{N})
-
\sum_{t=N}^{N' - 1} r(S_t, A_t)
+
\bar V_k(S_{N'})\right]
\\
&= \EE_{\pi}\left[\indicator{N < N'}\left(
\bar V_k(S_{N})
-
\sum_{t=N}^{N' - 1} r(S_t, A_t)
+
\bar V_k(S_{N'})\right)\right]\\
& \overset{(i)}{\geq}
\EE_{\pi}\left[ \indicator{N < N'}\left(
\bar Q_k(S_{N}, A_N)
-
\sum_{t=N}^{N' - 1} r(S_t, A_t)
+
\bar V_k(S_{N'})\right)\right] \\
& =
\EE_{\pi}\left[ \indicator{N < N'}\left(
E_k(S_N, A_N)
+ \bar V_k(S_{N+1}) -
\sum_{t=N+1}^{N' - 1} r(S_t, A_t)
+
\bar V_k(S_{N'}) \right)\right] \\
& =
\EE_{\pi}\left[ \indicator{N < N'}
E_k(S_N, A_N)\right]\\
& \qquad +
\EE_{\pi}\left[ \indicator{N + 1 < N'}
\left(\bar V_k(S_{N+1}) -
\sum_{t=N+1}^{N' - 1} r(S_t, A_t)
+
\bar V_k(S_{N'}) \right)\right] \\
& \overset{(ii)}{\geq}
\EE_{\pi}\left[\sum_{t=N}^{N' - 1} E_k(S_t, A_t) \right].
\end{align*}
where step $(i)$ follows from how the greedy assumption of $\bar V_k$ and step $(ii)$ is a recursive application of the previous steps. Finally, $\EE_{\pi}\left[\sum_{t=N}^{N' - 1} E_k(S_t, A_t) \right] \geq 0$ holds because of strong optimism.
\end{proof}
\textcolor{olive}{JZ: I think the proof of the Lemma above is flawed, here is an alternative which might be simpler.}
\begin{lemma}
\label{lem:gap_surp_bound_jul}
Assume strong optimism and greedy $\bar V_k$, i.e., $\bar V_k(s) \geq \max_{a} \bar Q_k(s,a)$ for all $s \in \mathcal{S}$.
Let $\mathcal{O} \subseteq [H]$ be the set of time steps such that there is an optimal policy $\pi^\star$ which satisfies $\PP_{\pi^\star}(S_h, A_h) \geq \PP_{\pi_k}(S_h, A_h)$ for every $h \in \mathcal{O}$. Then
\begin{align}
\EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h)\right] \leq
\EE_{\pi_k} \left[ \sum_{h = 1}^{H} \indicator{h \notin \mathcal{O}} E_k(S_h, A_h)\right].
\end{align}
\end{lemma}
\begin{proof}
Define the following value function:
\begin{align*}
\bar V_{k,h_0}^\pi(s) &= \EE_{\pi}\left[\sum_{h=h_0}^H E_k(S_h,A_h)+r(S_h,A_h)\,\middle\vert\,S_{h_0}=s\right]\\
&= E_k(s_0,\pi(s_0))+r(s_0,\pi(s_0)) + \langle P(\cdot|s_0,\pi(s_0)), \bar V^\pi_{k,h_0+1}\rangle\,,
\end{align*}
where
$\bar V_{k,H+1}^\pi=0$ by convention.
First we show that for all $\pi$: $\bar V_{k}^\pi\leq \bar V_k$.
This holds trivially for $H+1$, so by induction given that $\bar V_{k,h+1}^\pi\leq \bar V_{k,h+1}$
\begin{align*}
\bar V_{k,h}^\pi(s) &= E_k(s,\pi(s))+r(s,\pi(s)) + \langle P(\cdot|s,\pi(s)), \bar V^\pi_{k,h+1}\rangle\\
&\leq E_k(s,\pi(s))+r(s,\pi(s)) + \langle P(\cdot|s,\pi(s)), \bar V_{k,h+1}\rangle\\
&=\bar Q_{k,h}(s,\pi(s))\leq \bar V_{k,h}(s)\,.
\end{align*}
Having established $\bar V_{k}^\pi\leq \bar V_k$, we have
\begin{align*}
\EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h)\right] &= V^{\pi^*}_0-V^{\pi_k}_0\\
&\leq V^{\pi^*}_0-\bar V^{\pi^*}_{k,0}+\bar V^{\pi_k}_{k,0}-V^{\pi_k}_0\\
&= \EE_{\pi_k} \left[ \sum_{h = 1}^{H} E_k(S_h, A_h)\right]-\EE_{\pi^*} \left[ \sum_{h = 1}^{H} E_k(S_h, A_h)\right]\\
&\leq \EE_{\pi_k} \left[ \sum_{h = 1}^{H} \indicator{h \notin \mathcal{O}} E_k(S_h, A_h)\right]\,.
\end{align*}
\end{proof}
Using this result, we can get a version of \pref{lem:Vdd_lb1} that takes $\mathcal{O}$ into account. It holds in any MDP but is never worse than \pref{lem:Vdd_lb1} in deterministic MDPs. In stochastic MDPs this may yield a worse bound because the stopping time $B$ is missing. We may be able also add it though.
\begin{lemma}
Let $\epsilon_k : \mathcal{S} \times \mathcal{A} \rightarrow \RR^+_0$ be arbitrary and let $\mathcal{O} \subseteq H$ be the set of time steps such that there is an optimal policy $\pi^\star$ which satisfies $\PP_{\pi^\star}(S_h, A_h) \geq \PP_{\pi_k}(S_h, A_h)$ for every $h \in \mathcal{O}$. Then for any strongly and greedily optimistic algorithm (which satisfies $\bar V_k$, i.e., $\bar V_k(s) \geq \max_{a} \bar Q_k(s,a)$ for all $s \in \mathcal{S}$), it holds that
\begin{align*}
\label{eq:clipped_ineq_tight_det}
\ddot V_k(s_1) - V^{\pi_k}(s_1) &\geq
\EE_{\pi_k}\left[
\sum_{h=1}^H \indicator{h \notin \mathcal{O}} \left( \gap(S_{h},A_h) - \epsilon_{k}(S_{h}, A_h)\right)
\right]\\
&= \EE_{\pi_k}\left[
\sum_{h=1}^H \left( \gap(S_{h},A_h) -
\indicator{h \notin \mathcal{O}}\epsilon_{k}(S_{h}, A_h)\right)
\right]
\end{align*}
\end{lemma}
\tm{We can most likely prove something stronger from Lemma~\ref{lem:gap_surp_bound_jul} which also simplifies a lot or removes a lot of the proofs. Sketch below:}
Let $\tilde \mathcal{O}_k = \{(s,a) \in \mathcal{S}\times\mathcal{A}: \PP_{\pi_k}(S_{\kappa(s)} = s, A = a) \leq \PP_{\pi^*}(S_{\kappa(s)} = s, A = a)\}$ be the set of state-action pairs which have higher probability to visited by $\pi^*$ than $\pi_k$. By Lemma~\ref{lem:Vdd_lb1} we have
\begin{align*}
\ddot V_k(s_1) - V^{\pi_k}(s_1) &
\geq \EE_{\pi_k}\left[\sum_{h=B}^{H} \ddot E_k(S_h,A_h)\right]\\
& \geq \EE_{\pi_k}\left[\sum_{h=B}^{H}
\indicator{(S_h,A_h) \not\in \tilde\mathcal{O}_k}
\ddot E_k(S_h,A_h)\right]\\
&\geq \EE_{\pi_k}\left[\sum_{h=B}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k) E_k(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=B}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k) \epsilon_k(S_h,A_h)\right],
\end{align*}
where $B$ is the stopping time for the process defined by $\pi_k$ matching $\pi^*$.
Next we use Lemma~\ref{lem:gap_surp_bound_jul} to lower bound $\EE_{\pi_k}\left[\sum_{h=B}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k) E_k(S_h,A_h)\right]$. Recall that from the proof of the lemma it follows that
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=1}^H\gap(S_h,A_h)\right] \leq \EE_{\pi_k}\left[\sum_{h=1}^HE_k(S_h,A_h)\right] - \EE_{\pi^*}\left[\sum_{h=1}^H E_k(S_h,A_h)\right].
\end{align*}
Fix $h$ and consider the difference \begin{align*}
\EE_{\pi_k}[E_k(S_h,A_h)] - \EE_{\pi^*}[E_k(S_h,A_h)] &= \sum_{(s,a): \kappa(s)=h} \left(\PP_{\pi_k}(S_h=s,A_h=a) - \PP_{\pi^*}(S_h=s,A_h=a)\right)E_k(s,a)\\
&\leq \sum_{(s,a) \not\in \tilde \mathcal{O}_k} \PP_{\pi_k}(S_h=s,A_h=a)E_k(s,a)\\
&= \EE_{\pi_k}[\chi((S_h,A_h)\not\in \tilde\mathcal{O}_k)E_k(S_h,A_h)].
\end{align*}
\tm{Unfortunately I don't know how to introduce the stopping time in the difference of expectations because $\mathcal{A}_h$ depends on $\pi_k$ but is independent of the randomness in $\pi^*$. Further if we do not have the stopping time then Lemma~\ref{lem:clipping_gaps_rel} will fail.}
\begin{align*}
\ddot V_k(s_1) - V^{\pi_k}(s_1) &= \EE_{\pi_k}\left[\sum_{h=1}^{H}\ddot E_k(S_h,A_h)\right] \geq \EE_{\pi_k}\left[\sum_{h=1}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k)\ddot E_k(S_h,A_h)\right]\\
&\geq \EE_{\pi_k}\left[\sum_{h=1}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k) E_k(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=1}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k) \epsilon_k(S_h,A_h)\right]\\
&\geq \EE_{\pi_k}\left[\sum_{h=1}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k) E_k(S_h,A_h)\right] - \EE_{\pi_k}\left[\sum_{h=B}^{H} \epsilon_k(S_h,A_h)\right],
\end{align*}
where the last inequality follows because if $\chi(\mathcal{A}_h) = 1$ this implies that $\forall h' < h$, $\chi((S_{h'},A_{h'}) \not\in \tilde O_k) = 0$ and hence
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=1}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k) \epsilon_k(S_h,A_h)\right] \leq \EE_{\pi_k}\left[\sum_{h'=h}^{H}\chi(\mathcal{A}_{h}) \epsilon_k(S_{h'},A_{h'})\right].
\end{align*}
The above implies
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^{H} \epsilon_k(S_h,A_h)\right] = \sum_{h=1}^{H}\EE_{\pi_k}\left[\sum_{h'=h}^{H}\chi(\mathcal{A}_{h}) \epsilon_k(S_{h'},A_{h'})\right] \geq \EE_{\pi_k}\left[\sum_{h=1}^{H}\chi((S_h,A_h) \not\in \tilde\mathcal{O}_k) \epsilon_k(S_h,A_h)\right].
\end{align*}
\cd{Fill in the proof for this which follows the proof of \pref{lem:Vdd_lb1} for the most part.}
\cd{This lemma is sufficient for us to be able to clip all states-action pairs at times in $\mathcal{O}$ to whatever we want (e.g. $\infty$).}
\cd{Could formulate $\mathcal{O}$ as a set of state-action pairs and have $\underline h$ and $\bar h$ just be stopping times of when a pair in $\mathcal{O}$ is reached?}
\cd{The current result doesn't quite give us the optimal regret rate for deterministic MDPs. For this, we would need a stronger version of \pref{lem:Oclipping_perfect}} where $\mathcal{O}$ can be covered by multiple optimal policies.
\subsection{Negative result for deterministic transitions?}
Consider the MDP in Figure~\ref{fig:mdp_det_lower}. This MDP has $2n+9$ states and $4n+8$ actions. The rewards for each action are either $1/12$ or $1/12+\epsilon/2$ and can be found next to the transitions from the respective states. We are going to label the states according to their layer and their position in the layer so that the first state is $s_{1,1}$ the state which is to the left of $s_{1,1}$ in layer 2 is $s_{2,1}$ and to the right $s_{2,2}$. In general the $i$-th state in layer $h$ is denoted as $s_{h,i}$. The rewards in all states but $s_{4,1},s_{4,2}$ and $s_{5,i}$ for all $i>1$ are deterministic. The rewards for $s_{4,2}$ and $s_{5,i}$ for all $i>1$ are Bernoulli with mean $1/12$ according to the figure. Further the rewards at state $s_{4,1}$ are Bernoulli with mean $1/12+\epsilon/2$ for the action leading to $s_{5,1}$ and $1/12$ for the action leading to $s_{5,2}$. From the construction it is clear that $V^*(s_{1,1}) = 1/2+\epsilon$. Further there are two sets of optimal policies with the above value function -- the $n$ optimal policies which visit state $s_{2,2}$ and the $n$ optimal policies which visit $s_{4,1}$. Notice that the information theoretic lower bound for this MDP is in $O(\log(K)/\epsilon)$ as only the state $s_{4,2}$ does not belong to an optimal policy. In particular, there is no dependence on $n$. Next we try to show that the class of optimistic algorithms will incur regret at least $\Omega(n\log(K)/\epsilon)$.
\paragraph{Class of algorithms.}
We adopt the class of algorithms from Section G.2 in \citep{simchowitz2019non} with an additional assumption which we clarify momentarily. Recall that the class of algorithms assumes access to an optimistic value function $\bar V_k(s) \geq V^*(s)$ and optimistic Q-functions.
In particular the algorithms construct optimistic Q and value functions as
\begin{align*}
\bar V_k(s) &= \max_{a\in\mathcal{A}} \bar Q_k(s,a)\\
Q_k(s,a) &= \hat r_k(s,a) + b_k^{rw}(s,a) + \hat p_k(s,a)^\top \bar V_k + b_k(s,a).
\end{align*}
We assume that $b_k^{rw}(s,a) \sim \sqrt{\frac{\log(M(1\lor n_k(s,a)))/\delta}{(1\lor n_k(s,a))}}$, where $M = \theta(n)$ and $b_k(s,a) \sim \sqrt{S}f_k(s,a)b_k^{rw}(s,a)$, where $f_k$ is a decreasing function in the number of visits to $(s,a)$ given by $n_k(s,a)$. One can verify that this is true for the the Q and value functions of StrongEuler.
\paragraph{Lower bound.}
We first show that for any $k$ it holds that $s_{2,1}$ is visited at least a constant fraction of $k$ times. Next, in a similar way we claim that $s_{5,i}, i>1$ are also visited a constant fraction of times. Finally this allows us to argue that for $\Omega(K)$ times the optimistic algorithm will try to solve the MDP problem starting from state $s_{2,1}$. Combining the fact that for this sub-MDP any state $s_{5,i},i>1$ is sub-optimal and the information theoretic lower bound for deterministic MDPs we see that the incurred regret would be at least $\Omega(n\log(K)/\epsilon)$.
\begin{lemma}
\label{lem:const_pulls}
It holds that $n_k(s_{2,1}) \geq \Omega(K)$ with probability at least $1-O(\delta\log(K))$.
\end{lemma}
\begin{proof}
We first show that there exists constants $c,c'$ such that for all $k$ large enough it holds that $c n_k(s_{5,i}) \geq n_k(s_{5,j}) \geq c' n_k(s_{5,i})$ with high probability for $i,j>1$. Fix a $t>3$ and suppose that $s_{5,i}$ has been visited $t$ times. Let $\pi_{s_{5,i}}$ be the policy with highest optimistic value function after the $t$ times $s_{5,i}$ is visited. Suppose that there exists some $j$ which is only played $\gamma t$ times for some $\gamma < 1$ to be determined later. We can now consider the policy $\pi_{s_{5,j}}$ which follows $\pi_{s_{5,i}}$ up to layer $4$ and then chooses the action which transitions to state $s_{5,j}$. The probability that after $t$ plays of $\pi_{s_{5,i}}$, $\pi_{s_{5,i}}$ is played again before $\pi_{s_{5,j}}$ is bounded by
\begin{align*}
&\PP\left(\hat r_{s_{5,i}}(t) + 2\sqrt{\frac{\log(Mt/\delta)}{t}} \geq \hat r_{s_{5,j}}(\gamma t) + 2\sqrt{\frac{\log(\gamma Mt/\delta)}{\gamma t}}\right)\\
&\leq \PP\left(\hat r_{s_{5,i}}(t) - \sqrt{\frac{\log(Mt/\delta)}{t}} \geq \hat r_{s_{5,j}}(\gamma t) + \sqrt{\frac{\log(\gamma Mt/\delta)}{\gamma t}}\right)\\
&\leq \PP\left(\hat r_{s_{5,i}}(t) - \sqrt{\frac{\log(Mt/\delta)}{t}} \geq 1/10\right) + \PP\left(\hat r_{s_{5,j}}(\gamma t) + \sqrt{\frac{\log(\gamma Mt/\delta)}{\gamma t}} \leq 1/10\right) \leq \frac{2\delta}{M\gamma t},
\end{align*}
\cd{Should $\hat r_{s_{5,i}}$ be $\hat r_{s_{5,j}}$ on the RHS?}
where we have chosen $\gamma$ sufficiently small (e.g. 0.001) so that $3\sqrt{\frac{\log(Mt/\delta)}{t}} \leq \sqrt{\frac{\log(\gamma Mt/\delta)}{\gamma t}}$ \cd{Should this be $3\sqrt{\frac{\log(Mt/\delta)}{t}} \leq \sqrt{\frac{\log(\gamma Mt/\delta)}{\gamma t}}$?} . Now if we assume that there exists a $k$ during which $c n_k(s_{5,i}) \geq n_k(s_{5,j})$ for sufficiently small $c$ this implies that there exists a $t$ at which $\hat r_{s_{5,i}}(t) + 2\sqrt{\frac{\log(Mt/\delta)}{t}} \geq \hat r_{s_{5,j}}(\gamma t) + 2\sqrt{\frac{\log(\gamma Mt/\delta)}{\gamma t}}$ and so $c n_k(s_{5,i}) < n_k(s_{5,j})$ with probability $1- O(\log(K)\delta)$. The other direction of the inequality follows in a similar way. For the rest of the proof we can now assume that at any $k$ the number of times two policies visiting $s_{2,2}$ have been played differs by at most a constant, otherwise this would be a contradiction with $c n_k(s_{5,i}) \geq n_k(s_{5,j}) \geq c' n_k(s_{5,i})$ for $i>1$. Further, in a similar way we can show the same result for all optimal policies which visit $s_{2,1}$.
Fix $tn \in [K]$ and $\gamma > 1$ to be determined later. Suppose that $s_{2,1}$ has been visited $tn$ times and $s_{2,2}$ has been visited $\gamma tn$ times. We now compute the probability that $s_{2,2}$ is visited before $s_{2,1}$. Let $\pi^*_{1}$ be the optimal policy visiting $s_{2,1}$, which has been played most often out of all optimal policies visiting $s_{2,1}$. Under our assumption we know that $\pi^*_{1}$ has been visited at most $c_1 t$ times for some constant $c_1$. Let $\pi^*_2$ be the optimal policy visiting $s_{2,2}$ which has been visited least often among all policies visiting $s_{2,2}$. Again, under our assumption $\pi^*_2$ has been played at most $c_2 \gamma t$ times.
Denote by $\hat V^{\pi^*_{1}}(t)$ the optimistic value function after visiting $s_{2,1}$, $t$ times. Because the rewards and transitions are deterministic for $\pi^*_{1}$, except on $s_{4,1}$, it holds that $\hat V^{\pi^*_{1}}(c_1 t) \geq 1/2 + \epsilon + b_{\pi^*_1}(c_1 t) - \sqrt{\frac{\log(Mc_1 t/\delta)}{c_1 t}}$ with probability $1-\delta$ over all rounds. Here $b_{\pi^*_{1}}(c_1 t)$ is the sum of all the bonuses for states visited by $\pi^*_{1}$ after $c_1 t$ rounds in which $s_{2,1}$ has been visited and $a$ is the action at $s_{4,1}$ which transitions to $s_{5,1}$.
Our choice of $\pi^*_2$ implies
\begin{align*}
\hat V^{\pi^*_2}(c_2 \gamma t) \leq 3/12 + \epsilon + \hat r_{s_{4,1},\pi^*_2}(c_2\gamma t)+ \hat r_{s_{5,i},\pi^*_2}(c_2\gamma t) + b_{\pi^*_2}(c_2\gamma tn),
\end{align*}
where $\hat r_{s,\pi}(c_2 t)$ is the empirical mean of the reward for state-action pair $(s,\pi(s))$ after visiting the state-action pair $c_2 t$ times.
We now compute the probability that $s_{2,2}$ is visited before $s_{2,1}$ is visited $n$ times. This probability is bounded by the probability to play $\pi^*_2$ before $\pi^*_1$
\begin{align*}
&\PP(\hat V^{\pi^*_1}(c_1 t) \leq \hat V^{\pi^*_2}(c_2\gamma t)) \leq\\
&\PP\left(1/2 + \epsilon + b_{\pi^*_1}(c_1 t) - \sqrt{\frac{\log(Mc_1 t/\delta)}{c_1 t}} \leq 3/12 + \epsilon + \hat r_{s_{4,1},\pi^*_2}(c_2 \gamma t)+ \hat r_{s_{5,i},\pi^*_2}(c_2 \gamma t) + b_{\pi^*_2}(c_2 \gamma t)\right).
\end{align*}
Choose $\gamma$ so that the bonuses satisfy
\begin{align*}
\sqrt{\frac{\log(c_1Mt/\delta)}{c_1t}} \geq 3\sqrt{\frac{\log(c_2\gamma Mt/\delta)}{c_2\gamma t}}.
\end{align*}
This implies
\begin{align*}
&\PP(\hat V^{\pi^*_1}(c_1 t) \leq \hat V^{\pi^*_2}(c_2 t))\\
&\leq \PP\left(1/6 \leq \hat r_{s_{4,1},\pi^*_2}(c_2\gamma t)+ \hat r_{s_{5,i},\pi^*_2}(c_2\gamma t) - 4\sqrt{\frac{\log(c_2\gamma Mt/\delta)}{c_2\gamma t}}\right) \leq \frac{\delta}{c_2\gamma tM},
\end{align*}
where the first inequality follows by just considering the difference of bonuses for the rewards at each state-action pair visited by $\pi^*_1$ and $\pi^*_2$ and our choice of $\gamma$ and the second inequality follows because $\hat r_{s_{4,1},\pi^*_2}(c_2\gamma t)+ \hat r_{s_{5,i},\pi^*_2}(c_2\gamma t)$ is a sum of $2\gamma t$ independent Bernoulli variables. The above implies that $s_{2,1}$ is selected at least $n$ times before $s_{2,2}$ with probability $1-O(\delta\log(K))$. In particular for every $O(n)$ iterations of the optimistic algorithm we must have that $s_{2,1}$ is visited at least $O(n)$ times, otherwise there exists a $t$ for which $\hat V^{\pi^*_1}(c_1 t) \leq \hat V^{\pi^*_2}(c_2\gamma t)$.
\end{proof}
We can now show the lower bound.
\begin{theorem}
\label{thm:lower_bound_det_opt}
There exists an MDP instance with deterministic transitions on which any optimistic algorithm with $\delta \leq O(1/\log(K))$ will incur expected regret at least $\Omega(S\log(K)/\epsilon)$ while it is information theoretically possible to achieve $O(\log(K)/\epsilon)$ regret.
\end{theorem}
\begin{proof}
Lemma~\ref{lem:const_pulls} implies that with constant probability the optimistic algorithm will visit $s_{2,1}$ in the MDP in Figure~\ref{fig:mdp_det_lower} at least $\Omega(K)$. For the rest of the proof condition on this event. Because optimistic algorithms disregard all states which can not be visited from $s_{2,1}$ once they are in $s_{2,1}$ we can restrict our attention on the sub-MDP with starting state $s_{2,1}$. Further Lemma~\ref{lem:const_pulls} implies that the total number of times we have played a policy visiting $s_{2,2}$ as opposed to one visiting $s_{2,1}$ is at most a multiplicative constant $c$ more. This, together with the proof of Theorem~\ref{thm:lower_bound_gen} implies that the expected regret of any uniformly good algorithm on the sub-MDP is governed by the following LP
\begin{align*}
\minimize{\alpha(\pi)\geq 0,\eta(\pi)\geq 0}{\sum_{\pi \in \Pi} \eta(\pi)\left(\return{*}_{\theta} - \return{\pi}_{\theta}\right)}
{
\sum_{\pi \in \Pi} \alpha(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1 \qquad \textrm{for all } \,\,\lambda \in \Lambda(\theta)\\
&&&\sum_{\pi\in\Pi} \alpha(\pi) \leq c\sum_{\pi\in \Pi}\eta(\pi)
},
\end{align*}
i.e., we can assume that every policy is played roughly $c\eta(\pi)$ times instead of $\eta(\pi)$ times. Now following the proof of Theorem~\ref{thm:lower_bound_deterministic}, we can reduce the above problem to finding a feasible solution for the dual LP on the restricted set of confusing environments $\breve \Lambda(\theta)$ which only changes the rewards at states $s_{5,i},i>1$ or $s_{4,2}$. This dual LP is
\begin{align*}
\maximize{\mu(\lambda)\geq 0,\mu_\alpha\geq 0}{\sum_{\lambda \in \breve \Lambda(\theta)} \mu(\lambda)}
{
c\mu_\alpha \leq v^*_\theta - v^{\pi}\qquad \textrm{for all } \,\,\pi \in \Pi\\
&&&\sum_{\lambda \in \breve\Lambda(\theta)}\mu(\lambda)KL(\PP^{\pi}_\theta, \PP^{\pi}_\lambda) \leq \mu_\alpha\qquad \textrm{for all } \,\,\pi \in \Pi
}.
\end{align*}
Arguing in the same way about feasibility as in Theorem~\ref{thm:lower_bound_deterministic} shows that the expected regret of any uniformly good strategy on the sub-MDP for large enough $K$ is lower bounded by $\Omega(n\log(K)/\epsilon)$. This implies the claim of the theorem and completes the proof.
\end{proof}
\section{Experimental results}
\label{app:experiments}
In this section we present experiments based on the following deterministic LP which can be found in \pref{fig:mdp_experiments}.
\begin{figure}[ht]
\centering
\includegraphics[width=.6\textwidth]{imgs/experiments/MDP_for_exp.png}
\caption{Deterministic MDP used in experiments}
\label{fig:mdp_experiments}
\end{figure}
In short the MDP has only deterministic transitions and 3 layers. The starting state is denoted by $s_0$ and the $j$-th state at layer $i$ by $s_{i,j}$. There are $n+1$ possible actions at $s_0$, two possible actions at $s_{1,j},\forall j\in [n+1]$, and a single possible action at $s_{2,j}, \forall j\in[4]$. The only non-negative rewards are at state-action pairs in the final layer. The unique optimal policy reaches state $s_{2,1}$ and has return equal to $0.5$. We distinguish between two types of sub-optimal policies given by $\pi_1$ which visists $s_{1,1}$ and all other sub-optimal policies which visit $s_{1,j},j\geq 2$. The return of policy $\pi_1$ determines the $gap$ parameter in our experiments and the reward at state $s_{2,4}$ determines the $\epsilon$ parameter.
We run two sets of experiments using the \textsc{UCBVI} algorithm~\citep{azar2017minimax}. We have chosen this algorithm over Strong-\textsc{Euler} since \textsc{UCBVI} is slightly easier to implement and their differences are orthogonal to the issues studied here. The rewards in both experiments are Bernoulli with the respective mean provided below the state in \pref{fig:mdp_experiments}. In the first set of experiments we let the gap parameter to be equal to $0.5$ and in the second set of experiments we let the gap parameter to be $\sqrt{\frac{S}{K}}$. We let $\epsilon = \frac{4^{\epsilon_{pow}}}{\sqrt{K}}$, where $\epsilon_{pow}$ takes integer values between $0$ and $\lfloor0.5*\log_4(K)\rfloor$. We have two settings for $n$ (respectively $S$) which are $n=1$ and $n=250$. In all experiments we have set $K=500000$ and the topology of the MDP implies $H=3$. Each experiment is repeated $5$ times and we report the average regret of the algorithm, together with standard deviation of the regret. We note that in the first set of experiments we should observe regret which is close to $\Theta(\frac{SA\log(T)}{gap})$, this is because with our parameter choices the return gap is $gap/2$ for all settings of $\epsilon$. In the second set of experiments we should observe regret which is close to $\Theta(\sqrt{SAK})$ as the min-max regret bounds dominate.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=1.\textwidth]{imgs/experiments/MDP_small_UCB.png}
\caption{$n=1$}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=1.\textwidth]{imgs/experiments/MDP_large_UCB.png}
\caption{$n=250$}
\end{subfigure}
\caption{Large gap experiments}
\label{fig:large_gap}
\end{figure}
The first set of experiments can be found in Figure~\ref{fig:large_gap}. We plot $S^2A + \frac{SA\log(T)}{gap}$ in purple and $S^2A + \sqrt{SAK}$ in brown for reference. We include the additive term of $S^2A$ as this is what the theoretical regret bounds suggest. We see that for $n=1$ our experiments almost perfectly match theory, including the observations made regarding Opportunity~\ref{enum_prob_1} and Opportunity~\ref{enum_prob_3}. In particular there is no obvious dependence on $1/\gap_{\min} = 1/\epsilon$, especially when $\epsilon = O(1/\sqrt{K})$, which in the plot is reflected by $\epsilon_{pow} = 0$.
In the case for $n=250$ the algorithm performs better than what our theory suggests. We expect that our bounds do not accurately capture the dependence on $S$ and $A$, at least for deterministic transition MDPs.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=1.\textwidth]{imgs/experiments/MDP_small_UCB_sqrt.png}
\caption{$n=1$}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=1.\textwidth]{imgs/experiments/MDP_large_UCB_sqrt.png}
\caption{$n=250$}
\end{subfigure}
\caption{Small gap experiments}
\label{fig:small_gap}
\end{figure}
The second set of experiments can be found in Figure~\ref{fig:small_gap}. Similar observations hold as in the large gap experiment.
\section{Introduction}
Reinforcement Learning (RL) is a general scenario where learners, or agents, interact with
the environment to achieve some goal. The environment and an agent's interactions
are typically modeled as a Markov decision process (MDP) \citep{puterman1994markov},
which can represent a rich variety of tasks. But, for which MDPs can an agent or an RL algorithm succeed? This requires a theoretical analysis of the complexity of an MDP.
This paper studies this question in the tabular episodic setting, where an agent interacts with the environment in episodes of fixed length $H$ and where the size of the state and action space is finite ($S$ and $A$ respectively).
While the performance of RL algorithms in tabular Markov decision processes has been
the subject of many studies in the past \citep[e.g.][]{fiechter1994efficient,kakade2003sample,osband2013more,dann2015sample,dann2017unifying,azar2017minimax,jin2018q,zanette2019tighter, dann2019strategic}, the vast majority of existing analyses focuses on worst-case problem-independent regret bounds, which only take into account the size of the MDP, the horizon $H$ and the number of episodes $K$.
Recently, however, some significant progress has been achieved towards deriving
more optimistic problem-dependent guarantees. This includes
more refined regret bounds for the tabular episodic setting that depend on
structural properties of the specific MDP considered \citep{simchowitz2019non, lykouris2019corruption, jin2020simultaneously}. Motivated by instance-dependent analyses in multi-armed bandits \citep{lai1985asymptotically}, these analyses derive
gap-dependent regret-bounds of the form
$O\left(\sum_{(s,a)\in\mathcal{S}\times\mathcal{A}} \frac{H\log(K)}{\gap(s,a)}\right)$,
where the sum is over state-actions pairs $(s, a)$ and
where the gap notion is defined as the difference of the optimal value function $V^{*}$ of the Bellman optimal policy $\pi^*$ and the $Q$-function of $\pi^*$ at a sub-optimal action:
$\gap(s,a) = V^{*}(s) - Q^{*}(s,a)$. We will refer to this gap definition as \emph{value-function gap} in the following. We note that a similar notion of gap has been used in the infinite horizon setting to achieve instance-dependent bounds \citep{auer2007logarithmic,tewari2008optimistic,auer2009near,filippi2010optimism,ok2018exploration}, however, a strong assumption about irreducability of MDP is required.
The actual regret bounds are more complicated and include additional $(s, a)$-dependent factors that only depend on the rewards-to-go from $(s,a)$ and that are therefore irrelevant to this discussion.
While this notion of gap generalizes that of the multi-armed bandit setting, we argue that it has a major limitation: each gap only depends on the rewards-to-go from the state-action pair and does not take into account how this state-action pair can be reached. Hence, existing gap-dependent regret bounds treat all state-action pairs equally and ignore their topological ordering in the MDP which can have a major impact on the performance of RL. In this paper, we address this issue and formalize the following key observation about the difficulty of RL in an episodic MDP through improved instance-dependent regret bounds:
\begin{center}
\begin{minipage}{0.9\linewidth}
\emph{Learning a policy with optimal return does not require an RL agent to distinguish between actions with similar outcomes (small value-function gap) in states that can only be reached by taking highly suboptimal actions (large value-function gap).}
\end{minipage}
\end{center}
To illustrate this insight, consider autonomous driving, where each episode corresponds to driving from a start to a destination. If the RL agent decides to run a red light on a crowded intersection, then a car crash is inevitable. Even though the agent could slightly affect the severity of the car crash by steering, this effect is small and, hence, a good RL agent does not need to learn how to best steer after running a red light. Instead, it would only need a few samples to learn to obey the traffic light in the first place as the action of disregarding a red light has a very large value-function gap.
\begin{figure}[t]
\begin{minipage}{0.5\textwidth}
\centerline{\includegraphics[scale=.7]{imgs/gap_example_tikz.pdf}}
\end{minipage}
\resizebox{.7\textwidth}{!}{\begin{minipage}{1\textwidth}
\begin{tabular}{l|l|}
\cline{2-2}
& Regret bounds for deterministic transition MDPs: \\ \hline
\multicolumn{1}{|l|}{Prior work:} & $R_{prior}(K) \leq O\big(\sum_{(s,a) \not\in\pi^*} \frac{H\log(K)}{\gap(s,a)}\big)$ \\ \hline
\multicolumn{1}{|l|}{This work:} & $R_{our}(K) \leq O\bigg(\sum_{(s,a) \not\in\pi^*} \frac{H\log(K)}{\return{*} - \return{\pi^*_{(s,a)}}}\bigg)$ \\ \hline
\multicolumn{1}{|l|}{Comparison:} & $\return{*} - \return{\pi^*_{(s,a)}} = \sum_{(s',a') \in \pi^*_{(s,a)}\setminus \pi^*} \gap(s',a')$ \\ \hline
\end{tabular}
\end{minipage}}
\caption{Illustration of the limitations of existing gap-dependent bounds for episodic Markov decision processes}
\label{fig:summary}
\end{figure}
To understand how this observation translates into regret bounds, consider the small MDP in \prettyref{fig:summary}. The MDP has deterministic transitions and only terminal rewards. There are two decision points, $s_1$ and $\tilde s$, with two actions each, and all other states have only a single action. There are only three policies in this MDP: $\pi^*$, which plays action $a_1$ in state $s_1$ and then proceeds along the red path; $\pi_1$, which takes action $a_2$ at $s_1$ and $a_3$ at $\tilde s$ and proceeds along the blue path; and $\pi_3$ which takes action $a_2$ at $s_1$ and $a_4$ at $\tilde s$. All state-action pairs have zero reward except for the state-action pairs in the final layer. These rewards are illustrated in the figure in their respective colors and imply $V^{\pi^*}(s_1) > V^{\pi_1}(s_1) > V^{\pi_2}(s_1)$, $\gap(s_1,a_2) = c$ and $\gap(\tilde s,a_4) = \epsilon$ where $\epsilon \ll c$. Finally, the Bellman optimality condition implies that $\pi^*(\tilde s) = a_3$. On all other states the gap is zero. The idea behind the gap-dependent bounds is now to capture the necessary, from an information theoretic perspective, number of rounds to distinguish $\pi^*$ from any other sub-optimal $\pi$ on all states $s \in \mathcal{S}$. In the example MDP of \pref{fig:summary}, this amounts to distinguishing $\pi_1$ from $\pi^*$ on state $s_1$ and $\pi_2$ from $\pi_1$ in state $\tilde s$. Thus, the accumulated regret would be of the order $O(\log(T)(1/c + 1/\epsilon))$. However, a reasonable algorithm would only need to determine that $a_2$ is a sub-optimal action at state $s_1$, which would eliminate both $\pi_1$ and $\pi_2$ after only $\log(T)/c^2$ rounds. The incurred regret would then be only $O(\log(T)/c)$. Notice that $\epsilon$
and $c$ can be selected arbitrarily in this example. In particular, if we take $c=0.5$ and
$\epsilon = 1/\sqrt{K}$ this bound remains only logarithmic in the number of episodes, while
the recently derived gap-dependent regret bounds scale with $\sqrt{K}$.
This work is motivated by the insight and observations just discussed.
Our main contributions are two-fold. First, we investigate in what episodic
MDPs an algorithm could benefit from that insight by deriving information-theoretic regret lower-bounds. We show that existing gap-dependent bounds indeed cannot be improved in problems where each state is visited by an optimal policy with some probability. However, we also prove a new lower bound when the transitions of the MDP are deterministic that depends on a new notion of gap, called \emph{total-return gap}, and which indicate that a more favorable regret may be possible to achieve when there are states outside of the support of optimal policies.
Second, we show that improved regret bounds are indeed possible by proving a tighter regret bound for \textsc{StrongEuler}, an existing algorithm based on the optimism-in-the-face-of-uncertainty (OFU) principle \citep{simchowitz2019non}. While it is difficult to give a closed-form expression of our bound for general stochastic MDPs, we can express our bound in terms of return gaps in problems with deterministic transitions.
In fact, we show that it matches our lower-bound up to factors in the episode length $H$.
Our results suggest that value-function gaps are not sufficient to capture all the relevant problem structure that determines the regret of RL agents and that new forms of regret bounds are necessary. They also show that, at least for some MDPs, return gaps provide a finer alternative for quantifying the difficulty of episodic RL.
\section{Introduction}
Reinforcement Learning (RL) is a general scenario where agents interact with
the environment to achieve some goal. The environment and an agent's interactions
are typically modeled as a Markov decision process (MDP) \citep{puterman1994markov},
which can represent a rich variety of tasks. But, for which MDPs can an agent or an RL algorithm succeed? This requires a theoretical analysis of the complexity of an MDP.
This paper studies this question in the tabular episodic setting, where an agent interacts with the environment in episodes of fixed length $H$ and where the size of the state and action space is finite ($S$ and $A$ respectively).
While the performance of RL algorithms in tabular Markov decision processes has been
the subject of many studies in the past \citep[e.g.][]{fiechter1994efficient,kakade2003sample,osband2013more,dann2017unifying,azar2017minimax,jin2018q,zanette2019tighter, dann2019strategic}, the vast majority of existing analyses focuses on worst-case problem-independent regret bounds, which only take into account the size of the MDP, the horizon $H$ and the number of episodes $K$.
Recently, however, some significant progress has been achieved towards deriving
more optimistic (problem-dependent) guarantees. This includes
more refined regret bounds for the tabular episodic setting that depend on
structural properties of the specific MDP considered \citep{simchowitz2019non, lykouris2019corruption, jin2020simultaneously,foster2020instance,he2020logarithmic}. Motivated by instance-dependent analyses in multi-armed bandits \citep{lai1985asymptotically}, these analyses derive
gap-dependent regret-bounds of the form
$O\left(\sum_{(s,a)\in\mathcal{S}\times\mathcal{A}} \frac{H\log(K)}{\gap(s,a)}\right)$,
where the sum is over state-actions pairs $(s, a)$ and
where the gap notion is defined as the difference of the optimal value function $V^{*}$ of the Bellman optimal policy $\pi^*$ and the $Q$-function of $\pi^*$ at a sub-optimal action:
$\gap(s,a) = V^{*}(s) - Q^{*}(s,a)$. We will refer to this gap definition as \emph{value-function gap} in the following. We note that a similar notion of gap has been used in the infinite horizon setting to achieve instance-dependent bounds \citep{auer2007logarithmic,tewari2008optimistic,auer2009near,filippi2010optimism,ok2018exploration}, however, a strong assumption about irreducibility of the MDP is required.
While regret bounds based on these value function gaps generalize the bounds available in the multi-armed bandit setting, we argue that they have a major limitation. The bound at each state-action pair depends only on the gap at the pair and treats all state-action pairs equally, ignoring their topological ordering in the MDP. This can have a major impact on the derived bound.
In this paper, we address this issue and formalize the following key observation about the difficulty of RL in an episodic MDP through improved instance-dependent regret bounds:
\begin{center}
\begin{minipage}{0.9\linewidth}
\emph{Learning a policy with optimal return does not require an RL agent to distinguish between actions with similar outcomes (small value-function gap) in states that can only be reached by taking highly suboptimal actions (large value-function gap).}
\end{minipage}
\end{center}
To illustrate this insight, consider autonomous driving, where each episode corresponds to driving from a start to a destination. If the RL agent decides to run a red light on a crowded intersection, then a car crash is inevitable. Even though the agent could slightly affect the severity of the car crash by steering, this effect is small and, hence, a good RL agent does not need to learn how to best steer after running a red light. Instead, it would only need a few samples to learn to obey the traffic light in the first place as the action of disregarding a red light has a very large value-function gap.
\begin{figure}[t]
\begin{minipage}[c]{0.3\textwidth}
\includegraphics[width=\textwidth, trim=0 0cm 0cm 0, clip]{imgs/gap_example_tikz2.png}
\end{minipage}
\scalebox{0.9}{
\begin{minipage}[r]{0.7\textwidth}
\bgroup
\vspace*{-5mm}
\def2.1{2.1}
\begin{tabular}{r|c|c|}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{Value-function gap (prior)} &
\multicolumn{1}{c}{Return gap (ours)}
\\ \cline{2-3}
\multirow{2}{*}{
\begin{minipage}{1.4cm}\raggedleft General Regret bounds\end{minipage}
& $\displaystyle O\Big(\sum_{s,a} \frac{H\log(K)}{\gap(s,a)}\Big)$
& $\displaystyle O\Big(\sum_{s,a} \frac{\log(K)}{\overline{\gap}(s,a)}\Big)$ \\%\hline
& $\displaystyle \Omega\Big(\sum_{\substack{s,a \colon s \in \pi^*}} \frac{\log(K)}{\gap(s,a)}\Big)$
& $\displaystyle \Omega\Big(\sum_{s,a} \frac{\log(K)}{H\overline{\gap}(s,a)}\Big)$ \\ \cline{2-3}
\multirow{2}{*}{
\begin{minipage}{1.6cm}\raggedleft Example on the left\end{minipage}
}
&
\begin{minipage}{2.5cm}
\begin{center}
$\gap(s_1, a_2) = c$\\
$\gap(s_2, a_4) = \epsilon$
\end{center}
\end{minipage}
&
\begin{minipage}{4cm}
\begin{center}
$\overline{\gap}(s_1, a_2) = c$\\
$\overline{\gap}(s_2, a_4) = \frac{c + \epsilon}{H} \approx c$
\end{center}
\end{minipage}
\\
&
$\displaystyle O\Big( \frac{SH\log(K)}{\epsilon}\Big)$
&
$\displaystyle O\Big( \frac{SH\log(K)}{c}\Big)$
\\ \cline{2-3}
\end{tabular}\egroup
\end{minipage}}
\caption{Comparison of our contributions in MDPs with deterministic transitions. Bounds only include the main terms and all sums over $(s,a)$ are understood to only include terms where the respective gap is nonzero. $\overline{\gap}$ is our alternative \emph{return gap} definition introduced later (\pref{def:return_gap}).
}
\label{fig:summary}
\end{figure}
To understand how this observation translates into regret bounds, consider the toy example in \prettyref{fig:summary}. This MDP has deterministic transitions and only terminal rewards with $c \gg \epsilon > 0$. There are two decision points, $s_1$ and $s_2$, with two actions each, and all other states have a single action. There are three policies which govern the regret bounds: $\pi^*$ (red path) which takes action $a_1$ in state $s_1$;
$\pi_1$ which takes action $a_2$ at $s_1$ and $a_3$ at $s_2$ (blue path); and $\pi_2$ which takes action $a_2$ at $s_1$ and $a_4$ at $s_2$ (green path).
Since $\pi^*$ follows the red path, it never reaches $s_2$ and achieves optimal return $c+\epsilon$, while $\pi_1$ and $\pi_2$ are both suboptimal with return $\epsilon$ and $0$ respectively.
Existing value-function gaps evaluate to $\gap(s_1,a_2) = c$ and $\gap(s_2,a_4) = \epsilon$ which yields a regret bound of order $H\log(K)(1/c + 1/\epsilon)$. The idea behind these bounds is to capture the necessary number of episodes to distinguish the value of the optimal policy $\pi^*$ from the value of any other sub-optimal policy \emph{on all states}.
However, since $\pi^*$ will never reach $s_2$ it is not necessary to distinguish it from any other policy at $s_2$.
A good algorithm only needs to determine that $a_2$ is sub-optimal in $s_1$, which eliminates both $\pi_1$ and $\pi_2$ as optimal policies after only $\log(K)/c^2$ episodes. This suggests a regret of order $O(\log(K)/c)$.
The bounds presented in this paper achieve this rate up to factors of $H$ by replacing the gaps at every state-action pair with the average of all gaps along certain paths containing the state action pair. We call these averaged gaps \emph{return gaps}. The return gap at $(s,a)$ is denoted as $\overline{\gap}(s,a)$. Our new bounds replace $\gap(s_2,a_4) = \epsilon$ by $\overline{\gap}(s_2,a_4) \approx \frac{1}{2}\gap(s_1,a_2) + \frac{1}{2}\gap(s_2,a_4) = \Omega(c)$.
Notice that $\epsilon$ and $c$ can be selected arbitrarily in this example. In particular, if we take $c=0.5$ and $\epsilon = 1/\sqrt{K}$ our bounds remain logarithmic $O(\log(K))$, while prior regret bounds scale as $\sqrt{K}$.
This work is motivated by the insight just discussed. First, we show that improved regret bounds are indeed possible by proving a tighter regret bound for \textsc{StrongEuler}, an existing algorithm based on the optimism-in-the-face-of-uncertainty (OFU) principle \citep{simchowitz2019non}.
Our regret bound is stated in terms of our new return gaps that capture the problem difficulty more accurately and avoid explicit dependencies on the smallest value function gap $\gap_{\min}$. Our technique applies to optimistic algorithms in general and as a by-product improves the dependency on episode length $H$ of prior results.
Second, we investigate the difficulty of RL in episodic MDPs from an information-theoretic perspective by deriving regret lower-bounds. We show that existing value-function gaps are indeed sufficient to capture difficulty of problems but only when each state is visited by an optimal policy with some probability.
Finally, we prove a new lower bound when the transitions of the MDP are deterministic that depends only on the difference in return of the optimal policy and suboptimal policies, which is closely related to our notion of return gap.
\section{Novel upper bounds for optimistic algorithms}
\label{sec:upper_bounds}
Before presenting improved our upper bounds for general MDPs, we first discuss and state our results for the special case when transitions are deterministic to motivate our definitions in the more general stochastic transition case.
\subsection{Special case: Markov decision processes with deterministic transitions}
As hinted to in the introduction, the way in which prior regret bounds treat value-function gaps independently at each state-action pair can lead to excessively loose regret bounds.
Prior regret bounds~\citep{simchowitz2019non,lykouris2019corruption,jin2020simultaneously} that use value-function gaps
scale at least as
$$\sum_{s,a \colon \gap(s,a) >0 } \frac{H\log(K)}{\gap(s,a)} +
\sum_{s,a \colon \gap(s,a) =0 }
\frac{H\log(K)}{\gap_{\min}} \quad \textrm{ where }\quad \gap_{\min} = \!\!\!\min_{s, a \colon \gap(s,a) > 0}\! \gap(s,a).
$$
To illustrate where these bounds are loose, let us revisit the example MDP in \pref{fig:summary}. Here, the bounds evaluate to $ \frac{H\log(K)}{c}+\frac{H\log(K)}{\epsilon}+\frac{SH\log(K)}{\epsilon}$, where the first two terms come form the positive gaps and the last term comes from all the state-action pairs with zero gaps which are accounted for with $1 / \gap_{\min}$. There are several places for improvement:
\begin{enumerate}[label = \textbf{P.\arabic*}]
\item\label{enum_prob_1} \textbf{State-action pairs that can only be visited by taking optimal actions:} We should not pay the $1/\gap_{\min}$ factor for such $(s,a)$ as there are no other suboptimal policies $\pi$ to distinguish from $\pi^*$ in such states.
\item\label{enum_prob_3}
\textbf{State-action pairs that can only be visited by taking at least one suboptimal action:}
We should not pay the $1 / \gap(s_2, a_3)$ factor for state-action pair $(s_2, a_3)$ and the $1 / \gap_{\min}$ factor for $(s_2, a_4)$ because no optimal policy visits $s_2$. Such state-action pairs should only be accounted for with the price to learn that $a_2$ is not optimal in state $s_1$. After all, learning to distinguish between $\pi_1$ and $\pi_2$ is unnecessary for optimal return.
\end{enumerate}
\ref{enum_prob_1} and \ref{enum_prob_3} suggest that the price at which a state-action pair $(s,a)$ occurs in the regret bound should not only be determined by the rewards achievable after visiting $(s,a)$ or $s$, as is the case in value function gaps $\gap(s,a) = V^*(s) - Q^*(s,a)$. Therefore, in addition to the value function gap which measures how much reward to go is lost by playing $a$ instead of the optimal action in $s$, we also consider the difference of entire sum of rewards. In particular, we look at the difference of optimal return $v^*$ and the return $v^*_{s,a}$ of the best suboptimal policy that visits $(s,a)$,
\begin{align*}
v^* - v^{*}_{s,a} \qquad \textrm{where} \qquad
v^{*}_{s,a} \equiv \max_{\substack{\pi \in \Pi \setminus \Pi^* \colon\\ w^\pi(s,a) > 0}} v^\pi~.
\end{align*}
The intuition behind this term is that $v^* - v^{*}_{s,a}$ measures the cost
\paragraph{Regret bounds for deterministic transition MDPs.}
We now interpret \pref{def:return_gap} for deterministic MDPs and derive a novel regret bound which is essentially tight for optimistic algorithms. For any state-action pair $(s,a)$ let $\pi^\dagger_{(s,a)}$ denote the policy with highest value function which is not an optimal policy. Formally $\pi^\dagger_{(s,a)} = \argmin_{\pi: (s,a)\in\pi, \pi\not\in \Pi^*}v^* - v^\pi$. \pref{def:return_gap} now evaluates to $\overline{\gap}(s,a) = \gap(s,a) + \frac{v^* - v^{\pi^\dagger_{(s,a)}}}{H}$ and Theorem~\ref{thm:reg_bound_gen_informal} yields the following regret bound.
\begin{corollary}
\label{cor:det_trans_reg}
Assume \textsc{StrongEuler} is applied to an MDP with deterministic transitions and optimal value function bounded by $1$. Then its regret is bounded with high probability for all $K \geq 1$ as
\begin{align*}
\mathfrak{R}(K) &\leq O\Bigg(\sum_{s,a \colon \overline{\gap}(s,a)> 0} \frac{H\log(K)}{v^* - v^{\pi^\dagger_{(s,a)}}}\Bigg).
\end{align*}
\end{corollary}
We now compare this bound again to the existing bound stated in \pref{eq:cor_1b_simchowitz}. In \citet{simchowitz2019non}, $\alpha \in [0,1]$ is defined as the smallest value such that for all $(s,a,s') \in \mathcal{S}\times\mathcal{A}\times\mathcal{S}$ it holds that
$P(s'|s,a) - P(s'|s,\pi^*(s)) \leq \alpha P(s'|s,a)$, where $\pi^*$ is a Bellman-optimal policy.
Hence, unless all state-action pairs are optimal it must hold that $\alpha=1$. For any $\gap(s,a)>0$ it holds that $v^* - v^{\pi^\dagger_{(s,a)}} \geq \sum_{(s',a') \in v^{\pi^\dagger_{(s,a)}}}\gap(s',a') \geq \gap(s,a)$. Further for every $(s,a) \in \mathcal{Z}_{opt}$ for which $\overline{\gap}(s,a) >0$ it holds that $v^* - v^{\pi^\dagger_{(s,a)}} \geq \gap_{\min}$. Combining these two observations it holds that the bound in Corollary~\ref{cor:det_trans_reg} is no worse than the one in \citep{simchowitz2019non}.
We would further like to emphasize that this bound achieves the $O(SH \log(K) / c)$ regret we hoped to achieve in the example MDP in Figure~\ref{fig:summary}.
is addressed by the fact that at any episode, to reach states only visited by an optimal policy we have had to play an optimal policy and hence we have not incurred any regret during the respective episode. This observation can actually be taken a little further and allows us to not include state-action pairs $(s,a)\in \mathcal{S}\times\mathcal{A}$ such that $\gap(s,a) = 0$ and all preceding state-action pairs in the topology of the MDP also have zero gaps. This observation will also motivate our formal definition of the return gap, $\overline{\gap}$.
Problem~\ref{enum_prob_3} is
addressed in a slightly more subtle manner.
Our intuition for deterministic MDPs is the following and is based on the gap decomposition for the instantaneous regret with respect to any suboptimal policy $\pi$:
\begin{align*}
v^* - v^{\pi} = \sum_{h=1}^H \gap(s_h^{\pi},a_h^{\pi}).
\end{align*}
In the above $(s_h^{\pi},a_h^{\pi})$ is the state-action pair visited by $\pi$ at layer $h$. We can replace the gap at each state-action pair in the support of $\pi$ by a normalized sum (by $H$) of all strictly positive gaps at states in the support of $\pi$ while still preserving the regret decomposition of $v^* - v^{\pi}$. This suggests that the regret we have to pay at every $(s,a)\in\pi$ is actually dependent on $(v^* - v^{\pi})/H$ rather than $\gap(s,a)$ (in fact it turns out we can take the maximum of the two quantities).
\subsection{General case: Markov decision processes with stochastic transitions}
Our intuition can be extended to arbitrary MDPs by replacing the normalized sum by an appropriate expectation.
Formally we define the return gaps as
\begin{definition}[Return gap]
\label{def:return_gap}
For any state-action pair $(s,a)\in\mathcal{S}\times\mathcal{A}$ let $\mathcal{B}(s,a)\equiv \{B \leq \kappa(s), S_{\kappa(s)} = s, A_{\kappa(s)} = a\}$ denote the event that state-action pair $(s,a)$ is visited and that a suboptimal action was played any time up to visiting $(s,a)$.
We define the return gap as
\begin{align*}
\overline{\gap}(s,a) \equiv \gap(s,a)\lor\min_{\substack{\pi \in \Pi \colon \\\PP_{\pi}(\mathcal{B}(s,a)) > 0}}&~
\frac{1}{H}
\EE_{\pi}\left[ \sum_{h=1}^H \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a) \right]
\end{align*}
if there is a policy $\pi \in \Pi$ with $\PP_\pi(\mathcal{B}(s,a)) > 0$ and $\overline{\gap}(s,a) \equiv 0$ otherwise.
\end{definition}
The intuition behind event $\mathcal{B}(s,a)$ follows from what we discussed regarding Problem~\ref{enum_prob_1}. In particular if the event does not occur, then we have followed an optimal policy $\pi^*$ up to and including $(s,a)$, and hence no regret has been incurred. The benefit of $\mathcal{B}(s,a)$ is apparent in non-deterministic MDPs. Consider the case when we are faced with a policy $\pi$ which with probability $1-\delta$ follows $\pi^*$ and with probability $\delta$ visits some highly sub-optimal state action pair $(s,a)$ at which $\gap(s,a)$ is large. Taking the expected sum of gaps, rather than conditioning on $\mathcal{B}(s,a)$ would result in a regret bound which $\gap(s,a)$ by $\delta$, effectively using $v^* - v^{\pi}$ as the gap. The event $\mathcal{B}$ allows us to only pay a cost inversely proportional to $\overline{\gap}(s,a) \geq \gap(s,a)$. Our (informal) main upper bound pertains to the \textsc{StrongEuler} algorithm proposed in~\citep{simchowitz2019non} and is stated below.
\begin{theorem}[Informal]
\label{thm:reg_bound_gen_informal}
The regret $\mathfrak{R}(K)$ of \textsc{StrongEuler} is bounded with high probability as
\begin{align*}
\mathfrak{R}(K) &\lessapprox \sum_{\substack{(s,a)\in\mathcal{S}\times\mathcal{A} \colon \\\overline{\gap}(s,a) > 0}} \frac{\mathcal{V}^*(s,a)}{\overline{\gap}(s,a)}
\log\frac{MK}{\delta}
\quad + \quad SAH^3 (S \vee H)\log \frac{MH}{\overline{\gap}_{\min}} \log\frac{MK}{\delta}
.
\end{align*}
\end{theorem}
In the above $\mathcal{V}^*(s,a)$ is a term capturing the variance of the $Q$ function of $\pi^*$ at $(s,a)$ and we have restricted the bound to only the terms which have inverse polynomial dependence on the gaps. \tm{We could include discussion about the bound here or defer to the end of the next subsection as it is now.}
\paragraph{Comparison with existing gap-dependent bounds.} We now proceed to compare our bound to the existing gap-dependent bound for \textsc{StrongEuler} in \citet[Corollary B.1]{simchowitz2019non}
\begin{align}
\label{eq:cor_1b_simchowitz}
\mathfrak{R}(K) \leq O\Bigg(\sum_{(s,a) \notin \mathcal{Z}_{opt}} \frac{\alpha H\mathcal{V}^*(s,a)\log(K)}{\gap(s,a)} + |\mathcal{Z}_{opt}|\frac{H\mathcal{V}^*\log(K)}{\gap_{\min}}\Bigg),
\end{align}
where $\mathcal{V}^* = \max_{(s,a)}\mathcal{V}^*(s,a)$ and $\mathcal{Z}_{opt} = \{ s, a \colon \gap(s,a) = 0\}$ is the set state-action pair with zero $\gap(s,a) = 0$ according to a Bellman optimal policy $\pi^*$.
We here focus only on terms that have a dependency on $K$ and an inverse-polynomial dependency on gaps as all other terms are comparable. Our bound in \pref{thm:reg_bound_gen_informal} is no worse than this bound by noting that on $\mathcal{Z}_{opt}$ it always holds that $\overline{\gap}(s,a) \geq \frac{\gap_{\min}}{H}$. Further, outside of $\mathcal{Z}_{opt}$ our bound has no dependence on $H$ and $\overline{\gap}(s,a) \geq \gap(s,a)$.
However, to see that our bound can be much tighter, consider the case where a state $s\in \mathcal{Z}_{opt}$ is only reachable after making suboptimal actions, i.e., where the sum of value-function gaps is much larger than $\gap_{\min}$. By our definition of return gaps, we only pay the inverse of these larger gaps instead of $\gap_{\min}$.
This improvement is already apparent in our motivating example in Figure~\ref{fig:summary} where \pref{thm:reg_bound_gen_informal} achieves the desired $\log(K)/c$ regret bound as opposed to the $\log(K)/\epsilon$ bound suggested by prior work.
\subsection{Improved clipping: from minimum gap to average gap}
\label{sec:gen_clipping}
We now sketch how to derive our regret guarantees for \textsc{StrongEuler}. Our framework is also applicable to other algorithms that are \emph{optmistic}, that is at every episode they maintain a Q-function, $\bar Q_k$, which overestimates the Q-function of $\pi^*$ on every state-action pair, $\bar Q_k(s,a) \geq Q^*(s,a)$, with high probability.
We would like to point out that our results match the ones in \citep{simchowitz2019non} without requiring \emph{strong optimism}.
Strongly optimistic algorithms are optimistic and they further guarantee that their surpluses which, roughly speaking, quantify the local amount of optimism in $s,a$ and formally defined as
\begin{align}
\label{eq:surpl_def}
E_k(s,a) = \bar Q_k(s,a) - r(s,a) - \langle P(\cdot|s,a), \bar V_k\rangle,
\end{align}
are guaranteed to be non-negative as well (with high probability).
We now formalize how to exchange the gap at each state-action pair for the return gap. We begin by quickly recalling the key steps in existing analyses.
At the core of gap-dependent analyses is the following bound on the instantaneous regret by the \emph{clipped surpluses}
\begin{align*}
V^*(s_1) - V^{\pi_k}(s_1) \lessapprox \sum_{s,a} w^{\pi_k}(s,a)\clip[E_{k}(s,a) \vert \gap(s,a) \lor \gap_{\min}],
\end{align*}
where $\clip[ a | b] = \indicator{a \geq b} a$ is the clipping operator.
This is tighter than the usual bound achievable by just using optimism and the definition of surpluses which is $V^*(s_1) - V^{\pi_k}(s_1) \leq \sum_{(s,a) \in \mathcal{S}\times\mathcal{A}}w_{\pi_k}(s,a)E_{k}(s,a)$ and is the key to the instance dependent bounds.
Next, using concentration arguments it is possible to show that $E_{k}(s,a) \lessapprox \sqrt{\frac{\mathcal{V}(s,a)}{k}}$, for sub-optimal state-action pairs which are visited often, where $\mathcal{V}(s,a)$ is a variance term for $(s,a)$. We can intuitively ignore other state-action pairs as they do not generate too much regret. This implies that we can not incur regret for more than $\frac{\mathcal{V}(s,a)}{(\gap(s,a)\lor \gap_{\min})^2}$ rounds as otherwise we would have $\sqrt{\frac{\mathcal{V}(s,a)}{\sum_{j=1}^k w_{\pi_j}(s,a)}} < \gap(s,a)\lor \gap_{\min}$, which implies that $E_k(s,a)$ would be clipped to zero.
\paragraph{Improving on $\gap_{\min}$.}
The issues described with prior work upper bounds occur precisely because of clipping to $\gap_{\min}$. Suppose we replaced $\gap_{\min}$ with a generic clipping function $\epsilon_{k} :\mathcal{S}\times\mathcal{A} \rightarrow \mathbb{R}$. How large can we hope for this function to be? The following self-bounding trick reveals some intuition. By definition of gaps we can write $\frac{1}{4}(V^*(s_1) - V^{\pi_k}(s_1)) = \frac{1}{4}\sum_{(s,a)\in\mathcal{S}\times\mathcal{A}} w_{\pi_k}(s,a)\gap(s,a) = \frac{1}{4}\EE_{\pi_k}[\sum_{h=1}^H\gap(S_h,A_h)]$. By definition of surpluses and optimism we also have an upper bound on the instantaneous regret as $V^*(s_1) - V^{\pi_k}(s_1) \leq \sum_{h=1}^H\EE_{\pi_k}[ E_k(S_h,A_h)]$. Subtracting $\frac{1}{4}(V^*(s_1) - V^{\pi_k}(s_1))$ from both sides implies $\frac{3}{4}(V^*(s_1) - V^{\pi_k}(s_1)) \leq \sum_{h=1}^H\EE_{\pi_k}\left[\clip[E_k(S_h,A_h)|\frac{1}{4}\gap(S_h,A_h)]\right]$. Further, if we were able to lower bound $\EE_{\pi_k}[\sum_{h=1}^H\gap(S_h,A_h)]$ by our clipping function as $\sum_{h=1}^H\EE_{\pi_k}[\gap(S_h,A_h)] \geq \sum_{h=1}^H\EE_{\pi_k}[\epsilon_{k}(S_h)]$ we could repeat the self bounding trick again and arrive at the desired $V^*(s_1) - V^{\pi_k}(s_1) \leq 4\sum_{h=1}^H\EE_{\pi_k}\left[\clip[E_k(S_h,A_h)|\frac{1}{4}(\gap(S_h,A_h)]\lor \epsilon_{k}(S_h))\right]$. Thus the key in deriving improved regret bounds is to lower bound the expected sum of gaps as tightly as possible by the clipping function. The above discussion is made formal by the next proposition.
\begin{restatable}[Improved surplus clipping bound]{proposition}{surplusclippingbound}
\label{prop:surplus_clipping_bound}
Let the surpluses $E_k(s,a)$ be generated by a strongly optimistic algorithm and consider clipping thresholds $\epsilon_k \colon \mathcal{S} \times \mathcal{A} \rightarrow \RR^+_0$ that satisfy
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^H \epsilon_k(S_h, A_h) \right]
\leq \frac{1}{2} \EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h) \right]
\end{align*}
where $B = \min\{ h \in [H+1] \colon \gap(S_h, A_h) > 0 \}$ is the first time a non-zero gap was encountered.
Then the instantaneous regret of $\pi_k$ can be bounded as
\begin{align*}
V^*(s_1) - V^{\pi_k}(s_1)
\leq 4 \sum_{s,a} w^{\pi_k}(s,a) \clip\left[ E_k(s,a) ~\bigg| ~ \frac 1 4 \gap(s,a) \vee \epsilon_{k}(s, a) \right].
\end{align*}
\end{restatable}
We now make the quick observation that taking $\epsilon_{k} \equiv \frac{\gap_{\min}}{2H}$ will satisfy the condition of Proposition~\ref{prop:surplus_clipping_bound}, because on the event $\mathcal{B}$ there exists at least one positive gap in the sum $\sum_{h=1}^H\gap(S_h,A_h)$, which, by definition, is at least $\gap_{\min}$. This shows that our results already can recover the bounds in prior work, with significantly less effort. The condition in Proposition~\ref{prop:surplus_clipping_bound} suggests that one can set $\epsilon_{k}(S_h,A_h)$ proportional to the average expected gap under policy $\pi_k$. Formally, the clipping function we use is
\begin{align}
\epsilon_{k}(s,a) =
\frac{1}{2H}
\EE_{\pi_k}\left[ \sum_{h=1}^H \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a)\right].
\label{eqn:epsilon_choice}
\end{align}
In the corner case where the event $\mathcal{B}$ has zero probability under $\pi_k$, we set $\epsilon_{k}(s,a) = \infty$.
\pref{lem:clipping_gaps_rel} in \pref{app:upper_bounds} shows that the above definition satisfies the condition in \pref{prop:surplus_clipping_bound}.
This definition is also never worse than the minimum-gap clipping of existing analysis since $\epsilon_{k}(s,a) \geq \frac{\gap_{\min}}{2H}$ follows from the fact that in event $\mathcal{B}$ at least one gap is non-zero.
Further, $\epsilon_{k}(s,a)$ takes a simple form in MDPs with deterministic transitions: $\epsilon_{k}(s,a) = \frac{1}{2H}\sum_{s\in supp(\pi_k)}\gap(s,\pi_k(s)) = \frac{\return{*}-\return{\pi_k}}{2H}$, if $(s,a) \in \pi_k$ and $\epsilon_{k}(s,a)=\infty$ otherwise (because $\mathcal{B}$ has zero probability).
\subsection{Policy-dependent regret bound for \textsc{StrongEuler}}
We will now show how our results from the previous sections translate into a stronger regret bound on the concrete example of \textsc{StrongEuler}. Ignoring lower-order terms, \citet{simchowitz2019non} showed that the surpluses of this algorithm are bounded as $E_k(s,a) \lessapprox \sqrt{\frac{\mathcal{V}^*(s,a)\log(\bar n_k(s,a))}{\bar n_k(s,a)}}$ where $\bar n_k(s,a) = \sum_{j=1}^{k-1} w^{\pi_j}(s,a)$ are the expected number of samples for $(s,a)$ up to episode $k$ and $\mathcal{V}^*(s,a) = \VV[R(s,a)] + \VV_{s'\sim P(\cdot|s,a)}[V^{*}(s')]$ is the one-step variance term w.r.t. the optimal value function. \pref{prop:surplus_clipping_bound} with the clipping to the average gap gives in this case
\begin{equation}
\label{eq:reg_bound_fake}
\begin{aligned}
&\mathfrak{R}(K)
\lessapprox \sum_{k=1}^K\sum_{s,a} w^{\pi_k}(s,a)\clip\left[\sqrt{\frac{\mathcal{V}^*(s,a)\log(\bar n_k(s,a))}{\bar n_k(s,a)}} ~\bigg|~ \gap(s,a) \lor \epsilon_{k}(s,a)\right].
\end{aligned}
\end{equation}
Existing analysis now translate such a clipping bound into a $\log(K)$ regret bound using an integration argument \citep[e.g. ][]{dann2019policy,simchowitz2019non}. However, we cannot rely on these arguments since our clipping thresholds are not necessarily constant across episodes. To address this technical challenge, we derive the following lemma based on an optimization view on such terms
\begin{lemma}\label{lem:opt_lemma_simpler}
For any sequence of thresholds $\gamma_1, \dots \gamma_K > 0$, consider the problem with $x_0 = 1$
\begin{align*}
\underset{x_1, \dots x_K \in [0,1]}{\operatorname{maximize}}
\sum_{k=1}^K \frac{ x_k\sqrt{\log(\sum_{j=0}^k x_j)}}{\sqrt{\sum_{j=0}^k x_j}}
\quad \mathrm{s.t.~~ for~~ all~~ }k \in [K]: \quad
\frac{\sqrt{\log(\sum_{j=0}^k x_j)}}{\sqrt{\sum_{j=0}^k x_j}} \geq \gamma_k.
\end{align*}
The optimal value is bounded for any $t \in [K]$ from above as
$\frac{\log(t)}{\epsilon_t} + \sqrt{(K-t) \log K}$.
\end{lemma}
Applying this lemma for each $(s,a)$ with $x_k = w^{\pi_k}(s,a)$ and appropriate clipping thresholds $\gamma_k \approx (\gap(s,a) + \epsilon_{k}(s,a))/\sqrt{\mathcal{V}^*(s,a)}$ we can derive our main result:
\begin{theorem}[Informal]
The regret $\mathfrak{R}(K)$ of \texttt{StrongEuler} is bounded with high probability as
\begin{align*}
\mathfrak{R}(K) \lessapprox
\sum_{s,a} \min_{t \in [K_{(s,a)}]} \left\{
\frac{\mathcal{V}^*(s,a)}{\gap(s,a) + \epsilon_{t}(s,a)}
+ \sqrt{\mathcal{V}^*(s,a) (K_{(s,a)} - t)}\right\} \log(K),
\end{align*}
where we only show terms with (inverse-)polynomial dependency on gaps and $K_{(s,a)}$ is the last episode during which a policy supported on $(s,a)$ was played. For the full bound, see \pref{thm:reg_bound_gen_se} in the appendix.
\end{theorem}
\tm{Possibly state the full version of the result here as we already have an informal version above.}
Note that our regret bound depends through $\epsilon_{t}(s,a)$, the average gap encountered by $\pi_t$, on the policies that the algorithm played. Note that this dependency is generally mild and can be removed by taking a maximum over all possible policies $\pi_t$ (see discussion on deterministic MDPs for an example).
Our regret bound can smoothly interpolate between the worst-case rate of $\sqrt{K}$ achieved for $t \ll K$ when all gaps and average gaps are $O(\sqrt{\mathcal{V}^*(s,a) K})$ and the gap-dependent regret rate achieved when $t \approx K$ that scales inversely with gaps. Note that our bound depends on the gaps in all episodes and can benefit from choices for $t$ that yields a large gap or average gap in the late episodes.
\section{Conclusion}
In this work, we prove that optimistic algorithms such as \textsc{StrongEuler}, can suffer substantially less regret compared to what prior work had shown. We do this by introducing a new notion of gap, while greatly simplifying and generalizing existing analysis techniques.
We further investigated the information-theoretic limits of learning episodic layered MDPs.
We provide two new closed-form lower bounds in the special case where the MDP has either
deterministic transitions or the optimal policy is supported on all states.
These lower bounds suggest that our notion of gap better captures the difficulty of an episodic MDP for RL.
\section{Novel upper bounds for optimistic algorithms}
\label{sec:upper_bounds}
As hinted to in the introduction, the way prior regret bounds treat value-function gaps independently at each state-action pair can lead to excessively loose guarantees.
Bounds that use value-function gaps \citep{simchowitz2019non,lykouris2019corruption,jin2020simultaneously}
scale at least as
$$\sum_{s,a \colon \gap(s,a) >0 } \frac{H\log(K)}{\gap(s,a)} +
\sum_{s,a \colon \gap(s,a) =0 }
\frac{H\log(K)}{\gap_{\min}} ,
$$
where state-action pairs with zero gap appear, with $\gap_{\min} = \min_{s, a \colon \gap(s,a) > 0} \gap(s,a)$, the smallest positive gap.
To illustrate where these bounds are loose, let us revisit the example in \pref{fig:summary}. Here, these bounds evaluate to $ \frac{H\log(K)}{c}+\frac{H\log(K)}{\epsilon}+\frac{SH\log(K)}{\epsilon}$, where the first two terms come from state-action pairs with positive value-function gaps and the last term comes from all the state-action pairs with zero gaps. There are several opportunities for improvement:
\begin{enumerate}[label = \textbf{O.\arabic*}]
\item\label{enum_prob_1} \textbf{State-action pairs that can only be visited by taking optimal actions:} We should not pay the $1/\gap_{\min}$ factor for such $(s,a)$ as there are no other suboptimal policies $\pi$ to distinguish from $\pi^*$ in such states.
\item\label{enum_prob_3}
\textbf{State-action pairs that can only be visited by taking at least one suboptimal action:}
We should not pay the $1 / \gap(s_2, a_3)$ factor for state-action pair $(s_2, a_3)$ and the $1 / \gap_{\min}$ factor for $(s_2, a_4)$ because no optimal policy visits $s_2$. Such state-action pairs should only be accounted for with the price to learn that $a_2$ is not optimal in state $s_1$. After all, learning to distinguish between $\pi_1$ and $\pi_2$ is unnecessary for optimal return.
\end{enumerate}
Both opportunities suggest that the price $\frac{1}{\gap(s,a)}$ or $\frac{1}{\gap_{\min}}$ that each state-action pair $(s,a)$ contributes to the regret bound can be reduced by taking into account the regret incurred by the time $(s,a)$ is reached. Opportunity~\ref{enum_prob_1} postulates that if no regret can be incurred up to (and including) the time step $(s,a)$ is reached, then this state-action pair should not appear in the regret bound. Similarly, if this regret is necessarily large, then the agent can learn this with few observations and stop reaching $(s,a)$ earlier than $\gap(s,a)$ may suggest. Thus, as claimed in \ref{enum_prob_3}, the contribution of $(s,a)$ to the regret should be more limited in this case.
Since the total regret incurred during one episode by a policy $\pi$ is simply the expected sum of value-function gaps visited (\pref{lem:gap_decomp_pi} in the appendix),
\begin{align}
v^* - v^\pi = \EE_{\pi}\left[ \sum_{h=1}^H \gap(S_h, A_h) \right],
\label{eq:reg_decomp}
\end{align}
we can measure the regret incurred up to reaching $(S_{t}, A_{t})$ by the sum of value function gaps $\sum_{h=1}^t \gap(S_h, A_h)$ up to this point $t$. We are interested in the regret incurred up to visiting a certain state-action pair $(s,a)$ which $\pi$ may visit only with some probability. We therefore need to take the expectation of such gaps conditioned on the event that $(s,a)$ is actually visited. We further condition on the event that this regret is nonzero, which is exactly the case when the agent encounters a positive value-function gap within the first $\kappa(s)$ time steps. We arrive at
\begin{align*}
\EE_{\pi}\left[ \sum_{h=1}^{\kappa(s)} \gap(S_h, A_h) ~ \bigg| ~S_{\kappa(s)} = s, A_{\kappa(s)} = a, B \leq \kappa(s) \right],
\end{align*}
where $B = \min \{ h \in [H+1] \colon \gap(S_h, A_h) > 0\}$ is the first time a non-zero gap is visited. This quantity measures the regret incurred up to visiting $(s,a)$ through suboptimal actions. If this quantity is large for all policies $\pi$, then a learner will stop visiting this state-action pair after few observations because it can rule out all actions that lead to $(s,a)$ quickly. Conversely, if the event that we condition on has zero probability under any policy, then $(s,a)$ can only be reached through optimal action choices (including $a$ in $s$) and incurs no regret. This motivates our new definition of gaps that combines value function gaps with the regret incurred up to visiting the state-action pair:
\begin{definition}[Return gap]
\label{def:return_gap}
For any state-action pair $(s, a)\in\mathcal{S}\times\mathcal{A}$ define $\mathcal{B}(s,a)\equiv \{B \leq \kappa(s), S_{\kappa(s)} = s, A_{\kappa(s)} = a\}$, where $B$ is the first time a non-zero gap is encountered. $\mathcal{B}(s,a)$ denotes the event that state-action pair $(s,a)$ is visited
and that a suboptimal action was played at any time up to visiting $(s,a)$.
We define the return gap as
\begin{align*}
\overline{\gap}(s,a) \equiv \gap(s,a)\lor\min_{\substack{\pi \in \Pi \colon \\\PP_{\pi}(\mathcal{B}(s,a)) > 0}}&~
\frac{1}{H} \,
\EE_{\pi} \bracket*{ \sum_{h = 1}^{\kappa(s)} \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a) }
\end{align*}
if there is a policy $\pi \in \Pi$ with $\PP_\pi(\mathcal{B}(s,a)) > 0$ and $\overline{\gap}(s,a) \equiv 0$ otherwise.
\end{definition}
The additional $1/H$ factor in the second term is a required normalization suggesting that it is the average gap rather than their sum that matters.
Equipped with this definition, we are ready to state our main upper bound which pertains to the \textsc{StrongEuler} algorithm proposed by \citet{simchowitz2019non}.
\begin{theorem}[Main Result (Informal)]
\label{thm:reg_bound_gen_informal}
The regret $\mathfrak{R}(K)$ of \textsc{StrongEuler} is bounded with high probability for all number of episodes $K$ as
\begin{align*}
\mathfrak{R}(K) &\lessapprox \sum_{\substack{(s,a)\in\mathcal{S}\times\mathcal{A} \colon \\\overline{\gap}(s,a) > 0}} \frac{\mathcal{V}^*(s,a)}{\overline{\gap}(s,a)}
\log K
.
\end{align*}
\end{theorem}
In the above,
we have restricted the bound to only those terms that have inverse polynomial dependence on the gaps.
\paragraph{Comparison with existing gap-dependent bounds.} We now compare our bound to the existing gap-dependent bound for \textsc{StrongEuler} by \citet[Corollary B.1]{simchowitz2019non}
\begin{align}
\label{eq:cor_1b_simchowitz}
\mathfrak{R}(K) \lessapprox \sum_{\substack{(s,a)\in\mathcal{S}\times\mathcal{A} \colon \\\gap(s,a) > 0}} \frac{ H\mathcal{V}^*(s,a)}{\gap(s,a)}\log K + \sum_{\substack{(s,a)\in\mathcal{S}\times\mathcal{A} \colon \\\gap(s,a) = 0}} \frac{H\mathcal{V}^*}{\gap_{\min}}\log K .
\end{align}
We here focus only on terms that admit a dependency on $K$ and an inverse-polynomial dependency on gaps as all other terms are comparable.
Most notable is the absence of the second term of \pref{eq:cor_1b_simchowitz} in our bound in \pref{thm:reg_bound_gen_informal}. Thus, while state-action pairs with $\overline{\gap}(s, a) = 0$ do not contribute to our regret bound, they appear with a $1/\gap_{\min}$ factor in existing bounds. Therefore, our bound addresses \ref{enum_prob_1} because it does not pay for state-action pairs that can only be visited through optimal actions.
Further, state-action pairs that do contribute to our bound satisfy $\frac{1}{\overline{\gap}(s,a)} \leq \frac{1}{\gap(s,a)} \wedge \frac{H}{\gap_{\min}}$ and thus never contribute more than in the existing bound in \pref{eq:cor_1b_simchowitz}. Therefore, our regret bound is never worse.
In fact, it is significantly tighter when there are states that are only reachable by taking severely suboptimal actions, i.e., when the average value-function gaps are much larger than $\gap(s,a)$ or $\gap_{\min}$. By our definition of return gaps, we only pay the inverse of these larger gaps instead of $\gap_{\min}$.
Thus, our bound also addresses \ref{enum_prob_3} and achieves the desired $\log(K)/c$ regret bound in the motivating example of \pref{fig:summary} as opposed to the $\log(K)/\epsilon$ bound of prior work.
\paragraph{Regret bound when transitions are deterministic.}
We now interpret \pref{def:return_gap} for MDPs with deterministic transitions and derive an alternative form of our bound in this case.
Let $\Pi_{s,a}$ be the set of all policies that visit $(s,a)$ and have taken a suboptimal action up to that visit, that is,
$$
\Pi_{s,a} \equiv \curl*{\pi \in \Pi ~\colon s^\pi_{\kappa(s)} = s ,a^\pi_{\kappa(s)} = a, \exists~ h \leq \kappa(s), \gap(s^\pi_{h},a^\pi_{h})>0}.
$$
where $(s^\pi_1, a^{\pi}_1, s^\pi_2, \dots, s^\pi_H, a^{\pi}_H)$ are the state-action pairs visited (deterministically) by $\pi$.
Further, let $v^{*}_{s,a} = \max_{\pi \in \Pi_{s,a}} v^\pi$ be the best return of such policies.
\pref{def:return_gap} now evaluates to $\overline{\gap}(s,a) = \gap(s,a) \vee \frac{1}{H}(v^* - v^{*}_{s,a})$ and the bound in \pref{thm:reg_bound_gen_informal} can be written as
\begin{align}
\label{eq:det_trans_reg}
\mathfrak{R}(K) &\lessapprox \sum_{s,a \colon \Pi_{s,a} \neq \varnothing} \frac{H\log(K)}{v^* - v^*_{s,a}}~.
\end{align}
We show in \pref{app:unique_opt_pol}, that it is possible to further improve this bound when the optimal policy is unique by only summing over state-action pairs which are not visited by the optimal policy.
\subsection{Regret analysis with improved clipping: from minimum gap to average gap}
\label{sec:gen_clipping}
In this section, we present the main technical innovations of our tighter regret analysis.
Our framework applies to \emph{optimistic} algorithms that maintain a $Q$-function estimate, $\bar Q_k(s,a)$, which overestimates the optimal $Q$-function $Q^*(s,a)$ with high probability in all states $s$, actions $a$ and episodes $k$. \cd{Are we sure we don't need strong optimism?}
We first give an overview of gap-dependent analyses and then describe our approach.
\paragraph{Overview of gap-dependent analyses. }
A central quantity in regret analyses of optimistic algorithms are \emph{surpluses} $E_k(s,a)$ which, roughly speaking, quantify the local amount of optimism in $(s, a)$ and are formally defined as
\begin{align}
\label{eq:surpl_def}
E_k(s,a) = \bar Q_k(s,a) - r(s,a) - \langle P(\cdot|s,a), \bar V_k\rangle~.
\end{align}
Worst-case regret analyses bound the regret in episode $k$ as
$\sum_{(s,a) \in \mathcal{S}\times\mathcal{A}}w_{\pi_k}(s,a)E_{k}(s,a)$, the expected surpluses under the optimistic policy $\pi_k$ executed in that episode. Instead, gap-dependent analyses rely on a tighter version and bound the instantaneous regret by the \emph{clipped surpluses} \citep[e.g. Proposition 3.1][]{simchowitz2019non}
\begin{align}
V^*(s_1) - V^{\pi_k}(s_1) \leq 2e \sum_{s,a} w^{\pi_k}(s,a)\clip\left[E_{k}(s,a) ~\bigg\vert~ \frac{1}{4H}\gap(s,a) \lor \frac{\gap_{\min}}{2H}\right].
\label{eqn:old_surplusclipping}
\end{align}
Using concentration arguments one can show that $E_{k}(s,a)$ shrinks at a rate of $\sqrt{\frac{1}{n_k(s,a)}}$, with $n_k(s,a)$ the current number of times $(s,a)$ has been visited. Thus, roughly speaking, after $\left(\frac{\gap(s,a)\lor \gap_{\min}}{H}\right)^{-2}$ visits to $(s,a)$, the surplus falls below the threshold and does not contribute any more to the regret. Since each visit incurs at most $\frac{\gap(s,a)\lor \gap_{\min}}{H}$ regret, this leads to a $\frac{H}{\gap(s,a)\lor \gap_{\min}}$ dependency on gaps for each state-action pair.
\paragraph{Sharper clipping with general thresholds.}
Our main technical contribution for achieving a regret bound in terms of return gaps $\overline{\gap}(s,a)$ is the following improved surplus clipping bound:
\begin{restatable}[Improved surplus clipping bound]{proposition}{surplusclippingbound}
\label{prop:surplus_clipping_bound}
Let the surpluses $E_k(s,a)$ be generated by an optimistic algorithm.
Then the instantaneous regret of $\pi_k$ is bounded as follows:
\vspace{-1mm}
\begin{align*}
V^*(s_1) - V^{\pi_k}(s_1)
\leq 4 \sum_{s,a} w^{\pi_k}(s,a) \clip\left[ E_k(s,a) ~\bigg| ~ \frac 1 4 \gap(s,a) \vee \epsilon_{k}(s, a) \right]~,
\end{align*}
\vspace{-1mm}
where $\epsilon_k \colon \mathcal{S} \times \mathcal{A} \rightarrow \RR^+_0$
is any clipping threshold function that satisfies
\vspace{-1mm}
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^H \epsilon_k(S_h, A_h) \right]
\leq \frac{1}{2} \EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h) \right].
\end{align*}
\end{restatable}
Compared to previous surplus clipping bounds in \eqref{eqn:old_surplusclipping}, there are several notable differences. First, instead of $\gap_{\min}/2H$, we can now pair $\gap(s,a)$ with more general clipping thresholds $\epsilon_k(s,a)$, as long as their expected sum over time steps after the first non-zero gap was encountered is at most half the expected sum of gaps. We will provide some intuition for this condition below. Note that $\epsilon_{k}(s,a) \equiv \frac{\gap_{\min}}{2H}$ satisfies the condition because the LHS evaluates to $\frac{\gap_{\min}}{2H} \PP_{\pi_k}( B \leq H)$ and there must be at least one positive gap in the sum $\sum_{h=1}^H\gap(S_h,A_h)$ on the RHS in event $\{B \leq H\}$. Thus our bound recovers existing results.
In addition, the first term in our clipping thresholds is $\frac{1}{4}\gap(s,a)$ instead of $\frac{1}{4H}\gap(s,a)$. \citet{simchowitz2019non} are able to remove this spurious $H$ factor only if the problem instance happens to be a bandit instance and the algorithm satisfies a condition called \emph{strong optimism} where surpluses have to be non-negative. Our analysis does not require such conditions and therefore generalizes these existing results.\footnote{Our layered state space assumption affects the $H$ dependencies in lower-order terms in our final regret compared to \citet{simchowitz2019non}. However, \pref{prop:surplus_clipping_bound} directly applies to their setting without any penalty in $H$.}
\paragraph{Intuition for threshold condition in \pref{prop:surplus_clipping_bound}.}
The key to proving \pref{prop:surplus_clipping_bound} is the following self-bounding trick for the instantaneous regret $V^*(s_1) - V^{\pi_k}(s_1)$. The self-bounding trick works in the following way. If we have a $\rho > 0$ which is upper bounded by $\gamma \geq \rho$ and lower bounded by $0 < \beta \leq \rho$, we can further bound $(1 - c) \rho \leq \gamma - c\beta$ for any constant $c \in [0,1]$. We can now use this trick twice with $\rho = V^*(s_1) - V^{\pi_k}(s_1)$, $\gamma = \EE_{\pi_k}[\sum_{h=1}^H E_k(S_h,A_h)]$, $c=1/4$ and $\beta$ equal to either the expected sum of gaps or the assumed lower bound with clipping functions, that is $\beta = \EE_{\pi_k}[\sum_{h=1}^H \gap(S_h,A_h)]$ or $\beta=\EE_{\pi_k}[\sum_{h=1}^H \epsilon_k(S_h,A_h)]$. This implies that one half of the instantaneous regret is bounded as
\begin{align*}
\frac{1}{2}(V^*(s_1) - V^{\pi_k}(s_1)) \leq \sum_{h=1}^H \EE_{\pi_k}\left[E_k(S_h,A_h) - \frac{\gap(S_h,A_h) + \epsilon_k(S_h,A_h)}{4}\right].
\end{align*}
Using the fact that the $\clip$ operator satisfies $a - b \leq \clip[a | b]$ gives the desired statement. This implies that, in order to achieve the tightest regret bound, we should clip $E_k(S_h,A_h)$ to the largest possible value. Thus, the goal is to lower bound the expected sum of gaps as tightly as possible by the clipping function.
Besides this insight, introducing the stopping time $B$ in the condition is key to addressing \ref{enum_prob_1} and requires a careful treatment laid out in the full proof in \pref{app:upper_bounds}.
\paragraph{Choice of clipping thresholds for return gaps.}
The condition in \pref{prop:surplus_clipping_bound} suggests that one can set $\epsilon_{k}(S_h,A_h)$ to be proportional to the average expected gap under policy $\pi_k$:
\begin{align}
\epsilon_{k}(s,a) =
\frac{1}{2H}
\EE_{\pi_k}\left[ \sum_{h=1}^H \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a)\right].
\label{eqn:epsilon_choice}
\end{align}
if $\PP_{\pi_k}(\mathcal{B}(s,a)) > 0$ and $\epsilon_{k}(s,a) = \infty$ otherwise.
\pref{lem:clipping_gaps_rel} in \pref{app:upper_bounds} shows that this choice indeed satisfies the condition in \pref{prop:surplus_clipping_bound}.
If we now take the minimum over all policies for $\pi_k$, then we can proceed with the standard analysis and derive our main result in \pref{thm:reg_bound_gen_informal}. However, by avoiding the minimum over policies, we can derive a stronger regret bound that depends on the actual policies executed by the algorithm. We present this bound in the next section.
\subsection{Policy-dependent regret bound}
We will now show how our results from the previous sections translate into a stronger regret bound on the concrete example of \textsc{StrongEuler}. Ignoring lower-order terms, \citet{simchowitz2019non} showed that the surpluses of this algorithm are bounded as $E_k(s,a) \lessapprox \sqrt{\frac{\mathcal{V}^*(s,a)\log(\bar n_k(s,a))}{\bar n_k(s,a)}}$ where $\bar n_k(s,a) = \sum_{j=1}^{k-1} w^{\pi_j}(s,a)$ are the expected number of samples for $(s,a)$ up to episode $k$ and $\mathcal{V}^*(s,a) = \VV[R(s,a)] + \VV_{s'\sim P(\cdot|s,a)}[V^{*}(s')]$ is the one-step variance term w.r.t.\ the optimal value function. In this case, \pref{prop:surplus_clipping_bound} with the clipping to the average gap gives:
\begin{equation}
\label{eq:reg_bound_fake}
\begin{aligned}
&\mathfrak{R}(K)
\lessapprox \sum_{k=1}^K\sum_{s,a} w^{\pi_k}(s,a)\clip\left[\sqrt{\frac{\mathcal{V}^*(s,a)\log(\bar n_k(s,a))}{\bar n_k(s,a)}} ~\bigg|~ \gap(s,a) \lor \epsilon_{k}(s,a)\right].
\end{aligned}
\end{equation}
An existing analysis now translates such a clipping bound into a $\log(K)$ regret bound using an integration argument \citep[e.g.][]{simchowitz2019non}. However, we cannot rely on these arguments since our clipping thresholds may not be constant across episodes. To address this technical challenge, we derive the following lemma based on an optimization view on such terms
\begin{lemma}\label{lem:opt_lemma_simpler}
For any sequence of thresholds $\gamma_1, \dots \gamma_K > 0$, consider the problem with $x_0 = 1$
\vspace{-2mm}
\begin{align*}
\underset{x_1, \dots x_K \in [0,1]}{\operatorname{maximize}}
\sum_{k=1}^K x_k\sqrt{\frac{ \log \paren*{\sum_{j = 0}^k x_j} }{\sum_{j=0}^k x_j}}
\quad \mathrm{s.t.~~ for~~ all~~ }k \in [K]: \quad
\sqrt{\frac{\log \paren*{\sum_{j = 0}^k x_j}}{\sum_{j = 0}^k x_j}} \geq \gamma_k.
\end{align*}
\vspace{-3mm}
The optimal value is bounded for any $t \in [K]$ from above as
$\frac{\log(t)}{\epsilon_t} + \sqrt{(K-t) \log K}$.
\end{lemma}
Applying this lemma for each $(s,a)$ with $x_k = w^{\pi_k}(s,a)$ and appropriate clipping thresholds $\gamma_k \approx (\gap(s,a) \vee \epsilon_{k}(s,a))/\sqrt{\mathcal{V}^*(s,a)}$ we can derive our main result:
\begin{theorem}
When \texttt{StrongEuler} is run with confidence parameter $\delta$, then with probability at least $1 - \delta$, its regret is bounded for all number of episodes $K$ as
\begin{align*}
\mathfrak{R}(K) &\lessapprox
\sum_{s,a} \min_{t \in [K_{(s,a)}]} \curl*{
\frac{\mathcal{V}^*(s,a)\log\left(\frac{M}{\delta}\right)\log\left(\frac{\mathcal{V}^*(s,a) \log(M/\delta)}{\gap(s,a) \vee \epsilon_{t}(s,a)}\right)}{\gap(s,a) \vee \epsilon_{t}(s,a)}+ \sqrt{\mathcal{V}^*(s,a) (K_{(s,a)} - t)} } \log(K)\\
&\qquad + S^2AH^4\log\left(\frac{MK}{\delta}\right)\log\left(\frac{MH}{\overline{\gap}_{\min}}\right),
\end{align*}
where $M\leq (SAH)^3$, $\overline{\gap}_{\min} = \min_{(s,a)}\overline{\gap}(s,a)$ and $K_{(s,a)}$ is the last episode during which a policy that may visit $(s,a)$ was played.
\end{theorem}
A slightly more refined version of this bound is stated \pref{thm:reg_bound_gen_se} in \pref{app:upper_bounds}.
Note that our regret bound depends through $\epsilon_{t}(s,a)$, the average gap encountered by $\pi_t$, on the policies that the algorithm played. We can smoothly interpolate between the worst-case rate of $\sqrt{K}$ achieved for $t \ll K$ when all gaps and average gaps are $O(\sqrt{\mathcal{V}^*(s,a) K})$ and the gap-dependent regret rate achieved when $t \approx K$ that scales inversely with gaps. Our bound depends on the gaps in all episodes and can benefit from choices for $t$ that yields a large gap or average gap in late episodes.
\section{Novel upper bounds for optimistic algorithms}
\label{sec:upper_bounds}
In this section, we present tighter regret upper-bounds for optimistic algorithms through a novel analysis technique.
Our technique can be generally applied to model-based optimistic algorithms such as \textsc{StrongEuler} \citep{simchowitz2019non}, \textsc{Ucbvi} \citep{azar2012sample}, \textsc{ORLC} \citep{dann2019policy} or \textsc{Euler} \citep{zanette2019tighter}.
In the following, we will first give a brief overview of this class of algorithms (see \pref{app:opt_algs} for more details) and then state our main results for the \textsc{StrongEuler} algorithm \cite{simchowitz2019non}. We focus on this algorithm for concreteness and ease of comparison.
Optimistic algorithms maintain estimators of the $Q$-functions at every state-action pair such that there exists at least one policy $\pi$ for which the estimator, $\bar Q^{\pi}$, overestimates the $Q$-function of the optimal policy, that is $\bar Q^{\pi}(s,a) \geq Q^*(s,a),\forall (s,a)\in\mathcal{S}\times\mathcal{A}$. During episode $k\in[K]$, the optimistic algorithm selects the policy $\pi_k$ with highest optimistic value function $\bar V_k$. By definition, it holds that $\bar V_k(s) \geq V^*(s)$. The optimistic value and $Q$-functions are constructed through finite-sample estimators of the true rewards $r(s,a)$ and the transition kernel $P(\cdot|s,a)$ plus bias terms, similar to estimators for the UCB-I multi-armed bandit algorithm. Careful construction of these bias terms is crucial for deriving min-max optimal regret bounds in $S,A$ and $H$ \citep{azar2017minimax}. Bias terms which yield the tightest known bounds come from concentration of martingales results such as Freedman's inequality~\citep{Freedman1975} and empirical Bernstein's inequality for martingales~\citep{maurer2009empirical}.
The \textsc{StrongEuler} algorithm not only satisfies optimism, i.e., $\bar V_k \geq V^*$, but also a stronger version called \emph{strong optimism}. To define strong optimism we need the notion of \emph{surplus} which roughly measures the optimism at a fixed state-action pair. Formally the surplus at $(s,a)$ during episode $k$ is defined as
\begin{align}
\label{eq:surpl_def}
E_k(s,a) = \bar Q_k(s,a) - r(s,a) - \langle P(\cdot|s,a), \bar V_k\rangle~.
\end{align}
We say that an algorithm is strongly optimistic if $E_k(s,a) \geq 0,\forall (s,a) \in \mathcal{S}\times\mathcal{A}, k\in[K]$. Surpluses are also central to our new regret bounds and we will carefully discuss their use in Appendix~\ref{app:upper_bounds}.
As hinted to in the introduction, the way prior regret bounds treat value-function gaps independently at each state-action pair can lead to excessively loose guarantees.
Bounds that use value-function gaps \citep{simchowitz2019non,lykouris2019corruption,jin2020simultaneously}
scale at least as
$$\sum_{s,a \colon \gap(s,a) >0 } \frac{H\log(K)}{\gap(s,a)} +
\sum_{s,a \colon \gap(s,a) =0 }
\frac{H\log(K)}{\gap_{\min}} ,
$$
where state-action pairs with zero gap appear, with $\gap_{\min} = \min_{s, a \colon \gap(s,a) > 0} \gap(s,a)$, the smallest positive gap.
To illustrate where these bounds are loose, let us revisit the example in \pref{fig:summary}. Here, these bounds evaluate to $ \frac{H\log(K)}{c}+\frac{H\log(K)}{\epsilon}+\frac{SH\log(K)}{\epsilon}$, where the first two terms come from state-action pairs with positive value-function gaps and the last term comes from all the state-action pairs with zero gaps. There are several opportunities for improvement:
\begin{enumerate}[label = \textbf{O.\arabic*}]
\item\label{enum_prob_1} \textbf{State-action pairs that can only be visited by taking optimal actions:} We should not pay the $1/\gap_{\min}$ factor for such $(s,a)$ as there are no other suboptimal policies $\pi$ to distinguish from $\pi^*$ in such states.
\item\label{enum_prob_3}
\textbf{State-action pairs that can only be visited by taking at least one suboptimal action:}
We should not pay the $1 / \gap(s_2, a_3)$ factor for state-action pair $(s_2, a_3)$ and the $1 / \gap_{\min}$ factor for $(s_2, a_4)$ because no optimal policy visits $s_2$. Such state-action pairs should only be accounted for with the price to learn that $a_2$ is not optimal in state $s_1$. After all, learning to distinguish between $\pi_1$ and $\pi_2$ is unnecessary for optimal return.
\end{enumerate}
Both opportunities suggest that the price $\frac{1}{\gap(s,a)}$ or $\frac{1}{\gap_{\min}}$ that each state-action pair $(s,a)$ contributes to the regret bound can be reduced by taking into account the regret incurred by the time $(s,a)$ is reached. Opportunity~\ref{enum_prob_1} postulates that if no regret can be incurred up to (and including) the time step $(s,a)$ is reached, then this state-action pair should not appear in the regret bound. Similarly, if this regret is necessarily large, then the agent can learn this with few observations and stop reaching $(s,a)$ earlier than $\gap(s,a)$ may suggest. Thus, as claimed in \ref{enum_prob_3}, the contribution of $(s,a)$ to the regret should be more limited in this case.
Since the total regret incurred during one episode by a policy $\pi$ is simply the expected sum of value-function gaps visited (\pref{lem:gap_decomp_pi} in the appendix),
\begin{align}
v^* - v^\pi = \EE_{\pi}\left[ \sum_{h=1}^H \gap(S_h, A_h) \right],
\label{eq:reg_decomp}
\end{align}
we can measure the regret incurred up to reaching $(S_{t}, A_{t})$ by the sum of value function gaps $\sum_{h=1}^t \gap(S_h, A_h)$ up to this point $t$. We are interested in the regret incurred up to visiting a certain state-action pair $(s,a)$ which $\pi$ may visit only with some probability. We therefore need to take the expectation of such gaps conditioned on the event that $(s,a)$ is actually visited. We further condition on the event that this regret is nonzero, which is exactly the case when the agent encounters a positive value-function gap within the first $\kappa(s)$ time steps. We arrive at
\begin{align*}
\EE_{\pi}\left[ \sum_{h=1}^{\kappa(s)} \gap(S_h, A_h) ~ \bigg| ~S_{\kappa(s)} = s, A_{\kappa(s)} = a, B \leq \kappa(s) \right],
\end{align*}
where $B = \min \{ h \in [H+1] \colon \gap(S_h, A_h) > 0\}$ is the first time a non-zero gap is visited. This quantity measures the regret incurred up to visiting $(s,a)$ through suboptimal actions. If this quantity is large for all policies $\pi$, then a learner will stop visiting this state-action pair after few observations because it can rule out all actions that lead to $(s,a)$ quickly. Conversely, if the event that we condition on has zero probability under any policy, then $(s,a)$ can only be reached through optimal action choices (including $a$ in $s$) and incurs no regret. This motivates our new definition of gaps that combines value function gaps with the regret incurred up to visiting the state-action pair:
\begin{definition}[Return gap]
\label{def:return_gap}
For any state-action pair $(s, a)\in\mathcal{S}\times\mathcal{A}$ define $\mathcal{B}(s,a)\equiv \{B \leq \kappa(s), S_{\kappa(s)} = s, A_{\kappa(s)} = a\}$, where $B$ is the first time a non-zero gap is encountered. $\mathcal{B}(s,a)$ denotes the event that state-action pair $(s,a)$ is visited
and that a suboptimal action was played at any time up to visiting $(s,a)$.
We define the return gap as
\begin{align*}
\overline{\gap}(s,a) \equiv \gap(s,a)\lor\min_{\substack{\pi \in \Pi \colon \\\PP_{\pi}(\mathcal{B}(s,a)) > 0}}&~
\frac{1}{H} \,
\EE_{\pi} \bracket*{ \sum_{h = 1}^{\kappa(s)} \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a) }
\end{align*}
if there is a policy $\pi \in \Pi$ with $\PP_\pi(\mathcal{B}(s,a)) > 0$ and $\overline{\gap}(s,a) \equiv 0$ otherwise.
\end{definition}
The additional $1/H$ factor in the second term is a required normalization suggesting that it is the average gap rather than their sum that matters.
We emphasize that Definition~\ref{def:return_gap} is independent of the choice of RL algorithm and in particular does not depend on the algorithm being optimistic. Thus, we expect our main ideas and techniques to be useful beyond the analysis of optimistic algorithms.
Equipped with this definition, we are ready to state our main upper bound which pertains to the \textsc{StrongEuler} algorithm proposed by \citet{simchowitz2019non}.
\begin{theorem}[Main Result (Informal)]
\label{thm:reg_bound_gen_informal}
The regret $\mathfrak{R}(K)$ of \textsc{StrongEuler} is bounded with high probability for all number of episodes $K$ as
\begin{align*}
\mathfrak{R}(K) &\lessapprox \sum_{\substack{(s,a)\in\mathcal{S}\times\mathcal{A} \colon \\\overline{\gap}(s,a) > 0}} \frac{\mathcal{V}^*(s,a)}{\overline{\gap}(s,a)}
\log K
.
\end{align*}
\end{theorem}
In the above,
we have restricted the bound to only those terms that have inverse polynomial dependence on the gaps.
\paragraph{Comparison with existing gap-dependent bounds.} We now compare our bound to the existing gap-dependent bound for \textsc{StrongEuler} by \citet[Corollary B.1]{simchowitz2019non}
\begin{align}
\label{eq:cor_1b_simchowitz}
\mathfrak{R}(K) \lessapprox \sum_{\substack{(s,a)\in\mathcal{S}\times\mathcal{A} \colon \\\gap(s,a) > 0}} \frac{ H\mathcal{V}^*(s,a)}{\gap(s,a)}\log K + \sum_{\substack{(s,a)\in\mathcal{S}\times\mathcal{A} \colon \\\gap(s,a) = 0}} \frac{H\mathcal{V}^*}{\gap_{\min}}\log K .
\end{align}
We here focus only on terms that admit a dependency on $K$ and an inverse-polynomial dependency on gaps as all other terms are comparable.
Most notable is the absence of the second term of \pref{eq:cor_1b_simchowitz} in our bound in \pref{thm:reg_bound_gen_informal}. Thus, while state-action pairs with $\overline{\gap}(s, a) = 0$ do not contribute to our regret bound, they appear with a $1/\gap_{\min}$ factor in existing bounds. Therefore, our bound addresses \ref{enum_prob_1} because it does not pay for state-action pairs that can only be visited through optimal actions.
Further, state-action pairs that do contribute to our bound satisfy $\frac{1}{\overline{\gap}(s,a)} \leq \frac{1}{\gap(s,a)} \wedge \frac{H}{\gap_{\min}}$ and thus never contribute more than in the existing bound in \pref{eq:cor_1b_simchowitz}. Therefore, our regret bound is never worse.
In fact, it is significantly tighter when there are states that are only reachable by taking severely suboptimal actions, i.e., when the average value-function gaps are much larger than $\gap(s,a)$ or $\gap_{\min}$. By our definition of return gaps, we only pay the inverse of these larger gaps instead of $\gap_{\min}$.
Thus, our bound also addresses \ref{enum_prob_3} and achieves the desired $\log(K)/c$ regret bound in the motivating example of \pref{fig:summary} as opposed to the $\log(K)/\epsilon$ bound of prior work.
One of the limitations of optimistic algorithms is their $\nicefrac{S}{\gap_{\min}}$ dependence even when there is only one state with a gap of $\gap_{\min}$ \citep{simchowitz2019non}. We note that even though our bound in \pref{thm:reg_bound_gen_informal} improves on prior work, our result does not aim to address this limitation.
Very recent concurrent work \citep{xu2021fine} proposed an action-elimination based algorithm that avoids the $\nicefrac{S}{\gap_{\min}}$ issue of optimistic algorithm but their regret bounds still suffer the issues illustrated in \pref{fig:summary} (e.g. \ref{enum_prob_3}).
We therefore view our contributions as complementary. In fact, we believe our analysis techniques can be applied to their algorithm as well and result similar improvements as for the example in \pref{fig:summary}.
\paragraph{Regret bound when transitions are deterministic.}
We now interpret \pref{def:return_gap} for MDPs with deterministic transitions and derive an alternative form of our bound in this case.
Let $\Pi_{s,a}$ be the set of all policies that visit $(s,a)$ and have taken a suboptimal action up to that visit, that is,
$$
\Pi_{s,a} \equiv \curl*{\pi \in \Pi ~\colon s^\pi_{\kappa(s)} = s ,a^\pi_{\kappa(s)} = a, \exists~ h \leq \kappa(s), \gap(s^\pi_{h},a^\pi_{h})>0}.
$$
where $(s^\pi_1, a^{\pi}_1, s^\pi_2, \dots, s^\pi_H, a^{\pi}_H)$ are the state-action pairs visited (deterministically) by $\pi$.
Further, let $v^{*}_{s,a} = \max_{\pi \in \Pi_{s,a}} v^\pi$ be the best return of such policies.
\pref{def:return_gap} now evaluates to $\overline{\gap}(s,a) = \gap(s,a) \vee \frac{1}{H}(v^* - v^{*}_{s,a})$ and the bound in \pref{thm:reg_bound_gen_informal} can be written as
\begin{align}
\label{eq:det_trans_reg}
\mathfrak{R}(K) &\lessapprox \sum_{s,a \colon \Pi_{s,a} \neq \varnothing} \frac{H\log(K)}{v^* - v^*_{s,a}}~.
\end{align}
We show in \pref{app:unique_opt_pol}, that it is possible to further improve this bound when the optimal policy is unique by only summing over state-action pairs which are not visited by the optimal policy.
\subsection{Regret analysis with improved clipping: from minimum gap to average gap}
\label{sec:gen_clipping}
In this section, we present the main technical innovations of our tighter regret analysis.
Our framework applies to \emph{optimistic} algorithms that maintain a $Q$-function estimate, $\bar Q_k(s,a)$, which overestimates the optimal $Q$-function $Q^*(s,a)$ with high probability in all states $s$, actions $a$ and episodes $k$.
We first give an overview of gap-dependent analyses and then describe our approach.
\paragraph{Overview of gap-dependent analyses. }
A central quantity in regret analyses of optimistic algorithms are the surpluses $E_k(s,a)$, defined in \pref{eq:surpl_def}, which, roughly speaking, quantify the local amount of optimism.
Worst-case regret analyses bound the regret in episode $k$ as
$\sum_{(s,a) \in \mathcal{S}\times\mathcal{A}}w_{\pi_k}(s,a)E_{k}(s,a)$, the expected surpluses under the optimistic policy $\pi_k$ executed in that episode. Instead, gap-dependent analyses rely on a tighter version and bound the instantaneous regret by the \emph{clipped surpluses} \citep[e.g. Proposition 3.1][]{simchowitz2019non}
\begin{align}
V^*(s_1) - V^{\pi_k}(s_1) \leq 2e \sum_{s,a} w^{\pi_k}(s,a)\clip\left[E_{k}(s,a) ~\bigg\vert~ \frac{1}{4H}\gap(s,a) \lor \frac{\gap_{\min}}{2H}\right].
\label{eqn:old_surplusclipping}
\end{align}
\paragraph{Sharper clipping with general thresholds.}
Our main technical contribution for achieving a regret bound in terms of return gaps $\overline{\gap}(s,a)$ is the following improved surplus clipping bound:
\begin{restatable}[Improved surplus clipping bound]{proposition}{surplusclippingbound}
\label{prop:surplus_clipping_bound}
Let the surpluses $E_k(s,a)$ be generated by an optimistic algorithm.
Then the instantaneous regret of $\pi_k$ is bounded as follows:
\vspace{-1mm}
\begin{align*}
V^*(s_1) - V^{\pi_k}(s_1)
\leq 4 \sum_{s,a} w^{\pi_k}(s,a) \clip\left[ E_k(s,a) ~\bigg| ~ \frac 1 4 \gap(s,a) \vee \epsilon_{k}(s, a) \right]~,
\end{align*}
\vspace{-1mm}
where $\epsilon_k \colon \mathcal{S} \times \mathcal{A} \rightarrow \RR^+_0$
is any clipping threshold function that satisfies
\vspace{-1mm}
\begin{align*}
\EE_{\pi_k}\left[\sum_{h=B}^H \epsilon_k(S_h, A_h) \right]
\leq \frac{1}{2} \EE_{\pi_k} \left[\sum_{h=1}^H \gap(S_h, A_h) \right].
\end{align*}
\end{restatable}
Compared to previous surplus clipping bounds in \eqref{eqn:old_surplusclipping}, there are several notable differences. First, instead of $\gap_{\min}/2H$, we can now pair $\gap(s,a)$ with more general clipping thresholds $\epsilon_k(s,a)$, as long as their expected sum over time steps after the first non-zero gap was encountered is at most half the expected sum of gaps. We will provide some intuition for this condition below. Note that $\epsilon_{k}(s,a) \equiv \frac{\gap_{\min}}{2H}$ satisfies the condition because the LHS is bounded between $\frac{\gap_{\min}}{2H} \PP_{\pi_k}( B \leq H)$ and $\gap_{\min} \PP_{\pi_k}( B \leq H)$, and there must be at least one positive gap in the sum $\sum_{h=1}^H\gap(S_h,A_h)$ on the RHS in event $\{B \leq H\}$. Thus our bound recovers existing results.
In addition, the first term in our clipping thresholds is $\frac{1}{4}\gap(s,a)$ instead of $\frac{1}{4H}\gap(s,a)$. \citet{simchowitz2019non} are able to remove this spurious $H$ factor only if the problem instance happens to be a bandit instance and the algorithm satisfies a condition called \emph{strong optimism} where surpluses have to be non-negative. Our analysis does not require such conditions and therefore generalizes these existing results.\footnote{Our layered state space assumption changes $H$ factors in lower-order terms of our final regret compared to \citet{simchowitz2019non}. However, \pref{prop:surplus_clipping_bound} directly applies to their setting with no penalty in $H$.}
\paragraph{Choice of clipping thresholds for return gaps.}
The condition in \pref{prop:surplus_clipping_bound} suggests that one can set $\epsilon_{k}(S_h,A_h)$ to be proportional to the average expected gap under policy $\pi_k$:
\begin{align}
\epsilon_{k}(s,a) =
\frac{1}{2H}
\EE_{\pi_k}\left[ \sum_{h=1}^H \gap(S_h, A_h) ~ \bigg| ~\mathcal{B}(s,a)\right].
\label{eqn:epsilon_choice}
\end{align}
if $\PP_{\pi_k}(\mathcal{B}(s,a)) > 0$ and $\epsilon_{k}(s,a) = \infty$ otherwise.
\pref{lem:clipping_gaps_rel} in \pref{app:upper_bounds} shows that this choice indeed satisfies the condition in \pref{prop:surplus_clipping_bound}.
If we now take the minimum over all policies for $\pi_k$, then we can proceed with the standard analysis and derive our main result in \pref{thm:reg_bound_gen_informal}. However, by avoiding the minimum over policies, we can derive a stronger policy-dependent regret bound which we discuss in the appendix.
\section{Instance-dependent lower bounds}
\label{sec:lower_bounds_main}
We here shed light on what properties on an episodic MDP determine the statistical difficulty of RL by deriving information-theoretic lower bounds on the asymptotic expected regret of any (good) algorithm. To that end, we first derive a general result that expresses a lower bound as the optimal value of a certain optimization problem and then derive closed-form lower-bounds from this optimization problem that depend on certain notions of gaps for two special cases of episodic MDPs.
Specifically, in those special cases, we assume that the rewards follow a Gaussian distribution with variance $1/2$. We further assume that the optimal value function is bounded in the same range as individual rewards, e.g.\ as $0 \leq V^*(s) < 1$ for all $s \in \mathcal{S}$. This assumption is common in the literature \citep[e.g.][]{krishnamurthy2016pac, jiang2017contextual, dann2018oracle} and can be considered harder than a normalization of $V^*(s) \in [0, H]$ \citep{jiang2018open}.
\subsection{General instance-dependent lower bound as an optimization problem}
The idea behind deriving instance-dependent lower bounds for the stochastic MAB problem~\citep{lai1985asymptotically,combes2017minimal,garivier2019explore} and infinite horizon MDPs~\citep{graves1997asymptotically,ok2018exploration} are based on first assuming that the algorithm studied is \emph{uniformly good}, that is, on any instance of the problem and for any $\alpha >0$, the algorithm incurs regret at most $o(T^\alpha)$, and then argue that, to achieve that guarantee, the algorithm must select a certain policy or action at least some number of times as it would otherwise not be able to distinguish the current MDP from another MDP that requires a different optimal strategy.
Since comparison between different MDPs is central to lower-bound constructions, it is convenient to make the problem-instance explicit in the notation. To that end, let $\Theta$ be the problem class of possible MDPs and we use subscripts $\theta$ and $\lambda$ for value functions, return, MDP parameters etc., to denote specific problem instances $\theta, \lambda \in \Theta$ of those quantities. Further, for a policy $\pi$ and MDP $\theta$, $\PP_\theta^{\pi}$ denotes the law of one episode, i.e., the distribution of $(S_1, A_1, R_1, S_2, A_2, R_2, \dots, S_{H+1})$. To state the general regret lower-bound we need to introduce the set of \emph{confusing} MDPs. This set consists of all MDPs $\lambda$ in which there is at least one optimal policy $\pi$ such that $\pi \not\in \Pi^*_\theta$, i.e., $\pi$ is not optimal for the original MDP and no policy in $\Pi^*_\theta$ has been changed.
\begin{definition}
\label{def:conf_MDP_set}
For any problem instance $\theta\in\Theta$ we define the set of confusing MDPs $\Lambda(\theta)$ as
\begin{align*}
\Lambda(\theta): = \{\lambda \in \Theta \colon \Pi^*_\lambda \setminus \Pi^*_\theta \neq \varnothing \textrm{ and } KL(\PP_\theta^{\pi}, \PP_\lambda^{\pi}) = 0 \,\,\forall \pi \in \Pi^*_\theta\}.
\end{align*}
\end{definition}
We are now ready to state our general regret lower-bound for episodic MDPs:
\begin{restatable}[General instance-dependent lower bound for episodic MDPs]{theorem}{generallb}
\label{thm:lower_bound_gen}
Let $\psi$ be a uniformly good RL algorithm for $\Theta$, that is, for all problem instances $\theta \in \Theta$ and exponents $\alpha > 0$, the regret of $\psi$ is bounded as $\EE[\mathfrak{R}_\theta(K)] \leq o(K^{\alpha})$, and assume that $\return{*}_\theta < H$. Then, for any $\theta \in \Theta$, the regret of $\psi$ satisfies
\begin{align*}
\liminf_{K \to \infty} \frac{\EE[\mathfrak{R}_\theta(K)]}{\log{K}} \geq C(\theta),
\end{align*}
where $C(\theta)$ is the optimal value of the following optimization problem
\begin{equation}
\label{eq:opt_prob}
\begin{aligned}
\minimize{\eta(\pi)\geq 0}{\sum_{\pi \in \Pi} \eta(\pi)\left(\return{*}_{\theta} - \return{\pi}_{\theta}\right)}
{
\sum_{\pi \in \Pi} \eta(\pi) KL(\PP_\theta^\pi,\PP_\lambda^\pi) \geq 1 \qquad \textrm{for all } \,\,\lambda \in \Lambda(\theta)
}.
\end{aligned}
\end{equation}
\end{restatable}
The optimization problem in \pref{thm:lower_bound_gen} can be interpreted as follows. The variables $\eta(\pi)$ are the (expected) number of times the algorithm chooses to play policy $\pi$ which makes the objective the total expected regret incurred by the algorithm.
The constraints encode that any uniformly good algorithm needs to be able to distinguish the true instance $\theta$ from all confusing instances $\lambda \in \Lambda(\theta)$, because otherwise it would incur linear regret. To do so, a uniformly good algorithm needs to play policies $\pi$ that induce different behavior in $\lambda$ and $\theta$ which is precisely captured by the constraints $\sum_{\pi\in\Pi}\eta(\pi) KL(\PP_\theta^{\pi}, \PP_\lambda^{\pi}) \geq 1$.
Although \pref{thm:lower_bound_gen} has the flavor of results in the bandit and RL literature, there are a few notable differences.
Compared to lower-bounds in the infinite-horizon MDP setting \citep{graves1997asymptotically,tewari2008optimistic,ok2018exploration}, we for example do not assume that the Markov chain induced by an optimal policy $\pi^*$ is irreducible. That irreducibility plays a key role in converting the semi-infinite linear program \pref{eq:opt_prob}, which typically has uncountably many constraints, into a linear program with only $O(SA)$ constraints. While for infinite horizon MDPs, irreducibility is somewhat necessary to facilitate exploration, this is not the case for the finite horizon setting and in general we cannot obtain a convenient reduction of the set of constraints $\Lambda(\theta)$ (see also \pref{app:lower_bounds_full_supp})
\subsection{Gap-dependent lower bound when optimal policies visit all states}
To derive closed-form gap-dependent bounds from the general optimization problem \pref{eq:opt_prob}, we need to identify a finite subset of confusing MDPs $\Lambda(\theta)$ that each require the RL agent to play a distinct set of policies that do not help to distinguish the other confusing MDPs. To do so, we restrict our attention to the special case of MDPs where every state is visited with non-zero probability by some optimal policy, similar to the irreducibility assumptions in the infinite-horizon setting \citep{tewari2008optimistic, ok2018exploration}. In this case, it is sufficient to raise the expected immediate reward of a suboptimal $(s,a)$ by $\gap_\theta(s,a)$ in order to create a confusing MDP, as shown in \pref{lem:non-empty_change_env}:
\begin{restatable}[]{lemma}{gaplemmafullsupp}
\label{lem:non-empty_change_env}
Let $\Theta$ be the set of all episodic MDPs with Gaussian immediate rewards and optimal value function uniformly bounded by 1 and let $\theta \in \Theta$ be an MDP in this class. Then for any suboptimal state-action pair $(s,a)$ with $\gap_\theta(s,a) > 0$ such that $s$ is visited by some optimal policy with non-zero probability, there exists a confusing MDP $\lambda \in \Lambda(\theta)$ with
\begin{itemize}[itemsep=1mm, topsep=1mm]
\item $\lambda$ and $\theta$ only differ in the immediate reward at $(s,a)$
\item $KL(\PP_\theta^{\pi}, \PP_\lambda^{\pi}) \leq \gap_\theta(s,a)^2$ for all $\pi \in \Pi$.
\end{itemize}
\end{restatable}
By relaxing the problem in \pref{eq:opt_prob} to only consider constraints from the confusing MDPs in \pref{lem:non-empty_change_env} with $KL(\PP_\theta^{\pi},\PP_\lambda^{\pi}) \leq \gap_\theta(s,a)^2$, for every $(s,a)$, we can derive the following closed-form bound:
\begin{restatable}[Gap-dependent lower bound when optimal policies visit all states]{theorem}{fullsupportlb}
\label{thm:lower_bound_all_states_supp}
Let $\Theta$ be the set of all episodic MDPs with Gaussian immediate rewards and optimal value function uniformly bounded by 1. Let $\theta \in \Theta$ be an instance where every state is visited by some optimal policy with non-zero probability. Then any uniformly good algorithm on $\Theta$ has expected regret on $\theta$ that satisfies
\begin{align*}
\liminf_{K\rightarrow \infty}\frac{\EE[\mathfrak{R}_\theta(K)]}{\log{K}} \geq \sum_{s,a \colon \gap_\theta(s,a) > 0} \frac{1}{\gap_\theta(s,a)}.
\end{align*}
\end{restatable}
\pref{thm:lower_bound_all_states_supp} can be viewed as a generalization of Proposition 2.2 in \citet{simchowitz2019non}, which gives a lower bound of order $\sum_{s,a \colon \gap_\theta(s,a) > 0} \frac{H}{\gap_\theta(s,a)}$ for a certain set of MDPs.\footnote{We translated their results to our setting where $V^* \leq 1$ which reduces the bound by a factor of $H$.} While our lower bound is a factor of $H$ worse, it is significantly more general and holds in any MDP where optimal policies visit all states and with appropriate normalization of the value function.
\pref{thm:lower_bound_all_states_supp} indicates that value-function gaps characterize the instance-optimal regret when optimal policies cover the entire state space.
\subsection{Gap-dependent lower bound for deterministic-transition MDPs}
\label{sec:lower_bound_def}
We expect that optimal policies do not visit all states in most MDPs of practical interest (e.g. because certain parts of the state space can only be reached by making an egregious error). We therefore now consider the general case where $\bigcup_{\pi \in \Pi^*_\theta} supp(\pi) \subsetneq \mathcal{S}$ but restrict our attention to MDPs with deterministic transitions where we are able to give an intuitive closed-form lower bound.
Note that deterministic transitions imply $\forall \pi,s,a:\,w^\pi(s,a)\in\{0,1\}$.
Here, a confusing MDP can be created by simply raising the reward of any $(s,a)$ by
\begin{align}
\label{eq:return_gap}
\return{*}_\theta - \max_{\pi \colon w^\pi_\theta(s,a) > 0}\return{\pi}_\theta~,
\end{align}
the regret of the best policy that visits $(s,a)$, as long as it is positive and $(s,a)$ is not visited by any optimal policy. \pref{eq:return_gap} is positive when no optimal policy visits $(s,a)$ in which case suboptimal actions have to be taken to reach $(s,a)$ and $\overline{\gap}_\theta(s,a) > 0$. Let $\pi^*_{(s,a)}$ be any maximizer in \pref{eq:return_gap}, which has to act optimally after visiting $(s,a)$. From the regret decomposition in \pref{eq:reg_decomp} and the fact that $\pi^*_{(s,a)}$ visits $(s,a)$ with probability $1$, it follows that $v_{\theta}^* - v_{\theta}^{\pi^*_{(s,a)}} \geq \gap_{\theta}(s,a)$. We further have $v_{\theta}^* - v_{\theta}^{\pi^*_{(s,a)}} \leq H\overline{\gap}_\theta(s,a)$.
Equipped with the subset of confusing MDPs $\lambda$ that each raise the reward of a single $(s,a)$ as
$r_\lambda(s,a) = r_\theta(s,a) + v^*_{\theta} - v^{\pi^*_{(s,a)}}_{\theta}$, we can derive the following gap-dependent lower bound:
\begin{restatable}{theorem}{lowerbounddeterministic}
\label{thm:lower_bound_deterministic}
Let $\Theta$ be the set of all episodic MDPs with Gaussian immediate rewards and optimal value function uniformly bounded by 1. Let $\theta \in \Theta$ be an instance with deterministic transitions. Then any uniformly good algorithm on $\Theta$ has expected regret on $\theta$ that satisfies
\begin{align*}
\liminf_{K \rightarrow \infty} \frac{\EE[\mathfrak{R}_\theta(K)]}{\log K}
\geq \sum_{s, a \in \mathcal{Z}_\theta \colon \overline{\gap}_\theta(s,a) > 0 }
\frac{1}{H \cdot (v^*_{\theta}-v^{\pi^*_{(s,a)}}_{\theta})} \geq \sum_{s, a \in \mathcal{Z}_\theta \colon \overline{\gap}_\theta(s,a) > 0 }
\frac{1}{H^2 \cdot \overline{\gap}_\theta(s,a)},
\end{align*}
where $\mathcal{Z}_\theta = \{ (s,a) \in \mathcal{S} \times \mathcal{A} \colon \forall \pi^* \in \Pi^*_\theta ~~~ w^{\pi^*}_\theta(s,a) = 0\}$ is the set of state-action pairs that no optimal policy in $\theta$ visits.
\end{restatable}
We now compare the above lower bound to the upper bound guaranteed by \textsc{StrongEuler} in \pref{eq:det_trans_reg}. The comparison is only with respect to number of episodes and gaps{\footnote{We carry out the comparison in expectation, since our lower bounds do not apply with high probability.}}
\begin{align*}
\sum_{s,a \in \mathcal{Z}_\theta \colon \overline{\gap}_\theta(s,a) > 0} \frac{\log(K)}{H^2\overline{\gap}_\theta(s,a)} \leq \mathbb{E}_\theta[\mathfrak{R}(K)] \leq \sum_{s,a \colon \overline{\gap}_\theta(s,a) > 0}\frac{\log(K)}{\overline{\gap}_\theta(s,a)}.
\end{align*}
The difference between the two bounds, besides the extra $H^2$ factor, is the fact that $(s,a)$ pairs that are visited by any optimal policy ($s,a \neq \mathcal{Z}_\theta$) do not appear in the lower-bound while the upper-bound pays for such pairs if they can also be visited after playing a suboptimal action. This could result in cases where the number of terms in the lower bound is $O(1)$ but the number of terms in the upper bound is $\Omega(SA)$ leading to a large discrepancy. In \pref{thm:det_lower_bound} in the appendix we show that there exists an MDP instance on which it is information-theoretically possible to achieve $O(\log(K)/\epsilon)$ regret, however, any optimistic algorithm with confidence parameter $\delta$ will incur expected regret of at least $\Omega(S\log(1/\delta)/\epsilon)$. \pref{thm:det_lower_bound} has two implications for optimistic algorithms in MDPs with deterministic transitions. Specifically, optimistic algorithms
\begin{itemize}[topsep=1mm, itemsep=0mm]
\item cannot be asymptotically optimal if confidence parameter $\delta$ is tuned to the time horizon $K$;
\item cannot have an anytime bound that matches the information-theoretic lower bound.
\end{itemize}
\section{Problem setting and notation}
We consider reinforcement learning in episodic tabular MDPs with a fixed horizon. An MDP can be described as a tuple $(\mathcal{S}, \mathcal{A}, P, R, H)$, where $\mathcal{S}$ and $\mathcal{A}$ are state- and action-space of size $S$ and $A$ respectively, $P$ is the state transition distribution with $P(\cdot|s,a) \in \Delta^{S-1}$ the next state probability distribution, given that action $a$ was taken in the current state $s$. $R$ is the reward distribution defined over $\mathcal{S} \times \mathcal{A}$
and $r(s,a) = \mathbb{E}[R(s,a)] \in [0,1]$. Episodes
admit a fixed length or \emph{horizon} $H$.
We consider \emph{layered} MDPs: each state $s \in \mathcal{S}$ belongs to a layer $\kappa(s) \in [H]$ and the only non-zero transitions are between states $s, s'$ in consecutive layers, with $\kappa(s') = \kappa(s)+1$. This common assumption \citep[see e.g.][]{krishnamurthy2016pac} corresponds to MDPs with time-dependent transitions, as in \citep{jin2018q, dann2017unifying}, but allows us to omit an explicit time-index in value-functions and policies.
For ease of presentation, we assume there is a unique start state $s_1$ with $\kappa(s_1) = 1$ but our results can be generalized to multiple (possibly adversarial) start states.
Similarly, for convenience, we assume that all states are reachable by some policy with non-zero probability, but not necessarily all policies or the same policy.
We denote by $K$ the number of episodes during which the MDP is visited. Before each episode $k \in [K]$, the agent selects a deterministic policy
$\pi_k \colon \mathcal{S} \rightarrow \mathcal{A}$ out of a set of all policies $\Pi$ and $\pi_k$ is then executed for all $H$ time steps in episode $k$.
For each policy $\pi$, we denote by
$w^\pi(s,a) = \PP( S_{\kappa(s)} = s, A_{\kappa(s)} = a \mid A_h = \pi(S_h) \,\,\forall h \in [H])$
and $w^\pi(s) = \sum_a w^\pi(s,a)$
probability of reaching state-action pair $(s,a)$ and state $s$ respectively when executing $\pi$.
For convenience, $supp(\pi) = \{ s \in \mathcal{S} \colon w^\pi(s) > 0\}$ is the set of states visited by $\pi$ with non-zero probability. The Q- and value function of a policy $\pi$ are
\begin{align*}
Q^\pi(s, a) &= \EE_\pi \Bigg[ \sum_{h=\kappa(s)}^H r(S_h, A_h) ~\Bigg|~ S_{\kappa(s)} = s, A_{\kappa(s)} = a\Bigg],
& \textrm{and} \quad
V^\pi(s) &= Q^\pi(s, \pi(s))
\end{align*}
and the regret incurred by the agent is the sum of its regret over $K$ episodes
\begin{equation}
\mathfrak{R}(K) = \sum_{k=1}^K \return{*} - \return{\pi_k} = \sum_{k=1}^K V^{*}(s_1) - V^{\pi_k}(s_1),
\end{equation}
where $\return{\pi} = V^{\pi}(s_1)$ is the expected total sum of rewards or \emph{return} of $\pi$ and $V^*$ is the optimal value function $V^*(s) = \max_{\pi \in \Pi} V^\pi(s)$.
Finally, the set of optimal policies is denoted as $\Pi^* = \{\pi \in \Pi : V^{\pi} = V^*\}$. Note that we only call a policy optimal if it satisfies the Bellman equation in every state, as is common in literature, but there may be policies outside of $\Pi^*$ that also achieve maximum return because they only take suboptimal actions outside of their support. The variance of the $Q$ function at a state-action pair $(s,a)$ of the optimal policy is $\mathcal{V}^*(s,a) = \VV[R(s,a)] + \VV_{s'\sim P(\cdot|s,a)}[V^{*}(s')]$, where $\VV[X]$ denotes the variance of the r.v.\ $X$. The maximum variance over all state-action pairs is $\mathcal{V}^* = \max_{(s,a)}\mathcal{V}^*(s,a)$.
Finally, our proofs will make use of the following clipping operator $\clip[a|b] = \chi(a\geq b)a$ that sets $a$ to zero if it is smaller than $b$, where $\chi$ is the indicator function.
\section{Additional related work}
Instance dependent bounds for the simpler problem of MAB were first introduced in \citet{lai1985asymptotically}. The techniques were very recently generalized and simplified in the work of \citet{garivier2019explore}. \citet{graves1997asymptotically} extend the lower bounds to the setting of controlled Markov chains with infinite horizon and proposed an algorithm which achieves the bound asymptotically. Building on their work \citet{combes2017minimal} establish both lower and (asymptotic) upper bounds for the Structured Stochastic Bandit problem. For infinite horizon MDPs with additional assumptions the works of \citet{auer2007logarithmic,tewari2008optimistic,auer2009near,filippi2010optimism,ok2018exploration} establish logarithmic in horizon regret bounds of the form $O(D^2S^2A\log(T)/\delta)$, where $\delta$ is a gap-like quantity and $D$ is a diameter measure. \citet{tewari2008optimistic,ok2018exploration} propose asymptotically optimal algorithms based on a linear program (LP). Their regret upper bounds can also be converted to have a gap-like flavor. Very recently there has been a plethora of papers proposing instance dependent regret bounds for finite horizon tabular MDPs~\citep{simchowitz2019non,lykouris2019corruption, yang2020q, jin2020simultaneously,du2020agnostic}. All of these bounds scale at least as $\Omega(SH\log(T)/\gap_{\min})$. The dependence on number of actions in the above works scales with better gap-dependent factors than $1/\gap_{\min}$. \citet{simchowitz2019non} show an MDP instance on which no optimistic algorithm can hope to do better.
\section*{Checklist}
\begin{enumerate}
\item For all authors...
\begin{enumerate}
\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
\answerYes{See \pref{sec:upper_bounds}, \pref{sec:lower_bounds_main} and corresponding sections in the appendix.}
\item Did you describe the limitations of your work?
\answerYes{See lower bounds, discussion after Equation~\ref{eq:det_trans_reg}, \pref{app:lower_bounds_issues}}
\item Did you discuss any potential negative societal impacts of your work?
\answerNA{Our work is theoretical and we do not see any potential negative societal impacts.}
\item Have you read the ethics review guidelines and ensured that your paper conforms to them?
\answerYes{}
\end{enumerate}
\item If you are including theoretical results...
\begin{enumerate}
\item Did you state the full set of assumptions of all theoretical results?
\answerYes{}
\item Did you include complete proofs of all theoretical results?
\answerYes{See \pref{app:lower_bounds} for lower bounds and \pref{app:upper_bounds} for upper bounds.}
\end{enumerate}
\item If you ran experiments...
\begin{enumerate}
\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
\answerNA{}
\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
\answerNA{}
\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
\answerNA{}
\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
\answerNA{}
\end{enumerate}
\item If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
\begin{enumerate}
\item If your work uses existing assets, did you cite the creators?
\answerNA{}
\item Did you mention the license of the assets?
\answerNA{}
\item Did you include any new assets either in the supplemental material or as a URL?
\answerNA{}
\item Did you discuss whether and how consent was obtained from people whose data you're using/curating?
\answerNA{}
\item Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
\answerNA{}
\end{enumerate}
\item If you used crowdsourcing or conducted research with human subjects...
\begin{enumerate}
\item Did you include the full text of instructions given to participants and screenshots, if applicable?
\answerNA{}
\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
\answerNA{}
\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
\answerNA{}
\end{enumerate}
\end{enumerate}
\input{appendix/app_related_work}
\input{appendix/optimistic_algs}
\input{appendix/experiments}
\section{Additional Notation}
We use the shorthand $(s, a) \in \pi$ to indicate that $\pi$ admits a non-zero probability of visiting the state-action pair $(s,a)$ and abusively use $\pi$ as the set of such state-action pairs, when convenient.
\input{appendix/app_lower_bounds}
\input{appendix/app_upper_bounds}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 668 |
Q: Uniformly Most Powerful Test Gamma Distribution In this worked-out solution, I'm convinced there is a typo:
In standarizing the variable, I understand how typically, we're supposed to subtract the mean from the variable in the numerator, so why are we adding (T+1/B_0)? Shouldn't it be subtraction? The rest makes sense. Thanks!
A: We can see from the information there that $E(X_i)=1/\beta_0$ (under the null, whence $E(\bar{X}_n) = 1/\beta_0$), and that $T=-\bar{X}_n$.
Consequently the expected value of $T$ is $-1/\beta_0$ and so $T-E(T)=T+1/\beta_0$.
There's no typo, it's doing the right thing there.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,670 |
\section{Introduction}
The emission of X-rays from massive stars is explained as due to strong stellar winds
and shocks in O stars through a mechanism of line deshadowing instability
(LDI; \citealp{Feldmeier1997a,Feldmeier1997b,Owocki1998}).
In the case of binarity, wind-wind collision gives an additional source of X-rays,
sometimes coupled with the presence of magnetic fields
that drive ionized winds and increase shock temperatures \citep{Babel1997,ud-Doula2002}.
The production of X-rays by stellar winds appears less realistic in B-type stars because of
their weaker winds with respect to those from O and WR stars.
Moving from O to early B-type stars, the rate of detection in X-rays among B stars
falls to about 50\%, where hard X-ray emission in a few cases are a signature of the
presence of strong magnetic fields or due to an unknown, low mass young and active companion.
X-rays from single early B stars are observed in a few cases, and their origin is likely
linked to their strong magnetism.
Cases of spots in young stars of NGC~2264, which presumably have a magnetic origin,
are given by \citet{Fossati2014}. Recent spectroscopic surveys of O-B stars have revealed
that about 7\% of these stars are magnetic \citep{Wade2014,Fossati2015}.
In a few cases magnetic fields of strength of a few kG are measured, likely accompanied
by peculiar photospheric abundances.
$\rho$~Oph\ is a multiple system of B-type stars. They are part of one of the closest
and densest sites of active star formation. In particular
$\rho$~Oph~A+B is a binary system of two B2 stars separated by about 310 AU at a distance
of $111\pm10$ pc from the Sun (parallax $9\pm0.9$ mas,
separation $\sim2.8\arcsec$; \citealp{VanLeeuwen2007,Malkov2012}),
the orbital period of the system is about $2400\pm330$ years \citep{Malkov2012}.
$\rho$~Oph~A rotates with $v\sim300$ km/s \citep{VanBelle2012,Glebocki2005,Uesugi1982},
has a mass of about 8-9 M$_\odot$, and radius of $\sim8 \mathrm{R}_\odot$.
In 2013 we observed $\rho$~Oph\ with {\em XMM-Newton}\ discovering that it emits X-rays.
In particular, we observed a significant rise of X-ray flux in the last 20-25 ks
of the exposure, accompanied by plasma temperatures around $\sim3$ keV
\citep[][hereafter Paper I]{Pillitteri2014c}.
We hypothesized that either an active spot was emerging on the stellar surface,
or an unknown low mass companion was causing the feature.
Along with $\rho$~Oph, we discovered a young cluster of about 25 pre-main
sequence (PMS) stars, which { are} mostly without circumstellar disks and of $5-10$ Myr in age.
These stars were likely born together with the $\rho$~Oph\ stars during the first event
of star formation in the cloud \citep{Pillitteri2016}.
In this paper we present the results of a follow-up observation of $\rho$~Oph\
with {\em XMM-Newton}, with a duration of 140 ks. The aim of the observation was to monitor the X-ray
emission of $\rho$~Oph\ for a full rotational period of the star ($\sim1.2$ days or 104 ks)
and understand the origin of its X-rays.
The structure of the paper is the following: in Sect. \ref{observations} we describe the
observations and data analysis; Sect. \ref{results} present the results;
and in Sect. \ref{discussion} we discuss them and present our
conclusions.
\begin{figure}
\resizebox{\columnwidth}{!}
{\includegraphics{zoom_rho_oph.png}}
\caption{\label{zoom} MOS 1 image of $\rho$~Oph, the positions of the two components from
SIMBAD database are indicated with colored boxes. The centroid of X-ray events is much closer
to the A component. The scale is linear, the events have been rebinned in blocks of 0.4\arcsec
and smoothed with a Gaussian of $\sigma = 1$ pixel. The separation between the two components
is $\sim2.8\arcsec$. The core of the point spread function of MOS1 is about 2\arcsec and
the astrometric precision is $\sim1.2\arcsec$.}
\end{figure}
\section{Observations and data analysis} \label{observations}
{\em XMM-Newton}\ observed $\rho$~Oph\ on February 22 2016 for 140 ks (ObsId 0760900101).
We used EPIC camera as the first instrument with the {\it Thick} filter to prevent UV
leakage due to the brightness of the target (U$=4.30$). For the same reason, OM exposures
were not taken for the safety of the instrument.
The observation data files (ODFs) were reduced with SAS ver. 15.0 to obtain tables of events
for MOS 1, 2 and {\em pn}\ filtered in 0.3-8.0 keV,
and with filters $\mathrm{FLAG}=0$ and $\mathrm{PATTERN}<=12$
as prescribed by the SAS guide. We used a circular
region of radius 35$\arcsec$ for both source and background events to select the events of $\rho$~Oph.
For {\em pn}\ we used a background extracted from a region at the same distance from the readout node of
the same chip of the source, as prescribed by the SAS guide book.
In Paper I we associated the X-ray emission with the A component of $\rho$~Oph; the MOS1 image
in Fig. \ref{zoom} shows that this is a reasonable assumption, given that the
centroid of the X-ray events { is} much closer to the position of $\rho$~Oph~A and completely offset
with respect to the position of $\rho$~Oph~B.
\begin{figure*}
\resizebox{\textwidth}{!}{
\includegraphics{lc_pn_segments.pdf}
}
\caption{ Plot of\label{lcpn} {\em pn}\ light curve of $\rho$~Oph: full band ($0.3-8.0$ keV, black),
soft band ($0.3-1.0$ keV, red), and hard band ($1.0-8.0$ keV, blue).
Segments used in time resolved spectroscopy are indicated by ticks on the bottom scale and
listed in the column to the right.}
\end{figure*}
Spectra, response matrices (rmf), ancillary files (arf), and light curves of $\rho$~Oph~A
were obtained with the specific SAS tasks ({\em evselect}, {\em rmfgen}, {\em arfgen}, {\em backscale}, and
{\em specgroup}). The spectra were analyzed with XSPEC ver. 12.8.0, while generic software R language
scripts and custom plotting routines were used to produce plots and calculate derived
quantities of interest.
To understand the physical changes in the emitting plasma of $\rho$~Oph~A, we divided
{ the {\em pn}\ } light curve in intervals and accumulated the spectra from each time interval.
We chose to obtain spectra with about 1600 counts each, which represents a trade off
to preserve details of the temporal changes of the spectrum and its count statistics.
In this way we divided the light curve in 37 time intervals of variable duration
($\sim1.2-6.0$ ks, see Fig. \ref{lcpn}). For each time bin we obtained {\em pn}\ spectra
and related calibration files (rmf and arf).
We used a thermal model with absorption composed of two {\em APEC} components
and a global {\em PHABS} photoelectric absorption ($N_H$).
We kept $N_H$ fixed to $3\times10^{21}$ cm$^{-2}$ (see Paper I), formally $Z = 0.3 Z_\odot$,
while temperatures and normalization factors were left free to vary.
Although the metallicity of Galactic B stars is well established \citep{Nieva2012}, { sub-solar}
abundances reflect a peculiar behavior observed in active stellar coronae (see Sect. \ref{timeresspec}).
{\em XMM-Newton}\ allows us to acquire simultaneously high resolution spectra of the central target
with RGS gratings, provided it is bright enough.
This was the case for $\rho$~Oph\ in the present 140 ks exposure.
The RGS spectra were obtained by first extracting the RGS1 and RGS2 spectra of $\rho$~Oph,
then we added together the first order spectra with the SAS task {\em rgscombine}.
We accumulated RGS
spectra in several time intervals: full exposure (140 ks), first event ($10-40$ ks since
start of exposure), second event ($115-140$ ks), the sum of first and second
event (high state spectrum), and the relatively low activity interval in between ($40-115$ ks).
For each time window we added together both RGS1 and RGS2 spectra of the first order.
We focused on the range $5-20$ \AA\ because this is the range over which
most of the flux and its changes are observed.
$\rho$~Oph\ has been observed with ESO-VLT and UVES spectrograph in 2001 and 2005
with different coverage of wavelengths (from 3000\AA\ to 9000\AA), different exposure times
(2s. to 15s.), and { with a spectral} resolution $R\sim 42,000$.
We used the reduced spectra from such observations to derive an estimate of the rotational
velocity along the line-of-sight v~sin$i$ and to check the presence of lines from a low mass companion and any
Doppler shift that could hint at the presence of such companion.
We used a combination of {\em Iraf/pyraf} and a set of custom scripts to read the spectra
in Iraf from the original {\em MIDAS} format, display the spectra, measure line absorption,
remove the heliocentric component of Doppler shift due to the motion of the Earth, and
calculate the cross-correlation function between spectra at two different epochs.
\section{Results} \label{results}
Fig. \ref{lcpn} shows the light curve of $\rho$~Oph~A in the full band ($0.3-8.0$ keV), soft band
($0.3-1.0$ keV) and hard band ($1.0-8.0$ keV). During the observation
$\rho$~Oph~A exhibited two main episodes of variability: one at the beginning of the observation and
another more powerful toward the end of the exposure.
The first episode was a cusp-shaped increase of rate occurring during the first
40 ks. The peak of this event occurred at $t\sim$ 25 ks and the decay phase appeared to be
slightly steeper than the rise phase.
The second large increase of the rate occurred at $t\sim116$ ks with a peak at around $t\sim129$ ks
and a decay that lasted until the end of the observation ($t\sim139$ ks). Likely, we missed the very
end of this decay.
Minor episodes of variability in the form of flares are visible at $t\sim55$ ks and $105$ ks.
However, in the following we refer to this interval as the quiescent state, while
the main two events of variability are referred to as the high state of $\rho$~Oph.
The characteristics of the plasma and its changes in temperature and emission
measure are discussed in detail in Sect. \ref{timeresspec}.
\subsection{Phased light curves and rotational velocity}
The rise of the rate in the first event is suspiciously
similar to the rise of the rate observed in 2013 (Paper I), so we wonder whether the same mechanism
is responsible for the increase of the rate observed twice in 2016. In particular,
we hypothesize that a spot or an unseen companion transited during the observation,
as speculated in Paper I.
The time elapsed between the two main peaks of the light curve in Fig. \ref{lcpn} is
approximately 104 ks (or 1.2 days).
If we consider this time as an initial guess for a phase-folding time,
only a small adjustment of the period is needed to aligning the rate peaks observed in 2013 and 2016,
respectively. Fig. \ref{phasedlc} shows the phase folded light curves of 2013 and 2016, where
as zero phase we used the beginning of the 2013 {\em pn}\ light curve. We thus find that a period of 104.11 ks
corresponding to 1.205 days aligns the three peaks, and this is our best estimate of the period
of rotation of $\rho$~Oph~A based on the periodic variations of its X-ray emission.
This period corresponds to a rotational velocity of $\sim340$ km/s at the equator when assuming
a stellar radius of $\sim8 R_\odot$; the { velocity derived from the 1.205 days period }
is roughly consistent with the rotational velocity of
$\rho$~Oph~A determined from optical observations \citep[see][]{VanBelle2012}.
The separation between the two epochs of observations is on the order of 1000 days and this
leads to a precision of the period of 1/1000 of day.
Additional X-ray observations with shorter cadence could validate the measurement of
the X-ray rotational period of $\rho$~Oph~A.
We refined the estimate of v$\sin i$ of $\rho$~Oph~A with an analysis of the
Fourier transform of the line profile of He line at 6678\AA\ from one of the available UVES
spectra
(Fig. \ref{fft}, \citealp{Gray88,Smith1976}). The first minimum of the transform is related to
the line broadening due to rotation and thus to v~sin$i$.
The He line was chosen because it is well isolated from nearby lines, has a good signal,
and offers an easy modeling of its profile.
We used a window of wavelengths that encompasses the line (6671\AA$-$6685.5\AA),
and we smoothed and normalized the profile with a {\em lowess}\footnote{As implemented in R,
\url{https://stat.ethz.ch/R-manual/R-devel/library/stats/html/lowess.html}} smoother.
The Fourier transform was obtained on a sample of 5000 points interpolated along
the line profile. The first minimum of the Fourier transform occurs at
$\sigma = 0.00275$c/\AA , which corresponds to v~sin$i=239.5$ km/s.
The uncertainty is estimated at $\pm10$ km/s.
From v~sin$i$ and the rotational velocity, derived from the period obtained from the X-ray
variability, we infer a $i$ angle between line of sight and rotational axis of $45\pm5\degr$,
taking into account an uncertainty of $0.5R_\odot$ on the stellar radius.
As first proposed in Paper I and discussed further in the next section, the presence of an active spot of
magnetic origin and its periodic appearance would imply a magnetic field misaligned with
the rotation axis.
On the other hand, if the X-rays were entirely produced by an active low mass companion, depending
on the mass ratio, its motion around $\rho$~Oph\ at a short distance would produce a wobble
detectable as a Doppler shift in the spectra. For example, supposing a mass ratio of 1:15
(a late-K star), a period of 1.2 days, and an orbital velocity of the companion of 300-350 km/s
would produce a Doppler shift of $\sim20$ km/s, which is easily detectable in the spectra of $\rho$~Oph.
We first { subtracted} the component due to the motion of the Earth from
the UVES spectra, { then we cross-correlated the spectra taken at different epochs and of similar wavelength}
coverage \citep{Tonry1979} avoiding regions with telluric lines. We did not detect any Doppler shift,
meaning that either there is no companion or that the spectra
were taken at approximately the same phase ($\mathrm{HJD_1}=2452132.6579; \mathrm{HJD_2}=2453618.4866$,
$\Delta \phi\sim 0.053$, for a period of 1.205 days).
The phase difference implies a displacement of $19\degr$ on the orbit. By supposing that $\rho$~Oph~A
has an average $v_{orb}\sim 20 km/s$, the sin$i$ factor would amount to a difference of velocity of
6.5 km/s at the two phases; this difference is very similar to the UVES spectral resolution.
A dedicated spectroscopic monitoring is thus required to detect a companion.
\begin{figure}
\resizebox{\columnwidth}{!}{
\includegraphics{phased_lc_notnorm.pdf}
}
\caption{\label{phasedlc} Light curves of the rate phase-folded with a period of 1.205 days.
The red dots indicate the 2013 observation, and the green and blue dots indicate the 2016 observation split in two
periods. Phase zero was set arbitrarily to the beginning of 2013 observation.
The three main peaks of X-ray emission can be put in phase with a period of 1.205 days,
which corresponds to the rotational period of the star.}
\end{figure}
\begin{figure}
\resizebox{\columnwidth}{!}{
\includegraphics{fft_profile.pdf}
}
\caption{\label{fft} Smoothed normalized profile of He line at 6678\AA\ (left) and normalized
amplitude of Fourier transform of the same profile (right, solid line). The dotted curve denotes
the transform of the rotational broadening profile alone.
The first minimum occurs at $\sigma\sim0.00275\ c/\mathrm\AA$ (vertical dotted line),
corresponding to v~sin$i\sim 239.5$ km/s. Uncertainty on v~sin$i$ is estimated at around $\pm10$ km/s.}
\end{figure}
\subsection{Time resolved spectroscopy}
\label{timeresspec}
Here we discuss the spectral variations of $\rho$~Oph~A with particular regard to the
two main episodes of variability that characterized the X-ray light curve of
$\rho$~Oph~A.
For each time interval, we performed a best fit of the spectrum with an absorbed thermal model composed
of the sum of two APEC models. The free parameters were the temperatures and the respective emission
measures of the thermal components, while absorption and abundances were kept fixed at
$N_H=3\times10^{21}$ cm$^{-2}$ and $Z=0.3 Z_\odot$ in agreement with what was found in Paper I.
By letting absorption vary, we obtained values of $N_H$ consistent with the assumed value,
and the variation of plasma temperatures were minimal with respect to the best fit with
a fixed absorption.
We discuss in detail the evolution of the hot component of
the plasma, since the cool component had little variation during the observation,
being comprised of { 0.7 keV $< kT <$ 1.2 keV} with a median of { $kT \sim 0.9$} keV
and standard deviation of 0.15 keV.
In Fig. \ref{epicspectra} we show the {\em pn}\ spectra of the quiescent phase at the peak of the two
main events of variability. The spectrum changes significantly, especially during the second
event when the temperature of the plasma was in excess of 5 keV.
Evidence of such high temperatures is provided by the appearance of the complex { of} lines
around 6.7 keV owing to highly ionized Fe.
\begin{figure}
\resizebox{\columnwidth}{!}{
\includegraphics{epic_spectra.pdf}
}
\caption{Plot of\label{epicspectra} {\em pn}\ spectra of $\rho$~Oph~A during quiescent phase (time bins $10-24$,
bin 13 excluded, black spectrum), peak of the first event (bins $5-8$, red spectrum),
and peak of the second event (bins $27-33$, green spectrum). This latter shows the complex { of} lines of
ionized Fe at 6.7 keV evidencing the high temperature of the plasma ($\ge5.4$ keV) reached
during the peak of the second event. }
\end{figure}
\noindent{{\em First event} (intervals $1-10$).}
The hot temperature during this event varied by 1.9 keV$<kT<$2.5 keV (Table \ref{tabfit},
Fig. \ref{logtem}) apart from interval number 6 when it had a sudden increase ($kT \sim3.4$ keV)
of about 3$\sigma$ higher than the average of the previous values such heating rapidly vanished
during the next interval (number 7).
{The spectrum corresponding to the intervals around the peak of the rate increase is shown in Fig. \ref{epicspectra}
(red curve).}
During intervals 8 through 10 the hot temperature
slightly cooled down toward values lower than those seen at the beginning of the observation.
We can conclude that, on average, only the plasma emission measure (EM) increased significantly
during intervals $2-7$, while the temperature was almost steady.
This behavior fits the scenario in which a hot spot gradually appears on the stellar surface
due to the stellar rotation (P$_\mathrm{rot}\sim1.205$ days $\sim$ 104.1 ks).
In interval number 6 either a short flare happened or the very hot core of the region
was visible for a short time.
The vanishing of the high rate is plausible with the gradual disappearance of the same spot
at the opposite limb of the star.
\noindent{\em Quiescent state.}
During intervals $11-24$ the star showed a relatively small flare during interval
number 13 ($\sim55$ ks since the start of the observation, visible in four bins of
the light curve of Fig. \ref{lcpn}) and with a duration of approximately 5 ks,
plus other small scale variability afterwards. This demonstrates that even during the
{\em quiescent} state some degree of X-ray variability was present.
When modeling the plasma with two absorbed thermal APEC components,
the cool component was around $0.8-0.9$ keV and the hot component in { 1.9 keV $< kT <2.7$ keV},
which is within 2$\sigma$ of the values of hot temperatures seen during the first event.
On average, the quiescent flux is about $1.35\times10^{-12}$ {erg~s$^{-1}$~cm$^{-2}$}, which is very similar
to the quiescent flux measured in 2013 ($1.5\times10^{-12}$ {erg~s$^{-1}$~cm$^{-2}$}).
We grouped together the time bins between 11 and 24 (interval 13 excluded)
to gain more count statistics and allow a more refined analysis of the corresponding spectrum
(black curve in Fig. \ref{epicspectra}).
We tried models with two, three, and four APEC components and models with two and three VAPEC components.
The 2T APEC model does not give us a statistically valid fit,
while the three APEC component model does. The model with four APEC components does not improve the
fit to data. For the three APEC components we find
$kT_1=0.26\pm0.03$ keV, $kT_2=0.98\pm0.04$ keV, and $kT_3=2.97\pm0.7$ keV with ratios of
emission measures (EM) $EM_3/ EM_1 \sim 0.12$, $EM_2/EM_1 \sim 0.4$.
We thus detect a soft component at 0.26 keV and there is evidence of a hot component at $\sim3$ keV
also during quiescence.
{ For the VAPEC models we left Fe, O, and Ne free to vary and linked all other elements to the Fe abundance.}
Both the 2T VAPEC model and the 3T VAPEC give a satisfactory fit to the data.
For the 2T VAPEC model the two temperatures are $kT_1 = (0.81\pm0.04)$ keV and $kT_2 =(2.6\pm0.5)$ keV,
respectively, with $EM_2/EM_1 \sim 0.5$. The abundances of Fe and other metals are found around
$0.13Z_\odot$, but with O and Ne around 1.1 and 0.75 times the solar value. This pattern is
similar to that observed in active stellar coronae (the so-called inverse FIP effect,
see, e.g., \citealp{Guedel2004} and references therein) where the abundance of Fe (an element with low
first ionization potential, FIP) is found depleted with respect to the value measured in the
solar corona, while the abundances of elements with high FIP, such as O and Ne, appear enhanced.
For the 3T VAPEC model, a pattern of abundances similar to that of the 2T VAPEC model fit
is found ($Fe/Fe_\odot = 0.15$,
$[O/Fe] \sim 1.3$, $[Ne/Fe]\sim0.8$). In this case the temperatures are $kT_1 = (0.1\pm0.1)$ keV,
$kT_2 = (0.80\pm 0.04)$ keV, and $kT_3 = (2.63\pm0.6)$ keV with EM ratios of $EM_2/ EM_1 \sim 4.5$,
$EM_3/EM_1 \sim 1.9$. We detect thus a cool component and a hot component in the quiescent
phase spectrum, whereas the main component remains that at $\sim0.8$ keV.
This also justifies the use of a simple 2T APEC model for time resolved spectroscopy,
and the focus on the hot component of such a model to study the two main increases of X-ray
flux of $\rho$~Oph~A.
\noindent{{\em Second event} (flare, intervals $25-37$).}
These intervals contain the second and most powerful event observed in $\rho$~Oph~A.
The {\em pn}\ spectrum relative to the time bins around the peak of the rate is shown in
Fig. \ref{epicspectra} (green curve). The spectrum is characterized by the appearance of lines
of highly ionized Fe at 6.7 keV.
The rise of the flare started at interval number 25,
and its evolution is best described by the hot component of the spectral fitting, which we discuss here.
An increase of EM and temperature is detected, starting from intervals 25 and 26,
and even more during intervals $27-29$. Interval number 28 shows the peak of $kT\sim5.4$ keV,
then $kT$ decreases steadily through intervals $29-31$, and more slowly through intervals $32-37$,
with some apparent reheating at interval 34.
At these times EM showed its maximum values (with a peak of $\log EM (\mathrm{cm^-3})\ge 53.9$ in
interval 31) then it decreased toward pre-flare values. The peak of the hot temperature was reached in interval 28
and does not coincide with the maximum of EM, which peaked during interval 31.
This fact is consistent with what is observed in the flares of cool stars and in the Sun, where the
initial heating and peak of temperature is followed by an increase of the density of the flaring loop,
filled up by plasma evaporating from the loop footpoints \citep{Reale2007}.
The behavior of the flare closely resembles that of solar and stellar flares observed in X-rays,
where energy is suddenly released with an impulse in plasma magnetically confined in loops,
followed by a longer cooling decay. We presume that {\em XMM-Newton}\ did not completely observe the full decay of
the flare and the disappearance of the hot region as in the first event.
This and the too shallow track of the decay in the $\rm{EM}-\rm{kT}$
diagram did not allow us to have reliable
diagnostics from the flare decay as in \citet{Reale1997}, and we preferred to derive loop diagnostics from
the analysis of the rise phase, as described in \citet{Reale2007}. The loop length can be derived
from measuring the duration of the rise of the light curve and/or the time delay of the emission measure peak
from the temperature peak \citep{Reale2007} as follows:
\begin{equation}
L_9 \approx 3 ~ \psi^2 T_{0,7}^{1/2} t_{M,3} \approx 190
\label{eq:lris}
\end{equation}
\begin{equation}
L_9 \approx 2.5 ~ \frac{\psi^2}{\ln \psi} T_{0,7}^{1/2} \Delta t_{0-M,3} \approx 140
\label{eq:ldelt}
,\end{equation}
where
\[
\psi = \frac{T_0}{T_{M}} \approx 1.9,
\]
$L_9$ is the loop half-length in units of $10^9$ cm, $t_{M,3} \approx 4.6$ ks is the time of the emission
measure maximum measured from the flare beginning, $T_{0}$ ($T_{0,7}$) the loop maximum temperature (in units of $10^7$ K),
$T_{M}$ the temperature at the emission measure maximum, and $\Delta t_{0-M,3} \approx 2.6$ the time elapsed between the
peak of the temperature and the peak of the emission measure in ks. The values $T_0 \approx 150$ MK (bin 28) and $T_M \approx 80$ MK (bin 31)
are maximum loop temperatures inferred from the measured temperatures, which average over the whole flaring loop
(see Appendix in \citealt{Reale2007}). In summary,
we inferred a loop half-length $L= 1.4-1.9\times10^{11}$ cm, which corresponds to about $25\%-33\%$ of the stellar radius
or about $2-2.7$ times the solar radius.
We also estimated that the coronal magnetic field must have an intensity $B \ge 300$ G to confine the plasma
within the loop.
\begin{figure}
\resizebox{\columnwidth}{!}{
\includegraphics{kt_logem.pdf}
}
\caption{\label{logtem} $kT_2$ vs. emission measure ($EM_2$) of the hot component of the 2T APEC
best-fit models for the time intervals (see text) relative to the first event (red points)\
and second event (blue points).
The point relative to the quiescent phase (time intervals $12-24$) corresponds to the hottest
component of the 3T VAPEC best-fit model. Ellipses denote the $1\sigma$ confidence ranges.}
\end{figure}
\section{RGS spectra}
The average spectrum of $\rho$~Oph~A is shown in Fig. \ref{rgs} (left panel), while
the spectra of the quiescent ($40-115$ ks, black) interval, the first ($10-40$ ks, orange)
and second ($115-140$ ks, blue)
event are shown in the right panel of Fig. \ref{rgs}.
These spectra are rebinned to have at least 25 counts per bin to enhance the signal,
although this reduces the original spectral resolution.
Qualitatively, hot lines from \ion{O}{VIII} and \ion{Ne}{X} are detected both in the quiescent interval
and during the variability events, indicating the presence of hot plasma ($kT \ge 1$ keV) that is already
detected in the {\em pn}\ spectra.
Another sign of high temperature plasma is the relatively high level of continuum,
below 10\AA, similar to what has been observed in stars with high activity and plasma
temperatures like, for example, UX Ari and $\epsilon$ Eri \citep{Ness2002}.
The best fit of the average RGS spectrum with two absorbed APEC thermal components results in
a temperature around 0.3 keV and a hot temperature around 5 keV for a fixed $Z=0.3Z_\odot$, in agreement
with the {\em pn}\ spectral analysis, even though the range of energy of the RGS is limited to $\sim 2.5$ keV.
Changes of line intensities are mostly observed during the second event. In particular,
we observed an increase of both continuum and line strength at wavelengths shorter than 12\AA\ during
the second (flare) event. Lines of \ion{Fe}{XXIV} are visible during the flare owing to temperatures in excess of
5 keV as determined from the analysis of the {\em pn}\ spectra.
We measured the line fluxes by modeling single lines with Gaussian profiles characterized by a line central
wavelength and a full width at half maximum (FWHM). In this way we take into account any intrinsic widths
of the lines. The model also comprises a continuum level in a window of about $0.7-1$\,\AA\
around each feature; we used CIAO Sherpa 4.8\footnote{\url{http://cxc.harvard.edu/sherpa4.8/}
to obtain the best fit parameters.}
Table \ref{lines} presents the measurements of the main lines identified in the spectra.
Determining a 1$\sigma$ confidence range was difficult for some lines as it depends on the statistics
and number of bins available for fitting. As a consequence, any quantitative
conclusions on the widths of the lines and the amount of change during the increase of rates are
impossible with these data. In general the line widths are consistent with zero even for the
spectra relative to the high rate events, when we presume that a line broadening should be present owing to the traveling of the source of X-rays across the surface, either this is an active spot or an unseen
companion during the interval in which the spectra were accumulated.
Qualitatively, RGS spectra show the potential for a deeper high resolution spectroscopic
follow up of $\rho$~Oph\ in order to improve the statistics and precisely measure line widths and
velocity fields.
\begin{figure*}
\resizebox{\textwidth}{!}{
\includegraphics[width=0.48\textwidth,angle=0]{rgs_spec_total.pdf}
\includegraphics[width=0.48\textwidth]{rgs_qu_1st_2nd.pdf}
}
\caption{\label{rgs} RGS spectra of $\rho$~Oph~A in the range $5-30$\AA.
Left panel: the average spectrum during the full exposure (140 ks). The spectrum
was binned to have at least 25 counts per bin. Right panel:
black refers the spectrum relative to the quiescent state between the two main variability events,
the orange spectrum refers to the first event, and the blue spectrum refers to
the second event. Changes in the intensities of lines are visible mostly during the flare
event at wavelengths below 12\AA, when both lines and continuum increased their strength.}
\end{figure*}
\section{Discussion and conclusions}\label{discussion}
We have presented an {\em XMM-Newton}\ observation that monitored $\rho$~Oph\ for 140 ks,
encompassing an entire rotation period of the primary star of the system ($P\sim 1.2$ days).
We observed a change of X-ray emission twice, each time lasting about $35$ks and separated by
about 104 ks.
This ``lighthouse'' effect has increased its strength in the 2016 observation. On the other hand,
the quiescent phase has remained at similar levels of emission with respect to the 2013 observation.
The cause of this phenomenon can be attributed either to a large active spot
on the stellar surface, or an unknown low mass companion, and both scenarios have strengths
and weaknesses.
\noindent{\bf Active spot.} In this scenario, the spot is a magnetospheric feature that could also have
a photospheric counterpart and has been steadily present on the stellar
surface for at least 2.5 yrs (i.e., since the epoch of the 2013 {\em XMM-Newton}\ observation, Paper I),
as the light curves of 2013 and 2016 can be put in phase with the stellar rotation period,
thus matching the periodic increases of rate. In this respect $\rho$~Oph~A would be a magnetic
early B star similar to those discovered by, for example, \citet{Fossati2014}.
From the duration of the rate change (about $30-35$ ks) compared to the period (104 ks)
and taking into account the inclination angle of the star ($i\sim45\degr$),
we infer a quite large size for the spot, amounting to about 1.5 stellar radii.
The cusp-like shape of the first episode of variability, free of major flares,
and the lack of a flat top level suggest that such a large spot could be plausible
because when the spot is completely on view after emerging from one limb it starts to disappear
at the opposite limb.
We attribute the origin of the spot to a strong magnetic field that is responsible for
creating the spot in the form of magnetic loops. The loops confine high temperature plasma that emits
X-rays. In this scenario the spot can be seen as a scaled-up version of loop-populated solar active regions.
Flares can occur within these loops that make the typical temperature of the plasma
rise above $kT\ge5$ keV, as observed in the second event.
No constraints on the magnetic field
in $\rho$~Oph~A could be found in the literature, therefore
magnetic field scenario remains hypothetical at the moment. In the future work we aim to obtain reliable magnetic field measurements of this interesting object.
If $\rho$~Oph~A has a magnetic field, then it would not be the only known early B star exhibiting surface
spots and a magnetic field.
In $\tau$~Sco \citet{Donati2006} discovered a magnetic field with a complex topology.
The magnetic loops should be responsible for emitting X-rays as in solar-type stars.
However, the lack of modulated X-ray emission hints that the magnetic loops are small and
distributed across the surface \citep{Oskinova2016,Ignace2010}.
However, in $\rho$~Oph\ we find well-defined periodic modulation, thus a strong magnetic field
must be concentrated in a large spot. We speculate that the magnetic field has a large scale
structure that is similar to a simple dipolar configuration and a stable configuration deduced by
the long lifetime of the spot ($\ge 2.5$ yrs).
\citet{Wade2016} have found a strong magnetic field, on the order of a few kG, in HD 164492C1, which is
an early B star similar to $\rho$~Oph\ rotating in 1.36 days and part of a hierarchical system with other
two A/Herbig Ae stars belonging to the Trifid nebula. However, one difference from $\rho$~Oph\ is that the
HD 164492 system is tighter than that of $\rho$~Oph\ and the origin of the magnetic field in such objects
could be associated with binarity and the stellar formation mechanism.
Another case of strong magnetism in an early B star (CPD~$-62\degr 2124$) is reported by \citet{Hubrig2017},
where a strong dipolar magnetic field with strength in excess of $\sim30$ kG has been detected.
Analogous to $\rho$~Oph~A, CPD~$-62\degr 2124$ is a fast rotator with rotational period of about 1.45 days.
Weak emission of X-rays has been detected in $\sigma$~Sgr (B2V, $\sim65$ pc) in a short 10 ks
{\em XMM-Newton}\ observation (PI Oskinova; XMM ObsId 0721210101).
For this star we derived a flux of about $7.4\times10^{-15}$ {erg~s$^{-1}$~cm$^{-2}$}\ in 0.3-8.0 keV and a luminosity
of $\sim3.7\times10^{27}$ {erg~s$^{-1}$}, which is about three orders of magnitude lower than the X-ray
luminosity of $\rho$~Oph~A. The count statistics are low but we can infer that the spectrum is almost
entirely below 1 keV and peaks around 0.7 keV, pointing to a substantial difference in the
mechanisms that produce X-rays in $\sigma$~Sgr and $\rho$~Oph.
B stars do not possess magnetic coronae { such as solar-type stars},
and in this respect they are similar to A type stars,
which are mostly dark in X-rays and lack strong magnetic fields. However, even among
A type stars, which are presumed to be less active stars during the main sequence lifetime
because they lack a solar-type corona, { \citet{Balona2013} found a number of stars with spots
in an analysis based on {\em Kepler} data.}
\noindent{\bf Low mass companion.}
Another explanation is the presence of an unknown low mass
stellar companion { as the source of X-rays}.
If such a companion is fully responsible for the X-rays, it means that it is not completely eclipsed
by $\rho$~Oph~A when orbiting around it. To support this we notice that $\rho$~Oph~B (another B2 star)
is undetected in the same data. During the quiescent state we observed the less active part
of the corona of the unknown companion, while during the recurrent increase of rate
we had observed the most active part of it,
i.e., perhaps the part of the surface of the companion facing the primary (its day side).
From the combined analysis of the X-ray derived period and the v~sin$i$ from the line profile,
it is suggested that the primary is inclined by $\sim45\degr$. If the plane of the orbit of the
companion has the same inclination it is plausible that during the superior conjunction part of its
surface (its day side) is partially obscured by the primary, dark in X-rays.
Two factors can boost the X-ray activity of the hypothetical companion to such a high level:
its young age, coeval with $\rho$~Oph\ and with the
other solar-type stars discovered around it \citep{Pillitteri2016}, and being part of a tight
binary system. This would enhance the stellar activity even more than in a configuration of single star,
as observed for example in the RS CVn systems.
In fact, if the companion rotates in 1.2 days, as $\rho$~Oph s does, such a period would correspond
to a very close separation of $\sim9.8 \mathrm{R}\odot$ that is similar to the radius of $\rho$~Oph~A.
This results in an extreme system with the companion almost on the verge of collapsing
onto the primary. The system should be in a 1:1 spin-orbit locked configuration reached
in the first 5 Myr, making its evolution fast.
The plasma temperatures, the type of impulsive variability and the pattern of abundances
of the quiescent spectrum are fully consistent with the X-ray properties of
young, active pre-main sequence stars.
If a young unseen companion emits such X-rays, the derived loop length ($1.4-1.9\times10^{11}$ cm
or $\sim2-2.7 R_\odot$) would be a few of its stellar radius.
This implies an extended structure that perhaps could connect the surface of $\rho$~Oph~A
with the companion and hints to some form of magnetic interaction between the two stars.
At the same time the hottest part of the corona of the companion should be eclipsed at some
phases of the orbital motion to reproduce the features observed in the light curve.
In the literature we find cases of X-rays from massive stars that are attributed to a low mass companion,
such as $\sigma$ Ori E (B2Vp; \citealp{Sanz-Forcada2004}), which was detected in a flaring state during a {\em XMM-Newton}\ observation.
$\sigma$ Ori E has a hard spectrum similar to $\rho$~Oph~A and remarkably harder than that of $\sigma$ Ori AB
(O9.5V). However, in contrast to $\sigma$ Ori E we did not detect a 6.4 keV line from neutral Fe,
even during interval { 28} when the spectrum reached the maximum temperature.
The absence of a circumstellar disk on the companion of $\rho$~Oph~A is perhaps linked to the lack of
the 6.4 keV line as seen in $\sigma$ Ori E. There, the neutral Fe could emit by fluorescence
during the flare as the relatively cold material of the disk is hit by high energy X-ray photons
created during the flare, while in $\rho$~Oph~A there is no circumstellar disk from where
fluorescence of neutral Fe can arise.
The analysis of the optical spectra from the ESO archive did not reveal any Doppler shift
due to the presence of the companion. This does not exclude its presence but rather calls for
an ad hoc spectroscopic campaign on $\rho$~Oph~A.
A mixture of both scenarios could be present as well; that is, X-rays produced by the coronal
activity of an unseen companion coupled to and enhanced by an intrinsic magnetism of
$\rho$~Oph~A.
Among other cases of uncertain origin of X-rays we cite IQ Aur, which is an A0p
bright in X-rays and flaring during a {\em XMM-Newton}\ observation with temperatures up to 6 keV
\citep{Robrade2011}. In the case of IQ Aur a magnetized wind model and the presence of an unseen
low mass companion are discussed by Robrade et al. Qualitatively the X-rays from IQ Aur are
similar to those from $\rho$~Oph\ except for the fact that the flare has been observed once and
never with periodic recurrence. In the case of a magnetized wind emission the occurrence of
a flare does not easily fit the current models.
Other mechanisms of production of X-rays observed in binary systems of
massive stars are less probable because the separation between $\rho$~Oph~A and B is about 300 AU, the short period
of the phenomenon appears to be linked to the rotation of the primary, and because the winds from
B2 stars are weaker than those from O and Wolf Rayet stars.
A clear case of periodic variation of X-ray flux of a massive star has been reported
in $\chi^1$ CMa by \citet{Oskinova2014}.
$\chi^1$ CMa is a variable Beta Cep type star (B0.5-1 V-IV) characterized
by a strong magnetic field ($B>5000$ G). $\chi^1$ CMa exhibits X-ray and H-band mag variability
in phase, however its X-ray spectrum is softer than $\rho$~Oph\ and it does not show flares.
The mechanism that produces X-rays in $\chi^1$ CMa is still not completely clear;
it is likely linked to the compression phase and the plasma heating in the consequent shock, as the
maximum rate of X-rays happens at the minimum stellar radius. This scenario is not applicable
to $\rho$~Oph\ because it cannot explain the much higher plasma temperatures observed in it and its flaring
activity.
The present work indicates the peculiarity of $\rho$~Oph\ in the X-ray band,
as it represents perhaps the best example of an X-ray active early B star and
one of the most favorable targets for studying magnetism in B stars.
On the other hand, $\rho$~Oph~A could be an extreme system with a low mass companion on
the verge of collapsing onto the primary. In both cases $\rho$~Oph~A deserves more investigation
to understand the origin of its X-rays.
\begin{acknowledgements}
We would like to thank the anonymous referee for constructive comments.
IP is grateful to dr. Javier Lopez-Santiago and dr. Ines Crespo-Chacon for providing useful information
on the analysis of the line profile and the derivation of v~sin$i$.
S.J.W. was supported by NASA contract NAS8-03060.
\end{acknowledgements}
| {
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} | 9,030 |
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'use strict';
var should = require('should'),
request = require('supertest'),
path = require('path'),
mongoose = require('mongoose'),
User = mongoose.model('User'),
FbMarketing = mongoose.model('FbMarketing'),
express = require(path.resolve('./config/lib/express'));
/**
* Globals
*/
var app,
agent,
credentials,
user,
fbMarketing;
/**
* Fb marketing routes tests
*/
describe('Fb marketing CRUD tests', function () {
before(function (done) {
// Get application
app = express.init(mongoose);
agent = request.agent(app);
done();
});
beforeEach(function (done) {
// Create user credentials
credentials = {
username: 'username',
password: 'M3@n.jsI$Aw3$0m3'
};
// Create a new user
user = new User({
firstName: 'Full',
lastName: 'Name',
displayName: 'Full Name',
email: 'test@test.com',
username: credentials.username,
password: credentials.password,
provider: 'local'
});
// Save a user to the test db and create new Fb marketing
user.save(function () {
fbMarketing = {
name: 'Fb marketing name'
};
done();
});
});
it('should be able to save a Fb marketing if logged in', function (done) {
agent.post('/api/auth/signin')
.send(credentials)
.expect(200)
.end(function (signinErr, signinRes) {
// Handle signin error
if (signinErr) {
return done(signinErr);
}
// Get the userId
var userId = user.id;
// Save a new Fb marketing
agent.post('/api/fbMarketings')
.send(fbMarketing)
.expect(200)
.end(function (fbMarketingSaveErr, fbMarketingSaveRes) {
// Handle Fb marketing save error
if (fbMarketingSaveErr) {
return done(fbMarketingSaveErr);
}
// Get a list of Fb marketings
agent.get('/api/fbMarketings')
.end(function (fbMarketingsGetErr, fbMarketingsGetRes) {
// Handle Fb marketings save error
if (fbMarketingsGetErr) {
return done(fbMarketingsGetErr);
}
// Get Fb marketings list
var fbMarketings = fbMarketingsGetRes.body;
// Set assertions
(fbMarketings[0].user._id).should.equal(userId);
(fbMarketings[0].name).should.match('Fb marketing name');
// Call the assertion callback
done();
});
});
});
});
it('should not be able to save an Fb marketing if not logged in', function (done) {
agent.post('/api/fbMarketings')
.send(fbMarketing)
.expect(403)
.end(function (fbMarketingSaveErr, fbMarketingSaveRes) {
// Call the assertion callback
done(fbMarketingSaveErr);
});
});
it('should not be able to save an Fb marketing if no name is provided', function (done) {
// Invalidate name field
fbMarketing.name = '';
agent.post('/api/auth/signin')
.send(credentials)
.expect(200)
.end(function (signinErr, signinRes) {
// Handle signin error
if (signinErr) {
return done(signinErr);
}
// Get the userId
var userId = user.id;
// Save a new Fb marketing
agent.post('/api/fbMarketings')
.send(fbMarketing)
.expect(400)
.end(function (fbMarketingSaveErr, fbMarketingSaveRes) {
// Set message assertion
(fbMarketingSaveRes.body.message).should.match('Please fill Fb marketing name');
// Handle Fb marketing save error
done(fbMarketingSaveErr);
});
});
});
it('should be able to update an Fb marketing if signed in', function (done) {
agent.post('/api/auth/signin')
.send(credentials)
.expect(200)
.end(function (signinErr, signinRes) {
// Handle signin error
if (signinErr) {
return done(signinErr);
}
// Get the userId
var userId = user.id;
// Save a new Fb marketing
agent.post('/api/fbMarketings')
.send(fbMarketing)
.expect(200)
.end(function (fbMarketingSaveErr, fbMarketingSaveRes) {
// Handle Fb marketing save error
if (fbMarketingSaveErr) {
return done(fbMarketingSaveErr);
}
// Update Fb marketing name
fbMarketing.name = 'WHY YOU GOTTA BE SO MEAN?';
// Update an existing Fb marketing
agent.put('/api/fbMarketings/' + fbMarketingSaveRes.body._id)
.send(fbMarketing)
.expect(200)
.end(function (fbMarketingUpdateErr, fbMarketingUpdateRes) {
// Handle Fb marketing update error
if (fbMarketingUpdateErr) {
return done(fbMarketingUpdateErr);
}
// Set assertions
(fbMarketingUpdateRes.body._id).should.equal(fbMarketingSaveRes.body._id);
(fbMarketingUpdateRes.body.name).should.match('WHY YOU GOTTA BE SO MEAN?');
// Call the assertion callback
done();
});
});
});
});
it('should be able to get a list of Fb marketings if not signed in', function (done) {
// Create new Fb marketing model instance
var fbMarketingObj = new FbMarketing(fbMarketing);
// Save the fbMarketing
fbMarketingObj.save(function () {
// Request Fb marketings
request(app).get('/api/fbMarketings')
.end(function (req, res) {
// Set assertion
res.body.should.be.instanceof(Array).and.have.lengthOf(1);
// Call the assertion callback
done();
});
});
});
it('should be able to get a single Fb marketing if not signed in', function (done) {
// Create new Fb marketing model instance
var fbMarketingObj = new FbMarketing(fbMarketing);
// Save the Fb marketing
fbMarketingObj.save(function () {
request(app).get('/api/fbMarketings/' + fbMarketingObj._id)
.end(function (req, res) {
// Set assertion
res.body.should.be.instanceof(Object).and.have.property('name', fbMarketing.name);
// Call the assertion callback
done();
});
});
});
it('should return proper error for single Fb marketing with an invalid Id, if not signed in', function (done) {
// test is not a valid mongoose Id
request(app).get('/api/fbMarketings/test')
.end(function (req, res) {
// Set assertion
res.body.should.be.instanceof(Object).and.have.property('message', 'Fb marketing is invalid');
// Call the assertion callback
done();
});
});
it('should return proper error for single Fb marketing which doesnt exist, if not signed in', function (done) {
// This is a valid mongoose Id but a non-existent Fb marketing
request(app).get('/api/fbMarketings/559e9cd815f80b4c256a8f41')
.end(function (req, res) {
// Set assertion
res.body.should.be.instanceof(Object).and.have.property('message', 'No Fb marketing with that identifier has been found');
// Call the assertion callback
done();
});
});
it('should be able to delete an Fb marketing if signed in', function (done) {
agent.post('/api/auth/signin')
.send(credentials)
.expect(200)
.end(function (signinErr, signinRes) {
// Handle signin error
if (signinErr) {
return done(signinErr);
}
// Get the userId
var userId = user.id;
// Save a new Fb marketing
agent.post('/api/fbMarketings')
.send(fbMarketing)
.expect(200)
.end(function (fbMarketingSaveErr, fbMarketingSaveRes) {
// Handle Fb marketing save error
if (fbMarketingSaveErr) {
return done(fbMarketingSaveErr);
}
// Delete an existing Fb marketing
agent.delete('/api/fbMarketings/' + fbMarketingSaveRes.body._id)
.send(fbMarketing)
.expect(200)
.end(function (fbMarketingDeleteErr, fbMarketingDeleteRes) {
// Handle fbMarketing error error
if (fbMarketingDeleteErr) {
return done(fbMarketingDeleteErr);
}
// Set assertions
(fbMarketingDeleteRes.body._id).should.equal(fbMarketingSaveRes.body._id);
// Call the assertion callback
done();
});
});
});
});
it('should not be able to delete an Fb marketing if not signed in', function (done) {
// Set Fb marketing user
fbMarketing.user = user;
// Create new Fb marketing model instance
var fbMarketingObj = new FbMarketing(fbMarketing);
// Save the Fb marketing
fbMarketingObj.save(function () {
// Try deleting Fb marketing
request(app).delete('/api/fbMarketings/' + fbMarketingObj._id)
.expect(403)
.end(function (fbMarketingDeleteErr, fbMarketingDeleteRes) {
// Set message assertion
(fbMarketingDeleteRes.body.message).should.match('User is not authorized');
// Handle Fb marketing error error
done(fbMarketingDeleteErr);
});
});
});
it('should be able to get a single Fb marketing that has an orphaned user reference', function (done) {
// Create orphan user creds
var _creds = {
username: 'orphan',
password: 'M3@n.jsI$Aw3$0m3'
};
// Create orphan user
var _orphan = new User({
firstName: 'Full',
lastName: 'Name',
displayName: 'Full Name',
email: 'orphan@test.com',
username: _creds.username,
password: _creds.password,
provider: 'local'
});
_orphan.save(function (err, orphan) {
// Handle save error
if (err) {
return done(err);
}
agent.post('/api/auth/signin')
.send(_creds)
.expect(200)
.end(function (signinErr, signinRes) {
// Handle signin error
if (signinErr) {
return done(signinErr);
}
// Get the userId
var orphanId = orphan._id;
// Save a new Fb marketing
agent.post('/api/fbMarketings')
.send(fbMarketing)
.expect(200)
.end(function (fbMarketingSaveErr, fbMarketingSaveRes) {
// Handle Fb marketing save error
if (fbMarketingSaveErr) {
return done(fbMarketingSaveErr);
}
// Set assertions on new Fb marketing
(fbMarketingSaveRes.body.name).should.equal(fbMarketing.name);
should.exist(fbMarketingSaveRes.body.user);
should.equal(fbMarketingSaveRes.body.user._id, orphanId);
// force the Fb marketing to have an orphaned user reference
orphan.remove(function () {
// now signin with valid user
agent.post('/api/auth/signin')
.send(credentials)
.expect(200)
.end(function (err, res) {
// Handle signin error
if (err) {
return done(err);
}
// Get the Fb marketing
agent.get('/api/fbMarketings/' + fbMarketingSaveRes.body._id)
.expect(200)
.end(function (fbMarketingInfoErr, fbMarketingInfoRes) {
// Handle Fb marketing error
if (fbMarketingInfoErr) {
return done(fbMarketingInfoErr);
}
// Set assertions
(fbMarketingInfoRes.body._id).should.equal(fbMarketingSaveRes.body._id);
(fbMarketingInfoRes.body.name).should.equal(fbMarketing.name);
should.equal(fbMarketingInfoRes.body.user, undefined);
// Call the assertion callback
done();
});
});
});
});
});
});
});
afterEach(function (done) {
User.remove().exec(function () {
FbMarketing.remove().exec(done);
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,487 |
{"url":"https:\/\/arxiv-download.xixiaoyao.cn\/list\/math.NT\/recent","text":"# Number Theory\n\n## Authors and titles for recent submissions\n\n[ total of 60 entries: 1-25 | 26-50 | 51-60 ]\n[ showing 25 entries per page: fewer | more | all ]\n\n### Thu, 25 Nov 2021\n\n[1]\nTitle: Sums of two squares are strongly biased towards quadratic residues\nAuthors: Ofir Gorodetsky\nSubjects: Number Theory (math.NT)\n[2]\nTitle: Algebraic integers with conjugates in a prescribed distribution\nAuthors: Alexander Smith\nSubjects: Number Theory (math.NT)\n[3]\nTitle: Un peu d'effectivit\u00e9 pour les vari\u00e9t\u00e9s modulaires de Hilbert-Blumenthal\nAuthors: Levent Alp\u00f6ge\nComments: 39 pages, title's in French (because I like Szpiro's modest phrase) but the paper's in English\nSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)\n[4]\nTitle: A generalised Roth's theorem\nSubjects: Number Theory (math.NT)\n[5]\nTitle: A note on Dirichlet spectrum\nSubjects: Number Theory (math.NT)\n\n### Wed, 24 Nov 2021\n\n[6]\nTitle: Some new results in effective diophantine approximation\nSubjects: Number Theory (math.NT)\n[7]\nTitle: Partitions Associated to Class Groups of Imaginary Quadratic Number Fields\nSubjects: Number Theory (math.NT)\n[8]\nTitle: Infinite order linear difference equation satisfied by a refinement of Goss zeta function\nAuthors: Su Hu, Min-Soo Kim\nSubjects: Number Theory (math.NT)\n[9]\nTitle: On a uniform bound for exponential sums modulo $p^m$ for Deligne polynomials\nAuthors: Kien Huu Nguyen\nSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG); Logic (math.LO)\n[10]\nTitle: On non-congruent numbers with $8a\\pm1$ type odd prime factors and tame kernels\nAuthors: Shenxing Zhang\nSubjects: Number Theory (math.NT); K-Theory and Homology (math.KT)\n[11]\nTitle: Multivariable de Rham representations, Sen theory and $p$-adic differential equations\nSubjects: Number Theory (math.NT)\n[12]\nTitle: Consecutive real quadratic fields with large class numbers\nSubjects: Number Theory (math.NT)\n[13]\nTitle: On the Borel complexity of continued fraction normal, absolutely abnormal numbers\nComments: Video talk by first author on this and related topics: this https URL\nSubjects: Number Theory (math.NT); Logic (math.LO)\n[14]\u00a0 arXiv:2111.11774 (cross-list from math.CO) [pdf, ps, other]\nTitle: Matrix Waring Problem\nAuthors: Krishna Kishore\nSubjects: Combinatorics (math.CO); Number Theory (math.NT)\n[15]\u00a0 arXiv:2111.11580 (cross-list from math.KT) [pdf, ps, other]\nTitle: Hilbert reciprocity using K-theory localization\nAuthors: Oliver Braunling\nSubjects: K-Theory and Homology (math.KT); Number Theory (math.NT)\n\n### Tue, 23 Nov 2021 (showing first 10 of 24 entries)\n\n[16]\nTitle: Growth of Odd Torsion Over Imaginary Quadratic Fields of Class Number 1\nAuthors: Irmak Bal\u00e7\u0131k\nSubjects: Number Theory (math.NT)\n[17]\nTitle: Non-cyclic Torsion Over Imaginary Quadratic Fields of Class Number 1\nAuthors: Irmak Bal\u00e7\u0131k\nSubjects: Number Theory (math.NT)\n[18]\nTitle: On a discriminator for the polynomial $f(x)=x^3+x$\nComments: Correct a print mistake in Theorem 1.1\nSubjects: Number Theory (math.NT)\n[19]\nTitle: Generalised Andr\u00e9-Pink-Zannier Conjecture for Shimura varieties of abelian type\nSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)\n[20]\nTitle: A note on mod-$p$ local-global compatibility via Scholze's functor\nSubjects: Number Theory (math.NT); Representation Theory (math.RT)\n[21]\nTitle: On the functional graph of $f(X)=c(X^{q+1}+aX^2)$ over quadratic extensions of finite fields\nSubjects: Number Theory (math.NT); Information Theory (cs.IT); Dynamical Systems (math.DS)\n[22]\nTitle: A $q$-multisum identity arising from finite chain ring probabilities\nSubjects: Number Theory (math.NT); Combinatorics (math.CO)\n[23]\nTitle: On special elements for $\\mathbb{G}_m$\nSubjects: Number Theory (math.NT)\n[24]\nTitle: Common values of a class of linear recurrence\nAuthors: Attila Peth\u0151\nSubjects: Number Theory (math.NT)\n[25]\nTitle: Quadratic Weyl group multiple Dirichlet series of Type $D_{\\scriptscriptstyle 4}^{\\scriptscriptstyle (1)}$","date":"2021-11-28 17:51:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4889119863510132, \"perplexity\": 14371.249029705767}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358570.48\/warc\/CC-MAIN-20211128164634-20211128194634-00415.warc.gz\"}"} | null | null |
\section{Introduction
\label{sec:intro
A numerical semigroup $S$ is an additively closed subset of $\ZZ_{\ge 0}$, usually specified using a generating set $r_1, \ldots, r_k$, i.e.,
$$S = \<r_1, \ldots, r_k\> = \{z_1r_1 + z_2r_2 + \cdots + z_kr_k \mid z_1, \ldots, z_k \in \ZZ_{\ge 0}\}.$$
Many classical problems surrounding numerical semigroups involve arithmetic invariants, such as the Frobenius number $\mathsf F(S)$, genus $\mathsf g(S)$, type $\mathsf t(S)$, and delta set $\Delta(S)$, each of which is difficult to compute when the generators of $S$ are large.
For a thorough introduction to numerical semigroups, see~\cite{numerical}.
This paper considers parametrized families of numerical semigroups of the form
$$P_n = \<f_1(n), \ldots, f_k(n)\>$$
for some functions $f_1(n), \ldots, f_k(n)$. Such families have arisen in two main settings in the last decade. First is the \emph{parametric Frobenius problem}, which asks under what conditions the function $n \mapsto \mathsf F(P_n)$ coincides with a quasipolynomial (that is, a polynomial with periodic coefficients) for large $n$. It was conjectured in~\cite{rouneparametricfrob} that this holds whenever the functions $f_i$ are themselves polynomials, where this was proven in the case where $\deg f_i = 1$ for all $i$, as well as in the case $k = 3$. This appears to have been proven in general~\cite{shenparametricfrob}, though the results have yet to appear outside the \texttt{arXiv}, and the authors of this manuscript have been unable to contact the author.
Separately, \emph{shifted} numerical semigroups, which have a specialized parametrization
$$M_n = \<n, n + r_2, \ldots, n + r_k\>$$
for positive integers $r_2, \ldots, r_k$, have been examined in numerous recent papers. It~is known that the delta set of $M_n$ is eventually periodic~\cite{shiftydelta}, and that the Frobenius number, genus, and type of $M_n$ are each eventually quasipolynomial~\cite{shiftedaperysets}. Additionally, the minimal relations between the generators of $M_n$, usually studied in the form of minimal presentations \cite{numerical} or syzygies of the defining toric ideal \cite{cca}, are known to satisfy a certain periodicity originally conjectured by Herzog and Srinivasan and proven by Vu~\cite{vu14}. These results were later improved in~\cite{shiftyminpres}, wherein several consequences for other semigroup invariants were also derived, and further specialized in~\cite{shiftedtangentcone,shifted3gen}.
The results mentioned above provide ample evidence of a more general phenomenon, which we now conjecture formally.
\begin{conj}\label{conj:main}
If $f_1, \ldots, f_k:\ZZ \to \ZZ$ are eventually increasing polynomials and
$$P_n = \<f_1(n), \ldots, f_k(n)\>,$$
then $\Betti(P_n)$ is eventually quasipolynomial in $n$. As a consequence, the Frobenius number, genus, and type of $P_n$ are each eventually quasipolynomial in $n$.
\end{conj}
Note that the word ``consequence'' in Conjecture~\ref{conj:main} is intended as an informal~claim. In~particular, the main results of~\cite{shiftyminpres,shiftedaperysets} for shifted numerical semigroups (where the conjecture is already proven) stem from a single underlying result (\cite[Theorem~3.4]{shiftyminpres}) regarding the Betti elements of $P_n$ (that is, elements whose factorizations encode the minimal relations between the generators of $P_n$). Conjecture~\ref{conj:main} claims this core behavior occurs more generally, and that the remaining claims follow as consequences.
\begin{remark}\label{r:conjproof}
After posting this manuscript, a proof of the ``eventually quasipolynomial'' claims in Conjecture~\ref{conj:main} appeared elsewhere on the arXiv~\cite{parametricpresburgerbigthm}. The results therein are broad, with most claims extended to parametrized families of affine semigroups, but the proofs are nonconstructive, relying on formal logic and Presburger arithmetic. As~such, the informal ``consequence'' claim discussed above remains open.
\end{remark}
In this paper, we prove Conjecture~\ref{conj:main} in the case where the functions $f_1(n), \ldots, f_k(n)$ are linear. The main results are in Sections~\ref{sec:minpres} and~\ref{sec:aperysets}, which generalize results for shifted numerical semigroups that appeared in~\cite{shiftyminpres} and~\cite{shiftedaperysets}, respectively. The results in those sections follow from a central result about Betti elements for large $n$ (Theorem~\ref{t:mesalemma}), providing the ``consequently'' part of Conjecture~\ref{conj:main}. As a necessary step in stating our main results, we develop the notion of ``weighted factorization length'' in Section~\ref{sec:weightedlengths}, and generalize several known results involving standard factorization length. As evidence of the generality in Conjecture~\ref{conj:main}, we close this paper with Example~\ref{e:nonlinear}, a non-linear example where Conjecture~\ref{conj:main} appears to hold.
\subsection*{Acknowledgements}
The authors would like to thank Scott Chapman and Pedro Garc\'ia-S\'anchez for their helpful comments and suggestions.
\section{Numerical semigroups and factorization length
\label{sec:background
In this section, we state some background definitions for factorizations of numerical semigroup elements; the books~\cite{nonuniq} and~\cite{numerical} contain thorough introductions to nonunique factorization and numerical semigroups, respectively. Several of the quantities in Definition~\ref{d:numerical} involving (unweighted) factorization length have a weighted generalization introduced in subsequent sections of this paper.
\begin{defn}\label{d:numerical}
A \emph{numerical semigroup} $S$ is an additive subsemigroup of $\ZZ_{\ge 0}$ (note, we do \textbf{not} require $S$ to have finite complement). We write
$$S = \<r_1, \ldots, r_k\> = \{z_1r_1 + \cdots + z_kr_k : z_1, \ldots, z_k \in \ZZ_{\ge 0}\}$$
for the semigroup generated by $r_1, \ldots, r_k$.
A \emph{factorization} of $n \in S$ is an expression
$$n = z_1r_1 + \cdots + z_kr_k$$
of $n$ as a sum of generators of $S$, and the \emph{length} of a factorization is the sum $z_1 + \cdots + z_k$. The \emph{set of factorizations} of $n$ is the set
$$\mathsf Z_S(n) = \{z \in \ZZ_{\ge 0}^k : n = z_1r_1 + \cdots + z_kr_k\}$$
viewed as a subset of $\ZZ_{\ge 0}^k$, and the \emph{length set} of $n$ is the set
$$\mathsf L_S(n) = \{z_1 + \cdots + z_k : z \in \mathsf Z_S(n)\},$$
of all possible factorization lengths of $n$. Writing $\mathsf L_S(n) = \{\ell_1 < \cdots < \ell_m\}$, define
$$\Delta_S(n) = \{\ell_i - \ell_{i-1} : 2 \le i \le m\} \qquad \text{and} \qquad \Delta(S) = \bigcup_{n \in S} \Delta_S(n)$$
as the \emph{delta sets} of $n$ and $S$, respectively.
The \emph{maximum} and \emph{minimum} factorization length functions are defined as
$$\mathsf M_S(n) = \max \mathsf L_S(n) \qquad \text{ and } \qquad \mathsf m_S(n) = \min \mathsf L_S(n),$$
respectively.
\end{defn}
We state two results from the literature (Theorems~\ref{t:deltaset} and~\ref{t:maxminorig}) that we will generalize in the next section. The first result depends on the following definition.
\begin{defn}\label{d:bettielement}
Given a numerical semigroup $S$ and an element $n \in S$, the \textit{factorization graph} of $n$, denoted $\nabla_n$, has vertex set $\mathsf Z(n)$, and two vertices $z,z' \in \mathsf Z(n)$ are connected by an edge whenever they have at least one generator in common. We say $n$ is a \emph{Betti element} of $S$ if $\nabla_n$ is disconnected. Define
$$\Betti(S) = \{n \in S \mid n \text{ is a Betti element of } S\}.$$
\end{defn}
\begin{example}\label{e:bettielements}
The Betti elements of $S = \<6, 9, 20\>$ are $\Betti(S) = \{18,60\}$, whose factorization graphs are depicted in Figure~\ref{f:bettielements}. As we will see in Section~\ref{sec:minpres}, these elements encode the minimal relations between the generators of $S$:\ $18$ is the smallest element that can be factored using $6$ and $9$, and $60$ is the smallest element that can be factored using $6$ and $9$ and separately using $20$.
\end{example}
\begin{figure}
\begin{center}
\includegraphics[height=1.2in]{betti-6-9-20--18.pdf}
\hspace{1.0in}
\includegraphics[height=1.2in]{betti-6-9-20--60.pdf}
\end{center}
\caption{The factorization graphs $\nabla_{18}$ (left) and $\nabla_{60}$ (right) in the numerical semigroup $S = \<6, 9, 20\>$ from Example~\ref{e:bettielements}.}
\label{f:bettielements}
\end{figure}
\begin{remark}\label{r:bettinumbers}
From a commutative algebra viewpoint, Betti elements coincide with graded degrees of the minimal generators of toric ideals. Given a numerical semigroup $S = \<r_1, \ldots, r_k\>$, the kernel $I = \ker\varphi$ of the ring homomorphism determined by
$$\begin{array}{r@{}c@{}l}
\varphi:\CC[x_1, \ldots, x_k] &{}\to{}& \CC[y] \\
x_i &{}\mapsto{}& y^{r_i}
\end{array}$$
is the defining toric ideal of $S$. As an example, if $S = \<6, 9, 20\>$, then the defining toric ideal $I \subset \CC[x,y,z]$ has precisely 4 minimal generating sets, namely
$$\{x^3 - y^2, x^{10} - z^3\}, \, \{x^3 - y^2, x^7y^2 - z^3\}, \,
\{x^3 - y^2, x^4y^4 - z^3\}, \, \text{and} \, \{x^3 - y^2, xy^6 - z^3\},$$
each of which has one homogeneous element of degree $18$ and one of degree $60$ (here, the graded degree of each monomial is determined by its image under $\varphi$). This matches the Betti elements $\Betti(S) = \{18, 60\}$ obtained in Example~\ref{e:bettielements}.
\end{remark}
\begin{thm}[{\cite{deltasets,bfdelta}}]\label{t:deltaset}
For any numerical semigroup $S = \<r_1, \ldots, r_k\>$, the set $\Delta(S)$ is~nonempty and finite, and $\gcd \Delta(S) = \min \Delta(S)$. Moreover,
$$\min \Delta(S) = \gcd\{r_i - r_{i-1} : 2 \le i \le k\}$$
and
$$\max \Delta(S) = \max_{n \in \Betti(S)} \max \Delta_S(n).$$
\end{thm}
\begin{thm}[{\cite{elastsets,shiftyminpres}}]\label{t:maxminorig}
For $n > r_k^2$ in a numerical semigroup $S = \<r_1, \ldots, r_k\>$, we have
$$\mathsf M(n + r_1) = \mathsf M(n) + 1 \qquad \text{and} \qquad \mathsf m(n + r_k) = \mathsf m(n) + 1.$$
\end{thm}
The functions in Theorem~\ref{t:maxminorig} are said to coincide for large $n$ with \emph{quasipolynomials}, that is, polynomial functions $\ZZ \to \RR$ with periodic coefficients. In particular,
$$\mathsf M(n) = \tfrac{1}{r_1}n + a(n) \qquad \text{and} \qquad \mathsf m(n) = \tfrac{1}{r_k}n + b(n)$$
for some periodic functions $a(n)$ and $b(n)$ with periods $r_1$ and $r_k$, respectively.
\section{Weighted factorization lengths
\label{sec:weightedlengths
Before examining parametrized families of numerical semigroups, we introduce a generalization of factorization length that independently weights each generator and plays a key role in the results of subsequent sections. We give two main results in this section, each of which generalizes existing results for the usual factorization length. The~first is Theorem~\ref{t:maxminquasi}, which generalizes \cite[Theorems~4.2 and~4.3]{elastsets} and joins a growing family of ``eventually quasipolynomial'' results concerning factorization length (see \cite{factorhilbert} and the references therein for an overview). The second is Theorem~\ref{t:weighteddelta}, which gives weighted versions of \cite[Lemma~3]{arithmeticintdom} and \cite[Theorem 2.5]{bfdelta}, both of which are central to the study of delta sets.
\begin{defn}\label{d:weightedlength}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$ and a rational vector $w = (w_1, \ldots, w_k) \in \QQ^k$ of \emph{weights}.
Given $n \in S$ and $z = (z_1, \ldots, z_k) \in \mathsf Z(n)$, the \textit{weighted length} of $z$ is
$$|z|_w = w \cdot z = w_1z_1 + \cdots + w_kz_k,$$
and the \emph{weighted length set} of $n$ is
$$\mathsf L_{S,w}(n) = \{|z|_w : z \in \mathsf Z(n)\}.$$
The maps $\mathsf M_w: S \mapsto \QQ$ and $\mathsf m_w: S \mapsto \QQ$ given by
$$\mathsf M_w(n) = \max \mathsf L_{S,w}(n) \qquad \text{and} \qquad \mathsf m_w(n) = \min \mathsf L_{S,w}(n)$$
are the \emph{maximum weighted length} and \emph{minimum weighted length} functions, respectively.
\end{defn}
\begin{defn}\label{d:worder}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$ and a weight vector $w =~(w_1, \ldots, w_k) \in \QQ^k$.
The \emph{$w$-ordering} $\le_w$ on $\{r_1, \ldots, r_k\}$ is defined so that
$$r_i \le_w r_j \qquad \text{whenever} \qquad w_i/r_i \ge w_j/r_j.$$
Note that the $w$-ordering is transitive, but need not be a total (or even partial) ordering, as $r_i =_w r_j$ is possible for $r_i \ne r_j$.
\end{defn}
\begin{remark}\label{r:standardlengthweighted}
The standard length $|\!\cdot\!|$ can be viewed as a special case of weighted length $|\cdot|_w$ with weight vector $w = (1,\ldots, 1)$. In this case, the $w$-ordering on $r_1, \ldots, r_k$ is the usual total ordering in $\ZZ$.
\end{remark}
\begin{example}\label{e:weightedlength}
Let $S = \<6, 9, 20\>$. For the weight vector $w = (3,1,4)$, the $w$-ordering on the generators of $S$ is $6 <_w 20 <_w 9$ since $\tfrac{3}{6} > \tfrac{4}{20} > \tfrac{1}{9}$. The same $w$-ordering is induced by $w = (3,-1,4)$, but some factorizations have negative weighted length, e.g.~$(2,12,1) \in \mathsf Z_S(140)$ has $|(2,12,1)|_w = -2$. Figure~\ref{f:weightedlength} depicts $\mathsf m_{S,w}(-)$ for both weight vectors; evident is the eventually quasilinear property implied by Theorem~\ref{t:maxminquasi}.
\end{example}
\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{6-9-20--3-1-4--minlen.pdf}
\hspace{0.2in}
\includegraphics[width=2.8in]{6-9-20--3-m1-4--minlen.pdf}
\end{center}
\caption{Plots depicting the minimum weighted factorization lengths of elements of $S = \<6, 9, 20\>$ for the weight vectors $w = (3,1,4)$ (left) and $w = (3,-1,4)$ (right) from Example~\ref{e:weightedlength}, created using \texttt{Sage} and the \texttt{GAP} package \texttt{numericalsgps}~\cite{numericalsgpsgap}.
}
\label{f:weightedlength}
\end{figure}
\begin{lemma}[{\cite[Lemma~4.1]{elastsets}}]\label{l:oldlemma41}
Suppose $q \ge 1$, and fix $c_1, \ldots, c_r \in \ZZ$ with $r \ge q$. There exists $T \subsetneq \{1, \ldots, r\}$ satisfying $\sum_{i \in T} c_i \equiv \sum_{i=1}^r c_i \bmod q$.
\end{lemma}
\begin{lemma}\label{l:maxminquasi}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$, a weight vector $w \in \QQ^k$, and suppose $r_1 \le_w r_2 \le_w \cdots \le_w r_k$.
\begin{enumerate}[(a)]
\item
If $a \in \mathsf Z_S(n)$ satisfies $a_1 + \cdots + a_k \ge r_1$, then there is some factorization $b \in \mathsf Z_S(n)$ with $|b|_w \ge |a|_w$ and $b_1 > 0$.
\item
If $a \in \mathsf Z_S(n)$ satisfies $a_1 + \cdots + a_k \ge r_k$, then there is some factorization $b \in \mathsf Z_S(n)$ with $|b|_w \le |a|_w$ and $b_k > 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
First, we claim if $a' = (0, a_2', \ldots, a_k'), b' = (b_1', 0, \ldots, 0) \in \mathsf Z(n)$, then $|b'|_w \ge |a'|_w$. Indeed, this follows from the fact that
$$|a'|_w
= \sum_{i=2}^k w_ia_i'
= \sum_{i=2}^k \frac{w_i}{r_i} r_ia_i'
\le \sum_{i=2}^k \frac{w_1}{r_1} r_ia_i'
= \frac{w_1}{r_1} r_1b_1'
= w_1b_1'
= |b'|_w.$$
Now, under the assumptions for part~(a), we see
$$a_1r_1 = n - a_2r_2 - \cdots - a_kr_k$$
implies $a_2r_2 + \ldots + a_kr_k \equiv n \bmod r_1$.
Lemma~\ref{l:oldlemma41} then guarantees the existence of integers $b_2, \ldots, b_k \ge 0$ such that (i)~$b_i \le a_i$ for each $i > 1$, (ii)~$\sum_{i =2}^ka_i >\sum_{i =2}^kb_i$, and (iii)~$b_2r_2 + \cdots + b_kr_k \equiv n \bmod r_1$.
This in particular means there exists $b_1 > 0$ so that $b = (b_1,\ldots, b_k) \in \mathsf Z(n)$. Rearranging the equation
$$n = a_1r_1 + \cdots + a_kr_k = b_1r_1 + \cdots + b_kr_k$$
yields
$$(b_1 - a_1)r_1 = (a_2 - b_2)r_2 + \cdots + (a_k - b_k)r_k$$
Applying the above claim to $(b_1-a_1, 0, \ldots, 0)$ and
$(0,a_2 - b_2, \ldots, a_k - b_k)$ implies
$$w_1(b_1 - a_1) \ge w_2(a_2 - b_2) + \cdots + w_k(a_k - b_k),$$
meaning
$$w_1b_1 + \cdots + w_kb_k \ge w_1a_1 + \cdots + w_ka_k,$$
so $|b|_w \ge |a|_w$. This proves part~(a).
The proof of part~(b) is analogous and thus omitted.
\end{proof}
\begin{thm}\label{t:maxminquasi}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$ and a weight vector $w \in \QQ^k$, and suppose $r_1 \le_w r_2 \le_w \cdots \le_w r_k$. Let $R = \max(r_1, \ldots, r_k)$.
\begin{enumerate}[(a)]
\item
\label{t:maxminquasi:maxlen}
For all $n > R^2$, the maximal weighted length function $\mathsf M_w: S \to \QQ$ satisfies
$$\mathsf M_w(n) = \mathsf M_w(n - r_1) + w_1.$$
\item
\label{t:maxminquasi:minlen}
For all $n > R^2$, the minimal weighted factorization length $\mathsf m_w: S \to \QQ$ satisfies
$$\mathsf m_w(n) = \mathsf m_w(n - r_k) + w_k.$$
\end{enumerate}
\end{thm}
\begin{proof}
Suppose $n > R^2$. First, we claim there is a factorization of $n$ with maximum weighted length with positive first coordinate. Indeed, fix any factorization $a \in \mathsf Z(n)$. If $a_2 + \cdots + a_k < r_1$, then $a_1 > 0$ by the assumption on $n$. On the other hand, if $a_2 + \cdots + a_k \ge r_1$ and $a_1 = 0$, then the claim follows from Lemma~\ref{l:maxminquasi}(a).
Now, by the above claim, let $a \in \mathsf Z(n)$ denote a maximum weighted length factorization with $a_1 > 0$. This means $a' = (a_1 - 1, a_2, \ldots, a_k) \in \mathsf Z(n - r_1)$ also has maximum weighted factorization length, so
$$\mathsf M_w(n - r_1)
= |a '|_w
= w_1(a_1 - 1) + w_2a_2 + \ldots w_ka_k
= |a|_w - w_1
= \mathsf M_w(n) - w_1,$$
thereby proving part~(a).
By a similar argument, some minimal weighted length factorization $a \in \mathsf Z(n)$ has $a_k > 0$. The proof of part~(b) then follows analogously.
\end{proof}
\begin{remark}\label{r:wtie}
For a given numerical semigroup $S = \<r_1, \ldots, r_k\>$, if a particular weight vector $w$ induces a ``tie'' $r_1 =_w \cdots =_w r_j$ in the $w$-ordering, then Theorem~\ref{t:maxminquasi} obtains an improved period $\gcd(r_1, \ldots, r_j)$ for the quasilinear function $\mathsf M_{S,w}$. For example, if~$S = \<6,9,10,14\>$ and $w = (2, 3, 5, 7)$, then $r_1 =_w r_2 >_w r_3 =_w r_4$, and for large $n$, $\mathsf M_{S,w}(n)$ and $\mathsf m_{S,w}(n)$ are each quasilinear with minimal periods $2$ and $3$, respectively. See Figure~\ref{f:wtie} for a depiction.
\end{remark}
\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{6-9-10-14--maxlen.pdf}
\hspace{0.2in}
\includegraphics[width=2.8in]{6-9-10-14--minlen.pdf}
\end{center}
\caption{Maximum (left) and minimum (right) weighted factorization lengths for $S = \<6, 9, 10, 14\>$ and $w = (2,3,5,7)$ from Remark~\ref{r:wtie}, created using \texttt{Sage} and the \texttt{GAP} package \texttt{numericalsgps}~\cite{numericalsgpsgap}.
}
\label{f:wtie}
\end{figure}
\begin{remark}\label{r:maxminwbound}
Much to our surprise, the bounds in Theorem~\ref{t:maxminquasi} do not depend on $w$, although it is worth noting that an optimal bound necessarily depends on $w$. Indeed, suppose $S = \<9,10,23\>$. If $w = (1,3,5)$, then $n = 64$ is the largest $n$ for which the first equality in Theorem~\ref{t:maxminquasi} fails to hold, and if $w = (6,9,5)$, then $n = 81$ is the largest such $n$. For both weight vectors, the generator $10$ is minimal under the $w$-ordering.
\end{remark}
The following corollary of Lemma~\ref{l:maxminquasi} and Theorem~\ref{t:maxminquasi} will be used in Section~\ref{sec:linearfamilies}. Note the additional assumption that $w$ has positive integer entries.
\begin{cor}\label{c:maxminquasi}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$, a weight vector $w \in \ZZ_{\ge 1}^k$, and suppose $r_1 \le_w r_2 \le_w \cdots \le_w r_k$. Fix $w_0 \in \ZZ_{\ge 1}$.
\begin{enumerate}[(a)]
\item
If $a \in \mathsf Z_S(n)$ satisfies $a_1 + \cdots + a_k \ge w_0r_1$, then there is some factorization $b \in \mathsf Z_S(n)$ with $|b|_w - |a|_w \in w_0\ZZ_{\ge 0}$ and $b_1 > 0$.
\item
If $a \in \mathsf Z_S(n)$ satisfies $a_1 + \cdots + a_k \ge w_0r_k$, then there is some factorization $b \in \mathsf Z_S(n)$ with $|a|_w - |b|_w \in w_0\ZZ_{\ge 0}$ and $b_k > 0$.
\end{enumerate}
\end{cor}
\begin{proof}
If $a_1 > 0$, then choosing $b = a$ proves part~(a), so suppose $a_1 = 0$. Fix $a' \in \ZZ_{\ge 0}^k$ such that $a_i' \le a_i$ for each $i$ and $a_1' + \cdots + a_k' \ge r_1$, and write $n' \in S$ so that $a' \in \mathsf Z(n')$. By Lemma~\ref{l:maxminquasi}(a), there exists $b' \in \mathsf Z(n')$ with $|b'|_w \ge |a'|_w$ and $b_1' > 0$. If $|b'|_w = |a'|_w$, then choosing $b = b' + (a - a')$ proves part~(a), so suppose $|b'|_w > |a'|_w$.
Now, fix a collection $c_1, \ldots, c_{w_0} \in \ZZ_{\ge 0}^k$ of vectors that sum to $a$.
Apply the above argument to each $c_i$ (in the role of $a'$) to obtain vectors $d_1, \ldots, d_{w_0} \in \ZZ_{\ge 0}^k$ (i.e., each corresponding vector $b'$ above), and let $\ell_i = |d_i|_w - |c_i|_w$. By Lemma~\ref{l:oldlemma41}, there exists a subset $T \subset \{1, \ldots, w_0\}$ so that $\sum_{i \in T} \ell_i \equiv 0 \bmod w_0$. Letting
$$b = \sum_{i \in T} d_i + a - \sum_{i \in T} c_i$$
we obtain
$$|b|_w - |a|_w = \biggl|\sum_{i \in T} d_i + a - \sum_{i \in T} c_i \biggr|_w - |a|_w = \sum_{i \in T} (|d_i|_w - |c_i|_w) \in w_0\ZZ_{\ge 0}$$
which completes the proof of part~(a).
As in the proof of Lemma~\ref{l:maxminquasi}, the proof of part~(b) is analogous.
\end{proof}
For the remainder of this section, we turn our attention to the weighted delta set. As~with weighted length sets, choosing the weight vector $w = (1, \ldots, 1)$ in the following definition recovers the usual delta set.
\begin{defn}\label{d:deltasets}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$, a weight vector $w \in \QQ^k$, and an element $n \in S$, and write
$$\mathsf L_{S,w}(n) = \{\ell_1 < \ell_2 < \cdots < \ell_r\}.$$
The \emph{weighted delta set} of $n$ is given by
$$\Delta_{S,w}(n) = \{\ell_i - \ell_{i-1}: i = 2, \ldots, r\},$$
and the \emph{weighted delta set} of $S$ is given by
$$\Delta_w(S) = \bigcup_{n \in S} \Delta_{S,w}(n).$$
Note that, unlike the usual delta set, it is possible to have $\Delta_w(S) = \emptyset$. Indeed, this happens when $w_i = r_i$ for every $i$, as $\mathsf L_w(n) = \{n\}$ for every $n \in S$ in this case.
\end{defn}
\begin{thm}\label{t:weighteddelta}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$ and a vector $w \in \QQ^k$.
\begin{enumerate}[(a)]
\item
If $\Delta_w(S) \ne \emptyset$, then $\Delta_w(S) \subset d\ZZ_{\ge 1}$, where $d = \min \Delta_w(S)$.
\item
We have
$$\min \Delta_w(S) = \gcd(\{w_ir_j - w_jr_i : 1 \le i < j \le r\}).$$
\item
The set $\Delta_w(S)$ is finite. Moreover,
$$\max \Delta_w(S) = \max_{n \in \Betti(S)} \max \Delta_w(n).$$
\end{enumerate}
\end{thm}
\begin{proof}
Each $w_i = t_i/u_i$ for some $t_i, u_i \in \ZZ$, so we must have $\Delta_w(S) \subset \delta\ZZ_{\ge 1}$, where $\delta = 1/(u_1 \cdots u_k)$. Fix $d' \in \Delta_w(S)$, and fix $c, c' \in \ZZ_{\ge 1}$ so that $d = c\delta$ and $d' = c'\delta$. Write $\gcd(c,c') = mc - m'c'$ for $m, m' \in \ZZ_{\ge 1}$. We must have elements $n, n' \in S$ and factorizations $a, b \in \mathsf Z(n)$ and $a', b' \in \mathsf Z(n')$ so that $|a|_w - |b|_w = d$ and $|a'|_w - |b'|_w = d'$. By the linearity of $|\cdot|_w$, the factorizations $ma + m'b', m'a' + mb \in \mathsf Z(mn + m'n')$ satisfy
$$|ma + m'b'|_w - |m'a' + mb|_w = m(|a|_w - |b|_w) - m'(|a'|_w - |b'|_w) = \gcd(c,c')\delta,$$
so by the minimality of $d$, we conclude $c = \gcd(c,c')$. This proves part~(a).
To prove part~(b), let
$$d' = \gcd(\{w_ir_j - w_jr_i : 1 \le i < j \le r\}).$$
Since $r_je_i, r_ie_j \in \mathsf Z(r_ir_j)$, the above argument implies
$$d \mid w_ir_j - w_jr_i = |r_je_i|_w - |r_ie_j|_w,$$
meaning $d \mid d'$. Conversely, suppose
$$a_1r_1 + \cdots + a_kr_k = b_1r_1 + \cdots + b_kr_k.$$
In order to show $d'$ divides $|a|_w - |b|_w = |a - b|_w$, by the linearity of $|\cdot|_w$ it suffices to express $a - b$ as an integer combination of the vectors $e_{ij} = r_je_i - r_ie_j$. Notice that
$$(a_1 - b_1)r_1 = (b_2 - a_2)r_2 + \cdots + (b_k - a_k)r_k$$
and since $\gcd(r_1, \ldots, r_k) = 1$, we must have $\gcd(r_2, \ldots, r_k) \mid (a_1 - b_1)$. As such, $a_1 - b_1 = c_2r_2 + \cdots + c_kr_k$ for some $c_i \in \ZZ$, meaning
$$a - b - c_2e_{12} - \cdots - c_ke_{1k} = (a_2 - b_2 + c_2r_1)e_2 + \cdots + (a_k - b_k + c_kr_1)e_k,$$
which has first coordinate $0$. Induction on $k$ concludes the proof of part~(b).
For part~(c), fix $n \in S$ and $x, y \in \mathsf Z(n)$ where $|x|_w < |y|_w$ are sequential in $\mathsf L_w(n)$. By~\cite[Lemma 2.1]{bfdelta}, there is a chain of factorizations $x_0, \ldots, x_t \in \mathsf Z(n)$ with $x_0 = x$, $x_t = y$, and $(x_i, x_{i+1}) = (a_i + c_i, b_i + c_i)$ for some $c_i \in \ZZ_{\ge 0}^k$ and factorizations $a_i, b_i \in \mathsf Z(n_i)$ lying in different connected components of the factorization graph~$\nabla_{n_i}$ of some Betti element~$\beta_i$. Since $|x|_w$ and $|y|_w$ are sequential in $\mathsf L_w(n)$, there must be some $i$ so that
$$|x_i|_w \le |x|_w < |y|_w \le |x_{i+1}|_w,$$
and no factorization $z \in \mathsf Z(\beta_i)$ can satisfy $|x|_w < |z + c_i|_w < |y|_w$. As such, we must have $|y|_w - |x|_w \le \max \Delta_w(\beta_i)$. This completes the proof.
\end{proof}
\section{Linear families of numerical semigroups
\label{sec:linearfamilies
In the remainder of this manuscript, we examine a particular parametrized family of numerical semigroups, of the form
$$P_n := \<w_1n + r_1, \ldots, w_kn + r_k\>$$
for fixed $r = (r_1, \ldots, r_k) \in \ZZ^k$ and $w = (w_1, \ldots, w_k) \in \ZZ_{\ge 1}^k$. The main result of this section is Theorem~\ref{t:mesalemma}, which describes the possible minimal generators that can occur for the defining toric ideal of $P_n$ for large $n$. This result is a generalization of \cite[Theorem 3.4]{shiftyminpres}, which sat at the center of the results in \cite{shiftyminpres,shiftedaperysets} for shifted numerical semigroups (see \cite[Remark~4.10]{shiftyminpres}). Likewise, Theorem~\ref{t:mesalemma} identifies the key structural changes that occur in $P_n$ for large~$n$ that are central to our results on Betti numbers, minimal presentations (Section~\ref{sec:minpres}) and Frobenius numbers (Section~\ref{sec:aperysets}).
We begin by imposing some assumptions on $r_1, \ldots, r_k$ and $w_1, \ldots, w_k$, all of which can be made without loss of generality.
\begin{notation}\label{n:parametrized}
Since $w_1, \ldots, w_k \in \ZZ_{\ge 1}$, we can reparametrize $n$ so $r_1, \ldots, r_k \in \ZZ_{\ge 0}$.
Reorder $r_1, \ldots, r_k$ (and correspondingly $w_1, \ldots, w_k$) so that $r_1 \le_w \cdots \le_w r_k$, that is,
$$r_1/w_1 \le \cdots \le r_k/w_k$$
(this is equivalent to Definition~\ref{d:worder} since $w$ has all positive entries). Note that if $r_i = 0$, then $r_j = 0$ for all $j \le i$ as well. Define
$$W = \max\{w_1, \ldots, w_k\}
\qquad \text{and} \qquad
R = \max\{r_1, \ldots, r_k\},$$
and reparametrize $n$ appropriately so that $0 \le r_1 < w_1$.
\end{notation}
\begin{remark}\label{r:rationalgens}
The proof of Theorem~\ref{t:mesalemma} begins by reparametrizing $P_n$ so that the first generator equals the input parameter. However, doing so forces the constant terms $t_2, \ldots, t_k$ to be (potentially) rational. Several times throughout the proof, Lemma~\ref{l:maxminquasi}(b) is carefully applied to the additive subsemigroup $T = \<t_2, \ldots, t_k\> \subset \QQ_{\ge 0}$ in the following sense:\ $T$ can be scaled by a unique rational value $\delta \in \QQ_{> 0}$ to obtain an isomorphic semigroup $\delta T \subset \ZZ_{\ge 0}$ with finite complement.
\end{remark}
\begin{thm}\label{t:mesalemma}
Let $z$ and $z'$ be factorizations of a Betti element $\beta \in P_n$ in different connected components of $\nabla_\beta$ with $|z|_w > |z'|_w$. If $n > w_1^2W\!R^2$, then
\begin{enumerate}[(a)]
\item
the connected components of $z$ and $z'$ in $\nabla_\beta$ contain every factorization of weighted length $|z|_w$ and $|z'|_w$, respectively;
\item
some factorization $y$ with $|z|_w = |y|_w$ has $y_1 > 0$; and
\item
some factorization $y'$ with $|z'|_w = |y'|_w$ has $y_k' > 0$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $m = w_1n + r_1$ so that
$$\begin{array}{r@{}c@{}l}
P_n
&{}={}& \big\<m, \, \tfrac{w_2}{w_1}m + \big(r_2 - w_2\tfrac{r_1}{w_1}\big), \, \ldots, \, \tfrac{w_k}{w_1}m + \big(r_k - w_k\tfrac{r_1}{w_1}\big)\big\> \\[0.2em]
&{}={}& \<m, \, v_2 m + t_2, \, \ldots, \, v_k m + t_k\>,
\end{array}$$
where each $t_i = r_i - w_i\tfrac{r_1}{w_1} \ge r_i - w_i\tfrac{r_i}{w_i} = 0$ and $v_i = \tfrac{w_i}{w_1}$. With this notation, we see $r_i \le_w r_j$ implies
$$\frac{t_i}{v_i}
= \frac{w_1r_i - w_ir_1}{w_i}
= w_1\frac{r_i}{w_i} - r_1
\le w_1\frac{r_j}{w_j} - r_1
= \frac{w_1r_j - w_jr_1}{w_j}
= \frac{t_j}{v_j}.$$
In particular, this implies (i) $t_1 = \cdots = t_{j-1} = 0$ for some $j < r$, and (ii) $t_j \le_v \cdots \le_v t_k$, viewing $v = (v_j, \ldots, v_k)$ as a weight vector for $T = \<t_j, \ldots, t_k\>$. For simplicity, given $t \in T$ and $a \in \mathsf Z_T(t)$, we write
$$|z|_v = v_1z_1 + \cdots + v_kz_k \qquad \text{and} \qquad |a|_v = v_ja_j + \cdots + v_ka_k$$
throughout the remainder of the proof. The key observation is that
$$\beta - |z|_v m
= z_1m + \sum_{i = j}^k z_i (v_i m + t_i) - z_1m - m \sum_{i = j}^k v_iz_i
= \sum_{i = j}^k z_i t_i
$$
yields a natural mapping of each factorization of $\beta \in P_n$ of weighted length $\ell$ to some factorization of $\beta - \ell m \in T$ of weighted length at most $\ell$. Let
$$a = (z_j, \ldots, z_k) \in \mathsf Z_T(\beta - |z |_v m) \qquad \text{and} \qquad a' = (z_j', \ldots, z_k') \in \mathsf Z_T(\beta - |z'|_v m)$$
denote the factorizations in $T$ corresponding to $z$ and $z'$, respectively. First, we claim some factorization in the same connected component of $\nabla_{\beta}$ as $z'$ has positive last coordinate. If $z_k' > 0$, then the claim is proven, so suppose $z'_k = 0$.
Since $w \in \ZZ_{\ge 1}^k$,
$$\begin{array}{r@{}c@{}l}
\beta - |z'|_v m
&{}={}& \displaystyle
\beta - \tfrac{1}{w_1}|z'|_w m
\ge \beta - \tfrac{1}{w_1}\big(|z|_w - 1\big)m
= \tfrac{1}{w_1}m + \beta - |z|_v m
\ge \displaystyle
\tfrac{1}{w_1}m
\ge n.
\end{array}$$
By assumption, $n > w_1^2\!R^2$, so writing $\delta \in \QQ_{> 0}$ for the unique rational value such that $\delta T \subset \ZZ_{\ge 0}$ has finite complement, this implies
$$a_j' + \cdots + a_k' \ge \tfrac{1}{R}(a_j't_j + \cdots + a_k't_k) = \tfrac{1}{R}(\beta - |z'|_v m) \ge \tfrac{1}{R}n > \tfrac{1}{R}w_1^2R^2 \ge w_1\delta t_k,$$
and thus Corollary~\ref{c:maxminquasi}(b) implies some factorization with positive last coordinate and weighted length having integer difference from $|a'|_v$ can be obtained from $a'$ by replacing all but at least one generator with copies of $t_k$. In particular, this factorization is in the same connected component as $a'$. Moreover, Corollary~\ref{c:maxminquasi}(b) implies some factorization $a'' \in \mathsf Z_T(\beta - |z'|_v m)$ whose weighted length is minimal among those satisfying $|a'|_v - |a''|_v \in \ZZ$ has $a_k'' > 0$.
Under the above factorization mapping, the factorization $z'' = (|z'|_v - |a''|_v, 0, \ldots, 0, a_j'', \ldots, a_k'') \in \mathsf Z_{P_n}(\beta)$ corresponds to $a''$ since
$$(|z'|_v - |a''|_v)m + \sum_{i = j}^k a_i''(v_i m + t_i) = (|z'|_v - |a''|_v)m + |a''|_v m + (\beta - |z'|_v m) = \beta.$$
The factorization $z''$ is thus in the same connected component of $\nabla_{\beta}$ as $z'$ and has $z_k'' > 0$, so the claim is proved.
Since $z$ and $z'$ are in different connected components of $\nabla_{\beta}$, we must have $z_k = 0$. This means $a_j + \cdots + a_k < w_1 \delta t_k$, as otherwise the above argument would yield a factorization of $\beta$ with positive last coordinate that is connected to $z$ in $\nabla_{\beta}$. Writing $V = \max(v_1, \ldots, v_k) = W/w_1$, the assumption $n > w_1^2W\!R^2$ implies
$$\begin{array}{r@{}c@{}l}
|z|_v
&{}>{}& \displaystyle |z'|_v \ge |a'|_v = \sum_{i = j}^k v_ia_i' = \sum_{i = j}^k \frac{v_i}{t_i}a_i't_i \ge \frac{v_k}{t_k}\sum_{i = j}^k a_i't_i = \frac{v_k}{t_k}(\beta - |z'|_v m) > \frac{v_k}{t_k} w_1^2W\!R^2 \\
&{}={}& \displaystyle \frac{w_k}{t_k}w_1W\!R^2 \ge w_1W\!R \ge w_1t_kW = w_1^2t_kV \ge w_1\delta t_kV > V \sum_{i = j}^k a_i \ge \sum_{i = j}^k v_ia_i = |a|_v,
\end{array}$$
so the factorization $y = (|z|_v - |a|_v, 0, \ldots, 0, a_j, \ldots, a_k)$ proves part~(b).
Additionally, either $y$ is connected to $z$ in $\nabla_\beta$, or $z_1 = 0$ and $z_j = \cdots = z_k = 0$. In the latter case, the preceeding inequalities imply $y_1 = |z|_v > w_kW\!R \ge W$, so one of the factorizations
$$y - w_i e_1 + w_1 e_i \in \mathsf Z_{P_n}(\beta) \qquad \text{for} \qquad 1 \le i \le j - 1$$
yields a path from $y$ to $z$ in $\nabla_\beta$ since one of the values $z_2, \ldots, z_{j-1}$ must be positive. This proves the first half of part~(a).
Lastly, suppose $z_k' = 0$. The above argument yielded a factorization $z''$ in the same connected component as $z'$ with $z_k'' > 0$ and corresponding factorization $a''$ having minimal weighted length. Since $z$ and $z'$ are in different connected components of~$\nabla_\beta$, the first half of part~(a) implies $z_1' = z_1'' = \cdots = z_{j-1}' = z_{j-1}'' = 0$ and thus $|a'|_v = |a''|_v$.
This proves part~(c), and applying the arguments thus far to any factorization of weighted length $|z'|_w$ yields a path in $\nabla_\beta$ to $z'$ through $z''$. This completes the proof.
\end{proof}
\begin{example}\label{e:positivecoords}
In Theorem~\ref{t:mesalemma}(c), we cannot ensure that all choices of factorizations $z$ and $z'$ has positive first and last coordinates, respectively. Indeed, if $r = (0,0,2,3)$ and $w = (5,7,2,3)$, then $\beta = 1980$ is a Betti element of $P_{44}$ with
$$(0, 0, 22, 0), \quad (0, 0, 19, 2), \quad (0, 0, 9, 0), \quad \text{and} \quad (0, 0, 2, 5)$$
among its factorizations. The key is that in the proof of Theorem~\ref{t:mesalemma}, there are ties in the $w$-ordering for both first and last place. As a consequence of Theorem~\ref{t:minpresmap}, this phenomenon also occurs for a Betti element of $P_{44+15m}$ for each $m \ge 0$.
\end{example}
\begin{example}\label{e:uglymapping}
In the proof of Theorem~\ref{t:mesalemma}, the natural mapping from factorizations of $\beta \in P_n$ of weighted length $\ell$ to factorizations of $\beta - \ell m \in T$ of weighted length at most $\ell$ need not be injective nor surjective. Let $r = (0,0,5,7,9)$ and $w = (2,3,5,7,8)$. Certainly, the factorizations $(8,0,0,0,0)$ and $(2,4,0,0,0)$ of $\beta = 704 \in P_{44}$ are mapped to the same factorization of $0 \in T = \<\frac{5}{2}, \frac{7}{2}, \frac{9}{2}\>$. What is perhaps more subtle is that $\beta = 1620$ has 2 factorizations, namely
$$\mathsf Z_{P_{44}}(1620) = \{(0, 0, 3, 3, 0), (2, 0, 0, 0, 4)\}$$
but the corresponding element $18 \in T$ has factorizations
$$\mathsf Z_T(18) = \{(3, 3, 0), (4, 1, 1), (0, 0, 4)\}$$
and the second does not correspond to any factorizations of $\beta$. The issue is that $v = (1,\frac{3}{2},\frac{5}{2},\frac{7}{2},4)$, so $a = (4,1,1)$ has non-integral weighted length $|a|_v = \frac{35}{2}$, so it is impossible to fill the first or second coordinates of a corresponding factorization of $\beta$ to obtain the necessary weighted length. This is why, when constructing factorizations of $\beta$ from factorizations of elements of $T$ at several locations in the proof of Theorem~\ref{t:mesalemma}, we must ensure that the first coordinate (a weighted length difference) is integral.
\end{example}
Now we can state a generalization of \cite[Corollary 3.5]{shiftyminpres} and \cite[Corollary 5.7]{shiftyminpres} which follows from Theorem~\ref{t:mesalemma}.
\begin{cor}\label{c:bettidelta}
If $n > w_1^2W\!R^2$, then $\Delta_w(P_n) = \{d\}$, where
$$d =
\gcd(w_1, \ldots, w_{j-1}, \min \Delta_w(S))\gcd(S)
$$
with $r_{j-1} = 0 < r_j$ and $S = \<r_j, \ldots, r_k\>$.
\end{cor}
\begin{proof}
Since
$$w_{i'}(w_in + r_i) - w_{i}(w_{i'}n + r_{i'}) = w_{i'}r_i - w_ir_{i'}$$
for any $i, i' \le k$,
we have
$$\begin{array}{r@{}c@{}l}
\min\Delta_w(P_n)
&{}={}& \gcd(\{w_ir_{i'} : 1 \le i < j \le i' \le k\} \cup \{w_ir_{i'} - w_{i'}r_i : j \le i < i' \le k\}) \\
&{}={}& \gcd(w_1, \ldots, w_{j-1}, \min \Delta_w(S))\gcd(r_1, \ldots, r_k),
\end{array}$$
so the first claim follows from Theorem~\ref{t:weighteddelta}(b).
Applying Theorem~\ref{t:weighteddelta}(c), we will show if two factorizations $z, z'\in \mathsf Z(m)$ satisfy $|z|_w - |z'|_w \ge 2d$, then $z$ and $z'$ must be in the same connected component of $\nabla_m$. Let $\ell = |z|_w - |z'|_w$. Just as in the proof of Theorem~\ref{t:mesalemma}, we know
$$m - |z|_wn = z_1r_1 + \cdots + z_kr_k \in S,$$
so since $n > w_1^2W\!R^2$, we have
$$(m - |z|_wn) + (\ell - d)n
= (m - |z'|_w n - \ell n) + (\ell - d)n
= m - (|z'|_w + d)n \in S.$$
Any factorization of the above element of $S$ corresponds to a factorization $z'' \in \mathsf Z(m)$ with $|z''|_w = |z'|_w + d$ that is connected to both $z$ and $z'$ in $\nabla_m$ by Theorem~\ref{t:mesalemma}.
\end{proof}
\section{Minimal presentations of parametrized semigroups
\label{sec:minpres
Let $\pi_n:\ZZ_{\ge 0}^k \to P_n$ denote the map
$$\pi_n(z) = \sum_{i = 1}^k z_i(w_in + r_i) = |z|_w n + \sum_{i = 1}^k z_ir_i,$$
called the \emph{factorization homomorphism} of $P_n$. The equivalence relation $\ker \pi_n$ on $\ZZ_{\ge 0}^k$, called the \emph{kernel congruence}, is given by $(z,z') \in \ker\pi_n$ whenever $\pi_n(z) = \pi_n(z')$, (that is, when $z$ and $z'$ are factorizations for the same element in $P_n$). Here, $\ker \pi_n$ is a congruence since it is closed under \emph{translation}, that is, $(z + u, z' + u) \in \ker \pi_n$ for every $(z,z') \in \ker\pi_n$ and $u \in \ZZ_{\ge 0}^k$.
A minimal presentation (Definition~\ref{d:minpres}) of a given semigroup $T$ encodes a particular choice of minimal relations (or \emph{trades}) between the generators of $T$.
They are one of the fundamental tools with which to study the factorization structure of numerical semigroups, and are closely connected to the defining toric ideal of $T$ (Remark~\ref{r:minprestoricideal}).
For a thorough introduction, we refer the reader to \cite[Chapter~9]{fingenmon} and \cite[Chapter~7]{numerical}.
The results in this section generalize those in \cite{shiftyminpres}, where a special (unweighted) case of the parametrization defining $P_n$ is considered.
At the heart of the main results in~\cite{shiftyminpres} is a map between kernel congruences, used to establish a correspondence between minimal presentations for large $n$ that restricts to a bijection on Betti elements. Our analogous map, $\Phi_n$, is defined in Proposition~\ref{p:mapwelldefined}, and its key properties (which closely mirror those in \cite{shiftyminpres}) are given in Proposition~\ref{p:mapproperties}. The main results are Theorem~\ref{t:minpresmap} and Corollary~\ref{c:bettiquasi}, which establish periodicity results for the minimal presentations and Betti elements of $P_n$, respectively, for large $n$. For our more general parametrization, the period turns out to be
$$p = w_1r_k - w_kr_1,$$
which specializes to a period of $r_k$ when $w_1 = 1$ and $r_1 = 0$ (as in \cite{shiftyminpres}).
In this section, we omit several proofs that are nearly identical to those in \cite{shiftyminpres}, including only those aspects that are different in our more general setting.
\begin{defn}\label{d:minpres}
Fix a numerical semigroup $T = \<t_1, \ldots, t_k\>$ and let $\pi:\ZZ_{\ge 0}^k \to T$ denote the factorization homomorphism of $T$. A \emph{presentation} for $T$ is a set of relations $\rho \subset \ker \pi$ such that $\ker \pi$ is the unique minimal (w.r.t.\ containment) congruence on $\ZZ_{\ge 0}^k$ containing $\rho$. Equivalently, between any two factorizations $(z, z') \in \ker \pi$, there exists a \emph{chain} $a_0, a_1, \ldots, a_r$ with $a_0 = z$, $a_r = z'$, and
$$(a_{i-1},a_i) = (b_i,b_i') + (u_i,u_i) \in \ker \pi$$
for some $(b_i, b_i') \in \rho$ and $u_i \in \ZZ_{\ge 0}^k$ for each $i \le r$. We say $\rho$ is \emph{minimal} if it is minimal with respect to containment among all presentations of $T$.
\end{defn}
\begin{remark}\label{r:minprestoricideal}
Returning to the commutative algebra viewpoint in Remark~\ref{r:bettinumbers}, minimal presentations encode minimal generating sets of toric ideals. Let $T = \<t_1, \ldots, t_k\>$, and write $I = \ker\varphi$ for the defining toric ideal of $T$, where $\varphi$ is the ring homomorphism
$$\begin{array}{r@{}c@{}l}
\varphi:\CC[x_1, \ldots, x_k] &{}\to{}& \CC[y] \\
x_i &{}\mapsto{}& y^{t_i}.
\end{array}$$
Each relation $(a, b) \in \ker\pi$ corresponds to a binomial
$$x_1^{a_1} \cdots x_k^{a_k} - x_1^{b_1} \cdots x_k^{b_k} \in I,$$
and each minimal presentation of $T$ corresponds to some minimal generating set of $I$. As an example, if $T = \<6, 9, 20\>$, then the minimal presentations of $T$ are
$$\begin{array}{ll}
\{((3,0,0), (0,2,0)), ((10,0,0), (0,0,3))\}, & \{((3,0,0), (0,2,0)), ((7,2,0), (0,0,3))\}, \\
\{((3,0,0), (0,2,0)), ((\phantom{0}4,4,0), (0,0,3))\}, & \{((3,0,0), (0,2,0)), ((1,6,0), (0,0,3))\},
\end{array}$$
each of which corresponds to one of the 4 minimal generating sets of the defining toric ideal $I \subset \CC[x,y,z]$ listed in Remark~\ref{r:bettinumbers}.
\end{remark}
\begin{example}\label{e:minpresmap}
Let $r = (1,2,4,6)$ and $w = (3,4,6,9)$, and consider the following minimal presentations for $P_n$ with $n$ identical modulo $p = 3 \cdot 6 - 9 \cdot 1 = 9$.
$$
\begin{array}{
r@{\,\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,\,\,}
}
P_{506}:
& (( & 0, & 0, & 3, & 0), & (0, & 0, & 0, & 2 & )),
& (( & 0, & 3, & 0, & 0), & (2, & 0, & 1, & 0 & )),
\\
& (( & 506, & 1, & 0, & 0), & (0, & 0, & 0, & 169 & )),
& (( & 508, & 0, & 0, & 0), & (0, & 2, & 2, & 167 & ))
\\[0.5em]
P_{515}:
& (( & 0, & 0, & 3, & 0), & (0, & 0, & 0, & 2 & )),
& (( & 0, & 3, & 0, & 0), & (2, & 0, & 1, & 0 & )),
\\
& (( & 515, & 1, & 0, & 0), & (0, & 0, & 0, & 172 & )),
& (( & 517, & 0, & 0, & 0), & (0, & 2, & 2, & 170 & ))
\\[0.5em]
P_{524}:
& (( & 0, & 0, & 3, & 0), & (0, & 0, & 0, & 2 & )),
& (( & 0, & 3, & 0, & 0), & (2, & 0, & 1, & 0 & )),
\\
& (( & 524, & 1, & 0, & 0), & (0, & 0, & 0, & 175 & )),
& (( & 526, & 0, & 0, & 0), & (0, & 2, & 2, & 173 & ))
\end{array}
$$
Each first-row relation $(z,z')$ satisfies $|z|_w = |z'|_w$, and each second-row relation $(z,z')$ satisfies $|z| = |z'| + 1$. In the latter case, each time $n$ is increased by $p = 9$, the value of $z_1$ increases by $w_4 = 9$ and $z_4'$ increases by $w_1 = 3$.
\end{example}
\begin{defn}\label{d:monotonechain}
A chain $a_0, a_1, \ldots, a_r$ of factorizations is \emph{$w$-monotone} if the sequence $|a_0|_w, |a_1|_w, \ldots, |a_r|_w$ is monotone.
\end{defn}
\begin{prop}\label{p:mapwelldefined}
The map $\Phi_n \colon \ker \pi_n \to \ker \pi_{n + p}$ given by
$$\Phi_n(z,z')
= \left\{\begin{array}{ll}
(z + \ell w_k e_1, z' + \ell w_1 e_k) & \text{if } |z|_w > |z'|_w \\
(z + \ell w_1 e_k, z' + \ell w_k e_1) & \text{if } |z|_w < |z'|_w \\
(z,z') & \text{if } |z|_w = |z'|_w
\end{array}\right.
$$
for $(z,z') \in \ker \pi_n$ and $\ell = \big| |z|_w - |z'|_w \big|$ is well defined.
\end{prop}
\begin{proof}
Fix $(z,z') \in \ker \pi_n$ with $z = (z_1, \dots, z_k)$ and $z' = (z_1', \dots, z_k')$. By symmetry, we can assume that $\ell = |z|_w - |z'|_w \ge 0$. Now, we simply use $\pi_n(z) = \pi_n(z')$ to verify
$$\begin{array}{r@{}c@{}l}
\displaystyle \pi_{n + p}(z + \ell w_ke_1)
&{}={}& \displaystyle
\ell w_k r_1 + \sum_{i = 1}^k z_i(w_i(n + p) + r_i)
= \pi_n(z) + |z|_wp + \ell w_k r_1
\\ &{}={}& \displaystyle
\pi_n(z') + |z'|_wp + \ell(p + w_k r_1)
= \pi_n(z') + |z'|_wp + \ell w_1 r_k
\\ &{}={}& \displaystyle
\ell w_1 r_k + \sum_{i = 1}^k z_i'(w_i(n + p) + r_i)
= \pi_{n + p}(z' + \ell w_1e_k),
\end{array}$$
as desired.
\end{proof}
\begin{prop}\label{p:mapproperties}
Fix $n \in \ZZ_{\ge 1}$, $\rho \subset \ker \pi_n$, and $(z, z') \in \rho$, and let $(y, y') = \Phi_n(z, z')$.
\begin{enumerate}[(a)]
\item
\label{p:mapproperties_injective}
The map $\Phi_n$ is injective.
\item
\label{p:mapproperties_lendiffs}
The map $\Phi_n$ preserves weighted length differences: $|z|_w - |z'|_w = |y|_w - |y'|_w$.
\item
\label{p:mapproperties_closures}
The map $\Phi_n$ preserves the reflexive, symmetric, and translation closure operations: if $\rho$ is reflexive, symmetric, or closed under translation, then so is $\Phi_n(\rho)$.
\item
\label{p:mapproperties_monotonechains}
The map $\Phi_n$ preserves $w$-monotone chain connectivity: if $\rho$ is translation-closed and there exists a $w$-monotone $\rho$-chain from $z$ to $z'$, then there exists a $w$-monotone $\Phi_n(\rho)$-chain from $y$ to $y'$.
\end{enumerate}
\end{prop}
\begin{proof}
The proof is nearly identical to that of \cite[Proposition~4.4]{shiftyminpres}.
\end{proof}
Just as in \cite{shiftyminpres}, the main obstruction to Theorem~\ref{t:minpresmap} for arbitrary $n$ is that $\Phi_n$ need only preserve connectivity by $w$-monotone chains. By Proposition~\ref{p:monotonechain}, any two factorizations $(z,z') \in \ker \pi_n$ are guaranteed to be connected by a $w$-monotone chain if $n$ is large enough.
\begin{prop}\label{p:monotonechain}
Fix $n > w_1^2W\!R^2$ and a minimal presentation $\rho \subseteq \ker \pi_n$. There exists a $w$-monotone $\rho$-chain between any $(z,z') \in \ker \pi_n$.
\end{prop}
\begin{proof}
Without loss of generality, assume $\gcd(z,z') = 0$. Since $\Cong(\rho) = \ker \pi_n$, there exists a chain $z = a_0, a_1, \ldots, a_r = z'$ of factorizations such that for each $i < r$, we have
$$(a_i,a_{i+1}) = (b_i,b_i') + (u_i,u_i), \qquad (b_i, b_i') \in \rho, u_i \in \ZZ_{\ge 0}^k,$$
where $b_i$ and $b_i'$ occur in distinct connected components of the graph $\nabla\!_\beta$ of $\beta = \pi_n(b_i)$. By Corollary~\ref{c:bettidelta}, $\Delta_w(P_n) = \{d\}$, so $|a_i|_w - |a_{i-1}|_w \in \{-d,0,d\}$ for each $i \le r$.
By induction on the chain length $r$, it suffices to assume $|a_1|_w = \cdots = |a_{r-1}|_w$ and $|z|_w = |a_0|_w = |a_r|_w = |z'|_w$, and prove there exists a weighted length preserving chain from $z$ to $z'$. Indeed, any non-monotone chain must contain such a subchain, which could then be ``flattened'' to a subchain with all equal weighted lengths.
First, suppose $|a_1|_w = |a_0|_w + d$. Applying Theorem~\ref{t:mesalemma} to $(b_0,b_0')$ and $(b_{r-1},b_{r-1}')$, we see that $z$ and $z'$ share support with some factorizations $y$ and $y'$, respectively, with positive last coordinates and weighted lengths equal to $|z|_w = |z'|_w$. By induction on the semigroup element $\pi_n(z)$, there exist weighted length preserving chains connecting $z$ to $y$, $y$ to $y'$, and $y'$ to $z'$. The case $|a_1|_w = |a_0|_w - d$ follows similarly.
\end{proof}
\begin{thm}\label{t:minpresmap}
For any $n > w_1^2W\!R^2$, the image of any minimal presentation $\rho$ of $P_n$ under the map $\Phi_n:\ker \pi_n \to \ker \pi_{n + p}$ is a minimal presentation of $P_{n + p}$.
\end{thm}
\begin{proof}
We must show that any minimal presentation $\rho \subset \ker \pi_n$ of $P_n$ satisfies
$$\Cong(\Phi_n(\rho)) = \ker \pi_{n + p},$$
that is, the image of $\rho$ under $\Phi_n$ is a presentation for $P_{n + p}$. Fix $(y,y') \in \ker \pi_{n + p}$, and let $m = \pi_{n + p}(y)$. By Proposition~\ref{p:monotonechain}, there exists a $w$-monotone $\rho$-chain from $y$ to $y'$, which we can assume is $w$-monotone decreasing by Proposition~\ref{p:mapproperties}\eqref{p:mapproperties_closures}. We can also assume each step in this chain has the form $(b,b') + (u,u)$ for some $u \in \ZZ_{\ge 0}^k$ and $b, b' \in \mathsf Z(\beta)$ lying in different connected components of $\nabla\!_\beta$. By Proposition~\ref{p:mapproperties}\eqref{p:mapproperties_closures}, it suffices to prove each $(b,b')$ lies in $\Cong(\Phi_n(\rho))$, so we can assume $y$ and $y'$ lie in different connected components of $\nabla\!_m$.
First, if $|y|_w = |y'|_w$, then $\Phi_n(y,y') = (y,y')$ by Proposition~\ref{p:mapwelldefined}, so applying $\Phi_n$ to any $w$-monotone (in this case, length preserving) $\rho$-chain from $y$ to $y'$ yields a $\Phi_n(\rho)$-chain from $y$ to $y'$ by Proposition~\ref{p:mapproperties}\eqref{p:mapproperties_monotonechains}. In particular, $\Phi_n(\rho)$ connects any two factorizations of equal weighted length. On the other hand, if $|y|_w > |y'|_w$, then Corollary~\ref{c:bettidelta} implies $|y|_w = |y'|_w + d$, where $\Delta_w(P_n) = \{d\}$. By Theorems~\ref{t:mesalemma}(b) and~(c), some factorizations $x$ and $x'$ in the same connected components as $y$ and $y'$, respectively, satisfy $x_1 \ge d$ and $x_k' \ge d$. Since
$$\Phi_n(x - dw_ke_1, x' - dw_1e_k) = (x,x'),$$
$\Phi_n(\rho)$ connects $x$ and $x'$ in $\nabla_m$. As Theorem~\ref{t:mesalemma}(a) implies there are length-preserving chains from $y$ to $x$ and from $x'$ to $y'$, Proposition~\ref{p:mapproperties}\eqref{p:mapproperties_monotonechains} completes the proof.
\end{proof}
We are now ready to prove that the Betti elements of $P_n$ are eventually periodic.
\begin{cor}\label{c:bettiquasi}
For $n > w_1^2W\!R^2$, the map $\varphi_n: \Betti(P_n) \to \Betti(P_{n+p})$ given by
$$\beta \mapsto \left\{\begin{array}{@{\,}ll}
\beta + \lambda p & \text{if } \mathsf L_{P_n,w}(\beta) = \{\lambda\} \\
\beta + \lambda p + dw_1(w_kn + r_k) & \text{if } \mathsf L_{P_n,w}(\beta) = \{\lambda, \lambda + d\}
\end{array}\right.$$
is a bijection, where $\Delta_w(P_n) = \{d\}$.
\end{cor}
\begin{proof}
Fix $z, z' \in \mathsf Z_{P_n}(\beta)$ with $|z|_w = \max\mathsf L_{P_n,w}(\beta)$ and $|z'|_w = \min\mathsf L_{P_n,w}(\beta)$. In the first case, $\lambda = |z|_w = |z'|_w$, so $\Phi_n(z,z') = (z,z')$, so Theorem~\ref{t:minpresmap} implies
$$\varphi_n(\beta) = \beta + \lambda p = \sum_{i = 1}^k z_i(w_in + r_i) + \sum_{i = 1}^k w_iz_i p = \sum_{i = 1}^k z_i(w_i(n + p) + r_i) \in \Betti(P_{n+p}).$$
In the second case, $\lambda = |z|_w - d = |z'|_w$ by Corollary~\ref{c:bettidelta}, so $\Phi_n(z,z') = (z + de_1, z' + de_k)$, and thus Theorem~\ref{t:minpresmap} implies
$$\varphi_n(\beta) = \beta + \lambda p + dw_1(w_kn + r_k) = dw_1(w_kn + r_k) + \sum_{i = 1}^k z_i'(w_i(n + p) + r_i) \in \Betti(P_{n+p}).$$
As such, $\varphi_n$ is a bijection by Theorem~\ref{t:minpresmap}. This completes the proof.
\end{proof}
\begin{remark}\label{r:bettidichotomy}
Just as in \cite{shiftyminpres}, Corollary~\ref{c:bettiquasi} implies the elements of $\Betti(M_n)$ fall into two distinct categories:\ those with minimal relations of equal length (which increase linearly in $n$ upon successive applications of $\Phi_n$), and those with minimal relations of different length (which increase quadratically in $n$ upon successive applications of $\Phi_n$).
As an additional consequence of Corollary~\ref{c:bettiquasi}, we see the function $n \mapsto |\!\Betti(P_n)|$ is $p$-periodic for $n > w_1^2W\!R^2$, including if the elements of $\Betti(P_n)$ are counted with multiplicity (that is,~if each element $\beta \in \Betti(P_n)$ appears once for each relation between factorizations of $\beta$ occuring in a minimal presentation for $P_n$). In the commutative algebra language of Remarks~\ref{r:bettinumbers} and~\ref{r:minprestoricideal}, this says the number of minimal generators of the defining toric ideal of $P_n$ is $p$-periodic in $n$.
\end{remark}
\section{Apery sets and the Frobenius number
\label{sec:aperysets
In the final section of this paper, we examine the Frobenius number (Corollary~\ref{c:frobquasi}), genus (Corollary~\ref{c:genusquasi}), and type (Corollary~\ref{c:pseudofrobshifted}) of $P_n$ for large $n$. Our results utilize the Ap\'ery set (Definition~\ref{d:apery}) of $P_n$, from which each of these quantities can be quickly obtained (indeed, in numerical semigroup computations, one often computes the Ap\'ery set first since doing so has roughly the same computational complexity). Most of the results in this section generalize those in \cite{shiftedaperysets}.
Throughout this section, we add the following assumptions on the parametrization of $P_n$; the difficulties in the general case are discussed in Remark~\ref{r:aperysetrestriction}.
\begin{notation}\label{n:parametrizedapery}
Throughout this section, we restrict to the case $w_1 = 1$ (and consequently $r_1 = 0$), so that
$$P_n = \<n, w_2n + r_2, \ldots, w_kn + r_k\>,$$
in addition to all existing assumptions from Notation~\ref{n:parametrized}. Moreover, let $S = \<r_2, \ldots, r_k\>$ and $d = \gcd(S)$.
\end{notation}
\begin{defn}\label{d:apery}
Fix an additive subsemigroup $T \subset (\ZZ_{\ge 0}, +)$, and let $d = \gcd(T)$. The~\emph{Ap\'ery set} of $m \in T$ is
$$\Ap(T; m) = \{t \in T : t - m \in \ZZ \setminus T\}.$$
The \emph{genus} of $T$ is the number $\mathsf g(T) = |d\ZZ_{\ge 0} \setminus T|$ of positive integer multiples of $d$ lying outside of $T$, and the \emph{Frobenius number} of $T$ is the largest integer multiple of $d$ outside of $T$, that is, $\mathsf F(T) = \max(d\ZZ_{\ge 0} \setminus T)$.
\end{defn}
\begin{remark}\label{r:aperyfacts}
The quantities in Definition~\ref{d:apery} are usually only defined for numerical semigroups (that is, in the case when $d = 1$). We will make use here of the following properties of the Ap\'ery set, each of which follows immediately from a known result in the usual setting~\cite{numerical}.
\begin{enumerate}[(a)]
\item
Each element of $\Ap(T; m)$ is distinct modulo $m$. In particular, $\left|\Ap(T; m)\right| = m/d$.
\item
We have
$$\mathsf F(S) = \max(\Ap(T; m)) - m
\qquad \text{and} \qquad
\mathsf g(S) = \sum_{t \in \Ap(T; m)} \bigg\lfloor \frac{t}{m} \bigg\rfloor,
$$
known in the literature as Selmer's formulas \cite{selmersformula}.
\end{enumerate}
\end{remark}
\begin{example} \label{e:apery}
If $T = \<6, 9, 20\>$, then $\Ap(T; 6) = \{0, 49, 20, 9, 40, 29\}$, where the elements are listed based on their equivalence class modulo $6$. From Selmer's formulas in Remark~\ref{r:aperyfacts}, we conclude $\mathsf F(S) = 43$ and $\mathsf g(S) = \tfrac{147}{6} - \tfrac{5}{2} = 22$.
\end{example}
\begin{thm}\label{t:parametrizedapery}
If $n > W\!R^2$, then
$$\Ap(P_n;n) = \{i + \mathsf m_{S,w}(i)n \mid i \in \Ap(S;dn)\}.$$
Moreover, we have
$$\mathsf L_{P_n,w}(i + \mathsf m_{S,w}(i)n) = \{\mathsf m_{S,w}(i)\}$$
for each $i \in \Ap(S;dn)$.
\end{thm}
\begin{proof}
Fix $a \in \Ap(n)$. Since $a - n \notin P_n$, no factorization of $a$ has positive first coordinate, so Theorem~\ref{t:mesalemma} implies $\mathsf L_{P_n,w}(a) = \{\ell\}$ for some $\ell \in \ZZ_{\ge 0}$. Let $i = a - \ell n$.
The proof of Theorem~\ref{t:mesalemma} establishes a natural mapping
$$\begin{array}{rcl}
\{z\in \mathsf Z_{P_n}(a) : |z|_w = \ell\} & \rightarrow & \{s \in \mathsf Z_{S}(a - \ell n): |s|_w \le \ell\} \\
(z_0,z_1, \ldots, z_k) & \mapsto & (z_1, \ldots, z_k)
\end{array}$$
between factorizations of $a \in P_n$ and factorizations of $i = a - \ell n \in S$ of weighted length at most $\ell$. Moreover, in this setting, the above map is a bijection, since for any factorization $(z_2, \ldots, z_k) \in \mathsf Z_S(i)$, letting $z_1 = \ell - |z|_w$ yields a factorization $(z_1, \ldots, z_k) \in \mathsf Z_{P_n}(a)$. Since $a \in \Ap(P_n; n)$, no factorization of $a$ has positive first coordinate, so we must have
$$\ell = |z|_w = \mathsf m_{S,w}(a - \ell n) = \mathsf m_{S,w}(i).$$
Observing that $|\!\Ap(P_n; n)| = |\!\Ap(S; dn)| = n$ and that the elements of $\Ap(P_n; n)$ are all distinct modulo $n$ completes the proof.
\end{proof}
\begin{remark}\label{r:aperysetrestriction}
The primary difficulty in generalizing Theorem~\ref{t:parametrizedapery} to the general setting considered in Sections~\ref{sec:linearfamilies} and~\ref{sec:minpres} is the ``non-surjectivity'' demonstrated in Example~\ref{e:uglymapping}. The mapping utilized in the proof of Theorem~\ref{t:parametrizedapery} is indeed a specialization of the one established in the proof of Theorem~\ref{t:mesalemma}, but it specializes to a bijection in this case (i.e., when $w_1 = 1$).
\end{remark}
Generalizations of \cite[Corollaries~4.2 and~4.3]{shiftedaperysets} follow immediately from Theorem~\ref{t:parametrizedapery}, and make use of the following observation from~\cite{shiftedaperysets}.
\begin{prop}[{\cite[Proposition~3.4]{shiftedaperysets}}]\label{p:largeapery}
If $dn > F(S)$, then $\Ap(S;dn) = \{a_0, \ldots, a_{n-1}\}$, where
$$a_i = \left\{\begin{array}{ll}
di & \text{ if } di \in S; \\
di + dn & \text{ if } di \notin S.
\end{array}\right.$$
In particular, this holds whenever $n > W\!R^2$ as in Theorem~\ref{t:parametrizedapery}. \end{prop}
\begin{cor}\label{c:frobquasi}
For $n > W\!R^2$, the function $n \to \mathsf F(P_n)$ has the form
$$\mathsf F(P_n) = \tfrac{w_k}{r_k}n^2 + a_1(n)n + a_0(n)$$
for some $r_k$-periodic functions $a_1(n)$ and $a_0(n)$.
\end{cor}
\begin{proof}
Let $a$ denote the element of $\Ap(S;dn)$ for which $\mathsf m_{S,w}(-)$ is maximal. Theorem~\ref{t:parametrizedapery} and Proposition~\ref{p:largeapery} imply
\begin{center}
$\begin{array}{r@{}c@{}l}
\mathsf F(P_n) &{}={}& \max(\Ap(P_n)) - n = a - n + \mathsf m_{S,w}(a) \cdot n,
\end{array}$
\end{center}
and Theorem~\ref{t:maxminquasi}\eqref{t:maxminquasi:minlen} implies $a + r_k$ is the element of $\Ap(S;dn + r_k)$ for which $\mathsf m_{S,w}(-)$ is maximal. The quasilinearity of $\mathsf m_{S,w}(-)$ proves $n \mapsto \mathsf F(P_n)$ is quasiquadratic in $n$ with period $r_k$, and since the only degree-2 term in the above expression is $\mathsf m_{S,w}(a) \cdot n$, we obtain a leading coefficient identical to that of $\mathsf m_{S,w}(n)$, namely $w_k/r_k$.
\end{proof}
\begin{cor}\label{c:genusquasi}
For $n > W\!R^2$, the function $n \mapsto \mathsf g(P_n)$ has the form
$$\mathsf g(P_n) = \tfrac{w_k}{2r_k}n^2 + b_1(n)n + b_0(n)$$
for some $r_k$-periodic functions $b_1(n)$ and $b_0(n)$.
\end{cor}
\begin{proof}
By Remark~\ref{r:aperyfacts}, we can write
$$\mathsf g(P_n) = \sum_{a \in \Ap(P_n)} \left\lfloor\frac{a}{n}\right\rfloor.$$
Theorem~\ref{t:parametrizedapery} and Proposition~\ref{p:largeapery} then yield
$$\begin{array}{r@{}c@{}l}
\mathsf g(P_n)
&{}={}& \displaystyle \sum_{i \in \Ap(S;dn)} \left\lfloor\frac{i+ \mathsf m_{S,w}(i)n}{n}\right\rfloor
= \sum_{i \in \Ap(S;dn)} \left\lfloor\frac{i}{n}\right\rfloor
+ \sum_{i \in \Ap(S;dn)} \mathsf m_{S,w}(i)
\\[0.1em]
&{}={}& \displaystyle \sum_{t=1}^{n-1} \left\lfloor \frac{dt}{n} \right\rfloor
+ d \cdot \mathsf g(S)
+ \sum_{\substack{i < n \\ di \in S}} \mathsf m_{S,w}(di)
+ \sum_{\substack{i \ge 0 \\ di \notin S}} \mathsf m_{S,w}(di+dn).
\end{array}$$
Each of the terms is eventually quasipolynomial in $n$. The first term is $d$-quasilinear in $n$, the second term is independent of $n$, and Theorem~\ref{t:parametrizedapery} guarantees that the last two terms are eventually $r_k$-quasiquadratic and $r_k$-quasilinear in $n$, respectively. Since $d \mid r_k$, we conclude $n \mapsto \mathsf g(P_n)$ is quasiquadratic in $n$ with period $r_k$. As for the leading term, the only degree-2 term in the above expression has successive $r_k$-differences
$$\sum_{\substack{i < n + r_k \\ di \in S}} \mathsf m_{S,w}(di) - \sum_{\substack{i < n \\ di \in S}} \mathsf m_{S,w}(di) = \sum_{j = 0}^{r_k-1} \mathsf m_{S,w}(dn + dj)$$
which are linear with leading coefficient $r_k(w_k/r_k) = w_k$. This yields a leading coefficient of $w_k/2r_k$ for $n \mapsto \mathsf g(P_n)$, as claimed.
\end{proof}
\begin{remark}\label{r:irreducibleasymptotic}
A numerical semigroup $S$ is called \emph{irreducible} if it is maximal with respect to containment among all numerical semigroups with Frobenius number $\mathsf F(S)$. If $\mathsf F(S)$ is odd, this happens precisely when $\mathsf g(S) = (\mathsf F(S) + 1)/2$, and if $\mathsf F(S)$ is even, this happens precisely when $\mathsf g(S) = (\mathsf F(S) + 2)/2$. Irreducible numerical semigroups have the smallest possible genus for their respective Frobenius number \cite[Chapter~3]{numerical}.
As a consequence of the leading coefficients in Corollaries~\ref{c:frobquasi} and~\ref{c:genusquasi}, we obtain
$$\lim_{n \to \infty} \frac{\mathsf g(P_n)}{\mathsf F(P_n)} = \frac{1}{2},$$
which can be interpreted as saying $P_n$ is ``nearly'' irreducible for large $n$.
\end{remark}
As a consequence, we obtain that for sufficiently large $n$, the numerical semigroup~$P_n$ satisfies Wilf's conjecture~\cite{wilfconjecture}, which is a longstanding open problem for numerical semigroups; see~\cite{wilfsurvey} for a survey of recent progress.
\begin{cor}\label{c:wilfshifted}
For $n > W\!R^2$, the Wilf number of $P_n$, defined in \cite{delgadoconj} as
$$\mathsf W(P_n) = k(F(P_n) - g(P_n)) - (F(P_n) + 1),$$
is $r_k$-quasiquadratic in $n$. In~particular, $\mathsf W(P_n)$ is positive for all suffiently large $n$, and thus $P_n$ satisfies Wilf's conjecture for each such $n$.
\end{cor}
\begin{proof}
Apply Corollaries~\ref{c:frobquasi} and~\ref{c:genusquasi}.
\end{proof}
Our final result concerns the (Cohen-Macaulay) type of $P_n$ for large $n$, which, just as in~\cite{shiftedaperysets}, we obtain from the pseudo-Frobenius numbers of $P_n$.
\begin{defn}\label{d:pseudofrob}
An integer $m \ge 0$ is a \emph{pseudo-Frobenius number} of a numerical semigroup $T$ if $m \notin T$ but $m + n \in T$ for all positive $n \in T$. Denote the set of pseudo-Frobenius numbers of $T$ by $\mathsf{PF}(T)$, and the \emph{type} of $T$ by $\mathsf t(T) = |\mathsf{PF}(T)|$.
\end{defn}
\begin{cor}\label{c:pseudofrobshifted}
Given $n \in \ZZ_{\ge 0}$, let $F_n$ denote the set
$$F_n = \{i \in \Ap(S;dn) : a \equiv i \bmod n \text{ for some } a \in \mathsf{PF}(P_n)\}.$$
For $n > W\!R^2$, the map $F_n \to F_{n + r_k}$ given by
$$\begin{array}{rcl}
i
&\mapsto&
\left\{\begin{array}{ll}
i & \text{if } i \le dn \\
i + r_k & \text{if } i > dn
\end{array}\right.
\end{array}
$$
is a bijection. In particular, there is a bijection $\mathsf{PF}(P_n) \to \mathsf{PF}(P_{n + r_k})$, meaning the function $n \mapsto t(P_n)$ is $r_k$-periodic for $n > W\!R^2$.
\end{cor}
\begin{proof}
The proof is identical to that of \cite[Theorem~4.8]{shiftedaperysets}.
\end{proof}
\section{Evidence of Conjecture~\ref{conj:main}
\label{sec:evidence
Now that we have seen the formal definition of a minimal presentation, we are ready to see an example of Conjecture~\ref{conj:main} for a more general (i.e., nonlinear) parametrized semigroup family. Note that computational evidence for nonlinear families is harder to obtain since the substantially larger generators result in computations taking much longer to complete.
\begin{example}\label{e:nonlinear}
Consider the parametrized family of semigroups
$$P_n = \<m^2, m^2 + m + 1, m^2 + 2m + 1, m^2 + 2m + 3\>$$
and the following minimal presentations.
\smaller
$$
\begin{array}{
@{}r@{\,\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,}
}
P_{52} :
& (( & 0, & 0, & 27, & 0), & (0, & 1, & 0, & 26 & )),
& (( & 0, & 3, & 26, & 0), & (2, & 0, & 0, & 27 & )),
& (( & 0, & 4, & 0, & 0), & (2, & 0, & 1, & 1 & )),
\\
& (( & 25, & 2, & 14, & 0), & (0, & 0, & 0, & 40 & )),
& (( & 25, & 3, & 0, & 0), & (0, & 0, & 13, & 14 & )),
& (( & 27, & 0, & 0, & 0), & (0, & 1, & 12, & 13 & ))
\\[0.5em]
P_{56} :
& (( & 0, & 0, & 29, & 0), & (0, & 1, & 0, & 28 & )),
& (( & 0, & 3, & 28, & 0), & (2, & 0, & 0, & 29 & )),
& (( & 0, & 4, & 0, & 0), & (2, & 0, & 1, & 1 & )),
\\
& (( & 27, & 2, & 15, & 0), & (0, & 0, & 0, & 43 & )),
& (( & 27, & 3, & 0, & 0), & (0, & 0, & 14, & 15 & )),
& (( & 29, & 0, & 0, & 0), & (0, & 1, & 13, & 14 & ))
\\[0.5em]
P_{60} :
& (( & 0, & 0, & 31, & 0), & (0, & 1, & 0, & 30 & )),
& (( & 0, & 3, & 30, & 0), & (2, & 0, & 0, & 31 & )),
& (( & 0, & 4, & 0, & 0), & (2, & 0, & 1, & 1 & )),
\\
& (( & 29, & 2, & 16, & 0), & (0, & 0, & 0, & 46 & )),
& (( & 29, & 3, & 0, & 0), & (0, & 0, & 15, & 16 & )),
& (( & 31, & 0, & 0, & 0), & (0, & 1, & 14, & 15 & ))
\\[0.5em]
P_{64} :
& (( & 0, & 0, & 33, & 0), & (0, & 1, & 0, & 32 & )),
& (( & 0, & 3, & 32, & 0), & (2, & 0, & 0, & 33 & )),
& (( & 0, & 4, & 0, & 0), & (2, & 0, & 1, & 1 & )),
\\
& (( & 31, & 2, & 17, & 0), & (0, & 0, & 0, & 49 & )),
& (( & 31, & 3, & 0, & 0), & (0, & 0, & 16, & 17 & )),
& (( & 33, & 0, & 0, & 0), & (0, & 1, & 15, & 16 & ))
\end{array}
$$
\normalsize
Unlike linear parametrized families, successive minimal presentations have more than just 2 coordinates consistently increasing, though the pattern in the relations is clear.
\end{example}
\section{Introduction
\label{sec:intro
A numerical semigroup $S$ is an additively closed subset of $\ZZ_{\ge 0}$, usually specified using a generating set $r_1, \ldots, r_k$, i.e.,
$$S = \<r_1, \ldots, r_k\> = \{z_1r_1 + z_2r_2 + \cdots + z_kr_k \mid z_1, \ldots, z_k \in \ZZ_{\ge 0}\}.$$
Many classical problems surrounding numerical semigroups involve arithmetic invariants, such as the Frobenius number $\mathsf F(S)$, genus $\mathsf g(S)$, type $\mathsf t(S)$, and delta set $\Delta(S)$, each of which is difficult to compute when the generators of $S$ are large.
For a thorough introduction to numerical semigroups, see~\cite{numerical}.
This paper considers parametrized families of numerical semigroups of the form
$$P_n = \<f_1(n), \ldots, f_k(n)\>$$
for some functions $f_1(n), \ldots, f_k(n)$. Such families have arisen in two main settings in the last decade. First is the \emph{parametric Frobenius problem}, which asks under what conditions the function $n \mapsto \mathsf F(P_n)$ coincides with a quasipolynomial (that is, a polynomial with periodic coefficients) for large $n$. It was conjectured in~\cite{rouneparametricfrob} that this holds whenever the functions $f_i$ are themselves polynomials, where this was proven in the case where $\deg f_i = 1$ for all $i$, as well as in the case $k = 3$. This appears to have been proven in general~\cite{shenparametricfrob}, though the results have yet to appear outside the \texttt{arXiv}, and the authors of this manuscript have been unable to contact the author.
Separately, \emph{shifted} numerical semigroups, which have a specialized parametrization
$$M_n = \<n, n + r_2, \ldots, n + r_k\>$$
for positive integers $r_2, \ldots, r_k$, have been examined in numerous recent papers. It~is known that the delta set of $M_n$ is eventually periodic~\cite{shiftydelta}, and that the Frobenius number, genus, and type of $M_n$ are each eventually quasipolynomial~\cite{shiftedaperysets}. Additionally, the minimal relations between the generators of $M_n$, usually studied in the form of minimal presentations \cite{numerical} or syzygies of the defining toric ideal \cite{cca}, are known to satisfy a certain periodicity originally conjectured by Herzog and Srinivasan and proven by Vu~\cite{vu14}. These results were later improved in~\cite{shiftyminpres}, wherein several consequences for other semigroup invariants were also derived, and further specialized in~\cite{shiftedtangentcone,shifted3gen}.
The results mentioned above provide ample evidence of a more general phenomenon, which we now conjecture formally.
\begin{conj}\label{conj:main}
If $f_1, \ldots, f_k:\ZZ \to \ZZ$ are eventually increasing polynomials and
$$P_n = \<f_1(n), \ldots, f_k(n)\>,$$
then $\Betti(P_n)$ is eventually quasipolynomial in $n$. As a consequence, the Frobenius number, genus, and type of $P_n$ are each eventually quasipolynomial in $n$.
\end{conj}
Note that the word ``consequence'' in Conjecture~\ref{conj:main} is intended as an informal~claim. In~particular, the main results of~\cite{shiftyminpres,shiftedaperysets} for shifted numerical semigroups (where the conjecture is already proven) stem from a single underlying result (\cite[Theorem~3.4]{shiftyminpres}) regarding the Betti elements of $P_n$ (that is, elements whose factorizations encode the minimal relations between the generators of $P_n$). Conjecture~\ref{conj:main} claims this core behavior occurs more generally, and that the remaining claims follow as consequences.
\begin{remark}\label{r:conjproof}
After posting this manuscript, a proof of the ``eventually quasipolynomial'' claims in Conjecture~\ref{conj:main} appeared elsewhere on the arXiv~\cite{parametricpresburgerbigthm}. The results therein are broad, with most claims extended to parametrized families of affine semigroups, but the proofs are nonconstructive, relying on formal logic and Presburger arithmetic. As~such, the informal ``consequence'' claim discussed above remains open.
\end{remark}
In this paper, we prove Conjecture~\ref{conj:main} in the case where the functions $f_1(n), \ldots, f_k(n)$ are linear. The main results are in Sections~\ref{sec:minpres} and~\ref{sec:aperysets}, which generalize results for shifted numerical semigroups that appeared in~\cite{shiftyminpres} and~\cite{shiftedaperysets}, respectively. The results in those sections follow from a central result about Betti elements for large $n$ (Theorem~\ref{t:mesalemma}), providing the ``consequently'' part of Conjecture~\ref{conj:main}. As a necessary step in stating our main results, we develop the notion of ``weighted factorization length'' in Section~\ref{sec:weightedlengths}, and generalize several known results involving standard factorization length. As evidence of the generality in Conjecture~\ref{conj:main}, we close this paper with Example~\ref{e:nonlinear}, a non-linear example where Conjecture~\ref{conj:main} appears to hold.
\subsection*{Acknowledgements}
The authors would like to thank Scott Chapman and Pedro Garc\'ia-S\'anchez for their helpful comments and suggestions.
\section{Numerical semigroups and factorization length
\label{sec:background
In this section, we state some background definitions for factorizations of numerical semigroup elements; the books~\cite{nonuniq} and~\cite{numerical} contain thorough introductions to nonunique factorization and numerical semigroups, respectively. Several of the quantities in Definition~\ref{d:numerical} involving (unweighted) factorization length have a weighted generalization introduced in subsequent sections of this paper.
\begin{defn}\label{d:numerical}
A \emph{numerical semigroup} $S$ is an additive subsemigroup of $\ZZ_{\ge 0}$ (note, we do \textbf{not} require $S$ to have finite complement). We write
$$S = \<r_1, \ldots, r_k\> = \{z_1r_1 + \cdots + z_kr_k : z_1, \ldots, z_k \in \ZZ_{\ge 0}\}$$
for the semigroup generated by $r_1, \ldots, r_k$.
A \emph{factorization} of $n \in S$ is an expression
$$n = z_1r_1 + \cdots + z_kr_k$$
of $n$ as a sum of generators of $S$, and the \emph{length} of a factorization is the sum $z_1 + \cdots + z_k$. The \emph{set of factorizations} of $n$ is the set
$$\mathsf Z_S(n) = \{z \in \ZZ_{\ge 0}^k : n = z_1r_1 + \cdots + z_kr_k\}$$
viewed as a subset of $\ZZ_{\ge 0}^k$, and the \emph{length set} of $n$ is the set
$$\mathsf L_S(n) = \{z_1 + \cdots + z_k : z \in \mathsf Z_S(n)\},$$
of all possible factorization lengths of $n$. Writing $\mathsf L_S(n) = \{\ell_1 < \cdots < \ell_m\}$, define
$$\Delta_S(n) = \{\ell_i - \ell_{i-1} : 2 \le i \le m\} \qquad \text{and} \qquad \Delta(S) = \bigcup_{n \in S} \Delta_S(n)$$
as the \emph{delta sets} of $n$ and $S$, respectively.
The \emph{maximum} and \emph{minimum} factorization length functions are defined as
$$\mathsf M_S(n) = \max \mathsf L_S(n) \qquad \text{ and } \qquad \mathsf m_S(n) = \min \mathsf L_S(n),$$
respectively.
\end{defn}
We state two results from the literature (Theorems~\ref{t:deltaset} and~\ref{t:maxminorig}) that we will generalize in the next section. The first result depends on the following definition.
\begin{defn}\label{d:bettielement}
Given a numerical semigroup $S$ and an element $n \in S$, the \textit{factorization graph} of $n$, denoted $\nabla_n$, has vertex set $\mathsf Z(n)$, and two vertices $z,z' \in \mathsf Z(n)$ are connected by an edge whenever they have at least one generator in common. We say $n$ is a \emph{Betti element} of $S$ if $\nabla_n$ is disconnected. Define
$$\Betti(S) = \{n \in S \mid n \text{ is a Betti element of } S\}.$$
\end{defn}
\begin{example}\label{e:bettielements}
The Betti elements of $S = \<6, 9, 20\>$ are $\Betti(S) = \{18,60\}$, whose factorization graphs are depicted in Figure~\ref{f:bettielements}. As we will see in Section~\ref{sec:minpres}, these elements encode the minimal relations between the generators of $S$:\ $18$ is the smallest element that can be factored using $6$ and $9$, and $60$ is the smallest element that can be factored using $6$ and $9$ and separately using $20$.
\end{example}
\begin{figure}
\begin{center}
\includegraphics[height=1.2in]{betti-6-9-20--18.pdf}
\hspace{1.0in}
\includegraphics[height=1.2in]{betti-6-9-20--60.pdf}
\end{center}
\caption{The factorization graphs $\nabla_{18}$ (left) and $\nabla_{60}$ (right) in the numerical semigroup $S = \<6, 9, 20\>$ from Example~\ref{e:bettielements}.}
\label{f:bettielements}
\end{figure}
\begin{remark}\label{r:bettinumbers}
From a commutative algebra viewpoint, Betti elements coincide with graded degrees of the minimal generators of toric ideals. Given a numerical semigroup $S = \<r_1, \ldots, r_k\>$, the kernel $I = \ker\varphi$ of the ring homomorphism determined by
$$\begin{array}{r@{}c@{}l}
\varphi:\CC[x_1, \ldots, x_k] &{}\to{}& \CC[y] \\
x_i &{}\mapsto{}& y^{r_i}
\end{array}$$
is the defining toric ideal of $S$. As an example, if $S = \<6, 9, 20\>$, then the defining toric ideal $I \subset \CC[x,y,z]$ has precisely 4 minimal generating sets, namely
$$\{x^3 - y^2, x^{10} - z^3\}, \, \{x^3 - y^2, x^7y^2 - z^3\}, \,
\{x^3 - y^2, x^4y^4 - z^3\}, \, \text{and} \, \{x^3 - y^2, xy^6 - z^3\},$$
each of which has one homogeneous element of degree $18$ and one of degree $60$ (here, the graded degree of each monomial is determined by its image under $\varphi$). This matches the Betti elements $\Betti(S) = \{18, 60\}$ obtained in Example~\ref{e:bettielements}.
\end{remark}
\begin{thm}[{\cite{deltasets,bfdelta}}]\label{t:deltaset}
For any numerical semigroup $S = \<r_1, \ldots, r_k\>$, the set $\Delta(S)$ is~nonempty and finite, and $\gcd \Delta(S) = \min \Delta(S)$. Moreover,
$$\min \Delta(S) = \gcd\{r_i - r_{i-1} : 2 \le i \le k\}$$
and
$$\max \Delta(S) = \max_{n \in \Betti(S)} \max \Delta_S(n).$$
\end{thm}
\begin{thm}[{\cite{elastsets,shiftyminpres}}]\label{t:maxminorig}
For $n > r_k^2$ in a numerical semigroup $S = \<r_1, \ldots, r_k\>$, we have
$$\mathsf M(n + r_1) = \mathsf M(n) + 1 \qquad \text{and} \qquad \mathsf m(n + r_k) = \mathsf m(n) + 1.$$
\end{thm}
The functions in Theorem~\ref{t:maxminorig} are said to coincide for large $n$ with \emph{quasipolynomials}, that is, polynomial functions $\ZZ \to \RR$ with periodic coefficients. In particular,
$$\mathsf M(n) = \tfrac{1}{r_1}n + a(n) \qquad \text{and} \qquad \mathsf m(n) = \tfrac{1}{r_k}n + b(n)$$
for some periodic functions $a(n)$ and $b(n)$ with periods $r_1$ and $r_k$, respectively.
\section{Weighted factorization lengths
\label{sec:weightedlengths
Before examining parametrized families of numerical semigroups, we introduce a generalization of factorization length that independently weights each generator and plays a key role in the results of subsequent sections. We give two main results in this section, each of which generalizes existing results for the usual factorization length. The~first is Theorem~\ref{t:maxminquasi}, which generalizes \cite[Theorems~4.2 and~4.3]{elastsets} and joins a growing family of ``eventually quasipolynomial'' results concerning factorization length (see \cite{factorhilbert} and the references therein for an overview). The second is Theorem~\ref{t:weighteddelta}, which gives weighted versions of \cite[Lemma~3]{arithmeticintdom} and \cite[Theorem 2.5]{bfdelta}, both of which are central to the study of delta sets.
\begin{defn}\label{d:weightedlength}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$ and a rational vector $w = (w_1, \ldots, w_k) \in \QQ^k$ of \emph{weights}.
Given $n \in S$ and $z = (z_1, \ldots, z_k) \in \mathsf Z(n)$, the \textit{weighted length} of $z$ is
$$|z|_w = w \cdot z = w_1z_1 + \cdots + w_kz_k,$$
and the \emph{weighted length set} of $n$ is
$$\mathsf L_{S,w}(n) = \{|z|_w : z \in \mathsf Z(n)\}.$$
The maps $\mathsf M_w: S \mapsto \QQ$ and $\mathsf m_w: S \mapsto \QQ$ given by
$$\mathsf M_w(n) = \max \mathsf L_{S,w}(n) \qquad \text{and} \qquad \mathsf m_w(n) = \min \mathsf L_{S,w}(n)$$
are the \emph{maximum weighted length} and \emph{minimum weighted length} functions, respectively.
\end{defn}
\begin{defn}\label{d:worder}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$ and a weight vector $w =~(w_1, \ldots, w_k) \in \QQ^k$.
The \emph{$w$-ordering} $\le_w$ on $\{r_1, \ldots, r_k\}$ is defined so that
$$r_i \le_w r_j \qquad \text{whenever} \qquad w_i/r_i \ge w_j/r_j.$$
Note that the $w$-ordering is transitive, but need not be a total (or even partial) ordering, as $r_i =_w r_j$ is possible for $r_i \ne r_j$.
\end{defn}
\begin{remark}\label{r:standardlengthweighted}
The standard length $|\!\cdot\!|$ can be viewed as a special case of weighted length $|\cdot|_w$ with weight vector $w = (1,\ldots, 1)$. In this case, the $w$-ordering on $r_1, \ldots, r_k$ is the usual total ordering in $\ZZ$.
\end{remark}
\begin{example}\label{e:weightedlength}
Let $S = \<6, 9, 20\>$. For the weight vector $w = (3,1,4)$, the $w$-ordering on the generators of $S$ is $6 <_w 20 <_w 9$ since $\tfrac{3}{6} > \tfrac{4}{20} > \tfrac{1}{9}$. The same $w$-ordering is induced by $w = (3,-1,4)$, but some factorizations have negative weighted length, e.g.~$(2,12,1) \in \mathsf Z_S(140)$ has $|(2,12,1)|_w = -2$. Figure~\ref{f:weightedlength} depicts $\mathsf m_{S,w}(-)$ for both weight vectors; evident is the eventually quasilinear property implied by Theorem~\ref{t:maxminquasi}.
\end{example}
\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{6-9-20--3-1-4--minlen.pdf}
\hspace{0.2in}
\includegraphics[width=2.8in]{6-9-20--3-m1-4--minlen.pdf}
\end{center}
\caption{Plots depicting the minimum weighted factorization lengths of elements of $S = \<6, 9, 20\>$ for the weight vectors $w = (3,1,4)$ (left) and $w = (3,-1,4)$ (right) from Example~\ref{e:weightedlength}, created using \texttt{Sage} and the \texttt{GAP} package \texttt{numericalsgps}~\cite{numericalsgpsgap}.
}
\label{f:weightedlength}
\end{figure}
\begin{lemma}[{\cite[Lemma~4.1]{elastsets}}]\label{l:oldlemma41}
Suppose $q \ge 1$, and fix $c_1, \ldots, c_r \in \ZZ$ with $r \ge q$. There exists $T \subsetneq \{1, \ldots, r\}$ satisfying $\sum_{i \in T} c_i \equiv \sum_{i=1}^r c_i \bmod q$.
\end{lemma}
\begin{lemma}\label{l:maxminquasi}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$, a weight vector $w \in \QQ^k$, and suppose $r_1 \le_w r_2 \le_w \cdots \le_w r_k$.
\begin{enumerate}[(a)]
\item
If $a \in \mathsf Z_S(n)$ satisfies $a_1 + \cdots + a_k \ge r_1$, then there is some factorization $b \in \mathsf Z_S(n)$ with $|b|_w \ge |a|_w$ and $b_1 > 0$.
\item
If $a \in \mathsf Z_S(n)$ satisfies $a_1 + \cdots + a_k \ge r_k$, then there is some factorization $b \in \mathsf Z_S(n)$ with $|b|_w \le |a|_w$ and $b_k > 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
First, we claim if $a' = (0, a_2', \ldots, a_k'), b' = (b_1', 0, \ldots, 0) \in \mathsf Z(n)$, then $|b'|_w \ge |a'|_w$. Indeed, this follows from the fact that
$$|a'|_w
= \sum_{i=2}^k w_ia_i'
= \sum_{i=2}^k \frac{w_i}{r_i} r_ia_i'
\le \sum_{i=2}^k \frac{w_1}{r_1} r_ia_i'
= \frac{w_1}{r_1} r_1b_1'
= w_1b_1'
= |b'|_w.$$
Now, under the assumptions for part~(a), we see
$$a_1r_1 = n - a_2r_2 - \cdots - a_kr_k$$
implies $a_2r_2 + \ldots + a_kr_k \equiv n \bmod r_1$.
Lemma~\ref{l:oldlemma41} then guarantees the existence of integers $b_2, \ldots, b_k \ge 0$ such that (i)~$b_i \le a_i$ for each $i > 1$, (ii)~$\sum_{i =2}^ka_i >\sum_{i =2}^kb_i$, and (iii)~$b_2r_2 + \cdots + b_kr_k \equiv n \bmod r_1$.
This in particular means there exists $b_1 > 0$ so that $b = (b_1,\ldots, b_k) \in \mathsf Z(n)$. Rearranging the equation
$$n = a_1r_1 + \cdots + a_kr_k = b_1r_1 + \cdots + b_kr_k$$
yields
$$(b_1 - a_1)r_1 = (a_2 - b_2)r_2 + \cdots + (a_k - b_k)r_k$$
Applying the above claim to $(b_1-a_1, 0, \ldots, 0)$ and
$(0,a_2 - b_2, \ldots, a_k - b_k)$ implies
$$w_1(b_1 - a_1) \ge w_2(a_2 - b_2) + \cdots + w_k(a_k - b_k),$$
meaning
$$w_1b_1 + \cdots + w_kb_k \ge w_1a_1 + \cdots + w_ka_k,$$
so $|b|_w \ge |a|_w$. This proves part~(a).
The proof of part~(b) is analogous and thus omitted.
\end{proof}
\begin{thm}\label{t:maxminquasi}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$ and a weight vector $w \in \QQ^k$, and suppose $r_1 \le_w r_2 \le_w \cdots \le_w r_k$. Let $R = \max(r_1, \ldots, r_k)$.
\begin{enumerate}[(a)]
\item
\label{t:maxminquasi:maxlen}
For all $n > R^2$, the maximal weighted length function $\mathsf M_w: S \to \QQ$ satisfies
$$\mathsf M_w(n) = \mathsf M_w(n - r_1) + w_1.$$
\item
\label{t:maxminquasi:minlen}
For all $n > R^2$, the minimal weighted factorization length $\mathsf m_w: S \to \QQ$ satisfies
$$\mathsf m_w(n) = \mathsf m_w(n - r_k) + w_k.$$
\end{enumerate}
\end{thm}
\begin{proof}
Suppose $n > R^2$. First, we claim there is a factorization of $n$ with maximum weighted length with positive first coordinate. Indeed, fix any factorization $a \in \mathsf Z(n)$. If $a_2 + \cdots + a_k < r_1$, then $a_1 > 0$ by the assumption on $n$. On the other hand, if $a_2 + \cdots + a_k \ge r_1$ and $a_1 = 0$, then the claim follows from Lemma~\ref{l:maxminquasi}(a).
Now, by the above claim, let $a \in \mathsf Z(n)$ denote a maximum weighted length factorization with $a_1 > 0$. This means $a' = (a_1 - 1, a_2, \ldots, a_k) \in \mathsf Z(n - r_1)$ also has maximum weighted factorization length, so
$$\mathsf M_w(n - r_1)
= |a '|_w
= w_1(a_1 - 1) + w_2a_2 + \ldots w_ka_k
= |a|_w - w_1
= \mathsf M_w(n) - w_1,$$
thereby proving part~(a).
By a similar argument, some minimal weighted length factorization $a \in \mathsf Z(n)$ has $a_k > 0$. The proof of part~(b) then follows analogously.
\end{proof}
\begin{remark}\label{r:wtie}
For a given numerical semigroup $S = \<r_1, \ldots, r_k\>$, if a particular weight vector $w$ induces a ``tie'' $r_1 =_w \cdots =_w r_j$ in the $w$-ordering, then Theorem~\ref{t:maxminquasi} obtains an improved period $\gcd(r_1, \ldots, r_j)$ for the quasilinear function $\mathsf M_{S,w}$. For example, if~$S = \<6,9,10,14\>$ and $w = (2, 3, 5, 7)$, then $r_1 =_w r_2 >_w r_3 =_w r_4$, and for large $n$, $\mathsf M_{S,w}(n)$ and $\mathsf m_{S,w}(n)$ are each quasilinear with minimal periods $2$ and $3$, respectively. See Figure~\ref{f:wtie} for a depiction.
\end{remark}
\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{6-9-10-14--maxlen.pdf}
\hspace{0.2in}
\includegraphics[width=2.8in]{6-9-10-14--minlen.pdf}
\end{center}
\caption{Maximum (left) and minimum (right) weighted factorization lengths for $S = \<6, 9, 10, 14\>$ and $w = (2,3,5,7)$ from Remark~\ref{r:wtie}, created using \texttt{Sage} and the \texttt{GAP} package \texttt{numericalsgps}~\cite{numericalsgpsgap}.
}
\label{f:wtie}
\end{figure}
\begin{remark}\label{r:maxminwbound}
Much to our surprise, the bounds in Theorem~\ref{t:maxminquasi} do not depend on $w$, although it is worth noting that an optimal bound necessarily depends on $w$. Indeed, suppose $S = \<9,10,23\>$. If $w = (1,3,5)$, then $n = 64$ is the largest $n$ for which the first equality in Theorem~\ref{t:maxminquasi} fails to hold, and if $w = (6,9,5)$, then $n = 81$ is the largest such $n$. For both weight vectors, the generator $10$ is minimal under the $w$-ordering.
\end{remark}
The following corollary of Lemma~\ref{l:maxminquasi} and Theorem~\ref{t:maxminquasi} will be used in Section~\ref{sec:linearfamilies}. Note the additional assumption that $w$ has positive integer entries.
\begin{cor}\label{c:maxminquasi}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$, a weight vector $w \in \ZZ_{\ge 1}^k$, and suppose $r_1 \le_w r_2 \le_w \cdots \le_w r_k$. Fix $w_0 \in \ZZ_{\ge 1}$.
\begin{enumerate}[(a)]
\item
If $a \in \mathsf Z_S(n)$ satisfies $a_1 + \cdots + a_k \ge w_0r_1$, then there is some factorization $b \in \mathsf Z_S(n)$ with $|b|_w - |a|_w \in w_0\ZZ_{\ge 0}$ and $b_1 > 0$.
\item
If $a \in \mathsf Z_S(n)$ satisfies $a_1 + \cdots + a_k \ge w_0r_k$, then there is some factorization $b \in \mathsf Z_S(n)$ with $|a|_w - |b|_w \in w_0\ZZ_{\ge 0}$ and $b_k > 0$.
\end{enumerate}
\end{cor}
\begin{proof}
If $a_1 > 0$, then choosing $b = a$ proves part~(a), so suppose $a_1 = 0$. Fix $a' \in \ZZ_{\ge 0}^k$ such that $a_i' \le a_i$ for each $i$ and $a_1' + \cdots + a_k' \ge r_1$, and write $n' \in S$ so that $a' \in \mathsf Z(n')$. By Lemma~\ref{l:maxminquasi}(a), there exists $b' \in \mathsf Z(n')$ with $|b'|_w \ge |a'|_w$ and $b_1' > 0$. If $|b'|_w = |a'|_w$, then choosing $b = b' + (a - a')$ proves part~(a), so suppose $|b'|_w > |a'|_w$.
Now, fix a collection $c_1, \ldots, c_{w_0} \in \ZZ_{\ge 0}^k$ of vectors that sum to $a$.
Apply the above argument to each $c_i$ (in the role of $a'$) to obtain vectors $d_1, \ldots, d_{w_0} \in \ZZ_{\ge 0}^k$ (i.e., each corresponding vector $b'$ above), and let $\ell_i = |d_i|_w - |c_i|_w$. By Lemma~\ref{l:oldlemma41}, there exists a subset $T \subset \{1, \ldots, w_0\}$ so that $\sum_{i \in T} \ell_i \equiv 0 \bmod w_0$. Letting
$$b = \sum_{i \in T} d_i + a - \sum_{i \in T} c_i$$
we obtain
$$|b|_w - |a|_w = \biggl|\sum_{i \in T} d_i + a - \sum_{i \in T} c_i \biggr|_w - |a|_w = \sum_{i \in T} (|d_i|_w - |c_i|_w) \in w_0\ZZ_{\ge 0}$$
which completes the proof of part~(a).
As in the proof of Lemma~\ref{l:maxminquasi}, the proof of part~(b) is analogous.
\end{proof}
For the remainder of this section, we turn our attention to the weighted delta set. As~with weighted length sets, choosing the weight vector $w = (1, \ldots, 1)$ in the following definition recovers the usual delta set.
\begin{defn}\label{d:deltasets}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$, a weight vector $w \in \QQ^k$, and an element $n \in S$, and write
$$\mathsf L_{S,w}(n) = \{\ell_1 < \ell_2 < \cdots < \ell_r\}.$$
The \emph{weighted delta set} of $n$ is given by
$$\Delta_{S,w}(n) = \{\ell_i - \ell_{i-1}: i = 2, \ldots, r\},$$
and the \emph{weighted delta set} of $S$ is given by
$$\Delta_w(S) = \bigcup_{n \in S} \Delta_{S,w}(n).$$
Note that, unlike the usual delta set, it is possible to have $\Delta_w(S) = \emptyset$. Indeed, this happens when $w_i = r_i$ for every $i$, as $\mathsf L_w(n) = \{n\}$ for every $n \in S$ in this case.
\end{defn}
\begin{thm}\label{t:weighteddelta}
Fix a numerical semigroup $S = \<r_1, \ldots, r_k\>$ and a vector $w \in \QQ^k$.
\begin{enumerate}[(a)]
\item
If $\Delta_w(S) \ne \emptyset$, then $\Delta_w(S) \subset d\ZZ_{\ge 1}$, where $d = \min \Delta_w(S)$.
\item
We have
$$\min \Delta_w(S) = \gcd(\{w_ir_j - w_jr_i : 1 \le i < j \le r\}).$$
\item
The set $\Delta_w(S)$ is finite. Moreover,
$$\max \Delta_w(S) = \max_{n \in \Betti(S)} \max \Delta_w(n).$$
\end{enumerate}
\end{thm}
\begin{proof}
Each $w_i = t_i/u_i$ for some $t_i, u_i \in \ZZ$, so we must have $\Delta_w(S) \subset \delta\ZZ_{\ge 1}$, where $\delta = 1/(u_1 \cdots u_k)$. Fix $d' \in \Delta_w(S)$, and fix $c, c' \in \ZZ_{\ge 1}$ so that $d = c\delta$ and $d' = c'\delta$. Write $\gcd(c,c') = mc - m'c'$ for $m, m' \in \ZZ_{\ge 1}$. We must have elements $n, n' \in S$ and factorizations $a, b \in \mathsf Z(n)$ and $a', b' \in \mathsf Z(n')$ so that $|a|_w - |b|_w = d$ and $|a'|_w - |b'|_w = d'$. By the linearity of $|\cdot|_w$, the factorizations $ma + m'b', m'a' + mb \in \mathsf Z(mn + m'n')$ satisfy
$$|ma + m'b'|_w - |m'a' + mb|_w = m(|a|_w - |b|_w) - m'(|a'|_w - |b'|_w) = \gcd(c,c')\delta,$$
so by the minimality of $d$, we conclude $c = \gcd(c,c')$. This proves part~(a).
To prove part~(b), let
$$d' = \gcd(\{w_ir_j - w_jr_i : 1 \le i < j \le r\}).$$
Since $r_je_i, r_ie_j \in \mathsf Z(r_ir_j)$, the above argument implies
$$d \mid w_ir_j - w_jr_i = |r_je_i|_w - |r_ie_j|_w,$$
meaning $d \mid d'$. Conversely, suppose
$$a_1r_1 + \cdots + a_kr_k = b_1r_1 + \cdots + b_kr_k.$$
In order to show $d'$ divides $|a|_w - |b|_w = |a - b|_w$, by the linearity of $|\cdot|_w$ it suffices to express $a - b$ as an integer combination of the vectors $e_{ij} = r_je_i - r_ie_j$. Notice that
$$(a_1 - b_1)r_1 = (b_2 - a_2)r_2 + \cdots + (b_k - a_k)r_k$$
and since $\gcd(r_1, \ldots, r_k) = 1$, we must have $\gcd(r_2, \ldots, r_k) \mid (a_1 - b_1)$. As such, $a_1 - b_1 = c_2r_2 + \cdots + c_kr_k$ for some $c_i \in \ZZ$, meaning
$$a - b - c_2e_{12} - \cdots - c_ke_{1k} = (a_2 - b_2 + c_2r_1)e_2 + \cdots + (a_k - b_k + c_kr_1)e_k,$$
which has first coordinate $0$. Induction on $k$ concludes the proof of part~(b).
For part~(c), fix $n \in S$ and $x, y \in \mathsf Z(n)$ where $|x|_w < |y|_w$ are sequential in $\mathsf L_w(n)$. By~\cite[Lemma 2.1]{bfdelta}, there is a chain of factorizations $x_0, \ldots, x_t \in \mathsf Z(n)$ with $x_0 = x$, $x_t = y$, and $(x_i, x_{i+1}) = (a_i + c_i, b_i + c_i)$ for some $c_i \in \ZZ_{\ge 0}^k$ and factorizations $a_i, b_i \in \mathsf Z(n_i)$ lying in different connected components of the factorization graph~$\nabla_{n_i}$ of some Betti element~$\beta_i$. Since $|x|_w$ and $|y|_w$ are sequential in $\mathsf L_w(n)$, there must be some $i$ so that
$$|x_i|_w \le |x|_w < |y|_w \le |x_{i+1}|_w,$$
and no factorization $z \in \mathsf Z(\beta_i)$ can satisfy $|x|_w < |z + c_i|_w < |y|_w$. As such, we must have $|y|_w - |x|_w \le \max \Delta_w(\beta_i)$. This completes the proof.
\end{proof}
\section{Linear families of numerical semigroups
\label{sec:linearfamilies
In the remainder of this manuscript, we examine a particular parametrized family of numerical semigroups, of the form
$$P_n := \<w_1n + r_1, \ldots, w_kn + r_k\>$$
for fixed $r = (r_1, \ldots, r_k) \in \ZZ^k$ and $w = (w_1, \ldots, w_k) \in \ZZ_{\ge 1}^k$. The main result of this section is Theorem~\ref{t:mesalemma}, which describes the possible minimal generators that can occur for the defining toric ideal of $P_n$ for large $n$. This result is a generalization of \cite[Theorem 3.4]{shiftyminpres}, which sat at the center of the results in \cite{shiftyminpres,shiftedaperysets} for shifted numerical semigroups (see \cite[Remark~4.10]{shiftyminpres}). Likewise, Theorem~\ref{t:mesalemma} identifies the key structural changes that occur in $P_n$ for large~$n$ that are central to our results on Betti numbers, minimal presentations (Section~\ref{sec:minpres}) and Frobenius numbers (Section~\ref{sec:aperysets}).
We begin by imposing some assumptions on $r_1, \ldots, r_k$ and $w_1, \ldots, w_k$, all of which can be made without loss of generality.
\begin{notation}\label{n:parametrized}
Since $w_1, \ldots, w_k \in \ZZ_{\ge 1}$, we can reparametrize $n$ so $r_1, \ldots, r_k \in \ZZ_{\ge 0}$.
Reorder $r_1, \ldots, r_k$ (and correspondingly $w_1, \ldots, w_k$) so that $r_1 \le_w \cdots \le_w r_k$, that is,
$$r_1/w_1 \le \cdots \le r_k/w_k$$
(this is equivalent to Definition~\ref{d:worder} since $w$ has all positive entries). Note that if $r_i = 0$, then $r_j = 0$ for all $j \le i$ as well. Define
$$W = \max\{w_1, \ldots, w_k\}
\qquad \text{and} \qquad
R = \max\{r_1, \ldots, r_k\},$$
and reparametrize $n$ appropriately so that $0 \le r_1 < w_1$.
\end{notation}
\begin{remark}\label{r:rationalgens}
The proof of Theorem~\ref{t:mesalemma} begins by reparametrizing $P_n$ so that the first generator equals the input parameter. However, doing so forces the constant terms $t_2, \ldots, t_k$ to be (potentially) rational. Several times throughout the proof, Lemma~\ref{l:maxminquasi}(b) is carefully applied to the additive subsemigroup $T = \<t_2, \ldots, t_k\> \subset \QQ_{\ge 0}$ in the following sense:\ $T$ can be scaled by a unique rational value $\delta \in \QQ_{> 0}$ to obtain an isomorphic semigroup $\delta T \subset \ZZ_{\ge 0}$ with finite complement.
\end{remark}
\begin{thm}\label{t:mesalemma}
Let $z$ and $z'$ be factorizations of a Betti element $\beta \in P_n$ in different connected components of $\nabla_\beta$ with $|z|_w > |z'|_w$. If $n > w_1^2W\!R^2$, then
\begin{enumerate}[(a)]
\item
the connected components of $z$ and $z'$ in $\nabla_\beta$ contain every factorization of weighted length $|z|_w$ and $|z'|_w$, respectively;
\item
some factorization $y$ with $|z|_w = |y|_w$ has $y_1 > 0$; and
\item
some factorization $y'$ with $|z'|_w = |y'|_w$ has $y_k' > 0$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $m = w_1n + r_1$ so that
$$\begin{array}{r@{}c@{}l}
P_n
&{}={}& \big\<m, \, \tfrac{w_2}{w_1}m + \big(r_2 - w_2\tfrac{r_1}{w_1}\big), \, \ldots, \, \tfrac{w_k}{w_1}m + \big(r_k - w_k\tfrac{r_1}{w_1}\big)\big\> \\[0.2em]
&{}={}& \<m, \, v_2 m + t_2, \, \ldots, \, v_k m + t_k\>,
\end{array}$$
where each $t_i = r_i - w_i\tfrac{r_1}{w_1} \ge r_i - w_i\tfrac{r_i}{w_i} = 0$ and $v_i = \tfrac{w_i}{w_1}$. With this notation, we see $r_i \le_w r_j$ implies
$$\frac{t_i}{v_i}
= \frac{w_1r_i - w_ir_1}{w_i}
= w_1\frac{r_i}{w_i} - r_1
\le w_1\frac{r_j}{w_j} - r_1
= \frac{w_1r_j - w_jr_1}{w_j}
= \frac{t_j}{v_j}.$$
In particular, this implies (i) $t_1 = \cdots = t_{j-1} = 0$ for some $j < r$, and (ii) $t_j \le_v \cdots \le_v t_k$, viewing $v = (v_j, \ldots, v_k)$ as a weight vector for $T = \<t_j, \ldots, t_k\>$. For simplicity, given $t \in T$ and $a \in \mathsf Z_T(t)$, we write
$$|z|_v = v_1z_1 + \cdots + v_kz_k \qquad \text{and} \qquad |a|_v = v_ja_j + \cdots + v_ka_k$$
throughout the remainder of the proof. The key observation is that
$$\beta - |z|_v m
= z_1m + \sum_{i = j}^k z_i (v_i m + t_i) - z_1m - m \sum_{i = j}^k v_iz_i
= \sum_{i = j}^k z_i t_i
$$
yields a natural mapping of each factorization of $\beta \in P_n$ of weighted length $\ell$ to some factorization of $\beta - \ell m \in T$ of weighted length at most $\ell$. Let
$$a = (z_j, \ldots, z_k) \in \mathsf Z_T(\beta - |z |_v m) \qquad \text{and} \qquad a' = (z_j', \ldots, z_k') \in \mathsf Z_T(\beta - |z'|_v m)$$
denote the factorizations in $T$ corresponding to $z$ and $z'$, respectively. First, we claim some factorization in the same connected component of $\nabla_{\beta}$ as $z'$ has positive last coordinate. If $z_k' > 0$, then the claim is proven, so suppose $z'_k = 0$.
Since $w \in \ZZ_{\ge 1}^k$,
$$\begin{array}{r@{}c@{}l}
\beta - |z'|_v m
&{}={}& \displaystyle
\beta - \tfrac{1}{w_1}|z'|_w m
\ge \beta - \tfrac{1}{w_1}\big(|z|_w - 1\big)m
= \tfrac{1}{w_1}m + \beta - |z|_v m
\ge \displaystyle
\tfrac{1}{w_1}m
\ge n.
\end{array}$$
By assumption, $n > w_1^2\!R^2$, so writing $\delta \in \QQ_{> 0}$ for the unique rational value such that $\delta T \subset \ZZ_{\ge 0}$ has finite complement, this implies
$$a_j' + \cdots + a_k' \ge \tfrac{1}{R}(a_j't_j + \cdots + a_k't_k) = \tfrac{1}{R}(\beta - |z'|_v m) \ge \tfrac{1}{R}n > \tfrac{1}{R}w_1^2R^2 \ge w_1\delta t_k,$$
and thus Corollary~\ref{c:maxminquasi}(b) implies some factorization with positive last coordinate and weighted length having integer difference from $|a'|_v$ can be obtained from $a'$ by replacing all but at least one generator with copies of $t_k$. In particular, this factorization is in the same connected component as $a'$. Moreover, Corollary~\ref{c:maxminquasi}(b) implies some factorization $a'' \in \mathsf Z_T(\beta - |z'|_v m)$ whose weighted length is minimal among those satisfying $|a'|_v - |a''|_v \in \ZZ$ has $a_k'' > 0$.
Under the above factorization mapping, the factorization $z'' = (|z'|_v - |a''|_v, 0, \ldots, 0, a_j'', \ldots, a_k'') \in \mathsf Z_{P_n}(\beta)$ corresponds to $a''$ since
$$(|z'|_v - |a''|_v)m + \sum_{i = j}^k a_i''(v_i m + t_i) = (|z'|_v - |a''|_v)m + |a''|_v m + (\beta - |z'|_v m) = \beta.$$
The factorization $z''$ is thus in the same connected component of $\nabla_{\beta}$ as $z'$ and has $z_k'' > 0$, so the claim is proved.
Since $z$ and $z'$ are in different connected components of $\nabla_{\beta}$, we must have $z_k = 0$. This means $a_j + \cdots + a_k < w_1 \delta t_k$, as otherwise the above argument would yield a factorization of $\beta$ with positive last coordinate that is connected to $z$ in $\nabla_{\beta}$. Writing $V = \max(v_1, \ldots, v_k) = W/w_1$, the assumption $n > w_1^2W\!R^2$ implies
$$\begin{array}{r@{}c@{}l}
|z|_v
&{}>{}& \displaystyle |z'|_v \ge |a'|_v = \sum_{i = j}^k v_ia_i' = \sum_{i = j}^k \frac{v_i}{t_i}a_i't_i \ge \frac{v_k}{t_k}\sum_{i = j}^k a_i't_i = \frac{v_k}{t_k}(\beta - |z'|_v m) > \frac{v_k}{t_k} w_1^2W\!R^2 \\
&{}={}& \displaystyle \frac{w_k}{t_k}w_1W\!R^2 \ge w_1W\!R \ge w_1t_kW = w_1^2t_kV \ge w_1\delta t_kV > V \sum_{i = j}^k a_i \ge \sum_{i = j}^k v_ia_i = |a|_v,
\end{array}$$
so the factorization $y = (|z|_v - |a|_v, 0, \ldots, 0, a_j, \ldots, a_k)$ proves part~(b).
Additionally, either $y$ is connected to $z$ in $\nabla_\beta$, or $z_1 = 0$ and $z_j = \cdots = z_k = 0$. In the latter case, the preceeding inequalities imply $y_1 = |z|_v > w_kW\!R \ge W$, so one of the factorizations
$$y - w_i e_1 + w_1 e_i \in \mathsf Z_{P_n}(\beta) \qquad \text{for} \qquad 1 \le i \le j - 1$$
yields a path from $y$ to $z$ in $\nabla_\beta$ since one of the values $z_2, \ldots, z_{j-1}$ must be positive. This proves the first half of part~(a).
Lastly, suppose $z_k' = 0$. The above argument yielded a factorization $z''$ in the same connected component as $z'$ with $z_k'' > 0$ and corresponding factorization $a''$ having minimal weighted length. Since $z$ and $z'$ are in different connected components of~$\nabla_\beta$, the first half of part~(a) implies $z_1' = z_1'' = \cdots = z_{j-1}' = z_{j-1}'' = 0$ and thus $|a'|_v = |a''|_v$.
This proves part~(c), and applying the arguments thus far to any factorization of weighted length $|z'|_w$ yields a path in $\nabla_\beta$ to $z'$ through $z''$. This completes the proof.
\end{proof}
\begin{example}\label{e:positivecoords}
In Theorem~\ref{t:mesalemma}(c), we cannot ensure that all choices of factorizations $z$ and $z'$ has positive first and last coordinates, respectively. Indeed, if $r = (0,0,2,3)$ and $w = (5,7,2,3)$, then $\beta = 1980$ is a Betti element of $P_{44}$ with
$$(0, 0, 22, 0), \quad (0, 0, 19, 2), \quad (0, 0, 9, 0), \quad \text{and} \quad (0, 0, 2, 5)$$
among its factorizations. The key is that in the proof of Theorem~\ref{t:mesalemma}, there are ties in the $w$-ordering for both first and last place. As a consequence of Theorem~\ref{t:minpresmap}, this phenomenon also occurs for a Betti element of $P_{44+15m}$ for each $m \ge 0$.
\end{example}
\begin{example}\label{e:uglymapping}
In the proof of Theorem~\ref{t:mesalemma}, the natural mapping from factorizations of $\beta \in P_n$ of weighted length $\ell$ to factorizations of $\beta - \ell m \in T$ of weighted length at most $\ell$ need not be injective nor surjective. Let $r = (0,0,5,7,9)$ and $w = (2,3,5,7,8)$. Certainly, the factorizations $(8,0,0,0,0)$ and $(2,4,0,0,0)$ of $\beta = 704 \in P_{44}$ are mapped to the same factorization of $0 \in T = \<\frac{5}{2}, \frac{7}{2}, \frac{9}{2}\>$. What is perhaps more subtle is that $\beta = 1620$ has 2 factorizations, namely
$$\mathsf Z_{P_{44}}(1620) = \{(0, 0, 3, 3, 0), (2, 0, 0, 0, 4)\}$$
but the corresponding element $18 \in T$ has factorizations
$$\mathsf Z_T(18) = \{(3, 3, 0), (4, 1, 1), (0, 0, 4)\}$$
and the second does not correspond to any factorizations of $\beta$. The issue is that $v = (1,\frac{3}{2},\frac{5}{2},\frac{7}{2},4)$, so $a = (4,1,1)$ has non-integral weighted length $|a|_v = \frac{35}{2}$, so it is impossible to fill the first or second coordinates of a corresponding factorization of $\beta$ to obtain the necessary weighted length. This is why, when constructing factorizations of $\beta$ from factorizations of elements of $T$ at several locations in the proof of Theorem~\ref{t:mesalemma}, we must ensure that the first coordinate (a weighted length difference) is integral.
\end{example}
Now we can state a generalization of \cite[Corollary 3.5]{shiftyminpres} and \cite[Corollary 5.7]{shiftyminpres} which follows from Theorem~\ref{t:mesalemma}.
\begin{cor}\label{c:bettidelta}
If $n > w_1^2W\!R^2$, then $\Delta_w(P_n) = \{d\}$, where
$$d =
\gcd(w_1, \ldots, w_{j-1}, \min \Delta_w(S))\gcd(S)
$$
with $r_{j-1} = 0 < r_j$ and $S = \<r_j, \ldots, r_k\>$.
\end{cor}
\begin{proof}
Since
$$w_{i'}(w_in + r_i) - w_{i}(w_{i'}n + r_{i'}) = w_{i'}r_i - w_ir_{i'}$$
for any $i, i' \le k$,
we have
$$\begin{array}{r@{}c@{}l}
\min\Delta_w(P_n)
&{}={}& \gcd(\{w_ir_{i'} : 1 \le i < j \le i' \le k\} \cup \{w_ir_{i'} - w_{i'}r_i : j \le i < i' \le k\}) \\
&{}={}& \gcd(w_1, \ldots, w_{j-1}, \min \Delta_w(S))\gcd(r_1, \ldots, r_k),
\end{array}$$
so the first claim follows from Theorem~\ref{t:weighteddelta}(b).
Applying Theorem~\ref{t:weighteddelta}(c), we will show if two factorizations $z, z'\in \mathsf Z(m)$ satisfy $|z|_w - |z'|_w \ge 2d$, then $z$ and $z'$ must be in the same connected component of $\nabla_m$. Let $\ell = |z|_w - |z'|_w$. Just as in the proof of Theorem~\ref{t:mesalemma}, we know
$$m - |z|_wn = z_1r_1 + \cdots + z_kr_k \in S,$$
so since $n > w_1^2W\!R^2$, we have
$$(m - |z|_wn) + (\ell - d)n
= (m - |z'|_w n - \ell n) + (\ell - d)n
= m - (|z'|_w + d)n \in S.$$
Any factorization of the above element of $S$ corresponds to a factorization $z'' \in \mathsf Z(m)$ with $|z''|_w = |z'|_w + d$ that is connected to both $z$ and $z'$ in $\nabla_m$ by Theorem~\ref{t:mesalemma}.
\end{proof}
\section{Minimal presentations of parametrized semigroups
\label{sec:minpres
Let $\pi_n:\ZZ_{\ge 0}^k \to P_n$ denote the map
$$\pi_n(z) = \sum_{i = 1}^k z_i(w_in + r_i) = |z|_w n + \sum_{i = 1}^k z_ir_i,$$
called the \emph{factorization homomorphism} of $P_n$. The equivalence relation $\ker \pi_n$ on $\ZZ_{\ge 0}^k$, called the \emph{kernel congruence}, is given by $(z,z') \in \ker\pi_n$ whenever $\pi_n(z) = \pi_n(z')$, (that is, when $z$ and $z'$ are factorizations for the same element in $P_n$). Here, $\ker \pi_n$ is a congruence since it is closed under \emph{translation}, that is, $(z + u, z' + u) \in \ker \pi_n$ for every $(z,z') \in \ker\pi_n$ and $u \in \ZZ_{\ge 0}^k$.
A minimal presentation (Definition~\ref{d:minpres}) of a given semigroup $T$ encodes a particular choice of minimal relations (or \emph{trades}) between the generators of $T$.
They are one of the fundamental tools with which to study the factorization structure of numerical semigroups, and are closely connected to the defining toric ideal of $T$ (Remark~\ref{r:minprestoricideal}).
For a thorough introduction, we refer the reader to \cite[Chapter~9]{fingenmon} and \cite[Chapter~7]{numerical}.
The results in this section generalize those in \cite{shiftyminpres}, where a special (unweighted) case of the parametrization defining $P_n$ is considered.
At the heart of the main results in~\cite{shiftyminpres} is a map between kernel congruences, used to establish a correspondence between minimal presentations for large $n$ that restricts to a bijection on Betti elements. Our analogous map, $\Phi_n$, is defined in Proposition~\ref{p:mapwelldefined}, and its key properties (which closely mirror those in \cite{shiftyminpres}) are given in Proposition~\ref{p:mapproperties}. The main results are Theorem~\ref{t:minpresmap} and Corollary~\ref{c:bettiquasi}, which establish periodicity results for the minimal presentations and Betti elements of $P_n$, respectively, for large $n$. For our more general parametrization, the period turns out to be
$$p = w_1r_k - w_kr_1,$$
which specializes to a period of $r_k$ when $w_1 = 1$ and $r_1 = 0$ (as in \cite{shiftyminpres}).
In this section, we omit several proofs that are nearly identical to those in \cite{shiftyminpres}, including only those aspects that are different in our more general setting.
\begin{defn}\label{d:minpres}
Fix a numerical semigroup $T = \<t_1, \ldots, t_k\>$ and let $\pi:\ZZ_{\ge 0}^k \to T$ denote the factorization homomorphism of $T$. A \emph{presentation} for $T$ is a set of relations $\rho \subset \ker \pi$ such that $\ker \pi$ is the unique minimal (w.r.t.\ containment) congruence on $\ZZ_{\ge 0}^k$ containing $\rho$. Equivalently, between any two factorizations $(z, z') \in \ker \pi$, there exists a \emph{chain} $a_0, a_1, \ldots, a_r$ with $a_0 = z$, $a_r = z'$, and
$$(a_{i-1},a_i) = (b_i,b_i') + (u_i,u_i) \in \ker \pi$$
for some $(b_i, b_i') \in \rho$ and $u_i \in \ZZ_{\ge 0}^k$ for each $i \le r$. We say $\rho$ is \emph{minimal} if it is minimal with respect to containment among all presentations of $T$.
\end{defn}
\begin{remark}\label{r:minprestoricideal}
Returning to the commutative algebra viewpoint in Remark~\ref{r:bettinumbers}, minimal presentations encode minimal generating sets of toric ideals. Let $T = \<t_1, \ldots, t_k\>$, and write $I = \ker\varphi$ for the defining toric ideal of $T$, where $\varphi$ is the ring homomorphism
$$\begin{array}{r@{}c@{}l}
\varphi:\CC[x_1, \ldots, x_k] &{}\to{}& \CC[y] \\
x_i &{}\mapsto{}& y^{t_i}.
\end{array}$$
Each relation $(a, b) \in \ker\pi$ corresponds to a binomial
$$x_1^{a_1} \cdots x_k^{a_k} - x_1^{b_1} \cdots x_k^{b_k} \in I,$$
and each minimal presentation of $T$ corresponds to some minimal generating set of $I$. As an example, if $T = \<6, 9, 20\>$, then the minimal presentations of $T$ are
$$\begin{array}{ll}
\{((3,0,0), (0,2,0)), ((10,0,0), (0,0,3))\}, & \{((3,0,0), (0,2,0)), ((7,2,0), (0,0,3))\}, \\
\{((3,0,0), (0,2,0)), ((\phantom{0}4,4,0), (0,0,3))\}, & \{((3,0,0), (0,2,0)), ((1,6,0), (0,0,3))\},
\end{array}$$
each of which corresponds to one of the 4 minimal generating sets of the defining toric ideal $I \subset \CC[x,y,z]$ listed in Remark~\ref{r:bettinumbers}.
\end{remark}
\begin{example}\label{e:minpresmap}
Let $r = (1,2,4,6)$ and $w = (3,4,6,9)$, and consider the following minimal presentations for $P_n$ with $n$ identical modulo $p = 3 \cdot 6 - 9 \cdot 1 = 9$.
$$
\begin{array}{
r@{\,\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,\,\,}
}
P_{506}:
& (( & 0, & 0, & 3, & 0), & (0, & 0, & 0, & 2 & )),
& (( & 0, & 3, & 0, & 0), & (2, & 0, & 1, & 0 & )),
\\
& (( & 506, & 1, & 0, & 0), & (0, & 0, & 0, & 169 & )),
& (( & 508, & 0, & 0, & 0), & (0, & 2, & 2, & 167 & ))
\\[0.5em]
P_{515}:
& (( & 0, & 0, & 3, & 0), & (0, & 0, & 0, & 2 & )),
& (( & 0, & 3, & 0, & 0), & (2, & 0, & 1, & 0 & )),
\\
& (( & 515, & 1, & 0, & 0), & (0, & 0, & 0, & 172 & )),
& (( & 517, & 0, & 0, & 0), & (0, & 2, & 2, & 170 & ))
\\[0.5em]
P_{524}:
& (( & 0, & 0, & 3, & 0), & (0, & 0, & 0, & 2 & )),
& (( & 0, & 3, & 0, & 0), & (2, & 0, & 1, & 0 & )),
\\
& (( & 524, & 1, & 0, & 0), & (0, & 0, & 0, & 175 & )),
& (( & 526, & 0, & 0, & 0), & (0, & 2, & 2, & 173 & ))
\end{array}
$$
Each first-row relation $(z,z')$ satisfies $|z|_w = |z'|_w$, and each second-row relation $(z,z')$ satisfies $|z| = |z'| + 1$. In the latter case, each time $n$ is increased by $p = 9$, the value of $z_1$ increases by $w_4 = 9$ and $z_4'$ increases by $w_1 = 3$.
\end{example}
\begin{defn}\label{d:monotonechain}
A chain $a_0, a_1, \ldots, a_r$ of factorizations is \emph{$w$-monotone} if the sequence $|a_0|_w, |a_1|_w, \ldots, |a_r|_w$ is monotone.
\end{defn}
\begin{prop}\label{p:mapwelldefined}
The map $\Phi_n \colon \ker \pi_n \to \ker \pi_{n + p}$ given by
$$\Phi_n(z,z')
= \left\{\begin{array}{ll}
(z + \ell w_k e_1, z' + \ell w_1 e_k) & \text{if } |z|_w > |z'|_w \\
(z + \ell w_1 e_k, z' + \ell w_k e_1) & \text{if } |z|_w < |z'|_w \\
(z,z') & \text{if } |z|_w = |z'|_w
\end{array}\right.
$$
for $(z,z') \in \ker \pi_n$ and $\ell = \big| |z|_w - |z'|_w \big|$ is well defined.
\end{prop}
\begin{proof}
Fix $(z,z') \in \ker \pi_n$ with $z = (z_1, \dots, z_k)$ and $z' = (z_1', \dots, z_k')$. By symmetry, we can assume that $\ell = |z|_w - |z'|_w \ge 0$. Now, we simply use $\pi_n(z) = \pi_n(z')$ to verify
$$\begin{array}{r@{}c@{}l}
\displaystyle \pi_{n + p}(z + \ell w_ke_1)
&{}={}& \displaystyle
\ell w_k r_1 + \sum_{i = 1}^k z_i(w_i(n + p) + r_i)
= \pi_n(z) + |z|_wp + \ell w_k r_1
\\ &{}={}& \displaystyle
\pi_n(z') + |z'|_wp + \ell(p + w_k r_1)
= \pi_n(z') + |z'|_wp + \ell w_1 r_k
\\ &{}={}& \displaystyle
\ell w_1 r_k + \sum_{i = 1}^k z_i'(w_i(n + p) + r_i)
= \pi_{n + p}(z' + \ell w_1e_k),
\end{array}$$
as desired.
\end{proof}
\begin{prop}\label{p:mapproperties}
Fix $n \in \ZZ_{\ge 1}$, $\rho \subset \ker \pi_n$, and $(z, z') \in \rho$, and let $(y, y') = \Phi_n(z, z')$.
\begin{enumerate}[(a)]
\item
\label{p:mapproperties_injective}
The map $\Phi_n$ is injective.
\item
\label{p:mapproperties_lendiffs}
The map $\Phi_n$ preserves weighted length differences: $|z|_w - |z'|_w = |y|_w - |y'|_w$.
\item
\label{p:mapproperties_closures}
The map $\Phi_n$ preserves the reflexive, symmetric, and translation closure operations: if $\rho$ is reflexive, symmetric, or closed under translation, then so is $\Phi_n(\rho)$.
\item
\label{p:mapproperties_monotonechains}
The map $\Phi_n$ preserves $w$-monotone chain connectivity: if $\rho$ is translation-closed and there exists a $w$-monotone $\rho$-chain from $z$ to $z'$, then there exists a $w$-monotone $\Phi_n(\rho)$-chain from $y$ to $y'$.
\end{enumerate}
\end{prop}
\begin{proof}
The proof is nearly identical to that of \cite[Proposition~4.4]{shiftyminpres}.
\end{proof}
Just as in \cite{shiftyminpres}, the main obstruction to Theorem~\ref{t:minpresmap} for arbitrary $n$ is that $\Phi_n$ need only preserve connectivity by $w$-monotone chains. By Proposition~\ref{p:monotonechain}, any two factorizations $(z,z') \in \ker \pi_n$ are guaranteed to be connected by a $w$-monotone chain if $n$ is large enough.
\begin{prop}\label{p:monotonechain}
Fix $n > w_1^2W\!R^2$ and a minimal presentation $\rho \subseteq \ker \pi_n$. There exists a $w$-monotone $\rho$-chain between any $(z,z') \in \ker \pi_n$.
\end{prop}
\begin{proof}
Without loss of generality, assume $\gcd(z,z') = 0$. Since $\Cong(\rho) = \ker \pi_n$, there exists a chain $z = a_0, a_1, \ldots, a_r = z'$ of factorizations such that for each $i < r$, we have
$$(a_i,a_{i+1}) = (b_i,b_i') + (u_i,u_i), \qquad (b_i, b_i') \in \rho, u_i \in \ZZ_{\ge 0}^k,$$
where $b_i$ and $b_i'$ occur in distinct connected components of the graph $\nabla\!_\beta$ of $\beta = \pi_n(b_i)$. By Corollary~\ref{c:bettidelta}, $\Delta_w(P_n) = \{d\}$, so $|a_i|_w - |a_{i-1}|_w \in \{-d,0,d\}$ for each $i \le r$.
By induction on the chain length $r$, it suffices to assume $|a_1|_w = \cdots = |a_{r-1}|_w$ and $|z|_w = |a_0|_w = |a_r|_w = |z'|_w$, and prove there exists a weighted length preserving chain from $z$ to $z'$. Indeed, any non-monotone chain must contain such a subchain, which could then be ``flattened'' to a subchain with all equal weighted lengths.
First, suppose $|a_1|_w = |a_0|_w + d$. Applying Theorem~\ref{t:mesalemma} to $(b_0,b_0')$ and $(b_{r-1},b_{r-1}')$, we see that $z$ and $z'$ share support with some factorizations $y$ and $y'$, respectively, with positive last coordinates and weighted lengths equal to $|z|_w = |z'|_w$. By induction on the semigroup element $\pi_n(z)$, there exist weighted length preserving chains connecting $z$ to $y$, $y$ to $y'$, and $y'$ to $z'$. The case $|a_1|_w = |a_0|_w - d$ follows similarly.
\end{proof}
\begin{thm}\label{t:minpresmap}
For any $n > w_1^2W\!R^2$, the image of any minimal presentation $\rho$ of $P_n$ under the map $\Phi_n:\ker \pi_n \to \ker \pi_{n + p}$ is a minimal presentation of $P_{n + p}$.
\end{thm}
\begin{proof}
We must show that any minimal presentation $\rho \subset \ker \pi_n$ of $P_n$ satisfies
$$\Cong(\Phi_n(\rho)) = \ker \pi_{n + p},$$
that is, the image of $\rho$ under $\Phi_n$ is a presentation for $P_{n + p}$. Fix $(y,y') \in \ker \pi_{n + p}$, and let $m = \pi_{n + p}(y)$. By Proposition~\ref{p:monotonechain}, there exists a $w$-monotone $\rho$-chain from $y$ to $y'$, which we can assume is $w$-monotone decreasing by Proposition~\ref{p:mapproperties}\eqref{p:mapproperties_closures}. We can also assume each step in this chain has the form $(b,b') + (u,u)$ for some $u \in \ZZ_{\ge 0}^k$ and $b, b' \in \mathsf Z(\beta)$ lying in different connected components of $\nabla\!_\beta$. By Proposition~\ref{p:mapproperties}\eqref{p:mapproperties_closures}, it suffices to prove each $(b,b')$ lies in $\Cong(\Phi_n(\rho))$, so we can assume $y$ and $y'$ lie in different connected components of $\nabla\!_m$.
First, if $|y|_w = |y'|_w$, then $\Phi_n(y,y') = (y,y')$ by Proposition~\ref{p:mapwelldefined}, so applying $\Phi_n$ to any $w$-monotone (in this case, length preserving) $\rho$-chain from $y$ to $y'$ yields a $\Phi_n(\rho)$-chain from $y$ to $y'$ by Proposition~\ref{p:mapproperties}\eqref{p:mapproperties_monotonechains}. In particular, $\Phi_n(\rho)$ connects any two factorizations of equal weighted length. On the other hand, if $|y|_w > |y'|_w$, then Corollary~\ref{c:bettidelta} implies $|y|_w = |y'|_w + d$, where $\Delta_w(P_n) = \{d\}$. By Theorems~\ref{t:mesalemma}(b) and~(c), some factorizations $x$ and $x'$ in the same connected components as $y$ and $y'$, respectively, satisfy $x_1 \ge d$ and $x_k' \ge d$. Since
$$\Phi_n(x - dw_ke_1, x' - dw_1e_k) = (x,x'),$$
$\Phi_n(\rho)$ connects $x$ and $x'$ in $\nabla_m$. As Theorem~\ref{t:mesalemma}(a) implies there are length-preserving chains from $y$ to $x$ and from $x'$ to $y'$, Proposition~\ref{p:mapproperties}\eqref{p:mapproperties_monotonechains} completes the proof.
\end{proof}
We are now ready to prove that the Betti elements of $P_n$ are eventually periodic.
\begin{cor}\label{c:bettiquasi}
For $n > w_1^2W\!R^2$, the map $\varphi_n: \Betti(P_n) \to \Betti(P_{n+p})$ given by
$$\beta \mapsto \left\{\begin{array}{@{\,}ll}
\beta + \lambda p & \text{if } \mathsf L_{P_n,w}(\beta) = \{\lambda\} \\
\beta + \lambda p + dw_1(w_kn + r_k) & \text{if } \mathsf L_{P_n,w}(\beta) = \{\lambda, \lambda + d\}
\end{array}\right.$$
is a bijection, where $\Delta_w(P_n) = \{d\}$.
\end{cor}
\begin{proof}
Fix $z, z' \in \mathsf Z_{P_n}(\beta)$ with $|z|_w = \max\mathsf L_{P_n,w}(\beta)$ and $|z'|_w = \min\mathsf L_{P_n,w}(\beta)$. In the first case, $\lambda = |z|_w = |z'|_w$, so $\Phi_n(z,z') = (z,z')$, so Theorem~\ref{t:minpresmap} implies
$$\varphi_n(\beta) = \beta + \lambda p = \sum_{i = 1}^k z_i(w_in + r_i) + \sum_{i = 1}^k w_iz_i p = \sum_{i = 1}^k z_i(w_i(n + p) + r_i) \in \Betti(P_{n+p}).$$
In the second case, $\lambda = |z|_w - d = |z'|_w$ by Corollary~\ref{c:bettidelta}, so $\Phi_n(z,z') = (z + de_1, z' + de_k)$, and thus Theorem~\ref{t:minpresmap} implies
$$\varphi_n(\beta) = \beta + \lambda p + dw_1(w_kn + r_k) = dw_1(w_kn + r_k) + \sum_{i = 1}^k z_i'(w_i(n + p) + r_i) \in \Betti(P_{n+p}).$$
As such, $\varphi_n$ is a bijection by Theorem~\ref{t:minpresmap}. This completes the proof.
\end{proof}
\begin{remark}\label{r:bettidichotomy}
Just as in \cite{shiftyminpres}, Corollary~\ref{c:bettiquasi} implies the elements of $\Betti(M_n)$ fall into two distinct categories:\ those with minimal relations of equal length (which increase linearly in $n$ upon successive applications of $\Phi_n$), and those with minimal relations of different length (which increase quadratically in $n$ upon successive applications of $\Phi_n$).
As an additional consequence of Corollary~\ref{c:bettiquasi}, we see the function $n \mapsto |\!\Betti(P_n)|$ is $p$-periodic for $n > w_1^2W\!R^2$, including if the elements of $\Betti(P_n)$ are counted with multiplicity (that is,~if each element $\beta \in \Betti(P_n)$ appears once for each relation between factorizations of $\beta$ occuring in a minimal presentation for $P_n$). In the commutative algebra language of Remarks~\ref{r:bettinumbers} and~\ref{r:minprestoricideal}, this says the number of minimal generators of the defining toric ideal of $P_n$ is $p$-periodic in $n$.
\end{remark}
\section{Apery sets and the Frobenius number
\label{sec:aperysets
In the final section of this paper, we examine the Frobenius number (Corollary~\ref{c:frobquasi}), genus (Corollary~\ref{c:genusquasi}), and type (Corollary~\ref{c:pseudofrobshifted}) of $P_n$ for large $n$. Our results utilize the Ap\'ery set (Definition~\ref{d:apery}) of $P_n$, from which each of these quantities can be quickly obtained (indeed, in numerical semigroup computations, one often computes the Ap\'ery set first since doing so has roughly the same computational complexity). Most of the results in this section generalize those in \cite{shiftedaperysets}.
Throughout this section, we add the following assumptions on the parametrization of $P_n$; the difficulties in the general case are discussed in Remark~\ref{r:aperysetrestriction}.
\begin{notation}\label{n:parametrizedapery}
Throughout this section, we restrict to the case $w_1 = 1$ (and consequently $r_1 = 0$), so that
$$P_n = \<n, w_2n + r_2, \ldots, w_kn + r_k\>,$$
in addition to all existing assumptions from Notation~\ref{n:parametrized}. Moreover, let $S = \<r_2, \ldots, r_k\>$ and $d = \gcd(S)$.
\end{notation}
\begin{defn}\label{d:apery}
Fix an additive subsemigroup $T \subset (\ZZ_{\ge 0}, +)$, and let $d = \gcd(T)$. The~\emph{Ap\'ery set} of $m \in T$ is
$$\Ap(T; m) = \{t \in T : t - m \in \ZZ \setminus T\}.$$
The \emph{genus} of $T$ is the number $\mathsf g(T) = |d\ZZ_{\ge 0} \setminus T|$ of positive integer multiples of $d$ lying outside of $T$, and the \emph{Frobenius number} of $T$ is the largest integer multiple of $d$ outside of $T$, that is, $\mathsf F(T) = \max(d\ZZ_{\ge 0} \setminus T)$.
\end{defn}
\begin{remark}\label{r:aperyfacts}
The quantities in Definition~\ref{d:apery} are usually only defined for numerical semigroups (that is, in the case when $d = 1$). We will make use here of the following properties of the Ap\'ery set, each of which follows immediately from a known result in the usual setting~\cite{numerical}.
\begin{enumerate}[(a)]
\item
Each element of $\Ap(T; m)$ is distinct modulo $m$. In particular, $\left|\Ap(T; m)\right| = m/d$.
\item
We have
$$\mathsf F(S) = \max(\Ap(T; m)) - m
\qquad \text{and} \qquad
\mathsf g(S) = \sum_{t \in \Ap(T; m)} \bigg\lfloor \frac{t}{m} \bigg\rfloor,
$$
known in the literature as Selmer's formulas \cite{selmersformula}.
\end{enumerate}
\end{remark}
\begin{example} \label{e:apery}
If $T = \<6, 9, 20\>$, then $\Ap(T; 6) = \{0, 49, 20, 9, 40, 29\}$, where the elements are listed based on their equivalence class modulo $6$. From Selmer's formulas in Remark~\ref{r:aperyfacts}, we conclude $\mathsf F(S) = 43$ and $\mathsf g(S) = \tfrac{147}{6} - \tfrac{5}{2} = 22$.
\end{example}
\begin{thm}\label{t:parametrizedapery}
If $n > W\!R^2$, then
$$\Ap(P_n;n) = \{i + \mathsf m_{S,w}(i)n \mid i \in \Ap(S;dn)\}.$$
Moreover, we have
$$\mathsf L_{P_n,w}(i + \mathsf m_{S,w}(i)n) = \{\mathsf m_{S,w}(i)\}$$
for each $i \in \Ap(S;dn)$.
\end{thm}
\begin{proof}
Fix $a \in \Ap(n)$. Since $a - n \notin P_n$, no factorization of $a$ has positive first coordinate, so Theorem~\ref{t:mesalemma} implies $\mathsf L_{P_n,w}(a) = \{\ell\}$ for some $\ell \in \ZZ_{\ge 0}$. Let $i = a - \ell n$.
The proof of Theorem~\ref{t:mesalemma} establishes a natural mapping
$$\begin{array}{rcl}
\{z\in \mathsf Z_{P_n}(a) : |z|_w = \ell\} & \rightarrow & \{s \in \mathsf Z_{S}(a - \ell n): |s|_w \le \ell\} \\
(z_0,z_1, \ldots, z_k) & \mapsto & (z_1, \ldots, z_k)
\end{array}$$
between factorizations of $a \in P_n$ and factorizations of $i = a - \ell n \in S$ of weighted length at most $\ell$. Moreover, in this setting, the above map is a bijection, since for any factorization $(z_2, \ldots, z_k) \in \mathsf Z_S(i)$, letting $z_1 = \ell - |z|_w$ yields a factorization $(z_1, \ldots, z_k) \in \mathsf Z_{P_n}(a)$. Since $a \in \Ap(P_n; n)$, no factorization of $a$ has positive first coordinate, so we must have
$$\ell = |z|_w = \mathsf m_{S,w}(a - \ell n) = \mathsf m_{S,w}(i).$$
Observing that $|\!\Ap(P_n; n)| = |\!\Ap(S; dn)| = n$ and that the elements of $\Ap(P_n; n)$ are all distinct modulo $n$ completes the proof.
\end{proof}
\begin{remark}\label{r:aperysetrestriction}
The primary difficulty in generalizing Theorem~\ref{t:parametrizedapery} to the general setting considered in Sections~\ref{sec:linearfamilies} and~\ref{sec:minpres} is the ``non-surjectivity'' demonstrated in Example~\ref{e:uglymapping}. The mapping utilized in the proof of Theorem~\ref{t:parametrizedapery} is indeed a specialization of the one established in the proof of Theorem~\ref{t:mesalemma}, but it specializes to a bijection in this case (i.e., when $w_1 = 1$).
\end{remark}
Generalizations of \cite[Corollaries~4.2 and~4.3]{shiftedaperysets} follow immediately from Theorem~\ref{t:parametrizedapery}, and make use of the following observation from~\cite{shiftedaperysets}.
\begin{prop}[{\cite[Proposition~3.4]{shiftedaperysets}}]\label{p:largeapery}
If $dn > F(S)$, then $\Ap(S;dn) = \{a_0, \ldots, a_{n-1}\}$, where
$$a_i = \left\{\begin{array}{ll}
di & \text{ if } di \in S; \\
di + dn & \text{ if } di \notin S.
\end{array}\right.$$
In particular, this holds whenever $n > W\!R^2$ as in Theorem~\ref{t:parametrizedapery}. \end{prop}
\begin{cor}\label{c:frobquasi}
For $n > W\!R^2$, the function $n \to \mathsf F(P_n)$ has the form
$$\mathsf F(P_n) = \tfrac{w_k}{r_k}n^2 + a_1(n)n + a_0(n)$$
for some $r_k$-periodic functions $a_1(n)$ and $a_0(n)$.
\end{cor}
\begin{proof}
Let $a$ denote the element of $\Ap(S;dn)$ for which $\mathsf m_{S,w}(-)$ is maximal. Theorem~\ref{t:parametrizedapery} and Proposition~\ref{p:largeapery} imply
\begin{center}
$\begin{array}{r@{}c@{}l}
\mathsf F(P_n) &{}={}& \max(\Ap(P_n)) - n = a - n + \mathsf m_{S,w}(a) \cdot n,
\end{array}$
\end{center}
and Theorem~\ref{t:maxminquasi}\eqref{t:maxminquasi:minlen} implies $a + r_k$ is the element of $\Ap(S;dn + r_k)$ for which $\mathsf m_{S,w}(-)$ is maximal. The quasilinearity of $\mathsf m_{S,w}(-)$ proves $n \mapsto \mathsf F(P_n)$ is quasiquadratic in $n$ with period $r_k$, and since the only degree-2 term in the above expression is $\mathsf m_{S,w}(a) \cdot n$, we obtain a leading coefficient identical to that of $\mathsf m_{S,w}(n)$, namely $w_k/r_k$.
\end{proof}
\begin{cor}\label{c:genusquasi}
For $n > W\!R^2$, the function $n \mapsto \mathsf g(P_n)$ has the form
$$\mathsf g(P_n) = \tfrac{w_k}{2r_k}n^2 + b_1(n)n + b_0(n)$$
for some $r_k$-periodic functions $b_1(n)$ and $b_0(n)$.
\end{cor}
\begin{proof}
By Remark~\ref{r:aperyfacts}, we can write
$$\mathsf g(P_n) = \sum_{a \in \Ap(P_n)} \left\lfloor\frac{a}{n}\right\rfloor.$$
Theorem~\ref{t:parametrizedapery} and Proposition~\ref{p:largeapery} then yield
$$\begin{array}{r@{}c@{}l}
\mathsf g(P_n)
&{}={}& \displaystyle \sum_{i \in \Ap(S;dn)} \left\lfloor\frac{i+ \mathsf m_{S,w}(i)n}{n}\right\rfloor
= \sum_{i \in \Ap(S;dn)} \left\lfloor\frac{i}{n}\right\rfloor
+ \sum_{i \in \Ap(S;dn)} \mathsf m_{S,w}(i)
\\[0.1em]
&{}={}& \displaystyle \sum_{t=1}^{n-1} \left\lfloor \frac{dt}{n} \right\rfloor
+ d \cdot \mathsf g(S)
+ \sum_{\substack{i < n \\ di \in S}} \mathsf m_{S,w}(di)
+ \sum_{\substack{i \ge 0 \\ di \notin S}} \mathsf m_{S,w}(di+dn).
\end{array}$$
Each of the terms is eventually quasipolynomial in $n$. The first term is $d$-quasilinear in $n$, the second term is independent of $n$, and Theorem~\ref{t:parametrizedapery} guarantees that the last two terms are eventually $r_k$-quasiquadratic and $r_k$-quasilinear in $n$, respectively. Since $d \mid r_k$, we conclude $n \mapsto \mathsf g(P_n)$ is quasiquadratic in $n$ with period $r_k$. As for the leading term, the only degree-2 term in the above expression has successive $r_k$-differences
$$\sum_{\substack{i < n + r_k \\ di \in S}} \mathsf m_{S,w}(di) - \sum_{\substack{i < n \\ di \in S}} \mathsf m_{S,w}(di) = \sum_{j = 0}^{r_k-1} \mathsf m_{S,w}(dn + dj)$$
which are linear with leading coefficient $r_k(w_k/r_k) = w_k$. This yields a leading coefficient of $w_k/2r_k$ for $n \mapsto \mathsf g(P_n)$, as claimed.
\end{proof}
\begin{remark}\label{r:irreducibleasymptotic}
A numerical semigroup $S$ is called \emph{irreducible} if it is maximal with respect to containment among all numerical semigroups with Frobenius number $\mathsf F(S)$. If $\mathsf F(S)$ is odd, this happens precisely when $\mathsf g(S) = (\mathsf F(S) + 1)/2$, and if $\mathsf F(S)$ is even, this happens precisely when $\mathsf g(S) = (\mathsf F(S) + 2)/2$. Irreducible numerical semigroups have the smallest possible genus for their respective Frobenius number \cite[Chapter~3]{numerical}.
As a consequence of the leading coefficients in Corollaries~\ref{c:frobquasi} and~\ref{c:genusquasi}, we obtain
$$\lim_{n \to \infty} \frac{\mathsf g(P_n)}{\mathsf F(P_n)} = \frac{1}{2},$$
which can be interpreted as saying $P_n$ is ``nearly'' irreducible for large $n$.
\end{remark}
As a consequence, we obtain that for sufficiently large $n$, the numerical semigroup~$P_n$ satisfies Wilf's conjecture~\cite{wilfconjecture}, which is a longstanding open problem for numerical semigroups; see~\cite{wilfsurvey} for a survey of recent progress.
\begin{cor}\label{c:wilfshifted}
For $n > W\!R^2$, the Wilf number of $P_n$, defined in \cite{delgadoconj} as
$$\mathsf W(P_n) = k(F(P_n) - g(P_n)) - (F(P_n) + 1),$$
is $r_k$-quasiquadratic in $n$. In~particular, $\mathsf W(P_n)$ is positive for all suffiently large $n$, and thus $P_n$ satisfies Wilf's conjecture for each such $n$.
\end{cor}
\begin{proof}
Apply Corollaries~\ref{c:frobquasi} and~\ref{c:genusquasi}.
\end{proof}
Our final result concerns the (Cohen-Macaulay) type of $P_n$ for large $n$, which, just as in~\cite{shiftedaperysets}, we obtain from the pseudo-Frobenius numbers of $P_n$.
\begin{defn}\label{d:pseudofrob}
An integer $m \ge 0$ is a \emph{pseudo-Frobenius number} of a numerical semigroup $T$ if $m \notin T$ but $m + n \in T$ for all positive $n \in T$. Denote the set of pseudo-Frobenius numbers of $T$ by $\mathsf{PF}(T)$, and the \emph{type} of $T$ by $\mathsf t(T) = |\mathsf{PF}(T)|$.
\end{defn}
\begin{cor}\label{c:pseudofrobshifted}
Given $n \in \ZZ_{\ge 0}$, let $F_n$ denote the set
$$F_n = \{i \in \Ap(S;dn) : a \equiv i \bmod n \text{ for some } a \in \mathsf{PF}(P_n)\}.$$
For $n > W\!R^2$, the map $F_n \to F_{n + r_k}$ given by
$$\begin{array}{rcl}
i
&\mapsto&
\left\{\begin{array}{ll}
i & \text{if } i \le dn \\
i + r_k & \text{if } i > dn
\end{array}\right.
\end{array}
$$
is a bijection. In particular, there is a bijection $\mathsf{PF}(P_n) \to \mathsf{PF}(P_{n + r_k})$, meaning the function $n \mapsto t(P_n)$ is $r_k$-periodic for $n > W\!R^2$.
\end{cor}
\begin{proof}
The proof is identical to that of \cite[Theorem~4.8]{shiftedaperysets}.
\end{proof}
\section{Evidence of Conjecture~\ref{conj:main}
\label{sec:evidence
Now that we have seen the formal definition of a minimal presentation, we are ready to see an example of Conjecture~\ref{conj:main} for a more general (i.e., nonlinear) parametrized semigroup family. Note that computational evidence for nonlinear families is harder to obtain since the substantially larger generators result in computations taking much longer to complete.
\begin{example}\label{e:nonlinear}
Consider the parametrized family of semigroups
$$P_n = \<m^2, m^2 + m + 1, m^2 + 2m + 1, m^2 + 2m + 3\>$$
and the following minimal presentations.
\smaller
$$
\begin{array}{
@{}r@{\,\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,}
l@{}r@{\,\,}r@{\,\,}r@{\,\,}r@{\,}r@{\,\,}r@{\,\,}r@{\,}r@{}l@{\,\,}
}
P_{52} :
& (( & 0, & 0, & 27, & 0), & (0, & 1, & 0, & 26 & )),
& (( & 0, & 3, & 26, & 0), & (2, & 0, & 0, & 27 & )),
& (( & 0, & 4, & 0, & 0), & (2, & 0, & 1, & 1 & )),
\\
& (( & 25, & 2, & 14, & 0), & (0, & 0, & 0, & 40 & )),
& (( & 25, & 3, & 0, & 0), & (0, & 0, & 13, & 14 & )),
& (( & 27, & 0, & 0, & 0), & (0, & 1, & 12, & 13 & ))
\\[0.5em]
P_{56} :
& (( & 0, & 0, & 29, & 0), & (0, & 1, & 0, & 28 & )),
& (( & 0, & 3, & 28, & 0), & (2, & 0, & 0, & 29 & )),
& (( & 0, & 4, & 0, & 0), & (2, & 0, & 1, & 1 & )),
\\
& (( & 27, & 2, & 15, & 0), & (0, & 0, & 0, & 43 & )),
& (( & 27, & 3, & 0, & 0), & (0, & 0, & 14, & 15 & )),
& (( & 29, & 0, & 0, & 0), & (0, & 1, & 13, & 14 & ))
\\[0.5em]
P_{60} :
& (( & 0, & 0, & 31, & 0), & (0, & 1, & 0, & 30 & )),
& (( & 0, & 3, & 30, & 0), & (2, & 0, & 0, & 31 & )),
& (( & 0, & 4, & 0, & 0), & (2, & 0, & 1, & 1 & )),
\\
& (( & 29, & 2, & 16, & 0), & (0, & 0, & 0, & 46 & )),
& (( & 29, & 3, & 0, & 0), & (0, & 0, & 15, & 16 & )),
& (( & 31, & 0, & 0, & 0), & (0, & 1, & 14, & 15 & ))
\\[0.5em]
P_{64} :
& (( & 0, & 0, & 33, & 0), & (0, & 1, & 0, & 32 & )),
& (( & 0, & 3, & 32, & 0), & (2, & 0, & 0, & 33 & )),
& (( & 0, & 4, & 0, & 0), & (2, & 0, & 1, & 1 & )),
\\
& (( & 31, & 2, & 17, & 0), & (0, & 0, & 0, & 49 & )),
& (( & 31, & 3, & 0, & 0), & (0, & 0, & 16, & 17 & )),
& (( & 33, & 0, & 0, & 0), & (0, & 1, & 15, & 16 & ))
\end{array}
$$
\normalsize
Unlike linear parametrized families, successive minimal presentations have more than just 2 coordinates consistently increasing, though the pattern in the relations is clear.
\end{example}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,075 |
{"url":"https:\/\/readtiger.com\/wkp\/en\/Transition_metal","text":"Transition metal\n\nTransition metals in the periodic table\n Hydrogen Helium Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon Sodium Magnesium Aluminium Silicon Phosphorus Sulfur Chlorine Argon Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon Caesium Barium Lanthanum Cerium Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinum Gold Mercury (element) Thallium Lead Bismuth Polonium Astatine Radon Francium Radium Actinium Thorium Protactinium Uranium Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium Rutherfordium Dubnium Seaborgium Bohrium Hassium Meitnerium Darmstadtium Roentgenium Copernicium Nihonium Flerovium Moscovium Livermorium Tennessine Oganesson\n\nIn chemistry, the term transition metal (or transition element) has three possible meanings:\n\n\u2022 The IUPAC definition[1] defines a transition metal as \"an element whose atom has a partially filled d sub-shell, or which can give rise to cations with an incomplete d sub-shell\".\n\u2022 Many scientists describe a \"transition metal\" as any element in the d-block of the periodic table, which includes groups 3 to 12 on the periodic table.[2][3] In actual practice, the f-block lanthanide and actinide series are also considered transition metals and are called \"inner transition metals\".\n\u2022 Cotton and Wilkinson[4] expand the brief IUPAC definition (see above) by specifying which elements are included. As well as the elements of groups 4 to 11, they add scandium and yttrium in group 3, which have a partially filled d subshell in the metallic state. Lanthanum and actinium in group 3 are, however, classified as lanthanides and actinides respectively.\n\nEnglish chemist Charles Bury (1890\u20131968) first used the word transition in this context in 1921, when he referred to a transition series of elements during the change of an inner layer of electrons (for example n\u00a0=\u00a03 in the 4th row of the periodic table) from a stable group of 8 to one of 18, or from 18 to 32.[5][6][7] These elements are now known as the d-block.\n\nClassification\n\nIn the d-block the atoms of the elements have between 1 and 10 d electrons.\n\nTransition metals in the d-block\nGroup 3 4 5 6 7 8 9 10 11 12\nPeriod 4 21Sc 22Ti 23V 24Cr 25Mn 26Fe 27Co 28Ni 29Cu 30Zn\n5 39Y 40Zr 41Nb 42Mo 43Tc 44Ru 45Rh 46Pd 47Ag 48Cd\n6 57La 72Hf 73Ta 74W 75Re 76Os 77Ir 78Pt 79Au 80Hg\n7 89Ac 104Rf 105Db 106Sg 107Bh 108Hs 109Mt 110Ds 111Rg 112Cn\n\nThe elements of groups 4\u201311 are generally recognized as transition metals, justified by their typical chemistry, i.e. a large range of complex ions in various oxidation states, colored complexes, and catalytic properties either as the element or as ions (or both). Sc and Y in group 3 are also generally recognized as transition metals. However, the elements La\u2013Lu and Ac\u2013Lr and group 12 attract different definitions from different authors.\n\n1. Many chemistry textbooks and printed periodic tables classify La and Ac as group 3 elements and transition metals, since their atomic ground-state configurations are s2d1 like Sc and Y. The elements Ce\u2013Lu are considered as the \"lanthanide\" series (or \"lanthanoid\" according to IUPAC) and Th\u2013Lr as the \"actinide\" series.[8][9] The two series together are classified as f-block elements, or (in older sources) as \"inner transition elements\".\n2. Some inorganic chemistry textbooks include La with the lanthanides and Ac with the actinides.[4][10][11] This classification is based on similarities in chemical behaviour and defines 15 elements in each of the two series, even though they correspond to the filling of an f subshell, which can only contain 14 electrons.\n3. A third classification defines the f-block elements as La\u2013Yb and Ac\u2013No, while placing Lu and Lr in group 3.[5] This is based on the Aufbau principle (or Madelung rule) for filling electron subshells, in which 4f is filled before 5d (and 5f before 6d), so that the f subshell is actually full at Yb (and No), while Lu (and Lr) has an [\u00a0]s2f14d1 configuration. However La and Ac are exceptions to the Aufbau principle with electron configuration [\u00a0]s2d1 (not [\u00a0]s2f1 as the Aufbau principle predicts), so it is not clear from atomic electron configurations whether La or Lu (Ac or Lr) should be considered as transition metals.[12]\n\nZinc, cadmium, and mercury are generally excluded from the transition metals,[5] as they have the electronic configuration [\u00a0]d10s2, with no incomplete d shell.[13] In the oxidation state +2 the ions have the electronic configuration [\u00a0]\u2026d10. However, these elements can exist in other oxidation states, including the +1 oxidation state, as in the diatomic ion Hg2+\n2\n. The group 12 elements Zn, Cd and Hg may therefore, under certain criteria, be classed as post-transition metals in this case. However, it is often convenient to include these elements in a discussion of the transition elements. For example, when discussing the crystal field stabilization energy of first-row transition elements, it is convenient to also include the elements calcium and zinc, as both Ca2+\nand Zn2+\nhave a value of zero, against which the value for other transition metal ions may be compared. Another example occurs in the Irving\u2013Williams series of stability constants of complexes.\n\nThe recent (though disputed and so far not reproduced independently) synthesis of mercury(IV) fluoride (HgF\n4\n) has been taken by some to reinforce the view that the group 12 elements should be considered transition metals,[14] but some authors still consider this compound to be exceptional.[15]\n\nAlthough meitnerium, darmstadtium, and roentgenium are within the d-block and are expected to behave as transition metals analogous to their lighter congeners iridium, platinum, and gold, this has not yet been experimentally confirmed.\n\nSubclasses\n\nEarly transition metals are on the left side of the periodic table from group 3 to group 7. Late transition metals are on the right side of the d-block, from group 8 to 11 (and 12 if it is counted as transition metals).\n\nElectronic configuration\n\nThe general electronic configuration of the d-block elements is [Inert gas] (n\u00a0\u2212\u00a01)d1\u201310n s0\u20132. The period 6 and 7 transition metals also add (n\u00a0\u2212\u00a02)f0\u201314 electrons, which are omitted from the tables below.\n\nThe Madelung rule predicts that the typical electronic structure of transition metal atoms can be written as [inert gas]ns2(n\u00a0\u2212\u00a01)dm where the inner d orbital is predicted to be filled after the valence-shell's s orbital is filled. This rule is however only approximate \u2013 it only holds for some of the transition elements, and only then in their neutral ground state.\n\nThe d-sub-shell is the next-to-last sub-shell and is denoted as ${\\displaystyle (n-1)d}$-sub-shell. The number of s electrons in the outermost s sub-shell is generally one or two except palladium (Pd), with no electron in that s-sub shell in its ground state. The s-sub-shell in the valence shell is represented as the ns sub-shell, e.g. 4s. In the periodic table, the transition metals are present in eight groups (4 to 11), with some authors including some elements in groups 3 or 12.\n\nThe elements in group 3 have an ns2(n\u00a0\u2212\u00a01)d1 configuration. The first transition series is present in the 4th period, and starts after Ca (Z\u00a0=\u00a020) of group-2 with the configuration [Ar]4s2, or scandium (Sc), the first element of group 3 with atomic number Z\u00a0=\u00a021 and configuration [Ar]4s23d1, depending on the definition used. As we move from left to right, electrons are added to the same d-sub-shell till it is complete. The element of group 11 in the first transition series is copper (Cu) with an atypical configuration [Ar]4s13d10. Despite the filled d subshell in metallic copper it nevertheless forms a stable ion with an incomplete d subshell. Since the electrons added fill the ${\\displaystyle (n-1)d}$ orbitals, the properties of the d-block elements are quite different from those of s and p block elements in which the filling occurs either in s or in p-orbitals of the valence shell. The electronic configuration of the individual elements present in all the d-block series are given below:[16]\n\n Group Atomic nr Element Electronconfiguration 3 4 5 6 7 8 9 10 11 12 21 22 23 24 25 26 27 28 29 30 Sc Ti V Cr Mn Fe Co Ni Cu Zn 3d14s2 3d24s2 3d34s2 3d54s1 3d54s2 3d64s2 3d74s2 3d84s2 3d104s1 3d104s2\n Atomic nr Element Electronconfiguration 39 40 41 42 43 44 45 46 47 48 Y Zr Nb Mo Tc Ru Rh Pd Ag Cd 4d15s2 4d25s2 4d45s1 4d55s1 4d55s2 4d75s1 4d85s1 4d10 4d105s1 4d105s2\n Atomic nr Element Electronconfiguration 57 72 73 74 75 76 77 78 79 80 La Hf Ta W Re Os Ir Pt Au Hg 5d16s2 5d26s2 5d36s2 5d46s2 5d56s2 5d66s2 5d76s2 5d96s1 5d106s1 5d106s2\n Atomic nr Element Electronconfiguration 89 104 105 106 107 108 109 110 111 112 Ac Rf Db Sg Bh Hs Mt Ds Rg Cn 6d17s2 6d27s2 6d37s2 6d47s2 6d57s2 6d67s2 6d77s2 6d87s2 6d97s2 6d107s2\n\nA careful look at the electronic configuration of the elements reveals that there are certain exceptions, for example Cr and Cu. These are either because of the symmetry or nuclear-electron and electron-electron force.\n\nThe ${\\displaystyle (n-1)d}$ orbitals that are involved in the transition metals are very significant because they influence such properties as magnetic character, variable oxidation states, formation of colored compounds etc. The valence ${\\displaystyle s(ns)}$ and ${\\displaystyle p(np)}$ orbitals have very little contribution in this regard since they hardly change in the moving from left to the right in a transition series. In transition metals, there is a greater horizontal similarities in the properties of the elements in a period in comparison to the periods in which the d-orbitals are not involved. This is because in a transition series, the valence shell electronic configuration of the elements do not change. However, there are some group similarities as well.\n\nCharacteristic properties\n\nThere are a number of properties shared by the transition elements that are not found in other elements, which results from the partially filled d shell. These include\n\n\u2022 the formation of compounds whose color is due to dd electronic transitions\n\u2022 the formation of compounds in many oxidation states, due to the relatively low energy gap between different possible oxidation states[17]\n\u2022 the formation of many paramagnetic compounds due to the presence of unpaired d electrons. A few compounds of main group elements are also paramagnetic (e.g. nitric oxide, oxygen)\n\nMost transition metals can be bound to a variety of ligands, allowing for a wide variety of transition metal complexes.[18]\n\nColoured compounds\n\nFrom left to right, aqueous solutions of: Co(NO\n3\n)\n2\n(red); K\n2\nCr\n2\nO\n7\n(orange); K\n2\nCrO\n4\n(yellow); NiCl\n2\n(turquoise); CuSO\n4\n(blue); KMnO\n4\n(purple).\n\nColour in transition-series metal compounds is generally due to electronic transitions of two principal types.\n\n\u2022 charge transfer transitions. An electron may jump from a predominantly ligand orbital to a predominantly metal orbital, giving rise to a ligand-to-metal charge-transfer (LMCT) transition. These can most easily occur when the metal is in a high oxidation state. For example, the colour of chromate, dichromate and permanganate ions is due to LMCT transitions. Another example is that mercuric iodide, HgI2, is red because of a LMCT transition.\n\nA metal-to-ligand charge transfer (MLCT) transition will be most likely when the metal is in a low oxidation state and the ligand is easily reduced.\n\nIn general charge transfer transitions result in more intense colours than d-d transitions.\n\n\u2022 d-d transitions. An electron jumps from one d-orbital to another. In complexes of the transition metals the d orbitals do not all have the same energy. The pattern of splitting of the d orbitals can be calculated using crystal field theory. The extent of the splitting depends on the particular metal, its oxidation state and the nature of the ligands. The actual energy levels are shown on Tanabe-Sugano diagrams.\n\nIn centrosymmetric complexes, such as octahedral complexes, d-d transitions are forbidden by the Laporte rule and only occur because of vibronic coupling in which a molecular vibration occurs together with a d-d transition. Tetrahedral complexes have somewhat more intense colour because mixing d and p orbitals is possible when there is no centre of symmetry, so transitions are not pure d-d transitions. The molar absorptivity (\u03b5) of bands caused by d-d transitions are relatively low, roughly in the range 5-500 M\u22121cm\u22121 (where M = mol dm\u22123).[19] Some d-d transitions are spin forbidden. An example occurs in octahedral, high-spin complexes of manganese(II), which has a d5 configuration in which all five electron has parallel spins; the colour of such complexes is much weaker than in complexes with spin-allowed transitions. Many compounds of manganese(II) appear almost colourless. The spectrum of [Mn(H\n2\nO)\n6\n]2+\nshows a maximum molar absorptivity of about 0.04 M\u22121cm\u22121 in the visible spectrum.\n\nOxidation states\n\nA characteristic of transition metals is that they exhibit two or more oxidation states, usually differing by one. For example, compounds of vanadium are known in all oxidation states between \u22121, such as [V(CO)\n6\n]\n, and +5, such as VO3\u2212\n4\n.\n\nMain group elements in groups 13 to 18 also exhibit multiple oxidation states. The \"common\" oxidation states of these elements typically differ by two. For example, compounds of gallium in oxidation states +1 and +3 exist in which there is a single gallium atom. No compound of Ga(II) is known: any such compound would have an unpaired electron and would behave as a free radical and be destroyed rapidly. The only compounds in which gallium has a formal oxidation state of +2 are dimeric compounds, such as [Ga\n2\nCl\n6\n]2\u2212\n, which contain a Ga-Ga bond formed from the unpaired electron on each Ga atom.[20] Thus the main difference in oxidation states, between transition elements and other elements is that oxidation states are known in which there is a single atom of the element and one or more unpaired electrons.\n\nThe maximum oxidation state in the first row transition metals is equal to the number of valence electrons from titanium (+4) up to manganese (+7), but decreases in the later elements. In the second row the maximum occurs with ruthenium (+8), and in the third row, the maximum occurs with iridium (+9). In compounds such as [MnO\n4\n]\nand OsO\n4\nthe elements achieve a stable configuration by covalent bonding.\n\nThe lowest oxidation states are exhibited in metal carbonyl complexes such as Cr(CO)\n6\n(oxidation state zero) and [Fe(CO)\n4\n]2\u2212\n(oxidation state \u22122) in which the 18-electron rule is obeyed. These complexes are also covalent.\n\nIonic compounds are mostly formed with oxidation states +2 and +3. In aqueous solution the ions are hydrated by (usually) six water molecules arranged octahedrally.\n\nMagnetism\n\nTransition metal compounds are paramagnetic when they have one or more unpaired d electrons.[21] In octahedral complexes with between four and seven d electrons both high spin and low spin states are possible. Tetrahedral transition metal complexes such as [FeCl\n4\n]2\u2212\nare high spin because the crystal field splitting is small so that the energy to be gained by virtue of the electrons being in lower energy orbitals is always less than the energy needed to pair up the spins. Some compounds are diamagnetic. These include octahedral, low-spin, d6 and square-planar d8 complexes. In these cases, crystal field splitting is such that all the electrons are paired up.\n\nFerromagnetism occurs when individual atoms are paramagnetic and the spin vectors are aligned parallel to each other in a crystalline material. Metallic iron and the alloy alnico are examples of ferromagnetic materials involving transition metals. Anti-ferromagnetism is another example of a magnetic property arising from a particular alignment of individual spins in the solid state.\n\nCatalytic properties\n\nThe transition metals and their compounds are known for their homogeneous and heterogeneous catalytic activity. This activity is ascribed to their ability to adopt multiple oxidation states and to form complexes. Vanadium(V) oxide (in the contact process), finely divided iron (in the Haber process), and nickel (in catalytic hydrogenation) are some of the examples. Catalysts at a solid surface (nanomaterial-based catalysts) involve the formation of bonds between reactant molecules and atoms of the surface of the catalyst (first row transition metals utilize 3d and 4s electrons for bonding). This has the effect of increasing the concentration of the reactants at the catalyst surface and also weakening of the bonds in the reacting molecules (the activation energy is lowered). Also because the transition metal ions can change their oxidation states, they become more effective as catalysts.\n\nAn interesting type of catalysis occurs when the products of a reaction catalyse the reaction producing more catalyst (autocatalysis). One example is the reaction of oxalic acid with acidified potassium permanganate (or manganate (VII)).[22] Once a little Mn2+ has been produced, it can react with MnO4 forming Mn3+. This then reacts with C2O4 ions forming Mn2+ again.\n\nPhysical properties\n\nAs implied by the name, all transition metals are metals and thus conductors of electricity.\n\nIn general, transition metals possess a high density and high melting points and boiling points. These properties are due to metallic bonding by delocalized d electrons, leading to cohesion which increases with the number of shared electrons. However the group 12 metals have much lower melting and boiling points since their full d subshells prevent d\u2013d bonding, which again tends to differentiate them from the accepted transition metals. Mercury has a melting point of \u221238.83\u00a0\u00b0C (\u221237.89\u00a0\u00b0F) and is a liquid at room temperature.\n\nReferences\n\n1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the \"Gold Book\") (1997). Online corrected version: \u00a0(2006\u2013) \"transition element\". doi:10.1351\/goldbook.T06456\n2. ^ Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002). General chemistry: principles and modern applications (8th ed.). Upper Saddle River, N.J: Prentice Hall. pp.\u00a0341\u2013342. ISBN\u00a0978-0-13-014329-7. LCCN\u00a02001032331. OCLC\u00a046872308.\n3. ^ Housecroft, C.\u00a0E. and Sharpe, A.\u00a0G. (2005) Inorganic Chemistry, 2nd ed, Pearson Prentice-Hall, pp. 20\u201321.\n4. ^ a b Cotton, F.\u00a0A. and Wilkinson, G. (1988) Inorganic Chemistry, 5th ed., Wiley, pp. 625\u2013627. ISBN\u00a0978-0-471-84997-1.\n5. ^ a b c Jensen, William B. (2003). \"The Place of Zinc, Cadmium, and Mercury in the Periodic Table\" (PDF). Journal of Chemical Education. 80 (8): 952\u2013961. Bibcode:2003JChEd..80..952J. doi:10.1021\/ed080p952.\n6. ^ Bury, C. R. (1921). \"Langmuir's theory of the arrangement of electrons in atoms and molecules\". J. Am. Chem. Soc. 43 (7): 1602\u20131609. doi:10.1021\/ja01440a023.\n7. ^ Bury, Charles Rugeley. Encyclopedia.com Complete dictionary of scientific biography (2008).\n8. ^ Petrucci, Harwood & Herring 2002, pp.\u00a049\u201350, 951.\n9. ^ Miessler, G. L. and Tarr, D. A. (1999) Inorganic Chemistry, 2nd edn, Prentice-Hall, p. 16. ISBN\u00a0978-0-13-841891-5.\n10. ^ Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. ISBN\u00a0978-0-08-037941-8.\n11. ^ Housecroft, C. E. and Sharpe, A. G. (2005) Inorganic Chemistry, 2nd ed., Pearson Prentice-Hall, p. 741.\n12. ^ Scerri, E.\u00a0R. (2011) A Very Short Introduction to the Periodic Table, Oxford University Press.\n13. ^ Cotton, F. Albert; Wilkinson, G.; Murillo, C. A. (1999). Advanced Inorganic Chemistry (6th ed.). New York: Wiley, ISBN\u00a0978-0-471-19957-1.\n14. ^ Wang, Xuefang; Andrews, Lester; Riedel, Sebastian; Kaupp, Martin (2007). \"Mercury Is a Transition Metal: The First Experimental Evidence for HgF4\". Angew. Chem. Int. Ed. 46 (44): 8371\u20138375. doi:10.1002\/anie.200703710. PMID\u00a017899620.\n15. ^ Jensen, William B. (2008). \"Is Mercury Now a Transition Element?\". J. Chem. Educ. 85 (9): 1182\u20131183. Bibcode:2008JChEd..85.1182J. doi:10.1021\/ed085p1182.\n16. ^ Miessler, G. L. and Tarr, D. A. (1999) Inorganic Chemistry, 2nd edn, Prentice-Hall, p. 38 ISBN\u00a0978-0-13-841891-5\n17. ^ Matsumoto, Paul S (2005). \"Trends in Ionization Energy of Transition-Metal Elements\". Journal of Chemical Education. 82 (11): 1660. Bibcode:2005JChEd..82.1660M. doi:10.1021\/ed082p1660.\n18. ^ Hogan, C. Michael (2010). \"Heavy metal\" in Encyclopedia of Earth. National Council for Science and the Environment. E. Monosson and C. Cleveland (eds.) Washington DC.\n19. ^ Orgel, L.E. (1966). An Introduction to Transition-Metal Chemistry, Ligand field theory (2nd. ed.). London: Methuen.\n20. ^ Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. ISBN\u00a0978-0-08-037941-8. p. 240\n21. ^ Figgis, B.N.; Lewis, J. (1960). Lewis, J.; Wilkins, R.G. (eds.). The Magnetochemistry of Complex Compounds. Modern Coordination Chemistry. New York: Wiley Interscience. pp.\u00a0400\u2013454.\n22. ^ Kovacs KA, Grof P, Burai L, Riedel M (2004). \"Revising the Mechanism of the Permanganate\/Oxalate Reaction\". J. Phys. Chem. A. 108 (50): 11026\u201311031. Bibcode:2004JPCA..10811026K. doi:10.1021\/jp047061u.","date":"2019-06-20 15:09:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 5, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7356297969818115, \"perplexity\": 4551.417440647126}, \"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-2019-26\/segments\/1560627999261.43\/warc\/CC-MAIN-20190620145650-20190620171650-00221.warc.gz\"}"} | null | null |
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Oligoaeschna kunigamiensis är en trollsländeart som först beskrevs av Ishida 1972. Oligoaeschna kunigamiensis ingår i släktet Oligoaeschna och familjen mosaiktrollsländor. IUCN kategoriserar arten globalt som starkt hotad. Inga underarter finns listade i Catalogue of Life.
Källor
Mosaiktrollsländor
kunigamiensis | {
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Weber Antal (Pest, 1823. október 9. – Budapest, 1889. augusztus 4.) magyar műépítész, a 19. század végének jelentős alkotója, akinek historizáló stílusú egyetemi középületeit a mai napig használják.
Életpályája
Iskolái elvégzése után Hild József építészhez került, akinek a vezetése alatt az esztergomi bazilika építkezésénél mint előmunkás dolgozott. Innen Bécsbe ment és a szépművészeti akadémia hallgatója lett. Az akadémia akkori nagy hírű igazgatója, Nobili gróf nagyon megkedvelte az önálló felfogású, szép ízlésű fiatal magyart és mindenáron Bécsben akarta marasztalni, de Weber csak haza vágyódott, itthon akarván tehetségét és tanulmányait érvényesíteni. 1846-ban kezdett Pesten mint önálló építész működni, de csakhamar a rajzeszközt a fegyverrel kellett felcserélnie. 1848-ban beállott mint közhonvéd a Zrinyi-zászlóaljba és a szabadságharc lezajlása után, hogy a besoroztatástól megmeneküljön, hosszú ideig bujdosni volt kénytelen. 1851-ben végre megnyugodhatott és ekkor kiváló tehetsége, rendkívüli szorgalma és becsületessége révén az ország több előkelő családjának lett építészévé. Az ország különböző részeiben több kastélyt, templomot épített vagy alakított át, amely működése 1867-ig tartott és ezen évtől kezdve főleg Budapesten épített. 1872-ben nagyobb tanulmányutat tett Olaszországban, de már előbb a hatvanas évek végén tevékeny részt vett a magyar mérnök- és építészegyesület megalapításában, amely egyesületnek egyik alapítója és haláláig választmányi tagja volt. 1873-ban, miután Budapest főváros bizottsági tagjává választották, kiváló tevékenységet fejtett ki úgy itt, mint a fővárosi közmunkák tanácsában a főváros rendezése körül. 1875-től 1885-ig ő vezette az összes magánépítkezési ügyeket. Az 1873. évi párizsi világkiállításon terveivel és a már végrehajtott építkezéseit feltüntető rajzaival kiváló elismerést aratott és a becsületrend lovagjává nevezték ki. 1881-ben Trefort Ágoston felkérésére átvette a vallás- és közoktatásügyi minisztérium építészeti osztályának vezetését és ezen működési körében rendkívüli munkásságot fejtett ki, amelynek elismeréséül a vaskoronarendet nyerte el.
Utolsó éveiben tüdő- és mellhártyagyulladással küzdött, 66 éves korában, 1889. augusztus 4-én hunyt el.
Ismert épületei
Budapesten
1870s: Halász-ház, Budapest, 1067 Eötvös utca 13.
1870s: Bellevue-villa, Budapest, 1062 Andrássy út 141. (később lebontották)
1872–1873: Egyesült Budapesti Fővárosi Takarékpénztár, 1051 Budapest, Dorottya utca 6.
1874: Szerb-ház, 1056 Budapest, Váci utca 66.
1874–1876: Bérpalota, 1065 Budapest, Bajcsy-Zsilinszky út 37.
1875–1876: Ádám-villa, Budapest, 1088 Bródy Sándor u. 4.
1877–1878: Erdődy-villa, Budapest, 1062 Andrássy út 104. (ma: Orosz Nagykövetség)
1879–1884/85 a II. számú Belklinika (ma: Semmelweis Egyetem II. Számú Belgyógyászati Klinika "A" épülete), 1088 Budapest, Szentkirályi u. 46. – Kolbenheyer Ferenc tervei szerint kezdték el az építését, de a tervező időközben elhunyt, így Weber fejezte be az épületet
1881–1884: Semmelweis Egyetem Központi Igazgatási Épület, 1085 Budapest, Üllői út 26.
1883–1885: Állat- és Ásványtani Intézet, ma: ELTE BTK, 1088 Budapest, Múzeum körút 4.
1883–1885: Természettani Intézet, ma: ELTE BTK, 1088 Budapest, Puskin utca 5.
1884: Angyalföldi Elme- és Ideggyógyintézet épületei, 1135 Budapest, Róbert Károly körút 82-84. (A régi központi épületet 1984-85-ben lebontották, az egykori intézetet a Nyírő Gyula Kórház magába olvasztotta és a kórház elme- és idegosztályaként működik tovább.)
?: egyetemi gazdászati épület
számos magánház.
Vidéken
Zichy-kastély, Nagyhörcsökpuszta
Erdődy-kastély, Vörösvár
Esterházy-kastély, Galánta
az ún. vörös torony
Római katolikus templom, Újszász
Római katolikus templom, Agárd
Restaurálta a cinkotai Beniczky-kastélyt.
Képtár
Jegyzetek
Források
Egyéb külső hivatkozások
https://repozitorium.omikk.bme.hu/handle/10890/608
http://lechnerkozpont.hu/cikk/egy-elfelejtett-nagy-magyar-epitomuveszrol
További információk
Nendtvich Gusztáv: Weber Antal In: Építő Ipar, 1889
Jakabffy Ferenc: Weber Antal és művei In: Magyar Építőművészet, 1907. m. sz.
Ybl Ervin: Weber Antal, Budapest, 1958
Marótzy Katalin: Wéber Antal építészete a magyar historizmusban; Terc, Budapest, 2009,
1823-ban született személyek
1889-ben elhunyt személyek
Magyar építészek
Pestiek
A francia Becsületrend magyar kitüntetettjei | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,640 |
Working class street punk/Oi! from Plymouth, UK.
Arch Rivals formed in 2014, based in Plymouth, UK, they made their live debut in 2015 supporting Peter and the Test Tube Babies. Later that year they made their European debut at Oi! this is Tegelen.
Their debut album "One More Round" was well received and opened the door to gigs up and down the UK and into Germany, France and Switzerland alongside some of the biggest names in punk and Oi!.
With their new album "A Constant State Of War" receiving rave reviews Arch Rivals are bringing their own brand of melodic Oi!/street punk for everyone to sing along to.
Mike Brands - Vocals
Tom Murphy - Guitar & Backing Vocals
Si Foulkes - Guitar & Backing Vocals
Kev Jones - Bass & Backing Vocals
Alex Kanchev - Drums
info@oithisistegelen.nl | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,149 |
Q: How to convert all string in Dart list to lowercase? I want check if Dart list contain string with list.contains so must convert string in array to lowercase first.
How to convert all string in list to lowercase?
For example:
[example@example.com, Example@example.com, example@Example.com, example@example.cOm, EXAMPLE@example.cOm]
A: You can map through the entire list and convert all the items to lowercase. Please see the code below.
List<String> emails = ["example@example.com", "Example@example.com", "example@Example.com", "example@example.cOm", "EXAMPLE@example.cOm"];
emails = emails.map((email)=>email.toLowerCase()).toList();
print(emails);
A: Right now with Dart you can use extension methods to do this kind of conversions
extension LowerCaseList on List<String> {
void toLowerCase() {
for (int i = 0; i < length; i++) {
this[i] = this[i].toLowerCase();
}
}
}
When you import it, you can use it like
List<String> someUpperCaseList = ["QWERTY", "UIOP"];
someUpperCaseList.toLowerCase();
print(someUpperCaseList[0]); // -> qwerty
A: You can try:
List<String> emails = ["example@example.com", "Example@example.com", "example@Example.com", "example@example.cOm", "EXAMPLE@example.cOm"];
for(String email in emails){
print(email.toLowerCase());
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,840 |
Q: Publishing verification result back to pact Broker Does anyone have an example of how to publish verification result back to the pact broker ?
I'm using maven implementation for all phases (Generate/Publish and verify)
The only page I found is this one: https://github.com/pact-foundation/pact_broker/wiki/Provider-verification-results
but it is not clear to me how to implement it by maven provider plugin
A: I believe this is the documentation you are looking for.
https://github.com/DiUS/pact-jvm/tree/master/pact-jvm-provider-maven#publishing-verification-results-to-a-pact-broker-version-354
Be aware that there is currently no configuration option to turn off the publishing of verifications when running verifications from your local machine (Ron will be adding it soon) so you need to ensure that your CI always runs after your local tests!
A: Thanks @Beth
I had to add the following section into my plugin setup in order to publish the results back:
<pactBroker>
<url></url>
<authentication>
<username></username>
<password></password>
</authentication>
</pactBroker>
This is how my plugin looks like:
<plugin>
<groupId>au.com.dius</groupId>
<artifactId>pact-jvm-provider-maven_2.11</artifactId>
<version>3.5.5</version>
<configuration>
<pactBrokerUrl></pactBrokerUrl>
<pactBrokerUsername></pactBrokerUsername>
<pactBrokerPassword></pactBrokerPassword>
<projectVersion>1.0.0</projectVersion>
<serviceProviders>
<serviceProvider>
<name>${project.artifactId}</name>
<protocol>http</protocol>
<host>${K8S_APP_URL}</host>
<port>${K8S_NODE_PORT}</port>
<path>/</path>
<pactFileDirectory>target/pacts</pactFileDirectory>
<pactBroker>
<url></url>
<authentication>
<username></username>
<password></password>
</authentication>
</pactBroker>
</serviceProvider>
</serviceProviders>
</configuration>
</plugin>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,730 |
<!DOCTYPE html>
<html>
<head>
<title>auto-height multicol with break</title>
<style>
.mc > div { padding:0 0.5em; }
</style>
</head>
<body style="min-width:40em;">
<div class="mc" style="-webkit-columns:3; -webkit-column-gap:1em; columns:3; column-gap:1em; orphans:1; widows:1; width:32em; background:olive;">
<div>
first column<br>
first column<br>
first column<br>
first column<br>
second column<br>
second column<br>
second column<br>
</div>
<div style="-webkit-column-break-before:always; break-before:column;">
third column<br>
</div>
</div>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,655 |
Camille Danguillaume (Châteaulin, 4 juni 1919 – Arpajon, 26 juni 1950) was een Frans wielrenner.
Danguilaume was profwielrenner van 1942 tot aan zijn dood in 1950 en won onder andere Luik-Bastenaken-Luik. Hij kwam uit een grote wielerfamilie. Hij was de oudste van vijf fietsende broers en een oom van twee wielrenners, onder wie Jean-Pierre Danguillaume die in de jaren '70 succesvol was. Tijdens het Frans kampioenschap van 1950 werd hij aangereden door een motorrijder. Hij bezweek een aantal dagen later aan zijn verwondingen.
Belangrijkste resultaten
1943
3e Parijs-Camembert
1946
Internationaal Wegcriterium (gedeeld winnaar met Kléber Piot)
1948
Internationaal Wegcriterium
1949
Luik-Bastenaken-Luik
2e Kampioenschap van Zürich
2e Frans kampioenschap
Resultaten in voornaamste wedstrijden
|
|}
Bronnen
Danguillaume, Camille | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 799 |
As part of our member community, we offer many packages to suit different budgets, lifestyles and goals. With private sessions and group sessions, we have an option for every person. Reach out to us today so we can help you find the ideal option for you.
Contact Us For Special Membership Pricing!
Pre-registration for classes is highly encouraged. All classes have limited spots so that there is never a crowded, gym feel.
Classes require at least two attendees or the instructor may cancel the class with two hours notice.
Students may cancel registration for class without penalty as long as the cancellation occurs at least 4 hours prior to the start of class.
Many of our students find that adding a personal lesson to their yoga program is beneficial. The personal lesson is different than a standard class and can include alignment adjustments, as well as a physical assessment for range of motion, strength, flexibility and balance. We recommend the private instruction for ensuring proper alignment in the poses to meet individual needs beyond what can be offered in class alone. Please visit the Personal Lessons page for more information. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,288 |
Rainbow Eclipse
Rainbow Eclipse last won the day on May 17 2014
Rainbow Eclipse received the most brohooves!
raykv423
SpongeBobsLittlePony
Vefka
Lunaris Adamantine
Thunderbolt1000T
About Rainbow Eclipse
I still have no idea who best pony is. Pinkie Pie or Fluttershy?
You. Yeah, I'm watching you like Pinkie Pie.
raykv423 started following Rainbow Eclipse December 14, 2020
SpongeBobsLittlePony started following Rainbow Eclipse April 19, 2017
Open Cutie Mark Crusade!
replied to a topic in Everfree Roleplays
@, "Oh. I do believe I have heard of that game somewhere. Perhaps we could try it out sometime?" said Riley. "I have quite an assortment of games for the Super Neightendo and GameColt. Ponio, Pokémane, Zeldock, Maretroid, Final Foaltasy..." He continued rattling off names.
Open Where Star shines Hearth Warming
@, "We were both thirteen when we met," said Riley. "We're about the same age, except Anala is older than me by a few months. Now we're twenty." "See, it was the first day of a new school year," said Anala. "I saw a bully trying to take Riley's lunch money, and I recognised said bully as the one who had done that same thing to me before when I didn't have a cutie mark. I couldn't stand to see it happen to another pony." "I ran over and told the bully to leave Riley alone. The bully got all cocky until I showed him my flank and said, 'Don't mess with the pony with a bow and arrow fo
I wish I could talk to someone about something bad I did when I was twelve, but I'm too ashamed to even say what it is...
I'll think about it.
MozillaToast
It's OK, we all do things we regret.
Storm Shine
*Huggles Rainbow* I'll be here for you if you want buddy! =)
@, Anala listened to Tempest's backstory. She winced at the mention of having his wing broken. "I started out in a more rural area for the first few years of my life, but then my parents split and my mom left. My dad had to move to Manehattan to get a cheaper living space for me, my sister, and my brother." "I thought that it was going to be the end of all those days I spent frolicking in the fields, but fortunately, there was one in Manehattan. The one we're in as we speak." "I lived in Manehattan all my life," said Riley. "I do make trips to Ponyville monthly, though."
Do you know what Rule 63 is?
I do. Why?
I was wondering about Male!lEloquence (I imagine R63s of pretty much everything). Would he be sort of chivalrous? It seems like a good equivalent to Fem!Eloquence's 'feminist' demeanour (or something along those lines).
I wouldn't say that, per se. I think he'd be kind to everypony, without paying a pony's gender any mind. I suppose there wouldn't be much difference at all aside from looks
@, "That's what the spirit of Hearth's Warming will do to you," Riley told Tempest Shift with a smile. Anala smiled and shook her head at her friend's comment. She then turned to face Tempest. "So, Tempest Shift... where exactly did you say you were from?" she asked.
@, "Hey, it's okay man. If you practice, you won't always be this hapless," she teased Tempest. It was kind of a mean joke, sure, but at least she had said it in a jokey manner as opposed to a blunt one. You could get away with saying anything if you said it in a jokey manner.
@ ,@, @@Yoshi89, "Great idea, One Way!" said Riley. "Anypony here good at video games? We could play my Super Neighte- wait. It's back in Manehattan." Just then, a cardboard-coloured unicorn stallion walked by. "Hi! I'm Plot Device! I couldn't help but notice that you seem like you need the assistance of magic! Is that so?" he asked Riley. 'Well, that was convenient,' thought Riley. "Uh, yes please. Can you teleport me to Manehattan and back?" Plot Device lit his horn and the two stallions disappeared. ~~~Approximately half a minute later~~~ They reappeared, and Riley
@, "Just be grateful that it was the blunt side that hit you," said Anala, looking down at Tempest. She picked up the bow and the arrow, pulled back, and shot the arrow at the target, landing it in the ring directly around the bullseye. "Ready to try again?" she asked.
@, Giving the ten bits back to Tempest, Anala led him to the destination. "Do you know how to hold a bow and arrow?" she asked. "Since you're a Pegasus, you should be able to hold it with your front hooves while in mid-air." "I couldn't fly when Anala first taught me," said Riley. "So I sat on her back. She's plenty stronger than me, since she's an Earth Pony."
@, Anala was curious about what crazy story would be told about how Riley came to meet Tempest Shift, but she decided to ask later. "Nice to meet you, Tempest Shift. Anala Elderberry," she introduced herself. "And yup, I teach archery." She grabbed her bow and an arrow, stood up on her hind legs, and shot, landing a bullseye. "Wanna try it?" she asked. "The normal price is ten Bits for an hour's session, but you can have a go for free as a Hearth's Warming's Eve present."
@, Riley listened as Tempest gave his explanation. "Sounds pretty cool!" he said. "Oh, here we are!" They had arrived at the field on the outskirts of the city. There was an archery board, a bow, plenty of arrows, and a supply shed. A white mare with a dark grey braided mane and tail and a beret the same colour as her coat was sitting there, surveying the field and the sky. "Hey, Anala!" called Riley. Anala turned to the source of the voice. "Hey there, Rye," she greeted. "Who's this you have with you?"
@, (OOC: BAM PLOT TWIST) "Yup! I'm reddy to go!" answered Riley, gesturing a hoof towards his red coat. And with that, the two stallions began heading down towards their destination, Riley leading the way. "By the way, what's that glowing thing in your bag?" he asked Tempest.
Ask Soarin!
replied to a topic in Ask a Pony
@@TheScrewUpCrew, *points to restaurant*
@@TheScrewUpCrew, ...There was a bug on it. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,215 |
title: Verdipeing
date: 29/07/2022
---
## Extreme hitte
### 2 Korintiërs 11:22-30
#### IJsbrekervraag
`Als je op je leven tot nu toe terugkijkt, zijn er dan periodes geweest waarin andere mensen je het enorm moeilijk maakten, en zo ja, wat vond je het moeilijkst te verdragen?`
#### Ik Verken
Zie de verdiepingsles van 9 juli jl. voor gegevens over de achtergrond van 2 Korintiërs. In 2 Korintiërs 11:23-28 geeft Paulus details van de 'beledigingen, nood, vervolging en ellende' (12:10) die hij tot dan toe had ondervonden. Dat was zo intens en heftig, dat het vergelijkbaar is met de uitzonderlijke hitte van het vuur waarin goud wordt gelouterd, maar in dit geval ging het om Paulus die mentaal en geestelijk zware beproevingen te verduren had gehad. Vreemdelingen, volksgenoten, schijngelovigen en zelfs geloofsgenoten maakten het hem bij tijden heel moeilijk.
#### Ik Bestudeer
De opsomming in 2 Korintiërs 11:22 (Hebreeën – Israëlieten – nakomelingen van Abraham) duidt erop dat Paulus te maken had met judaïsten die onder de Korintiërs propaganda maakten voor de Joodse instellingen en godsdienst. In vers 23 brengt hij – tegen zijn zin – naar voren, dat hij Christus meer dient dan zij doen. Tot de volksgenoten die hem bedreigden (vers 26) behoorden ook de in vers 24 vermelde groep 'Joden' of 'Judeeërs.
`Hoe bedreigden de in Handelingen 9:23,29 en 13:50 vermelde Joden Paulus?`
Paulus ondervond ook tegenstand van vreemdelingen (vers 26).
`Welke dreiging ging er voor Paulus uit van het volksoproer in Handelingen 19:21-40?`
`Welke hulp kreeg Paulus, en van wie?`
Daarnaast ervoer Paulus dagelijks druk vanwege zijn zorg voor de gemeenten (vers 28). Dat betekent dat ook geloofsgenoten het hem moeilijk maakten.
`Waaruit bestond de moeite die Paulus in 1 Korintiërs 5:11 en 2 Korintiërs 2:4 beschrijft?`
Tot de in 2 Korintiërs 11:26 genoemde 'schijngelovigen' behoorden ook de 'schijnapostelen' waarnaar Paulus in 2 Korintiërs 11:13 verwijst.
`Wat zegt hij over hen in 2 Korintiërs 11:4,19?`
Vers 24: Dit vers komt overeen met de bepaling in Deuteronomium 25:3. Het gaat hier om een straf van ten hoogste 40 slagen, de Joodse Misjna schrijft in Makkot 3.10 ten hoogste 39 slagen voor, om ervoor te zorgen dat nauwkeurig volgens het voorschrift van Deuteronomium 25:3 werd gehandeld. In Marcus 13:3,9 waarschuwde Jezus al Petrus, Jakobus, Johannes en Andreas, dat zij voor het gerecht gesleept en in synagogen gegeseld zouden worden. Mogelijk werd Paulus niet alleen gegeseld, maar ook uit synagogen verbannen. Vers 25: Vgl. Handelingen16:22 (stokslagen), 14:19 (steniging) en 27:41 (schipbreuk). Vers 29: Paulus geeft in dit vers aan dat hij 'in woede ontsteekt' wanneer iemand tot iets verkeerds wordt verleid.
`Wat zou hij hiermee bedoelen?`
`Psalm 103:13-14; Matteüs 28:20; 1 Korintiërs 10:13; 1 Petrus 1:7 Wat zeggen deze verzen over God?`
#### Ik Pas Toe
Paulus offerde zijn leven op voor de verspreiding van het evangelie. Vergelijk 1 Korintiërs 4:11-13, 2 Korintiërs 6:4-10 en 1 Petrus 2:19-23 met elkaar.
`Wat kunnen we leren van deze Bijbelgedeelten, voor het geval we in een moeilijke situatie terechtkomen of dat anderen ons onverdiend kwaad aandoen?`
`Welk nut kunnen beproevingen en verwondingen hebben, als we daarmee te maken krijgen omdat we ons christen-zijn uitdragen in woord of daad?`
In elke gemeente kun je te maken krijgen met valse leer of onderlinge conflicten.
`Hoe reageer jij daarop, wanneer je zoiets in jouw gemeente signaleert?`
`Wat zegt Paulus in Galaten 1:6-9 over dwaalleraars?`
`Hoe omschrijven 1 Johannes 4:1-3 en 2 Johannes 7 dergelijke personen?`
We hebben gelezen hoe Jezus en Paulus omgingen met moeilijke situaties in hun leven.
`Wat herken je hiervan in jouw eigen ervaring als christen of in het leven van iemand uit je omgeving?`
`Op welke manier denk je dat jij een broeder of zuster kunt helpen, en waarbij heb je zelf hulp nodig?`
#### Ik bid …
Heer, leer ons in alle omstandigheden waarin wij terechtkomen een beroep te doen op het geduld, de kracht en volharding die alleen U ons kunt geven. Dank U, Here Jezus, voor alles wat U hebt ondergaan, om ons uitzicht te geven op een leven zonder strijd, pijn en moeite. Amen. | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,415 |
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