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{"url":"http:\/\/openstudy.com\/updates\/506074b9e4b02e139410c8f2","text":"1. andriod09 Group Title\n\n|dw:1348498599367:dw| ik what the answers are, i just need help finding how to get them They are 5\", 12\", and 13\"\n\nThe Theorem is a^2 + b^2 = c^2 Where a and b are sides of the right triangle and c is the hypotenuse of the right triangle.\n\n3. andriod09 Group Title\n\nik the thirum, i just havn't done it in about 3 years.\n\nOk. Let a = x and b = x+7 therefore, (x)^2 + (x+7)^2 = (8+x)^2\n\n5. andriod09 Group Title\n\nokay, then what?\n\n6. andriod09 Group Title\n\n7. andriod09 Group Title\n\n@cshalvey @Callisto @Yahoo!\n\n8. Yahoo! Group Title\n\n$(x+8)^2 = x^2 + (x+7)^2$\n\n9. Yahoo! Group Title\n\nnw expand this....... $(a+b)^2 = a^2 + 2ab + b^2$\n\n10. andriod09 Group Title\n\nplease do not use the equation button i have a mathjax error and it annoys me. i see the slashes and brackets and braces.\n\n11. erica.d Group Title\n\n(x+8)^2=x^2+(x+7)^2\n\n12. Yahoo! Group Title\n\nlol.... ) (x)^2 + (x+7)^2 = (8+x)^2\n\n13. andriod09 Group Title\n\nokay?? then you do??????\n\n14. Callisto Group Title\n\nExpand the terms for both sides of the question, then simplify it. As Yahoo! mentioned, you can expand the term (a+b)^2 by using the identity (a+b)^2 = a^2 + 2ab + b^2. If you don't know where this identity comes from, you can simply perform multiplication for the term, i.e. (a+b)^2 = (a+b)(a+b), now expand it. What have you got for now?\n\n15. andriod09 Group Title\n\n|dw:1348501546848:dw|\n\n16. andriod09 Group Title\n\n@Callisto that made no sense to me. :c\n\n17. Callisto Group Title\n\nMay I know what made no sense to you?\n\n18. andriod09 Group Title\n\n\"Expand the terms for both sides of the question, then simplify it. As Yahoo! mentioned, you can expand the term (a+b)^2 by using the identity (a+b)^2 = a^2 + 2ab + b^2. If you don't know where this identity comes from, you can simply perform multiplication for the term, i.e. (a+b)^2 = (a+b)(a+b), now expand it.\"\n\n19. Callisto Group Title\n\nWhich part you don't understand?\n\n20. andriod09 Group Title\n\nthe expansion parts.\n\n21. Callisto Group Title\n\nDo you know how to expand (a+b)^2?\n\n22. andriod09 Group Title\n\nno. im homeschooled, i havn't even done pythagereum theory much.\n\n23. erica.d Group Title\n\nx^2+16x+64=x^2+x^2+14x+49\n\n24. Callisto Group Title\n\nThat is not Pyth. Thm. It's just simple multiplication of some factors. For instance, $x^2 = x \\times x$$x^3 = x \\times x \\times x$ Similarly, $(a+b)^2 = (a+b) \\times (a+b)$It can be expanded in this way: $(a+b)^2 = (a+b) \\times (a+b) = a(a+b) + b(a+b)$Can you further expand and simplify it?\n\n25. andriod09 Group Title\n\ni can not see the equations from the eqtation button!!!!!!!!!!\n\n26. Callisto Group Title\n\nequation button is for you to type latex...\n\n27. Callisto Group Title\n\n*type in latex\n\n28. andriod09 Group Title\n\nyes, but i see: $\\$$\\\\$][\\\\]$\\$ thats what i see, i see evey single slash, all the brackets, the braces, everything\n\n29. Callisto Group Title\n\n\\sqrt{x} = $$\\sqrt{x}$$ and so on.. You can try these on the equation button. Though, back to your question, are you still having trouble with your question?\n\n30. andriod09 Group Title\n\nyes. and i see the \\sqrt{x} = $$\\sqrt{x}$$ like i see all the braces, the terms for them, and everything like that. my mathjax thing isn't working atm\n\n31. Callisto Group Title\n\nCan you see what I've typed in latex?\n\n32. andriod09 Group Title\n\nno. i see the equation as if you write it with out the button, like i don't see the (x)^2 i see the ()^{} and things like that. and yes, i still am having problems with my problem\n\n33. Callisto Group Title\n\n|dw:1348503516047:dw|\n\n34. andriod09 Group Title\n\nyes.\n\n35. Callisto Group Title\n\nNow, can you expand a(a+b) +b(a+b) ?\n\n36. andriod09 Group Title\n\nhow?\n\n37. Callisto Group Title\n\nDistributing a into a+b and b into a+b\n\n38. andriod09 Group Title\n\nthen??\n\n39. Callisto Group Title\n\nExpand it first.\n\n40. andriod09 Group Title\n\nhow?\n\n41. Callisto Group Title\n\na(a+b) = a(a) + a(b) = a^2 + ab Can you try b(a+b) now?\n\n42. andriod09 Group Title\n\nis it: b(a+b)=b(a)+b(b)=b^2+ba?\n\n43. Callisto Group Title\n\nYes! Now combine them What is a(a+b) + b(a+b) ?\n\n44. andriod09 Group Title\n\n(a(a+b) = a(a) + a(b) = a^2 + ab)+(b(a+b)=b(a)+b(b)=b^2+ba)?\n\n45. Callisto Group Title\n\nNope not really.. can you try again?\n\n46. andriod09 Group Title\n\nwhat would it be then?\n\n47. andriod09 Group Title\n\n@ash2326\n\n48. andriod09 Group Title\n\n@ganeshie8\n\n49. andriod09 Group Title\n\n@Callisto\n\n50. andriod09 Group Title\n\n@ash2326\n\n51. andriod09 Group Title\n\n52. andriod09 Group Title\n\n|dw:1348510329568:dw|","date":"2014-07-23 08:02:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8861767053604126, \"perplexity\": 8000.881079522854}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-23\/segments\/1405997877644.62\/warc\/CC-MAIN-20140722025757-00105-ip-10-33-131-23.ec2.internal.warc.gz\"}"}
| null | null |
Education to take spotlight at 17th COHSOD meeting
Oct 15, 2008 News 0
On 18-19 November, 2008, the Caribbean Community (CARICOM) Secretariat will again, after ten years, bring together Ministers of Education and their technical advisors for a comprehensive evaluation of the last ten years of the Community's work in education.
This Seventeenth Meeting of the CARICOM Council for Human and Social Development (COHSOD) with special focus on education has been described as "a meeting for frank discussion on how we can do things better in education within the Region."
The first COHSOD on education was held in 1998 under the Revised Treaty of Chaguaramas and, according to Dr Edward Greene, CARICOM Assistant Secretary-General for Human and Social Development, this COHSOD would be one with a difference.
"The difference is that we would not only be looking at the achievements over the last ten years but also looking at those things that did not get implemented for one reason or the other; and in so doing examine how we can expedite the rate of implementation…by understanding some of the reasons why some things did not get done in the first place," explained Dr Greene.
Dr Greene, who was briefing the Public Information Unit of the CARICOM Secretariat, further explained that since the first COHSOD on education in 1998, a number of policies and programmes in education had evolved and it was now time to assess their impact on the Community.
He added that every effort would be made to ensure that all Ministers of Education attend the meeting. He noted that with the change of Governments over the past ten years, there were seven new Ministers of Education who should view this as an opportune time to be engaged in action-planning for the future of education in the Region.
"There is a general perception that the Secretariat does not do things quickly," Dr Greene noted, "but sometimes the reasons are rooted at the country level, hence the necessity to engage the Ministers at the country level in meetings such as the COHSOD," he continued.
Besides the comprehensive evaluation to be undertaken, the COHSOD will engage Education Ministers in critically examining the implications of various components of the Economic Partnership Agreement (EPA) for education as well as ways in which the education sector could tap into the opportunities it holds for the Region.
Scheduled to be held here, the 17th COHSOD, Dr Greene stated, would also identify and discuss the elements in education that could help to promote the CARICOM Single Market and Economy, (CSME).
He cited the establishment of the Caribbean Vocation Qualification (CVQ) which was launched in 2007 as one of the standards which would help to facilitate the movement of skilled persons other than University graduates within the Region.
Teacher education and training will also be one of the agenda items Dr Greene anticipates will generate much discussion. The COHSOD is expected to examine the treatment of teacher education and training, especially in this age of New Information technologies: "We developed and promoted a science and technology policy in 1998 and we now need to assess how we have advanced on that and to address its relevance of those recommendations from 1998 to what we have to do in 2008," the Assistant Secretary-General explained.
In addition, Dr Greene noted that "the new dimension of teacher education means coming to grips with the sociology of the environment and exploring distance education to reach the perceived 'un-reachable." As a result, the 17th COHSOD will also review the work of the Caribbean Knowledge Learning Network (CKLN) which was established in 2004 to foster the upgrading of tertiary institutions across the Region in an effort to increase their ability to use modern approaches to learning; and make recommendations on how this tool could be further maximized in facilitating greater collaboration between tertiary institutions in reaching a wider cross-section of the Community's students.
Also at the COHSOD, the New Vision for the Caribbean Examinations Council (CXC) as proposed by the new Director Dr Didacus Jules will also be presented. According to Dr Greene, "we are hoping that this new vision will engender vigorous discussion among Ministers and provide useful insights on where the Council goes from here as a credible institution making a greater contribution to learning and education in the Region."
> Stakeholder engagement is important to tourism development
> Vendors and M&CC clean up Stabroek Bazaar and...
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"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"https:\/\/www.nature.com\/articles\/s41467-018-07085-1\/?error=cookies_not_supported&code=6757b816-b40e-40b0-97b1-ebe76ab1a9b8","text":"Article | Open | Published:\n\n# An information-theoretic framework for deciphering pleiotropic and noisy biochemical signaling\n\n## Abstract\n\nMany components of signaling pathways are functionally pleiotropic, and signaling responses are marked with substantial cell-to-cell heterogeneity. Therefore, biochemical descriptions of signaling require quantitative support to explain how complex stimuli (inputs) are encoded in distinct activities of pathways effectors (outputs). A unique perspective of information theory cannot be fully utilized due to lack of modeling tools that account for the complexity of biochemical signaling, specifically for multiple inputs and outputs. Here, we develop a modeling framework of information theory that allows for efficient analysis of models with multiple inputs and outputs; accounts for temporal dynamics of signaling; enables analysis of how signals flow through shared network components; and is not restricted by limited variability of responses. The framework allows us to explain how identity and quantity of type I and type III interferon variants could be recognized by cells despite activating the same signaling effectors.\n\n## Introduction\n\nBiochemical signaling is a key mechanism to coordinate an organism in all aspects of its function. In a typical example, cells detect extracellular stimuli (input), e.g., growth factors, cytokines, or chemokines, with specific transmembrane receptors binding a ligand, which results in a biochemical activity on the inside of the cell, for example, the activation of a receptor-associated kinase1. The initial stimuli are processed in an intracellular relay mechanism and culminate in effectors (output), which might be transcription factors. The effectors carry information about the identity and intensity of the stimuli in order to initiate distinct cellular responses, which might involve gene transcription, or any other cellular process. Biochemical descriptions do not directly lead to understanding how the stimuli are translated into distinct responses as signaling processes are immensely complex2,3. Many components of signaling pathways are functionally pleiotropic:4,5,6 (i) a single stimulus often activates multiple effectors, (ii) a distinct effector can be activated by numerous stimuli, and (iii) signals triggered by different stimuli often travel through shared network components. Besides, (iv) biochemical signaling processes are intrinsically stochastic and responding cells exhibit quite varied behaviors when examined individually7,8, and (v) temporal dynamics of signaling in individual cells is correlated with physiological responses6,9,10. In light of these observations, understanding of how information about a complex mixture of extracellular stimuli is processed and translated into distinct cellular responses remains deficient2,3. For instance, human type I and type III interferons (IFNs) signal through distinct cell-surface receptors that appear to induce a shared signaling pathway. Yet, they can evoke different physiological effects11. The mechanism mediating this differential activity and signaling through common pathways remains largely unknown11,12,13,14,15.\n\nUnderstanding how cellular signaling processes can derive a variety of distinct outputs from complex inputs appears to be beyond solely experimental treatment. Therefore, an adequate modeling formalism is required. Following Berg and Purcell16, probabilistic modeling has been applied to examine fidelity of receptors as well as more complex biochemical signaling systems19. Specifically, information theory has been deployed as an integrated measure of signaling accuracy, a term known as information capacity, C*. Information capacity is expressed in bits, and generaly speaking, $$2^{C^ \\ast }$$ represents the maximal number of different inputs that a system can effectively resolve (e.g. different ligand concentrations)17. So far, both experimental and computational analysis of biochemical signaling within information theory revealed several unique aspects of how signaling pathways transmit information18,19,20,21,22,23,24. A tangible obstacle to further utilize the potential of information theory is the lack of computationally efficient tools that can account for complexities of biochemical signaling. Existing techniques are based on Blahut\u2013Arimoto algorithm18,25, small noise approximation19,26, or density estimation22 and their application so far has been limited to relatively simple systems, usually with one input and one output only. As analysis of systems with multiple inputs and outputs appears to be essential for deciphering of biochemical signaling27,28,29, we currently need new tools to study such systems. Here, we developed a computational framework of information theory that alleviates several drawbacks of existing tools, primarily by allowing efficient analysis of complex models with multiple inputs and multiple outputs. The method allowed us to provide an insight to one of the long-standing problems in signaling: how type I and type III interferon signaling can be recognized by cells despite activating the same signaling effectors.\n\n## Results\n\n### Quantification of information transfer in signaling systems\n\nWithin information theory, a signaling system is typically considered as a probability distribution P(Y|X\u2009=\u2009x) that for a given level of input, x, elicits output, Y. In a typical example, the input is the concentration of a ligand that activates a receptor, and the output is the activity of a signaling effector, which might be the nuclear concentration of an activated transcription factor. The output, Y, carries information about the level of the input, x. How much information is transferred depends on the signaling system itself, i.e., on its noise levels and sensitivity to changes of input values, as well as on how frequently different input values are transmitted. To illustrate the latter, consider two possible sets of input values. One set of input values generates similar and\/or irreproducible outputs, while the other generates distinct and reproducible outputs. If a pathway encounters signals from the first set more frequently than from the second one, its information transfer will be on average lower. The mutual information, I(X,Y), quantifies information transfer of a given signaling system, P(Y|X\u2009=\u2009x), that encounters input values following a given distribution, P(X) (see Methods). The maximal mutual information, with respect to all input distributions, termed information capacity, C*, quantifies information transfer under the most favorable distribution of input values\n\n$$C^ \\ast = \\begin{array}{*{20}{c}} {} \\\\ {{\\mathrm {max}}} \\\\ {P(X)} \\end{array}I(X,Y).$$\n(1)\n\nThe distribution for which the maximum of mutual information is achieved is called the optimal input distribution and denoted as P*(X). The information capacity, C*, is expressed in bits, and $$2^{C^ \\ast }$$ can be interpreted, within the Shannon\u2019s coding theorem17,30,31, as the number of input values that the system can effectively resolve based on the information contained in the output. For instance, if C*\u2009=\u20092, there exist four concentrations that can be distinguished with, on average, negligible error. Available approaches to compute information capacity are briefly described in Methods, whereas more background on information theory is provided in Section 1 of Supplementary information (SI).\n\n### Efficient calculation of information capacity in complex models\n\nIn a general setting, calculation of the information capacity, C*, is computationally expensive, if not prohibitive. Here, we propose a framework to study information flow in biochemical signaling models that alleviate several of the important drawbacks related to available approaches19,22,25,32. Specifically, the proposed framework is based on analytical solutions. This, in turn, leads to an efficient computational algorithm that accounts for the complexity of signaling, most importantly multidimensional inputs and outputs.\n\nWe propose to calculate the information capacity, using an asymptotic approach. Precisely, we consider a system with an output, YN\u2009=\u2009(Y(1),...,Y(N)), that consists of N independent copies of $$Y\\sim P( \\cdot |X = x)$$, where Y itself can be multidimensional, e.g., a time series of induced levels of transcription factors. Biologically, N can be interpreted as the number of cells that independently sense the signal, X. The corresponding information capacity problem is then written as\n\n$$C_N^ \\ast = \\begin{array}{*{20}{c}} {} \\\\ {{\\mathrm {max}}} \\\\ {P_N(X)} \\end{array}I(X,Y_N).$$\n(2)\n\n$$C_N^ \\ast$$ quantifies information about the input, X, jointly stored in N cells. For large N, Eq. (2) has an exact and computationally efficient solution based on the Fisher information matrix (FIM),\n\n$${\\mathrm{FIM}}_{ij}(x) = {\\Bbb E}\\left[ {\\frac{{\\partial {\\mathrm{log}}P(Y|X = x)}}{{\\partial x_i}}\\frac{{\\partial {\\mathrm{log}}P(Y|X = x)}}{{\\partial x_j}}} \\right],$$\n(3)\n\nwhere i and j refer to elements of the vector x\u2009=\u2009(x1,\u2026,xk), i.e., multidimensional input. Specifically, it has been shown in the statistical theory of reference priors30,33 that if FIM is non-singular, i.e., all inputs have a non-redundant impact on the output, then\n\n$$P_N^ \\ast (x)\\mathop{\\longrightarrow}\\limits_{{N \\to \\infty }}P_{{\\mathrm{JP}}}^ \\ast (x),$$\n(4)\n\nwhere\n\n$$P_{{\\mathrm{JP}}}^ \\ast (x) \\propto \\sqrt {|{\\mathrm{FIM}}(x)|} ,$$\n(5)\n\nand || denotes the matrix determinant. The distribution $$P_{{\\mathrm{JP}}}^ \\ast (x)$$ is known as the Jeffrey prior (JP). Similarly, it can be shown, see Section 1.4 SI and ref. 30, that\n\n$$C_N^ \\ast - \\frac{k}{2}{\\mathrm{log}}_2(N)\\mathop{\\longrightarrow}\\limits_{{N \\to \\infty }}C_{\\mathrm{A}}^ \\ast ,$$\n(6)\n\nwhere k is the dimension of input, and\n\n$$C_{\\mathrm{A}}^ \\ast = {\\mathrm{log}}_2\\left( {(2\\pi e)^{ - \\frac{k}{2}}{\\int}_{\\hskip -5pt \\mathscr{X}} \\sqrt {|{\\mathrm{FIM}}(x)|} {\\mathrm d}x} \\right),$$\n(7)\n\nwhere $${\\mathscr{X}}$$ is the space of signal values, x.\n\nAs suggested by Eq. (6), we will call $$C_{\\mathrm{A}}^ \\ast$$ as the asymptotic information capacity, where asymptotics is meant with respect to the number of cells, N. Equation (6) implies that $$C_{\\mathrm{A}}^ \\ast$$ can be used to approximate the joint capacity of N cells\n\n$$C_N^ \\ast \\approx C_{\\mathrm{A}}^ \\ast + \\frac{k}{2}{\\mathrm{log}}_2(N).$$\n(8)\n\nThe approximation demonstrates that the joint capacity of N cells depends on the baseline, asymptotic, capacity, $$C_{\\mathrm{A}}^ \\ast$$, and on the number of cells via $$\\frac{k}{2}{\\mathrm{log}}_2(N)$$, where the latter term vanishes for N\u2009=\u20091. Therefore, in terms of Eq. (8), the asymptotic capacity $$C_{\\mathrm{A}}^ \\ast$$ can be interpreted as the contribution of an individual cell to the capacity of an ensemble of N cells. Equivalently, the number of inputs resolvable by N cells increases linearly with $$N^{\\frac{k}{2}}$$ at the rate $$2^{C_{\\mathrm{A}}^ \\ast }$$\n\n$$2^{C_N^ \\ast } \\approx 2^{C_{\\mathrm{A}}^ \\ast } \\cdot N^{\\frac{k}{2}}.$$\n(9)\n\nIn terms of Eq. (9), the asymptotic capacity $$C_{\\mathrm{A}}^ \\ast$$ defines a rate, at which the number of resolvable states increases with N.\n\nImportantly, asymptotic capacity, $$C_{\\mathrm{A}}^ \\ast$$, can take negative values, which has a precise interpretation. The scaling law of Eqs. (8) and (9), which is warranted to be correct by convergence in Eq. (6), implies that $$C_{\\mathrm{A}}^ \\ast$$ must be allowed to take negative values. If $$C_{\\mathrm{A}}^ \\ast$$ was guaranteed to be positive then, any signaling system composed of N cells would be guaranteed to have the capacity $$C_N^ \\ast$$ larger than $$\\frac{k}{2}{\\mathrm{log}}_2(N)$$, which obviously is not the case. In other words, if the number of resolvable inputs $$2^{C_N^ \\ast }$$ increases slowly with N then $$2^{C_{\\mathrm{A}}^ \\ast }$$ must be accordingly small, which means negative $$C_{\\mathrm{A}}^ \\ast$$. For illustration, consider two systems with asymptotic capacities, $$C_{\\mathrm{A}}^ \\ast$$, of, say, \u22121 bit and 1 bit. Then, the capacity $$C_N^ \\ast$$, of the first is smaller by 2 bits compared to the second, for large N. Equivalently, the number of resolvable inputs of the first systems increases at the fourth of the rate of the latter. Besides, Eq. (7), implies that for systems with low Fisher information the number of resolvable inputs increases slowly with N.\n\nConveniently, $$C_{\\mathrm{A}}^ \\ast$$ reduces the problem of calculating the information capacity to the problem of calculating the FIM and we propose to take advantage of this. Fisher information can be calculated for systems with multiple inputs and outputs, and therefore the above approach allows simple computation of information capacity for such systems. To the best of our knowledge, this method has not been used to analyze biochemical signaling, most likely due to technical difficulties in calculating the FIM, which was, to a considerable degree, alleviated by methods recently developed34,35. In Section 6 of SI we discuss in details how FIM can be calculated in different scenarios.\n\n### Asymptotic capacity does not deviate substantially from non-asymptotic capacity in the test model\n\nThe asymptotic capacity, $$C_{\\mathrm{A}}^ \\ast$$, and the capacity of an individual cell, $$C_1^ \\ast$$, are related but not the same quantities. As we discuss in Section 1 of SI, differences arise from non-identical optimal input distributions of single cells and population of cells as well as the way in which information from different cells adds up. In the literature, so far, the interest in $$C_1^ \\ast$$ is dominating. Therefore, even though $$C_{\\mathrm{A}}^ \\ast$$ has a meaningful interpretation on its own, we have compared values of $$C_{\\mathrm{A}}^ \\ast$$ and $$C_1^ \\ast$$ in a test model. Blahut\u2013Arimoto algorithm was used to calculate the exact $$C_1^ \\ast$$. In the comparison we have also included the established, and virtually the only available method to approximate $$C_1^ \\ast$$, i.e., the small noise approximation19, denoted here as $$C_{{\\mathrm{SN}}}^ \\ast$$. We have designed a test model, for which all methods are computationally feasible, and which challenges the assumption of our method, i.e., asymptotics, and of the small noise approximation, i.e., limited stochasticity. Precisely, we considered a model of a biochemical sensor described by the binomial distribution $$Y\\sim {\\mathrm{Bin}}(h(S),L)$$ with the output Y being the number of active sensors and L being the copy number of sensors. The probability of the sensor being active was assumed to be the Michaelis\u2013Menten function, h(S)\u2009=\u2009S\/H\/(1\u2009+\u2009S\/H), with S\u2009=\u2009X + XF\/\u03bb, where X is the concentration of a cognate and XF of a non-cognate ligand, and \u03bb is the selectivity factor (the ratio of the binding affinities, Kd\u2019s, of the non-cognate and cognate ligands, $$\\lambda = \\frac{{H_{\\mathrm{F}}}}{H}$$). The higher the value of \u03bb, the less likely the receptor binds the false ligand. We have assumed that the concentration of the true ligand, X, is the input of the system and varies according to the optimal input distribution, P*(X), whereas the variability of the non-cognate ligand, modeled as the probability distribution P(XF), is the source of noise that leads to information loss. For calculation of $$C_1^ \\ast$$ with Blahut\u2013Arimoto algorithm, we used a complete model without any approximations.\n\nChanging the settings of this model allowed us to challenge the tested methods thoroughly. In total, we have considered 27 different scenarios by combining different variants of the probability distributions P(XF); sensor copy number, L; and of the selectivity factor, \u03bb. In each scenario, we have calculated capacities as a function of the standard deviation of the distribution P(XF), denoted as $$\\sigma _{X_{\\mathrm{F}}}$$. Relative deviations of $$C_{\\mathrm{A}}^ \\ast$$ and $$C_{{\\mathrm {SN}}}^ \\ast$$ from $$C_1^ \\ast$$, averaged over all considered scenarios of the test model, are presented in Fig.\u00a01, whereas comparison for each scenario is presented in Supplementary Figures 1\u20133. For limited variability, i.e., small $$\\sigma _{X_{\\mathrm{F}}}$$, both methods have similar accuracy. When the variability increases both methods become less accurate; however, $$C_{\\mathrm{A}}^ \\ast$$ has half lower error compared to $$C_{\\mathrm{SN}}^ \\ast$$. High variability violates the assumption of the small noise approximation, which explains higher error for high $$\\sigma _{X_{\\mathrm{F}}}$$. Lower approximation accuracy of $$C_{\\mathrm{A}}^ \\ast$$ results from the lack of asymptotics, i.e., N\u2009=\u20091. When using $$C_{\\mathrm{A}}^ \\ast$$ or $$C_{\\mathrm{SN}}^ \\ast$$ as approximations of $$C_1^ \\ast$$, which is a positive quantity, one should monitor for negative values and set approximation to zero. Supplementary Figure\u00a01 shows that in the test model both approximations fell below zero for several model settings, specifically these corresponding to low copy number and highest considered $$\\sigma _{X_{\\mathrm{F}}}$$. In Section 1.5 of SI we present an auxiliary approximation of $$C_1^ \\ast$$ that is guaranteed to be positive, but is computationally not as efficient as $$C_{\\mathrm{A}}^ \\ast$$.\n\nIn summary, our numerical analysis demonstrates that $$C_{\\mathrm{A}}^ \\ast$$ provided a more accurate approximation of $$C_1^ \\ast$$ than $$C_{{\\mathrm {SN}}}^ \\ast$$. The approximation error is at the order of maximum 30%, which indicates that the asymptotic capacity, $$C_{\\mathrm{A}}^ \\ast$$, served as a reliable approximation of the capacity of an individual cell, $$C_1^ \\ast$$.\n\n### Information transmission is maximized when frequent signals are recognized with high precision\n\nHow much information is transferred in a given signaling system depends on three factors: (i) sensitivity of the output to changes in the input, (ii) variability of output given input, and (iii) how frequently do different inputs occur. The first two are modeled by the input\u2013output distribution, P(Y|X\u2009=\u2009x), and the third is represented by the maximization problem in Eq. (2). Here we show that our approach allows for an insightful interpretation of the input distribution that is optimal for signaling. Precisely, consider an asymptotically efficient estimator, $$\\hat x(Y_N)$$ value, x, i.e., an estimator that achieves lowest possible variance for large data, e.g., maximum likelihood estimator. Then, the variance of this estimator, $$\\Sigma (\\hat x(Y_N))$$, is asymptotically described by the inverse of the Fisher information\n\n$$\\Sigma (\\hat x(Y_N))\\mathop{\\longrightarrow}\\limits^{{N \\to \\infty }}(N \\cdot {\\mathrm{FIM}}(x))^{ - 1}.$$\n(10)\n\nGiven the above, the optimal distribution of inputs, $$P_{{\\mathrm{JP}}}^ \\ast (x) \\propto \\sqrt {|{\\mathrm {FIM}}(x)|}$$, is defined in terms of the uncertainty of inferences, $$\\Sigma \\left( {\\hat x(Y_N)} \\right)$$, that cells can draw about the input value, x. Precisely, for large N, $$P_{{\\mathrm{JP}}}^ \\ast (x) \\propto 1\/\\sqrt {\\left| {\\Sigma \\left( {\\hat x(Y_N)} \\right)} \\right|}$$. Therefore, the optimal distribution, $$P_{{\\mathrm{JP}}}^ \\ast (x)$$, states that the system performs best in terms of the information capacity if frequent values are recognized and processed with high precision, whereas more rarely occurring signals need not be transmitted with similarly high accuracy. This is visualized in Fig.\u00a02 for a scenario with a one-dimensional input: in the optimal scenario signals occur at a frequency that is proportional to the inverse of the uncertainty measured as the standard deviation of the estimate of the signal, $$\\sqrt {\\Sigma (\\hat x(Y_N))}$$.\n\nSignaling precision is also closely related to the discrimination error. This is relevant as the information capacity per se does not indicate which exact states can be effectively discriminated. Precisely, consider two close input values x0 and x1, and the probability, \u03b5(x0, x1, YN), of not detecting the change x0\u2009\u2192\u2009x1 based on observations YN. Within the statistical framework of hypothesis testing, the probability \u03b5(x0, x1, YN) is approximated as17\n\n$$\\varepsilon (x_0,x_1,Y_N) \\approx {\\mathrm e}^{- N(x_1 - x_0){\\mathrm {FIM}}(x_0)(x_1 - x_0)^{T}}.$$\n(11)\n\nTherefore, changes in input concentrations that are easily recognized are these along sensitive directions of the FIMs. These directions and can be determined in our framework.\n\n### Signaling dynamics allows discrimination between identity and quantity of type I and type III interferons\n\nIn order to demonstrate how our method can be applied to provide a unique insight regarding the functioning of signaling pathways, we have addressed the problem of the type I and type III interferons signaling. Both IFN types induce the same signaling effectors and it is currently not clear how their identity and quantity is recognized by cells to induce distinct physiological responses12,13,14,15,36. Both IFN types have several variants and here we have selected IFN-\u03b1 and IFN-\u03bb1 as representatives of type I and type III IFNs, respectively. IFN-\u03b1 exerts its action through cognate two subunits receptor complex IFNAR1\/IFNAR2, whereas IFN-\u03bb1 signals through two subunit receptor complex IFNLR1\/IL10R\u03b1. Simplistically, receptor ligand binding induces a cascade of events. The cascade culminates with phosphoryled forms of STAT1 and STAT2 proteins translocating to the nucleus as homodimers (p-STAT1\/1) and heterodimers (p-STAT1\/2), where they bind DNA to specific cognate sites (Fig.\u00a03a). The mechanism that explains the differential physiological effect of IFN-\u03b1 and IFN-\u03bb1 despite inducing the same signaling effectors is largely unknown12,13,14. Recent data12,13,14,37, however, support the hypothesis that a differential temporal profile, understood as time series of the copy numbers of nuclear p-STAT1\/1 homodimers and p-STAT1\/2 heterodimers, carries information about identity and quantity of both IFNs and is further propagated by the gene expression machinery into distinct physiological responses. For instance, western blot experiments show a prolonged phosphorylation in response to IFN-\u03bb1 compared to IFN-\u03b114.\n\nOur framework provides a natural, and computationally feasible, framework to address IFN discrimination problem. As four resolvable states are required to distinguish between presence and absence of two stimuli, the capacity $$C_N^ \\ast \\ge 2$$ can be interpreted as the potential of a population of N cells to distinguish both identity and quantity of the two IFNs. Moreover, Eqs. (8) and (11) imply that if FIM is non-singular the capacity $$C_N^ \\ast$$ can be arbitrarily increased and the discrimination error \u03b5(x0, x1, YN) arbitrarily decreased with the population size N. Therefore, information capacity and FIMs constitute suitable tools to determine how information about identity and quantity of both IFN is encoded in signaling responses.\n\nTo this end, we have built a probabilistic model of the pathway\u2019s input\u2013output relationship, P(Y|X\u2009=\u2009x). Construction of the model was accomplished by assembling and refining model components of the JAK-STAT signaling available in literature38,39,40 (Fig.\u00a03a and Section 3 of SI). The input x\u2009=\u2009(x\u03b1, x\u03bb1) consists of concentrations of IFN-\u03b1 and IFN-\u03bb1, respectively. We assumed that the pathway is exposed for 30\u2009min to stimulation with a mixture of IFNs specified by the input. The output is defined as Y\u2009=\u2009(Y1\/2(t1), Y1\/1(t1),...,Y1\/2(tn), Y1\/1(tn)), where Y1\/2(ti) and Y1\/1(ti) denote copy numbers of nuclear of p-STAT1\/2 heterodimers and p-STAT1\/1 homodimers, respectively, at time ti. Times t1,\u2026,tn serve as a proxy of the complete temporal profile. To account for signaling noise, we assumed that the stochasticity results from: (i) randomness of individual reactions and (ii) also cell-to-cell variability in the copy numbers of STAT1 and STAT2 molecules as well as type I, RI, and type III, RIII, receptor complexes. The two noise sources are seen as main contributors of cell-to-cell heterogeneity in general41 and IFN signaling, specifically42. The copy numbers of the above entities per cell was assumed variable with the same coefficient of variation\n\n$$c_{\\mathrm{v}} = \\frac{{\\sigma _i}}{{\\mu _i}},$$\n(12)\n\nwhere \u03bci is the mean copy number per cell, and \u03c3i is its standard deviation, for i{STAT1,STAT2,RI,RIII}. Further, we considered coefficient of variation from 0.3 to 1.5 to reflect typically measured values43. Importantly, the model is in line with the present biochemical knowledge14,37 by allowing the only difference in responses to arise from the different kinetics of both receptor complexes. We quantified the difference in receptor kinetics using the ratio of deactivation rates of the type III and type I receptor complexes, $$k_{{\\bar{R}}_{{\\mathrm {III}}}}$$ and $$k_{{\\bar{R}}_{\\mathrm I}}$$, respectively (see Sections 3.2\u20133.3 of SI),\n\n$$\\delta = \\frac{{k_{{\\bar{R}}_{III}} }}{{k_{{\\bar{R}}_I}}},$$\n(13)\n\nwhich we call the differential kinetics coefficient. For a given value of \u03b4 (e.g. 0.5), upon activation, the type I receptor complex remains active on average 1\/\u03b4 (e.g. 2) times shorter than the type III receptor. As responses to IFN-\u03bb1 are prolonged compared to IFN-\u03b114, we have considered \u03b4(0, 1). Values close to 0 and 1 denote dissimilar and similar receptor kinetics, respectively. The model was numerically simulated within the framework of the linear noise approximation that allows efficient calculation of the FIMs34,35 using literature values of kinetic parameters (see Supplementary Table\u00a02). As shown in Fig.\u00a03b the model provides responses qualitatively consistent with experiments that show prolonged responses to IFN-\u03bb1 compared to IFN-\u03b114. The strength of this effect is controlled by the parameter \u03b4. Low \u03b4 implies a significantly longer response to IFN-\u03bb1, whereas for \u03b4 close to 1 responses to both IFNs appear to be indistinguishable.\n\nFirst, we examined the potential of the differential signaling dynamics to discriminate between identity and quantity of IFNs under noise limited to stochasticity of individual reactions, cv\u2009=\u20090. To this end, we considered outputs, Y, with different end time points, tn\u2019s, so that they contain only the information available to the cell until time tn. For the values of \u03b4 used in Fig.\u00a03b, we plotted the information capacity, $$C_{\\mathrm{A}}^ \\ast$$, as a function of tn (Fig.\u00a04a) as well as corresponding representative FIMs (Fig.\u00a04b). For early times, the capacities, $$C_{\\mathrm{A}}^ \\ast$$ are below 0, further, with increasing tn, raise over 2 bits, and finally plateau. The time-windows of rapid increase coincide with times, at which stimulation with different combinations of the two IFNs generates distinguishable output trajectories (compare with Fig.\u00a03b and Supplementary Figures 4 and 5). Correspondingly, FIM is singular only for early tn and high \u03b4. Also, it becomes close to orthogonal for late tn and small \u03b4.\n\nThese results indicate that for limited signaling noise, differential signaling dynamics has a potential to ensure discriminability between the two IFNs. Precisely, for all \u03b4\u2019s and late tn, $$C_{\\mathrm{A}}^ \\ast$$ reaches high values, and FIMs are non-singular. Therefore, at the population level, both capacity, $$C_N^ \\ast$$, arbitrarily increases (Eq.\u00a08), and the discrimination error arbitrarily decreases (Eq.\u00a011), with N. Moreover, using $$C_{\\mathrm{A}}^ \\ast$$ as an approximation of $$C_1^ \\ast$$, which can be safely done for high values of $$C_{\\mathrm{A}}^ \\ast$$, we can also conclude that the differential signaling dynamics results with the single-cell capacity, $$C_1^ \\ast$$, significantly higher than 2 bits. Two bits is a minimum necessary condition to encode four input values, e.g., presence and absence of two stimuli. However, Shannon information alone does not tell us which exact input values can be discriminated. Therefore, we can conclude only that individual cells can resolve at least four combinations of both IFN concentrations.\n\n### Population level discrimination is possible even at high noise and with minor kinetic differences\n\nThe above analysis demonstrated that with limited noise signaling dynamics ensures discrimination between both IFNs. Interestingly, the discrimination is possible even with modest differences in the kinetics of both receptors, i.e., \u03b4\u2009=\u20090.9, which corresponds to 10% difference in the receptors deactivation rates. Noise in signaling processes is, however, not limited to stochasticity of signaling reactions. In mammalian signaling, the noise is thought to be dominated by the copy number variability of signaling components8. Therefore, we have considered several noise levels, and examined how the information content of the complete temporal profile, tn\u2009=\u2009180, depends on the values of the differential kinetics coefficient, \u03b4. Fig.\u00a04c presents the capacity, $$C_{\\mathrm{A}}^ \\ast$$, as a function of \u03b4 for a range of biologically feasible43 values of cv. Fig.\u00a04d shows corresponding FIMs. Not surprisingly, both noise and lack of kinetic differences can severely compromise the information transfer (Fig.\u00a04c). $$C_{\\mathrm{A}}^ \\ast$$ falls substantially below 2 bits, reaching negative values for high noise and similar kinetics. On the other hand, representative FIMs are non-singular for all noise levels and values of \u03b4.\n\nThe above results primarily show that discriminability at the population level can be achieved even with minor differences in kinetic rates, and despite high noise levels. This is implied by Eqs. (8) and (11). As indicated by Eq. (8), high population capacity, $$C_N^ \\ast$$, can be ensured by large N, as long as $$C_{\\mathrm{A}}^ \\ast$$ is not prohibitively low. Similarly, Eq. (11) shows that low discrimination error, \u03b5(x0, x1, YN), can be ensured by large N, as long as FIMs are non-singular. Both conditions are satisfied in the considered scenarios as shown in Fig.\u00a04c, d. In addition, negative values of $$C_{\\mathrm{A}}^ \\ast$$, for high \u03b4 and cv, indicate slow increase of the overall number of resolvable input values with N (Eq.\u00a09).\n\nMoreover, our analysis demonstrates that discriminability at the population level does not require discriminability at the single-cell level. This conclusion can be made on the following ground. The capacity of two bits is a necessary condition to encode four input values, e.g., presence or absence of both IFNs. In other words, a system with capacity lower than two bits does not have a sufficient discriminatory power to resolve presence and absence of the two IFNs. Therefore if $$C_1^ \\ast < 2$$ the discrimination at the single-cell level is not possible. Here, we calculated $$C_{\\mathrm{A}}^ \\ast$$ not $$C_1^ \\ast$$, which can only serve as an approximation of $$C_1^ \\ast$$. However, $$C_{\\mathrm{A}}^ \\ast$$ falls substantially below two bits. Therefore, even if $$C_{\\mathrm{A}}^ \\ast$$ was not a very accurate of approximation of $$C_1^ \\ast$$, low values of $$C_{\\mathrm{A}}^ \\ast$$ strongly indicate that $$C_1^ \\ast$$ is smaller than two bits, which demonstrates that there is no discriminability at the single-cell level. On the other hand, as argued in the previous paragraph, discriminability at the population can be achieved by increasing N.\n\nInterestingly, our analysis also highlights the role of kinetic rates in efficient information transfer. Primarily, the model predicts that at the population level, the discriminability between the two IFNs can be achieved even at high noise with differences in kinetic rates at the order of 10%. This suggest that even minor divergence of evolutionary related receptors might suffice to augment information transfer. This prediction is in line with the highly cross-wired architecture of signaling pathways29,44. Secondly, Fig.\u00a04c shows that loss of information due to noise can be compensated by stronger kinetic differences, and vice versa. This trade-off emphasizes the divergence of kinetic rates as an easily accessible evolutionary strategy of increasing information transfer. Reduction of noise level requires an increase in the number of signaling molecules or\/and sophisticated control mechanisms. On the other hand, alteration of receptor kinetic rates can be caused by a single mutation45.\n\nOverall, our model predicts that the population can correctly decode information even in cases where single cells cannot, due to high noise or similar receptor kinetics. The question, however, arises how the population should be able to make correct decisions based on low capacity in single cells. To illustrate this, consider the expression of IFNs induced genes as a downstream output. Both IFNs induce expression of hundreds of gene, including several chemokines from CXCL and CCL family12,46,47. Specifically, it has been shown that temporal profiles of CXCL10 expression differ in response to both considered IFNs12.\n\nOne of the main function of these chemokines is to attract different types of leukocytes to a site of an infection. Therefore, concentration and timing of these chemoattractants can be seen as a decision of cellular population regarding which and how many leukocytes are needed at a given time. Concentration and timing are controlled jointly by a large number of cells due to averaging of secretions in the intercellular space. In consequence, the chemokine concentration depends on the information encoded in nuclear levels of the p-STAT1\/1 and p-STAT1\/2 dimers in multiple cells. Therefore, even if the capacity of individual cells is low, and as a result expression of the chemokine can be only vaguely controlled by IFNs level, the high joint capacity of N cells may lead to a finely tuned level of the chemokine in the intercellular space, as the differences in secretion of individual cells would average out. To further hypothesize how the high capacity of a population and low individual capacity may be utilized, one can also anticipate a different scenario. An initial stimulus leads to subsequent rounds of cell-to-cell communication through paracrine signaling. Low individual capacity implies that initially the stimulus is recognized with low precision. In subsequent rounds of communication information is exchanged between cells, and as a result, the initial stimulus may lead to finely tuned responses at later times.\n\nAlthough the above hypothetical mechanisms are in line with current understanding of IFN signaling, they imply the need for more detailed experimental testing. They also rise an essential questions regarding signaling processes: does effective information transfer require discriminability of IFNs, and signaling ligands more generally, at the single-cell level, or population level suffices? So far, differential IFNs signaling dynamics have been observed at the population level12,13,14,15,36. Experimental confirmation of our theoretical prediction would be of high relevance for reconciling the single-cell stochasticity with fine-tuned tissue level responses.\n\n## Discussion\n\nInformation theory appears to have a potential to promote further understanding of how cells translate information about complex stimuli into distinct activities of the pathway\u2019s effectors using pleiotropic and stochastic mechanisms. Our theoretical methodology establishes a general and computationally efficient framework that enables analysis of models with multiple inputs and outputs. Importantly, it also accounts for the temporal aspect of signaling. Here, we have shown that even in the presence of significant noise information about identity and quantity of IFN-\u03b1 and IFN-\u03bb1 can be transmitted despite shared network components. Discriminability at the population level can be achieved without discriminability at the single-cell level, and with only small differences in receptor kinetic rates. So far, signaling responses to both IFNs have been measured at the population level only12,13,14,15,36. Our analysis suggests that further verification at the single-cell level could provide interesting conclusions regarding how information processing differs between single cells and cellular populations. Scenarios of pleiotropic signaling, similar to IFNs, are common in signaling, e.g., Wnt, BMP29, as well as in GPCR signaling48,49. Therefore, our framework seems to offer an attractive opportunity to gain further insight into the functioning of many complex signaling systems.\n\n## Methods\n\n### Mutual information\n\nWithin information theory, quantification of information transfer of a given signaling system, P(Y|X\u2009=\u2009x), is performed in reference to the distribution of input values P(X). Although, randomness of output, Y, prevents the system from resolving a precise value of the input, x, the uncertainty regarding input values cannot be higher than the uncertainty associated with the input distribution, P(X). Uncertainty is usually quantified by entropy\n\n$$H(X) = - {\\int}_{\\hskip -5pt \\mathscr{X}} \\mathop {{{\\mathrm {log}}}}\\nolimits_2 (P(x))P(x){\\mathrm d}x,$$\n(14)\n\nwhere $${\\mathscr{X}}$$ is the space of possible values of the signal, X.\n\nObservation of output has a potential to reduce uncertainty regarding input value. Via the Bayes formula, plausible inputs that generated a specific output value, y, are represented as the probability distribution $$P(X|Y = y) = \\frac{{P(Y = y|X)P(X)}}{{P(Y = y)}}$$. Uncertainty regarding input value can be then quantified by the entropy of the distribution P(X|Y\u2009=\u2009y)\n\n$$H(X|Y = y) = {\\int}_{\\hskip -5pt \\mathscr{X}} {\\mathrm{log}}_2(P(x|Y = y))P(x|Y = y){\\mathrm d}x.$$\n(15)\n\nAs the output is random, averaging H(X|Y\u2009=\u2009y) over all possible outputs quantifies the average uncertainty regarding the input, given the output, H(X|Y)\n\n$$H(X|Y) = - {\\int}_{\\hskip -5pt \\mathscr{Y}} H(X|Y = y)P(y){\\mathrm d}y,$$\n(16)\n\nwhere is $${\\mathscr{Y}}$$ the space of possible values of the output, Y. The difference between H(X) and H(X|Y) measures the average reduction in uncertainty regarding the input resulting from observing an output and is referred to as mutual information, I(X,Y), between the input and the output\n\n$$I(X,Y) = H(X) - H(X|Y).$$\n(17)\n\nMore background on information theory is provided in Section 1 of Supplementary methods.\n\n### Existing methods to compute information capacity\n\nThree main approaches are available to calculate C* and P*(X). The state-of-the-art Blahut\u2013Arimoto algorithm is based on convex optimization25,32 and for systems with continuous variables it requires discretization of input and output values18. Although it works efficiently for systems with one-dimensional input and output, optimization may become computationally prohibitive for higher dimensionalities. An alternative approach is offered by the small noise (SN) approximation method19, which offers an analytical solution, and therefore avoids heavy computations. However, it assumes a limited stochasticity within the analyzed system. Recently, a method based on density approximation was proposed in ref. 22 to account for temporarily resolved outputs.\n\n### Code availability\n\nComputer code used to generate reported results is available from authors upon request.\n\n## Data availability\n\nThe study did not involve any datasets.\n\nPublisher\u2019s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## References\n\n1. 1.\n\nKholodenko, B. N. Cell-signalling dynamics in time and space. Nat. Rev. Mol. Cell Biol. 7, 165 (2006).\n\n2. 2.\n\nNurse, P. Life, logic and information. Nature 454, 424\u2013426 (2008).\n\n3. 3.\n\nTawfik, D. S. Messy biology and the origins of evolutionary innovations. Nat. Chem. Biol. 6, 692 (2010).\n\n4. 4.\n\nRu\u00e9, P. & Arias, A. M. Cell dynamics and gene expression control in tissue homeostasis and development. Mol. Syst. Biol. 11, 792 (2015).\n\n5. 5.\n\nLim, W. A. Designing customized cell signalling circuits. Nat. Rev. Mol. Cell Biol. 11, 393\u2013403 (2010).\n\n6. 6.\n\nBehar, M., Barken, D., Werner, S. L. & Hoffmann, A. The dynamics of signaling as a pharmacological target. Cell 155, 448\u2013461 (2013).\n\n7. 7.\n\nRand, U. et al. Multi-layered stochasticity and paracrine signal propagation shape the type-I interferon response. Mol. Syst. Biol. 8, 584 (2012).\n\n8. 8.\n\nTay, S. et al. Single-cell NF-\u03baB dynamics reveal digital activation and analogue information processing. Nature 466, 267\u2013271 (2010).\n\n9. 9.\n\nPurvis, J. E. & Lahav, G. Encoding and decoding cellular information through signaling dynamics. Cell 152, 945\u2013956 (2013).\n\n10. 10.\n\nPurvis, J. E. et al. p53 dynamics control cell fate. Science 336, 1440\u20131444 (2012).\n\n11. 11.\n\nOlagnier, D. & Hiscott, J. Type I and type III interferon-induced immune response: it\u2019s a matter of kinetics and magnitude. Hepatology 59, 1225\u20131228 (2014).\n\n12. 12.\n\nBolen, C. R., Ding, S., Robek, M. D. & Kleinstein, S. H. Dynamic expression profiling of type I and type III interferon-stimulated hepatocytes reveals a stable hierarchy of gene expression. Hepatology 59, 1262\u20131272 (2014).\n\n13. 13.\n\nJilg, N. et al. Kinetic differences in the induction of interferon stimulated genes by interferon-\u03b1 and interleukin 28B are altered by infection with hepatitis C virus. Hepatology 59, 1250\u20131261 (2014).\n\n14. 14.\n\nMaher, S. G. et al. IFN-\u03b1 and IFN-\u03bb differ in their antiproliferative effects and duration of JAK\/STAT signaling activity. Cancer Biol. Ther. 7, 1109\u20131115 (2008).\n\n15. 15.\n\nVoigt, E. A. & Yin, J. Kinetic differences and synergistic antiviral effects between type I and type III interferon signaling indicate pathway independence. J. Interferon Cytokine Res. 35, 734\u2013747 (2015).\n\n16. 16.\n\nBerg, H. C. & Purcell, E. M. Physics of chemoreception. Biophys. J. 20, 193\u2013219 (1977).\n\n17. 17.\n\nCover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, New York, 2012).\n\n18. 18.\n\nCheong, R., Rhee, A., Wang, C. J., Nemenman, I. & Levchenko, A. Information transduction capacity of noisy biochemical signaling networks. Science 334, 354\u2013358 (2011).\n\n19. 19.\n\nTka\u010dik, G., Callan, C. G. & Bialek, W. Information flow and optimization in transcriptional regulation. Proc. Natl. Acad. Sci. USA 105, 12265\u201312270 (2008).\n\n20. 20.\n\nVoliotis, M., Perrett, R. M., McWilliams, C., McArdle, C. A. & Bowsher, C. G. Information transfer by leaky, heterogeneous, protein kinase signaling systems. Proc. Natl. Acad. Sci. USA 111, E326\u2013E333 (2014).\n\n21. 21.\n\nSuderman, R., Bachman, J. A., Smith, A., Sorger, P. K. & Deeds, E. J. Fundamental trade-offs between information flow in single cells and cellular populations. Proc. Natl. Acad. Sci. USA 114, 5755\u20135760 (2017).\n\n22. 22.\n\nSelimkhanov, J. et al. Accurate information transmission through dynamic biochemical signaling networks. Science 346, 1370\u20131373 (2014).\n\n23. 23.\n\nMartins, B. M. & Swain, P. S. Trade-offs and constraints in allosteric sensing. PLoS Comput. Biol. 7, e1002261 (2011).\n\n24. 24.\n\nMarzen, S., Garcia, H. G. & Phillips, R. Statistical mechanics of Monod-Wyman-Changeux (MWC) models. J. Mol. Biol. 425, 1433\u20131460 (2013).\n\n25. 25.\n\nArimoto, S. An algorithm for computing the capacity of arbitrary discrete memoryless channels. IEEE Trans. Inf. Theory 18, 14\u201320 (1972).\n\n26. 26.\n\nTka\u010dik, G. & Walczak, A. M. Information transmission in genetic regulatory networks: a review. J. Phys. Condens. Matter 23, 153102 (2011).\n\n27. 27.\n\nWaltermann, C. & Klipp, E. Information theory based approaches to cellular signaling. Biochim. Biophys. Acta 1810, 924\u2013932 (2011).\n\n28. 28.\n\nMehta, P., Goyal, S., Long, T., Bassler, B. L. & Wingreen, N. S. Information processing and signal integration in bacterial quorum sensing. Mol. Syst. Biol. 5, 325 (2009).\n\n29. 29.\n\nAntebi, Y. E. et al. Combinatorial signal perception in the BMP pathway. Cell 170, 1184\u20131196 (2017).\n\n30. 30.\n\nClarke, B. S. & Barron, A. R. Jeffreys\u2019 prior is asymptotically least favorable under entropy risk. J. Stat. Plan. Infer. 41, 37\u201360 (1994).\n\n31. 31.\n\nShannon, C. E. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379\u2013657 (1948).\n\n32. 32.\n\nBlahut, R. Computation of channel capacity and rate-distortion functions. IEEE Trans. Inf. Theory 18, 460\u2013473 (1972).\n\n33. 33.\n\nBernardo, J. M. Reference posterior distributions for Bayesian inference. J. R. Stat. Soc. Ser. B (Methodol.) 41, 113\u2013147 (1979).\n\n34. 34.\n\nKomorowski, M., Costa, M. J., Rand, D. A. & Stumpf, M. P. Sensitivity, robustness, and identifiability in stochastic chemical kinetics models. Proc. Natl. Acad. Sci. USA 108, 8645\u20138650 (2011).\n\n35. 35.\n\nKomorowski, M., \u017durauskien\u0117, J. & Stumpf, M. P. StochSens\u2014matlab package for sensitivity analysis of stochastic chemical systems. Bioinformatics 28, 731\u2013733 (2012).\n\n36. 36.\n\nWack, A., Terczy\u0144ska-Dyla, E. & Hartmann, R. Guarding the frontiers: the biology of type III interferons. Nat. Immunol. 16, 802\u2013809 (2015).\n\n37. 37.\n\nNovatt, H. et al. Distinct patterns of expression of transcription factors in response to interferon \u03b2 and interferon \u03bb1. J. Interferon Cytokine Res. 36, 589\u2013598 (2016).\n\n38. 38.\n\nVanlier, J., Tiemann, C. A., Hilbers, P. A. J. & van Riel, N. A. W. An integrated strategy for prediction uncertainty analysis. Bioinformatics 28, 1130\u20131135 (2012).\n\n39. 39.\n\nSmieja, J., Jamaluddin, M., Brasier, A. R. & Kimmel, M. Model-based analysis of interferon-induced signaling pathway. Bioinformatics 24, 2363\u20132369 (2008).\n\n40. 40.\n\nSwameye, I., M\u00fcller, T. G., Timmer, J., Sandra, O. & Klingm\u00fcller, U. Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling. Proc. Natl. Acad. Sci. USA 100, 1028\u20131033 (2003).\n\n41. 41.\n\nSymmons, O. & Raj, A. What?s luck got to do with it: single cells, multiple fates, and biological nondeterminism. Mol. Cell 62, 788\u2013802 (2016).\n\n42. 42.\n\nLevin, D., Harari, D. & Schreiber, G. Stochastic receptor expression determines cell fate upon interferon treatment. Mol. Cell Biol. 31, 3252\u20133266 (2011).\n\n43. 43.\n\nBar-Even, A. et al. Noise in protein expression scales with natural protein abundance. Nat. Genet. 38, 636\u2013643 (2006).\n\n44. 44.\n\nAmit, I., Wides, R. & Yarden, Y. Evolvable signaling networks of receptor tyrosine kinases: relevance of robustness to malignancy and to cancer therapy. Mol. Syst. Biol. 3, 151 (2007).\n\n45. 45.\n\nTawfik, D. S. Accuracy-rate tradeoffs: how do enzymes meet demands of selectivity and catalytic efficiency? Curr. Opin. Chem. Biol. 21, 73\u201380 (2014).\n\n46. 46.\n\nGarcin, G. et al. Differential activity of type I interferon subtypes for dendritic cell differentiation. PLoS ONE 8, e58465 (2013).\n\n47. 47.\n\nBauer, J. W. et al. Elevated serum levels of interferon-regulated chemokines are biomarkers for active human systemic lupus erythematosus. PLoS Med. 3, e491 (2006).\n\n48. 48.\n\nRajagopal, S., Rajagopal, K. & Lefkowitz, R. J. Teaching old receptors new tricks: biasing seven-transmembrane receptors. Nat. Rev. Drug. Discov. 9, 373 (2010).\n\n49. 49.\n\nKenakin, T. Theoretical aspects of GPCR\u2013ligand complex pharmacology. Chem. Rev. 117, 4\u201320 (2016).\n\n50. 50.\n\nZhou, Z. et al. Type III interferon (IFN) induces a type I IFN-like response in a restricted subset of cells through signaling pathways involving both the Jak-STAT pathway and the mitogen-activated protein kinases. J. Virol. 81, 7749\u20137758 (2007).\n\n51. 51.\n\nDoyle, S. E. et al. Interleukin-29 uses a type 1 interferon-like program to promote antiviral responses in human hepatocytes. Hepatology 44, 896\u2013906 (2006).\n\n## Acknowledgements\n\nT.J. was supported by his own funds and the European Commission Research Executive Agency under grant CIG PCIG12-GA-2012-334298, M.K. and K.N. by the Polish National Science Centre under grant 2015\/17\/B\/NZ2\/03692. We thank Stefan Gr\u00fcnert, Marek Kocha\u0144czyk, Margaritis Voliotis, and Christopher Zechner for their helpful comments during the preparation of this manuscript. The model of biochemical sensor exposed to non-cognate ligand was inspired by discussions with Prof. Dan S. Tawfik.\n\n## Author information\n\n### Affiliations\n\n1. #### Institute of Fundamental Technological Research, Polish Academy of Sciences, Warszawa, 02-106, Poland\n\n\u2022 Tomasz Jetka\n\u2022 ,\u00a0Karol Niena\u0142towski\n\u2022 \u00a0&\u00a0Micha\u0142 Komorowski\n2. #### Department of Mathematics and School of Public Health, Imperial College London, London, SW7 2AZ, UK\n\n\u2022 Sarah Filippi\n3. #### Melbourne Integrative Genomics, School of BioSciences and School of Mathematics and Statistics, University of Melbourne, Parkville, 3010, VIC, Australia\n\n\u2022 Michael P. H. Stumpf\n\n### Contributions\n\nT.J. and M.K. designed research; T.J., K.N., and M.K. performed research; T.J., K.N., S.F., M.P.H.S. and M.K. analyzed data; and T.J., K.N., M.P.H.S. and M.K. wrote the paper.\n\n### Competing interests\n\nThe authors declare no competing interests.\n\n### Corresponding author\n\nCorrespondence to Micha\u0142 Komorowski.","date":"2018-11-17 15:40:41","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\": 2, \"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\": 0.6536663174629211, \"perplexity\": 2007.9405009388408}, \"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\/1542039743714.57\/warc\/CC-MAIN-20181117144031-20181117170031-00530.warc.gz\"}"}
| null | null |
I wish some UK Government/Establishment figure would explain why these type of characters were, and still are, essential to the "National security" of the UK ???????????
What made them SO important that the MSM had to be silenced regarding them?
I can only presume this guy is essential to "National security" as well?
Is the alleged "Demon Pastor" really James Bond in disguise?
"National security!" Update 1 Murdered boys probe.
We really must ask the question, "How many of the present Kincora Old Boy Association aka the DUP MPs are essential to "National security" ??????
Perhaps Jeffrey Donaldson aka Jeffrey Lundy or Willie Mcrea aka "Willie McLundy could tell us?
The "secret" of "National security" is revealed below.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 4,516
|
Q: DNS_ANY working but DNS_TXT not working for DS_GET_RECORD php function I am trying to debug dns_get_record since it isn't working for me as expected.
I created a Text DNS record "laramon_59939919ec899.glibix.com." with value "dd678f947384ed8d3531465439ff852e01e6eb1d"
With:
$result=dns_get_record('laramon_59939919ec899.glibix.com.',DNS_TXT);
print_r($result);
I get:
Array
(
)
But with:
$result=dns_get_record('laramon_59939919ec899.glibix.com.',DNS_ANY);
print_r($result);
I get:
Array
(
[0] => Array
(
[host] => laramon_59939919ec899.glibix.com
[class] => IN
[ttl] => 86182
[type] => TXT
[txt] => dd678f947384ed8d3531465439ff852e01e6eb1d
[entries] => Array
(
[0] => dd678f947384ed8d3531465439ff852e01e6eb1d
)
)
)
The record I have added is of TXT type. Can someone help me understand why do I not get the correct record when I am specifically looking for TXT record?
A: Thanks to @NickCoons
The DNS record was being returned from cache. I fixed it by changing DNS_TXT to DNS_ALL. Somehow, it seems like only DNS_TXT is returning the cached result.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,508
|
{"url":"https:\/\/leetcode.ca\/2021-07-26-1898-Maximum-Number-of-Removable-Characters\/","text":"Formatted question description: https:\/\/leetcode.ca\/all\/1898.html\n\n# 1898. Maximum Number of Removable Characters\n\nMedium\n\n## Description\n\nYou are given two strings s and p where p is a subsequence of s. You are also given a distinct 0-indexed integer array removable containing a subset of indices of s (s is also 0-indexed).\n\nYou want to choose an integer k (0 <= k <= removable.length) such that, after removing k characters from s using the first k indices in removable, p is still a subsequence of s. More formally, you will mark the character at s[removable[i]] for each 0 <= i < k, then remove all marked characters and check if p is still a subsequence.\n\nReturn the maximum k you can choose such that p is still a subsequence of s after the removals.\n\nA subsequence of a string is a new string generated from the original string with some characters (can be none) deleted without changing the relative order of the remaining characters.\n\nExample 1:\n\nInput: s = \u201cabcacb\u201d, p = \u201cab\u201d, removable = [3,1,0]\n\nOutput: 2\n\nExplanation: After removing the characters at indices 3 and 1, \u201cabcacb\u201d becomes \u201caccb\u201d.\n\n\u201cab\u201d is a subsequence of \u201caccb\u201d.\n\nIf we remove the characters at indices 3, 1, and 0, \u201cabcacb\u201d becomes \u201cccb\u201d, and \u201cab\u201d is no longer a subsequence.\n\nHence, the maximum k is 2.\n\nExample 2:\n\nInput: s = \u201cabcbddddd\u201d, p = \u201cabcd\u201d, removable = [3,2,1,4,5,6]\n\nOutput: 1\n\nExplanation: After removing the character at index 3, \u201cabcbddddd\u201d becomes \u201cabcddddd\u201d.\n\n\u201cabcd\u201d is a subsequence of \u201cabcddddd\u201d.\n\nExample 3:\n\nInput: s = \u201cabcab\u201d, p = \u201cabc\u201d, removable = [0,1,2,3,4]\n\nOutput: 0\n\nExplanation: If you remove the first index in the array removable, \u201cabc\u201d is no longer a subsequence.\n\nConstraints:\n\n\u2022 1 <= p.length <= s.length <= 10^5\n\u2022 0 <= removable.length < s.length\n\u2022 0 <= removable[i] < s.length\n\u2022 p is a subsequence of s.\n\u2022 s and p both consist of lowercase English letters.\n\u2022 The elements in removable are distinct.\n\n## Solution\n\nUse binary search. Initially, low = 0 and high = removable.length. Each time, let mid be the mean of low and high and check whether p is still a subsequence of s after removing mid characters from s according to removable. The maximum possible k can be found in this way.\n\nclass Solution {\npublic int maximumRemovals(String s, String p, int[] removable) {\nint low = 0, high = removable.length;\nwhile (low < high) {\nint mid = (high - low + 1) \/ 2 + low;\nif (isPossible(s, p, removable, mid))\nlow = mid;\nelse\nhigh = mid - 1;\n}\nreturn low;\n}\n\npublic boolean isPossible(String s, String p, int[] removable, int k) {\nint[] removes = new int[k];\nfor (int i = 0; i < k; i++)\nremoves[i] = removable[i];\nArrays.sort(removes);\nStringBuffer sb = new StringBuffer(s);\nfor (int i = k - 1; i >= 0; i--)\nsb.deleteCharAt(removes[i]);\nreturn isSubsequence(p, sb.toString());\n}\n\npublic boolean isSubsequence(String s, String t) {\nif (s.length() == 0)\nreturn true;\nif (s.length() > t.length())\nreturn false;\nint sLength = s.length(), tLength = t.length();\nint sIndex = 0, tIndex = 0;\nwhile (sIndex < sLength && tIndex < tLength) {\nchar sChar = s.charAt(sIndex), tChar = t.charAt(tIndex);\nif (sChar == tChar)\nsIndex++;\ntIndex++;\n}\nreturn sIndex == sLength;\n}\n}","date":"2022-05-17 16:52:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.23440030217170715, \"perplexity\": 6080.53336580443}, \"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-2022-21\/segments\/1652662519037.11\/warc\/CC-MAIN-20220517162558-20220517192558-00184.warc.gz\"}"}
| null | null |
In organic chemistry, cyclopentanonide is a functional group which is composed of a cyclic ketal of a diol with cyclopentanone. It is seen in amcinonide (triamcinolone acetate cyclopentanonide).
See also
Acetonide
Acetophenide
Acroleinide
Aminobenzal
Pentanonide
References
Cyclopentanonides
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,406
|
Sometimes things don't go as planned. Sony released Driveclub for the PlayStation 4 in October of 2014. At the time, a companion came out for Android, but Sony quickly pulled the app after less than a day on the site. The servers struggled to handle the load of everyone trying to play, so Sony delayed the PlayStation Plus Edition and mobile companion app in order to reduce the strain.
Now it's March 2016, and version 1.0 of the Driveclub companion app has returned to Google Play. The software has undergone a name change (it was originally MyDriveclub), and the developers have updated the interface into something that looks more at home on Android Lollipop and Marshmallow.
The app provides a space to track challenges and take on new ones. You can also pick up fame points if you're looking for something else to brag about.
The app is free to use. Hopefully this time the servers are good to go.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 321
|
Q: Where all List values are equal to column List Hello I have a problem and i cant solve it for 3 days. I've red many posts here and in google, but I cant find the solution.
In db I have column "Gener" - nvarchar(350) which contains for example this:
row 1: 1,4,32,11
row 2: 32,11
row 3: 1
row 4: 4,56,1,23
row 5: 4
From checkboxlist I check this values: 1,4 which add to
List<string> gnr = new List<string>();
The result which I want is row 1 and row 4.
I've made (take from stackoverflow) code which result is row 3 and row 4:
var result = from m in db.Movies
where gnr.Contains(m.Gener)
select m;
And code which result is row 1, row 3, row 4 and row 5:
foreach (string term in gnr)
{
var trb = db.movies.Where(o => o.Gener.Contains(term));
}
With Ole DB I can make it, but with LINQ I can't here is the code there:
List<string> Gener = new List<string>();
Gener = Action,Comedy
StringBuilder builder = new StringBuilder();
string lastItem = Gener[Gener.Count - 1];
// Here I made string Which I'll add to cmd string
foreach (string safePrime in Gener)
{
if (safePrime != lastItem)
{
builder.Append("((gener LIKE '%" + safePrime + "%')) AND").Append(" ");
}
else
{
builder.Append("((gener LIKE '%" + safePrime + "%')) ORDER By ID DESC").Append(" ");
}
}
string dbSelect = builder.ToString();
//The result from loop
dbSelect = "((GenerLIKE '%Action%')) AND ((GenerLIKE '%Comedy%')) ORDER By ID"
//Add dbSelect to exist cmd
Cmd1.CommandText = "SELECT * FROM movies WHERE " + dbSelect;
And the result here is what I want with LINQ, select all movies that are Action and Comedy
Thanks for the time you red this, I'll be very thankful for some help. Sorry for my english I hope it is readable.
A: String.Split does not work with Entity Framework, so you can move splitting Gener column value in memory:
var result = from m in db.movies.ToList()
let movieGnr = m.Gener.Replace(" ", "").Split(',')
where m.Gener != null && !gnr.Except(movieGnr).Any()
select m;
Returns rows 1 and 4.
UPDATE: As stated above, this solution will load all movies data into memory. What I suggest to you is changing DB structure - create MovieGeners table, which will contain Geners for each movie. And add navigation property to Movie which will contain list of Geners. This solution will allow to move all query to the database side.
int[] gnr;
var result = from m in db.movies.Include("Geners")
where gnr.All(g => m.Geners.Any(x => x.Id == g))
select m;
A: Below used query would returns row 1 and row 4 data.We need to use where clause and check the row data length. row1 and row 4 data length is 9.This query would work.
var result= db.Movies.Where(mv => mv.Gener.Length > 8).Select(mv => mv.Gener);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 4,484
|
{"url":"http:\/\/www.math.unist.ac.kr\/news\/5746066179751936\/view","text":"When: Jan. 15th 2020, 16:00.\nWhere: Building 108, Room 318.\n\nSpeaker: Lee Wan (\uc774\uc644), Yonsei University.\n\nTitle: On capitulation map of global fields.\n\nAbstract: For a finite Galois extension L\/K of global fields, let C(L) and C(K) denote the Galois groups of maximal S-ramified T-split abelian extensions of L and M, respectively (where S and T are sets of places). There is a natural map Ver from C(K) to C(L) induced from the transfer homomorphism. Since Artin's principal ideal theorem, to determine the kernel and the cokernel of Ver has been an interesting problem. They can be described by Galois cohomology groups of unit groups. As a corollary, we get the strict cohomological dimension of the Galois group of the S-ramified T-split maximal extension of a global field is equal to 2 when S has Dirichlet density 1 and T is finite.","date":"2020-01-18 21:28:18","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8707513809204102, \"perplexity\": 1076.980315214245}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250593937.27\/warc\/CC-MAIN-20200118193018-20200118221018-00175.warc.gz\"}"}
| null | null |
Everybody's All-American is a 1988 American sports drama film, released internationally as When I Fall in Love, directed by Taylor Hackford and based on the 1981 novel Everybody's All-American by longtime Sports Illustrated contributor Frank Deford.
The film covers 25 years in the life of a college football hero. It stars Dennis Quaid, Jessica Lange, Timothy Hutton and John Goodman.
Plot
Gavin Grey is a 1950s star athlete known by the moniker "The Grey Ghost," who plays football at the [fictional] University of Louisiana. His campus girlfriend Babs Rogers, nephew Donnie "Cake" McCaslin, and teammate Ed Lawrence adore his personality and charm. During the Sugar Bowl game, Gavin's play, defining his competitiveness throughout his career, causes a player from the opposing team to fumble the ball, which he returns to score a game-winning touchdown.
As his college days come to an end, Gavin ends up marrying Babs, starts a family, and gets drafted by the Washington Redskins. Lawrence opens a popular sports bar in Baton Rouge. Everyone is pleased for Gavin, including his friendly rival Narvel Blue, who might have achieved professional stardom had he chosen an athletic career path. Reality quickly sets in for Gavin as life in the NFL is difficult, the competition is fierce, and the schedule is grueling. Gavin is a respectable running back for the Redskins, but hardly the idol worshipped by everyone back home during his school years. Concurrently, Lawrence has accrued a number of gambling debts. He is later murdered by unidentified attackers, creating more debts for Gavin and Babs, who had invested in Lawrence's business.
Babs does her best to keep up with her husband's career and mood swings, and in doing so inherits the role of the wage earner in their household after he briefly retires. A sympathetic Donnie finds her frustrated and lonely, as his lifetime attraction to her brings them together for a brief extramarital affair. Gavin's financial setbacks encourage Babs to seek a job from Narvel to manage his restaurant.
During his retirement, money issues convince Gavin to accept a comeback offer from the Denver Broncos. The new NFL has passed him by, though, and Gavin is forced to accept that his playing days are over. He enters a failed business relationship with entrepreneur Bolling Kiely, whom he despises, spending countless hours telling old college football stories to clients. Donnie moves on with his life, becoming an author and getting engaged to a sophisticated woman named Leslie Stone, while supporting Gavin and Babs through a marital breakdown. A lost and pathetic figure, in the end, Gavin mends his relationship with Babs as he spends his withdrawal from professional sports reminiscing about his famed athletic youth.
Cast
Jessica Lange as Babs Rogers Grey
Dennis Quaid as Gavin "Grey Ghost" Grey
Timothy Hutton as Donnie "Cake" McCaslin
John Goodman as Ed "Bull" Lawrence
Carl Lumbly as Narvel Blue
Ray Baker as Bolling Kiely
Savannah Smith Boucher as Darlene Kiely
Patricia Clarkson as Leslie Stone
Wayne Knight as Fraternity Pisser
Production
Filming was stopped for weeks when Dennis Quaid had his collarbone broken by former New England Patriots cornerback Tim Fox during Footage of Quaid rolling in pain on the sidelines of the snow game appears in the finished film. A key scene featuring a candlelight parade involving large numbers of extras was filmed, on the steps of the Louisiana State Capitol, when snow started falling. Despite the beauty of the scene, director Taylor Hackford elected to reshoot the scene, as snow in Baton Rouge in November was such a rare event that he was worried it would be seen as a special effects goof in the film.
The game scenes were shot in LSU's Tiger Stadium during the halftimes of actual LSU games in 1987. The goalposts were altered to resemble the vintage "H" posts as needed during filming. Vertical posts were moved in place for the bottom portion of the H, and a multi-colored fabric covering was used to conceal the "modern" center support post. Upon completion of filming, the vertical posts and fabric were retracted so as not to interfere with the LSU games. In late 1993, LSU installed an updated model of the vintage posts permanently in the stadium. Some of the filming of the football scenes took place during halftime of the LSU-Alabama game on The producers wanted to continue shooting some scenes following the game, so they requested that the LSU fans remain after the game so that they could finish the scenes. However, Alabama won in and ten minutes after the game, the only fans still in the bleachers were wearing crimson, forcing the producers to finish shooting the following week (November 14) following LSU's game with Mississippi State,
Michael Apted was all set to direct Thomas Rickman's script in 1982 until Warner Bros. balked at the $16 million price tag, leading man Tommy Lee Jones and the fact that American football movies never do any business overseas. During its six years in development hell, Warren Beatty, Robert Redford, and Robert De Niro all circled the project.
Despite the fact that the novel was written about the University of North Carolina (which refused to allow filming because they suspected the story defamed campus legend Charlie "Choo Choo" Justice), when it was filmed at LSU, rumors started that Gavin Grey was based on the former LSU All-American Billy Cannon. He won the Heisman Trophy in 1959 and played eleven seasons for three professional teams, but served two and a half years in federal prison in the mid-1980s for his role in a counterfeiting ring. Deford himself denies this, saying: "Never met Cannon and knew nothing about him personally," he says. "Gavin was strictly a composite of many athletes from several sports that I had covered."
The film contains a much more hopeful and upbeat ending than the book, where Gavin takes his own life after trying to kill Babs as well.
Reception
Reaction to the film was mostly mixed. Review aggregator Rotten Tomatoes gives it a 44% rating based on 32 reviews, with an average rating of 5.2/10. Audiences polled by CinemaScore gave the film an average grade of "B+" on an A+ to F scale.
References
External links
Interview with Dennis Quaid from Everybody's All-American press junket at Texas Archive of the Moving Image
1988 films
1988 drama films
1980s sports drama films
American football films
American sports drama films
Films based on American novels
Films directed by Taylor Hackford
Films scored by James Newton Howard
Films set in the 1950s
Films set in the 1960s
Films set in Louisiana
Films shot in Colorado
Films shot in Louisiana
Warner Bros. films
1980s English-language films
1980s American films
|
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| 2,523
|
On Campus: Franklin's Chris Rodgers carries on the family hoops legacy
By Tim Whelan Jr./Daily News Correspondent
The Boston Celtics brought the Rodgers family to Massachusetts in 1979. In 2017, the family legacy is still strong on the court.
Raised in Franklin, Chris Rodgers is a junior standout for the men's basketball team at Worcester Polytechnic Institute. He also happens to be the grandson of former Celtics assistant and head coach Jimmy Rodgers.
In the 1980s, members of the Rodgers family shined at Walpole High. Tim Rodgers, Chris' dad, went on to play football at Tufts University, while Tim's younger brother Matt played quarterback at Iowa.
Chris Rodgers is carving out his own niche. The industrial engineering major is a member of the NEWMAC Academic All-Conference team, and his on-court work has been an asset as well.
"I knew this school would be the perfect fit for me," Rodgers said last week. "It's hard to believe I'm already a junior looking ahead at my senior year."
This weekend, the WPI men's basketball team did what it does every year, finding itself among the final four in the conference.
It wasn't to be, however. On Saturday, the Engineers' season came to an end in a hard-fought 63-61 defeat to MIT in the NEWMAC semifinal. Rodgers played 38 minutes, scoring 12 points to go with five rebounds. The Engineers finish at 17-9, breaking a streak of 13 straight seasons finishing with 20 or more victories.
"We wanted to be playing our best basketball as we entered the NEWMAC tournament," Rodgers said just days after he recorded the first double-double of his college career (13 points, 10 rebounds) in a Feb. 15 win at Wheaton.
Rodgers has been integral to keeping the team near the top of the NEWMAC standings the last few years. A starting guard, his 9.5 points per game were third on the team. The 6-foot-2 swingman also averaged 4.8 rebounds and 2.2 assists per contest.
The second of Tim and Carolyn's four kids, with an older sister Sarah and younger brothers Patrick and Jack, inherited the basketball gene from a staple of the Larry Bird-era Celtics.
Jimmy Rodgers was quite a familiar face around the area in the 1980s. After arriving as an assistant with head coach Bill Fitch in 1979, Rodgers was a part of three NBA champions. He was the Celtics' head coach for two seasons, from 1988 through 1990.
He went on to be the head coach of the expansion Minnesota Timberwolves and was a Chicago Bulls assistant for five years, where he was part of the final three Michael Jordan-led title teams.
"Having somebody like my grandfather to talk to, him knowing how college basketball works, he's a great help," Rodgers said. "More than anyone, though, my dad's been so supportive."
His grandfather's time growing up in the Chicago suburbs, playing at the University of Iowa, then having a 34-year coaching career that began at the University of North Dakota and was followed by four NBA stops lends itself to some memorable tales.
For most of the year, Rodgers' grandparents live in Naples, Fla. For the summer, though, they come up and stay with the Franklin branch of the Rodgers clan.
"He's got some of the best stories," his grandson said. "Really funny stuff."
Growing up in Franklin, Rodgers felt the tight bond between the community and its student-athletes from an early age.
"As early as sixth grade, we were learning to run the offense that Franklin High ran," Rodgers said. "I took a lot of pride playing for Franklin, thinking about everybody who came before me, and we wanted to keep that success going."
Rodgers recalled a particularly fun moment when he was a junior at Franklin in 2013 and the Panthers beat a powerful Mansfield team at the old Franklin Field House on Senior Night.
As a senior during the last season of the field house before it was torn down, the late venue will always have a soft spot for him and other Panthers.
"I miss it so much," Rodgers said. "It was the best, especially for big games. Just an amazing environment to have for a home court."
The college game, as with any sport, provided a wake-up call for Rodgers. Division III basketball is a big jump up from the Hockomock League.
"You come to college, and you're not faster than others on the court anymore," he said. "You have to hone your technique, like getting two feet in paint to get a rebound. You're not jumping over everyone — they can jump with you.
"But the game has slowed down. I'm thinking the game even more."
Rodgers has one more college offseason to prepare to get WPI back near the top of the NEWMAC. A familiar name in New England basketball lore is carrying on a family tradition.
Natick's Young ends swimming career on high note
Several months back, we wrote about Natick's Alex Young in this space. The Catholic Memorial product was a part of a Holy Cross record in the 400-yard (4x100) freestyle relay in each of Young's first three years as a Crusader.
Last weekend, the senior co-captain Young and his teammates made it four-for-four.
At the Patriot League Championships at Bucknell University in Lewisburg, Pa., Young managed to swim his fastest-ever split in the 100 — 47.91 seconds — despite being treated for pneumonia. His performance was part of the team-record pace of 3:08.39, narrowly breaking last year's mark of 3:09.61.
Hopkinton's Bolick shines at New Englands
Bentley University senior Tim Bolick, a 2013 Hopkinton graduate, had one of the highest individual finishes for the Falcons' indoor track team during the New England Intercollegiate Amateur Athletic Association Championships on Saturday at the Reggie Lewis Center.
Bolick was 10th of 32 in the 800 meters (1:54.13), achieving a new personal best by 1.26 seconds. His performance elevated him to sixth on Bentley's all-time list. Bolick was also part of Bentley's 4x800 relay that placed ninth overall and second among Division II participants in 7:56.43.
Tim Whelan Jr. can be reached at whelan.timothy@gmail.com. Follow him on Twitter @thattimwhelan.
Millis
Milford Daily News ~ 197 Main St., Milford, MA 01757 ~ Do Not Sell My Personal Information ~ Cookie Policy ~ Do Not Sell My Personal Information ~ Privacy Policy ~ Terms Of Service ~ Your California Privacy Rights / Privacy Policy
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"redpajama_set_name": "RedPajamaCommonCrawl"
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| 6,237
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{"url":"https:\/\/math.stackexchange.com\/questions\/886909\/int-0-infty-frac-sin2xx2x21-dx","text":"$\\int_0^\\infty \\frac{\\sin^2(x)}{x^2(x^2+1)} dx$ =?\n\nAfter reading articles on differentiation under the integral sign, I hit this post from mit, where after introducing the power tool, it challenges reader to do\n\n$$\\int_0^\\infty \\frac{\\sin^2(x)}{x^2(x^2+1)} dx$$\n\nObviously I have no clue where to start. Could any one give a hint?\n\n\u2022 I think you can simplify first the integral using partial fractions since $\\frac{1}{x^2 \\left(x^2+1\\right)}=\\frac{1}{x^2}-\\frac{1}{x^2+1}$. The first integral is simple; the second one is more problematic to me. Good luck. \u2013\u00a0Claude Leibovici Aug 4 '14 at 7:18\n\u2022 The definite integral of $\\frac{1}{x^2+1}$ is simple: it is $\\arctan(x)$. Remember that $\\arctan(0)=0$ and $\\arctan(\\infty)=\\pi\/2$. \u2013\u00a0Steven Van Geluwe Aug 4 '14 at 7:27\n\u2022 This question is the same as the problem in this link math.stackexchange.com\/questions\/691798\/\u2026 \u2013\u00a0xpaul Aug 4 '14 at 23:46\n\nThis is a possible way to evaluate the integral. Partial fraction decomposition and the double angle formula yield $$\\int^\\infty_0\\frac{\\sin^2{x}}{x^2(1+x^2)}dx=\\frac{1}{2}\\int^\\infty_0\\frac{1-\\cos{2x}}{x^2}dx-\\frac{1}{2}\\int^\\infty_0\\frac{1-\\cos{2x}}{1+x^2}dx$$ The first integral can be evaluated in many ways, differentiation under the integral sign is one of them. I prefer to proceed with a simple fact that follows from the definition of the gamma function. $$\\int^{\\infty}_0t^{n-1}e^{-xt} \\ dt=\\frac{\\Gamma(n)}{x^n}$$ Hence the first integral is \\begin{align} \\frac{1}{2}\\int^\\infty_0\\frac{1-\\cos{2x}}{x^2}dx &=\\frac{1}{2}\\int^\\infty_0(1-\\cos{2x})\\int^\\infty_0te^{-xt} \\ dt \\ dx\\\\ &=\\frac{1}{2}\\int^\\infty_0t\\int^\\infty_0e^{-xt}(1-\\cos{2x}) \\ dx \\ dt\\\\ &=\\int^\\infty_0\\left(\\int^\\infty_0e^{-xt}\\sin{2x} \\ dx\\right)dt\\\\ &=\\int^\\infty_0\\frac{2}{t^2+4}dt\\\\ &=\\frac{\\pi}{2}\\\\ \\end{align} The second integral can be broken up further and evaluated using the residue theorem. \\begin{align} \\frac{1}{2}\\int^\\infty_0\\frac{1-\\cos{2x}}{1+x^2}dx &=\\frac{\\pi}{4}-\\frac{1}{4}\\Re\\oint_{\\Gamma}\\frac{e^{2iz}}{1+z^2}dz\\\\ &=\\frac{\\pi}{4}-\\frac{1}{2}\\Re\\left(\\pi i\\operatorname{Res}(f,i)\\right)\\\\ &=\\frac{\\pi}{4}-\\frac{1}{2}\\Re\\left(\\pi i\\frac{e^{-2}}{2i}\\right)\\\\ &=\\frac{\\pi}{4}-\\frac{\\pi}{4e^2} \\end{align} Hence $$\\int^\\infty_0\\frac{\\sin^2{x}}{x^2(1+x^2)}dx=\\frac{\\pi}{4}\\left(1+e^{-2}\\right)$$\n\n\u2022 thanks a lot and.. the trick with $\\int^{\\infty}_0t^{n-1}e^{-xt} \\ dt=\\frac{\\Gamma(n)}{x^n}$ is brilliant! i only saw $n=1$ case before, never realized that could utilize $n>1$! \u2013\u00a0athos Aug 4 '14 at 10:35\n\nCould any one give a hint?\n\nPartial fraction decomposition, together with the fact that\n\n\u2022 $\\displaystyle\\int_0^\\infty\\frac{\\sin^2x}{x^2}dx=\\frac\\pi2$\n\n\u2022 $\\sin^2x=\\dfrac{1-\\cos2x}2$\n\n\u2022 $\\displaystyle\\int_0^\\infty\\frac{\\cos x}{x^2+a^2}dx=\\frac\\pi{2a~e^a}$","date":"2019-06-17 12:35:45","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\": 2, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9855247735977173, \"perplexity\": 691.3001209810684}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998475.92\/warc\/CC-MAIN-20190617123027-20190617145027-00143.warc.gz\"}"}
| null | null |
Development notes
=================
In here are a few notes about how the code is organized, used concepts, etc.
The main code is all pure Python. It is highly modular. The main playing engine is implemented in C/C++ as a Python module ([`ffmpeg.c`](https://github.com/albertz/music-player/blob/master/ffmpeg.c) and related). It uses [FFmpeg](http://ffmpeg.org/) for decoding and [PortAudio](http://www.portaudio.com/) for output.
A basic principle is to keep the code as simple as possible so that it works. I really want to avoid to overcomplicate things.
The main entry point is [`main`](https://github.com/albertz/music-player/blob/master/main.py). It initializes all the modules. The list of modules is defined in [`State.modules`](https://github.com/albertz/music-player/blob/master/State.py). It contains for example `queue`, `tracker`, `mediakeys`, `gui`, etc.
## Module
A module is controlled by the `utils.Module` class. It refers to a Python module (for example `queue`).
When you start a module (`Module.start`), it starts a new thread and executes the `<modulename>Main` function.
A module is supposed to be reloadable. There is the function `Module.reload` and `State.reloadModules` is supposed to reload all modules. This is mostly only used for/while debugging, though and is probably not stable and not well tested.
## Multithreading and multiprocessing
The whole code makes heavy use of multithreading and multiprocessing. Every module already runs in its own thread. But some modules itself spawn also other threads. The GUI module spawns a new thread for most actions. Heavy calculations should be done in a seperate process so that the GUI and the playing engine (which run both in the main process) are always responsive. There is `utils.AsyncTask` and `utils.asyncCall` for an easy and stable way to do something in a seperate process.
## Playing engine
This is all the [Python native-C/C++ module](https://github.com/albertz/music-player-core/).
The `player` module creates the player object as `State.state.player`. It setups the queue as `queue.queue`. `State.state` provides also some functions to control the player state (`playPause`, `nextSong`).
## GUI
The basic idea is that Python objects are directly represented in the GUI. The main window corresponds to the `State.state` object. Attributes of an object which should be shown in the GUI are marked via the `utils.UserAttrib` decorator. There, you can specify some further information to specify more concretely how an attribute should be displayed.
The GUI has its own module [`gui`](https://github.com/albertz/music-player/blob/master/gui.py). At the moment, only an OSX Cocoa interface ([`guiCocoa`](https://github.com/albertz/music-player/blob/master/guiCocoa.py)) is implemented but a PyQt implementation is planned. There is some special handling for this module as it needs to be run in the main thread in most cases. See `main` for further reference.
## Database
This is the module [`songdb`](https://github.com/albertz/music-player/blob/master/songdb.py).
The database is intended to be an optional system which stores some extra data/statistics about a song and also caches some data which is heavy to calculate (e.g. the fingerprint).
It provides several ways to identify a song:
- By the SHA1 of its path name (relative to the users home dir).
- By the SHA1 of its file.
- By the SHA1 of its AcoustId fingerprint.
This is so that the database stays robust in case the user moves a song file around or changes its metadata.
It uses [SQLite](http://www.sqlite.org/) as its backend. (As it is used mostly as a key/value store with optional external indexing, a complex SQL-like DB is not strictly needed. Earlier, I tried other DBs. For a history, see the [comment in the source](https://github.com/albertz/music-player/blob/master/songdb.py).)
It uses [binstruct](https://github.com/albertz/binstruct) for the serialization.
## Song attribute knowledge system
Some of the initial ideas are presented in [`attribs.txt`](https://github.com/albertz/music-player/blob/master/attribs.txt). This is implemented now mostly for the [`Song` class](https://github.com/albertz/music-player/blob/master/Song.py).
There are several sources where we can get some song attribute from:
- The local `song.__dict__`.
- The database.
- The file metadata (e.g. artist, title, duration).
- Calculate it from the file (e.g. duration, fingerprint, ReplayGain).
- Look it up from some Internet service like MusicBrainz.
To have a generic attribute read interface which captures all different cases, there is the function:
Song.get(self, attrib, timeout, accuracy)
For each attrib, there might be functions:
- `Song._estimate_<attrib>`, which is supposed to be fast. This is called no matter what the `timeout` is, in case we did not get it from the database.
- `Song._calc_<attrib>`, which is supposed to return the exact value but is heavy to call. If this is needed, it will be executed in a seperate process.
See [`Song`](https://github.com/albertz/music-player/blob/master/Song.py) for further reference.
## Playlist queue
The playlist queue is managed by the [`queue`](https://github.com/albertz/music-player/blob/master/queue.py) module. It has the logic to autofill the queue if there are too less songs in it. The algorithm to automatically select a new song uses the random file queue generator. This is a lazy directory unfolder and random picker, implemented in [`RandomFileQueue`](https://github.com/albertz/music-player/blob/master/RandomFileQueue.py). Every time, it looks at a few songs and selects some song based on
- the song rating,
- the current recently played context (mostly the song genre / tag map).
## Debugging
The module [`stdinconsole`](stdinconsole.py), when started with `--shell`, provides a handy IPython shell to the running application (in addition to the GUI which is still loaded). This is quite helpful to play around. In addition, as said earlier, all the modules are reloadable. I made this so I don't need to interrupt my music playing when playing with the code.
|
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|
Proving it is possible to close the pay gap between women and men in the workforce, enterprise software company Salesforce, under chairman and CEO Marc Benioff, has effectively bridged that disparity for its 28,000 global employees.
Salesforce's cloud-based, customer relationship management program allows businesses to manage their sales, service and marketing online rather than spending on IT infrastructure. The San Francisco-headquartered company has a market value of more than $66 billion; Benioff is not only an entrepreneur, but also a philanthropist committed to using his and his company's business acumen for the greater good. He recently spoke at the U.N. and participated in the World Economic Forum's Sustainable Development Impact Summit.
It was a concerted effort to accomplish parity, says Cindy Robbins, Salesforce's president and chief people officer. Beginning in 2015, Benioff established a program within the company to identify female executives with promise and mandating that these "high-potential" women would be included in quarterly operational review meetings.
The next step became, "How can we be more overt about bringing women's issues to table?" says Robbins, who along with a colleague broached the issue of equal pay. Benioff quickly agreed that equality had to be a core value of the company. "It turns out, getting his support was the easy part," says Robbins.
A company-wide audit and evaluation began: compensation was analyzed by employee groups in comparable roles to determine pay scale differences. Initially the company spent close to $3 million to reduce salary differences to ensure that those employees performing similar work at the same level were paid consistently. In 2017, the company conducted a second assessment and spent another $3 million to address variances.
Robbins explains that pay gaps exist for a variety of reasons from hiring and promotional practices to women carrying a gap from previous employment. Per the U.S. Census Bureau's latest stats, women on average earn 21% less than male counterparts in the same job.
Champions for equal pay include actress Patricia Arquette, who famously called for wage equality during her supporting actress acceptance speech at the 2015 Oscars. Benioff is among the first CEOs to address the issue.
"Our employees and company have responded really well as a whole," says Robbins, who credits Benioff for doing the right thing. "The CEO sets the tone and vision and we could not have done this without Marc. He drives the message," says the HR exec. She explains other companies can follow suit by closely examining compensation data; despite numerous variables (such as job level), an inspection will most likely yield discrepancies.
Variety's EmPOWerment Award is given to a male executive who uses his influence to further gender equality in the workplace. Previous recipients include Lucian Grainge, CEO of Universal Music Group, and Jim Gianopulos, Paramount Pictures CEO.
|
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Q: Animating Fragment causes other view to "jump" Short Question:
When using animation on FragmentTransactions, how can I animate other views with the animation?
Long Question:
Hi,
I am new to fragments and so on, and i am trying to animate them i a single activity so I created the following xml file for the activity:
<LinearLayout
android:id="@+id/run_select_fragment_container"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:keepScreenOn="true"
android:orientation="vertical">
<FrameLayout
android:id="@+id/activity_run_search_fragmentHeaderPlaceholder"
android:layout_width="match_parent"
android:layout_height="wrap_content" />
<FrameLayout
android:id="@+id/activity_run_search_fragmentPlaceholder"
android:layout_width="match_parent"
android:layout_height="0dp"
android:layout_weight="1"/>
</LinearLayout>
Then I definded the res/anim files i want to use for the animation:
fade_in_from_top.xml
<?xml version="1.0" encoding="utf-8"?>
<set xmlns:android="http://schemas.android.com/apk/res/android"
android:shareInterpolator="false">
<alpha android:fromAlpha="0.0" android:toAlpha="1.0"
android:duration="@android:integer/config_longAnimTime" />
<translate
android:fromXDelta="0%" android:toXDelta="0%"
android:fromYDelta="-100%" android:toYDelta="0%"
android:duration="1000" />
</set>
fade_in_from_bottom.xml
<?xml version="1.0" encoding="utf-8"?>
<set xmlns:android="http://schemas.android.com/apk/res/android"
android:shareInterpolator="false">
<alpha android:fromAlpha="0.0" android:toAlpha="1.0"
android:duration="@android:integer/config_longAnimTime" />
<translate
android:fromXDelta="0%" android:toXDelta="0%"
android:fromYDelta="100%" android:toYDelta="0%"
android:duration="1000"/>
</set>
fade_out_to_bottom.xml
<?xml version="1.0" encoding="utf-8"?>
<set xmlns:android="http://schemas.android.com/apk/res/android"
android:shareInterpolator="false">
<alpha
android:duration="@android:integer/config_longAnimTime"
android:fromAlpha="1.0"
android:toAlpha="0.0" />
<translate
android:duration="700"
android:fromXDelta="0%"
android:fromYDelta="0%"
android:toXDelta="0%"
android:toYDelta="100%" />
</set>
fade_out_to_top.xml
<?xml version="1.0" encoding="utf-8"?>
<set xmlns:android="http://schemas.android.com/apk/res/android"
android:shareInterpolator="false">
<alpha
android:duration="@android:integer/config_longAnimTime"
android:fromAlpha="1.0"
android:toAlpha="0.0" />
<translate
android:duration="700"
android:fromXDelta="0%"
android:fromYDelta="0%"
android:toXDelta="0%"
android:toYDelta="-100%" />
</set>
In detail:
The lower fragment contains always a listview with several items. I do change the fragments based on what to search by a toggle button. thats no problem. When the user clicks on a specific listentry the headerFragment will be filled with additional data and displayed with a fade in animation like:
First show loading fragment while loading the data:
final RSBaseFragment headerFragment = (RSBaseFragment) getSupportFragmentManager().findFragmentById(R.id.activity_run_search_fragmentHeaderPlaceholder);
FragmentTransaction ft = getSupportFragmentManager().beginTransaction();
LoadingFragment fragment = new LoadingFragment();
if (headerFragment != null) {
ft.setCustomAnimations(R.anim.fade_in_from_bottom, R.anim.fade_out_to_top, R.anim.fade_in_from_top, R.anim.fade_out_to_bottom);
ft.replace(R.id.activity_run_search_fragmentHeaderPlaceholder, fragment, HEADER_FRAGMENT_TAG);
} else {
ft.setCustomAnimations(R.anim.fade_in_from_top, R.anim.fade_out_to_bottom, R.anim.fade_in_from_bottom, R.anim.fade_out_to_top);
ft.add(R.id.activity_run_search_fragmentHeaderPlaceholder, fragment, HEADER_FRAGMENT_TAG);
}
ft.commit();
After data was loaded show the data in a new fragment replacing the loading fragment
HeaderFragment fragment = new HeaderFragment();
ft.setCustomAnimations(R.anim.fade_in_from_top, R.anim.fade_out_to_bottom, R.anim.fade_in_from_bottom, R.anim.fade_out_to_top);
ft.replace(R.id.activity_run_search_fragmentHeaderPlaceholder, fragment, HEADER_FRAGMENT_TAG);
The problem ist that the headerfragment is fading and sliding in from the top or bottom direction, while the fragment holding the listview ist "jumping" to the bottom Y of the headerfragment before it's even there. How can I animate the lower fragment to slide with the header fragment to create a smooth user experience?
Sorry for the long question. If searched for 3 days and did not found anything helping with my problem.
A: I nearly solved it by adding to the LinearLayout (run_select_fragment_container)
android:animateLayoutChanges="true"
And in the onCreate of the Activity I have done this
LinearLayout mainLayout = (LinearLayout) findViewById(R.id.run_select_fragment_container);
LayoutTransition layoutTransition = mainLayout.getLayoutTransition();
layoutTransition.enableTransitionType(LayoutTransition.CHANGING);
|
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| 5,226
|
var Benchmark = require('benchmark');
var suite = new Benchmark.Suite();
var Mustache = require('mustache');
var Plates = require('../lib/plates');
suite
.add('mustache', function() {
var view = { "foo": "Hello, World" };
var template = '<div id="foo">{{foo}}</div><div class="foo">';
Mustache.to_html(template, view);
})
.add('plates', function() {
var view = { "foo": "Hello, World" };
var template = '<div id="foo"></div><div class="foo">';
Plates.bind(template, view);
})
.on('cycle', function(event, bench) {
console.log(String(bench));
})
.on('complete', function() {
console.log('Fastest is ' + this.filter('fastest').pluck('name'));
})
.add('mustache iterations', function() {
var view = {
"stooges": [
"Moe",
"Larry",
"Curly"
]
};
var template = '{{#stooges}}<b>{{name}}</b>{{/stooges}}';
Mustache.to_html(template, view);
})
.add('plates iterations', function() {
var view = {
"stooges": [
"Moe",
"Larry",
"Curly"
]
};
var template = '<b class="stooges">Name</b>';
Plates.bind(template, view);
})
.run(true);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,388
|
Classifieds Currency exchange Newsletter
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If you have refined your search by clicking on one of the terms in the left-hand panel you can remove the filter by clicking on the × beside the category used to filter the results.
Search for "" in the "French news" category returned 173 matches.
French town with Australia link raises €18k for fires
A small French town with links to Australia going back to World War One has raised more than €18,000 for the fire-ravaged country.
Author: Connexion journalist
French town with Australia link sends wildfire support
A French school with links to Australia dating back to World War One is raising money and organising events in solidarity as the country continues to be devastated by deadly wildfires.
Paris museum welcomes 'Instagram artist in residence'
The Musée d'Orsay in Paris is to welcome a French "artist in residence" on its Instagram social media account, who will each week highlight one of the museum's great artists as if they were still alive today.
France celebrates with traditional galette des rois
It is galette des rois season in France, as the country marks Epiphany on January 6, and bakers compete to make the best, most original versions of the popular patisserie.
Bacon, money, proud...9 words English took from French
Any English speaker who has tried to learn French will know that it is not always easy to master, but a new book has shown that some surprising words are more similar than you might think.
New Paris show tells history of France...in Playmobil
More than 3,000 figures from children's toy brand Playmobil are now on public display at the army museum at Les Invalides in Paris, arranged in scenes from French history.
French Bordeaux guide recreates 15th century monuments
Monuments dating back to the 15th century have "reappeared" in the city of Bordeaux (Gironde, Nouvelle-Aquitaine) thanks to a 3D virtual reality project created by a local guide.
French chocolatier marks Berlin Wall fall in chocolate
A French chocolatier has celebrated 30 years since the fall of the Berlin Wall by replicating part of the infamous structure - using one tonne of solid chocolate.
Planet Mercury Sun event to be visible from France
The planet Mercury is to pass in front of the Sun early next week, in a rare sight that will be partly visible from mainland France.
French family keepsake up for auction at €24m
Small artwork recognised as lost 13th-century masterpiece
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Home/Travel/3 most popular festivals of Kenya
3 most popular festivals of Kenya
Paul watson November 3, 2020
Kenya is one of the richest countries with strong cultural diversity, as the country possess more that seventy tribal group within it. Kenya is the only African land in which different people from different parts of the continent migrated and settled in throughout the African history. This makes the land more peculiar as each and every group of people have their own style of living including their culture, tradition and festivals. Kenya is the land of colorful festivals, that sprinkles the individual culture of each clan on you during each festival. Every Kenyan festival brings people together to celebrate with joy and energy which is the secret of the unity of the communities till date.
Lamu Cultural Festival (November)
Lamu Cultural festival is celebrated by the Lamu community people and the people living in the island of Lamu of Kenya including other community members to celebrate their peaceful lifestyle from the past to the present and to the future. The Lamu people have their own cultural heritage and beliefs that is loved and attracted by number of tourists who take part in the celebration. According to their belief, every year, Lamu gains life during the Lamu Cultural festival. During this festival, the entire group of Kenyans gets gathered in the place to celebrate their beliefs and life styles that the Lamu people consider their soul and life. The cultural festival takes over with numbers of competitions and different races like donkey races (the highlight of the Lamu festival including endearing symbol the culture and tradition of the community), henna painting, Swahili poetry, dhow sailing and so on, in which the people participate with enthusiasm and joy. One of the oldest games played during this festival is the Boa competition which has some archeological evidences of playing it from thousands of years ago by the African people. The Lamu Cultural festival teaches every one about the insight of life, how life was led by their ancestors along with peaceful and loving lifestyle and architecture. Visiting Kenya during the festival is considered as the best time to visit Lamu,as the lessons of life can be learned in simple and attractive manner.
The entire life style of the Lamu people can be seen during this festival, including Lamu weddings, special henna paintings, and special dishes of Swahili. Their beliefs are followed from their ancestral ages and are engraved deeply in the heart and soul of every local in Kenya. The Lamu Island is located off the coastal area of Kenya that gets alive every year during the Lamu cultural festival. Along with the culture and beliefs, the festival also promotes the beauty and uniqueness of the island of Lamu as well as its people. The festival is usually celebrated at the end of November, from 21 to 24th of the month on which thousands of tourists from all over the world enjoy their trip to Kenya. Being 700 years old, the Lamu Island is declared as one of the Heritage Sites of the World. As Lamu was ruled by the Oman, most of the people of the Island are Muslims, and Islam plays an important role in the Lamu cultural festival. The festival was launched in 2001 that is supported and sponsored by various private and international embassies. The old skills of the Swahili culture like reading and storytelling is also greatly encouraged during the festival. The festival astonishes the foreigners with its magic of new life, and things turning old as you can see an ancient form of Lamu during these days.
Lake Turkana Festival (11-16 June)
Celebrated in the month of May, the Lake Turkana Festival is celebrated for encouraging and strengthening the bond and unity of different communities of the people. Nearly all the main tribal groups, living in northwest Kenya like Borana, Burji, Elmolo, Garee, Dassanech, Gabbra, Somali, Konso, Samburu, Rendille, Turkana, Sakuye and Wata, come forward to celebrate this festival so that they can get a chance to overcome all the stereotypes among them and can come under mutual understanding with themselves. As these communities of people have a bitter history on their side, as they fought frequently with each other for their rights in the past, this festival was organized for enriching their understanding and unity. The festival is celebrated every year in a small town named Loiyangalani, located on the south eastern coastal region of Lake Turkana. The place is well known for its Desert Museum that you can see nowhere else except here. Peaceful coexistence between different cultures are promoted through this festival. Nearly ten local ethnic communities live on the coast of Lake Turkana and the Lake Turkana Festival displays the performances of the ten communities,which can be considered unique and attractive by the outsiders. These people have their own way of living, living in unique huts and with a different variety of foods that can be tasted and enjoyed by the people who attend the festival.
The life and customs of the ten communities of Lake Turkana, their tradition and culture, arts and crafts, and music and dance leaves you with a positive experience that gives the fascinating and positive perception of life along the region of Lake Turkana. Each year the celebration takes place on a full moon day, which increases the beauty of the festival. Wandering on the streets of Loiyangalani gives you a memorable experience as the streets are decorated with festival mood, lights and cheers of different tribal clans. The festival goes for three days and on the final day of the festival the celebration ends with some political speeches, traditional dances, dramas or stage shows, fashion shows in their original traditional costumes and discos. Being a tourist, it will be more enthusiastic to wander along the streets and among the crowd of local tribes and watching the participants, visiting their traditional house and tasting their native food. Traveling to Lake Turkana with proper tourist guidance at the end of April and in May will let you experience the wonderful event.
Maulidi Festival (9-10 November)
Maulidi is one of the historical festivals of Kenya that is organized every year. As most of the Kenyan population belongs to the Muslim religion, Maulidi Festival has become one of the permanent features of the Islamic activities. The festival is celebrated in Lamu that gathers numbers of Muslims from several parts of the African continent, including from East Africa, and also the Islamic people from many parts of the world. This festival is celebrated on the birth of Prophet Mohammed, which is considered as the most important day of Muslims. Maulidi Festival is celebrated on the third month of every year as per the Muslim Calendar. According to the normal calendar used by world, the festival takes place during the month of June. The festival is being sponsored by the National Museum of Kenya from 1990 and also many local and international sponsors. The festival is organized with many cultural competitions like swimming, donkey races, which are the unique sports of the people of Kenya, tug of war, and henna painting competitions. The streets are filled with Swahili music and traditional dances as Lamu is well known for its rich culture and history. This is the main reason for which the Muslims of East Africa choose Lamu as the center for this event. Thousands of Muslims from all over the world recite qasidas prayers together that bring goosebumps to the listeners, and the festival gives you one unforgettable experience.
As the towns of Lamu are set in stone, traveling to Lamu for the Maulidi festival gives an outside the world feeling with its old history, culture and tradition since the town exists from the early 7th century. The Riyadha mosque founded in 1866 in Lamu is the location for their prayers. The peaceful and relaxing environment of Lamu with its beaches and natural richness makes the festival mood fresher and relaxed. Goma dance is most popular in the festival, in which people with their traditional dress and with a walking stick dance accordingly to the sound of the drum. The chanting and prayers can be heard throughout the night around the mosque accompanied with dance and songs. In spite of being a religious festival, it is also a historical festival so that the visitors are also allowed to participate with the natives. Visiting the island of Lamu during the Maulidi festival will impress the tourists and make them fall in love with the place and circumstances including the warmth and hospitality of the Lamu people.
Visit the German website Backpackertrail to find out more about this beautiful country, and to find out what activities Kenya has to offer.
List of other major festivals
January – New Year's Day
March – Nairobi Film Festival
April – Good Friday on April 19 Easter Monday on April 22
May – Labor Day
June – Madaraka Day on June 1, Eid al Fitr on June 5, Lake Turkana Festival, Maulidi Festival
August – Idd-ul-Azha on August 12, Maralal Camel Derby
October – Mashujaa Day on October 21, Moi day on October 10, Diwali on October 27
November – Lamu Cultural Festival
December – Independence Day (Jamhuri) on December 12, Christmas Day on December 25th, Boxing day on December 26, Rusinga Festival, Pawa Festival
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{"url":"https:\/\/sikademy.com\/answer\/computer-science\/discrete-mathematics\/question-by-using-the-rules-of-logical-equivalences-show-49oj\/","text":"is below this banner.\n\nCan't find a solution anywhere?\n\nNEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?\n\nYou will get a detailed answer to your question or assignment in the shortest time possible.\n\n## Here's the Solution to this Question\n\na)\u00a0$\\left( {p \\to \\left( {q \\to r} \\right)} \\right) \\to \\left( {\\left( {p \\wedge q} \\right) \\to r} \\right) = \\overline {\\left( {p \\to \\left( {q \\to r} \\right)} \\right)} \\vee \\left( {\\left( {p \\wedge q} \\right) \\to r} \\right) = \\overline {\\left( {\\overline p \\vee \\left( {q \\to r} \\right)} \\right)} \\vee \\left( {\\overline {\\left( {p \\wedge q} \\right)} \\vee r} \\right) = \\overline {\\left( {\\overline p \\vee \\left( {\\overline q \\vee r} \\right)} \\right)} \\vee \\left( {\\overline {\\left( {p \\wedge q} \\right)} \\vee r} \\right) = \\overline {\\left( {\\overline p \\vee \\overline q \\vee r} \\right)} \\vee \\left( {\\overline p \\vee \\overline q \\vee r} \\right) = p \\wedge q \\wedge \\overline r \\vee \\overline p \\vee \\overline q \\vee r = \\left( {p \\vee \\overline p \\vee \\overline q \\vee r} \\right) \\wedge \\left( {q \\vee \\overline p \\vee \\overline q \\vee r} \\right) \\wedge \\left( {\\overline r \\vee \\overline p \\vee \\overline q \\vee r} \\right) = \\left( {T \\vee \\overline q \\vee r} \\right) \\wedge \\left( {T \\vee \\overline p \\vee r} \\right) \\wedge \\left( {T \\vee \\overline p \\vee \\overline q } \\right) = T \\wedge T \\wedge T = T$\n\nQ. 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D.\n\nb) 1)\u00a0$\\left( {p \\wedge q} \\right) \\wedge \\left( {\\left( {q \\wedge \\neg r} \\right) \\vee \\left( {p \\wedge r} \\right)} \\right) = \\left( {p \\wedge q} \\right) \\wedge \\left( {\\left( {q \\vee p} \\right) \\wedge \\left( {q \\vee r} \\right) \\wedge \\left( {\\neg r \\vee p} \\right) \\wedge \\left( {\\neg r \\vee r} \\right)} \\right) = \\left( {p \\wedge q} \\right) \\wedge \\left( {\\left( {q \\vee p} \\right) \\wedge \\left( {q \\vee r} \\right) \\wedge \\left( {\\neg r \\vee p} \\right) \\wedge T} \\right) = \\left( {p \\wedge q} \\right) \\wedge \\left( {\\left( {q \\vee p} \\right) \\wedge \\left( {q \\vee r} \\right) \\wedge \\left( {\\neg r \\vee p} \\right)} \\right) = \\left( {p \\wedge q} \\right) \\wedge \\left( {q \\vee \\left( {p \\wedge r} \\right)} \\right) \\wedge \\left( {\\neg r \\vee p} \\right) = \\left( {\\left( {p \\wedge q \\wedge q} \\right) \\vee \\left( {p \\wedge q \\wedge p \\wedge r} \\right)} \\right) \\wedge \\left( {\\neg r \\vee p} \\right) = \\left( {\\left( {p \\wedge q} \\right) \\vee \\left( {p \\wedge q \\wedge r} \\right)} \\right) \\wedge \\left( {\\neg r \\vee p} \\right) = \\left( {p \\wedge q} \\right) \\wedge \\left( {T \\vee r} \\right) \\wedge \\left( {\\neg r \\vee p} \\right) = \\left( {p \\wedge q} \\right) \\wedge T \\wedge \\left( {\\neg r \\vee p} \\right) = \\left( {p \\wedge q} \\right) \\wedge \\left( {\\neg r \\vee p} \\right) = p \\wedge q \\wedge \\neg r \\vee p \\wedge q \\wedge p = p \\wedge q \\wedge \\neg r \\vee p \\wedge q = p \\wedge q \\wedge \\left( {\\neg r \\vee T} \\right) = p \\wedge q \\wedge T = p \\wedge q$\n\n2)\u00a0$\\neg \\left( {p \\to \\neg q} \\right) = \\neg \\left( {\\neg p \\vee \\neg q} \\right) = \\neg \\neg p \\wedge \\neg \\neg q = p \\wedge q$\n\nSo,\u00a0$\\left( {p \\wedge q} \\right) \\wedge \\left( {\\left( {q \\wedge \\neg r} \\right) \\vee \\left( {p \\wedge r} \\right)} \\right) = p \\wedge q$\u00a0and\u00a0$\\neg \\left( {p \\to \\neg q} \\right) = p \\wedge q$\n\nThen\n\n$\\left( {p \\wedge q} \\right) \\wedge \\left( {\\left( {q \\wedge \\neg r} \\right) \\vee \\left( {p \\wedge r} \\right)} \\right) = \\neg \\left( {p \\to \\neg q} \\right)$\n\nQ. 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D.\n\nc)\u00a0$\\left( {\\left( {p \\vee q} \\right) \\wedge \\left( {p \\to r} \\right) \\wedge \\left( {q \\to r} \\right)} \\right) \\to r = \\overline {\\left( {\\left( {p \\vee q} \\right) \\wedge \\left( {p \\to r} \\right) \\wedge \\left( {q \\to r} \\right)} \\right)} \\vee r = \\overline {\\left( {p \\vee q} \\right)} \\vee \\overline {\\left( {p \\to r} \\right)} \\vee \\overline {\\left( {q \\to r} \\right)} \\vee r = \\overline {\\left( {p \\vee q} \\right)} \\vee \\overline {\\left( {\\overline p \\vee r} \\right)} \\vee \\overline {\\left( {\\overline q \\vee r} \\right)} \\vee r = \\left( {\\overline p \\wedge \\overline q } \\right) \\vee \\left( {p \\wedge \\overline r } \\right) \\vee \\left( {q \\wedge \\overline r } \\right) \\vee r = \\left( {\\overline p \\wedge \\overline q } \\right) \\vee \\overline r \\wedge \\left( {p \\vee q} \\right) \\vee r = \\overline {\\left( {p \\vee q} \\right)} \\vee \\overline r \\wedge \\left( {p \\vee q} \\right) \\vee r = \\left( {\\overline {\\left( {p \\vee q} \\right)} \\vee \\overline r } \\right) \\wedge \\left( {\\overline {\\left( {p \\vee q} \\right)} \\vee \\left( {p \\vee q} \\right)} \\right) \\vee r = \\left( {\\overline {\\left( {p \\vee q} \\right)} \\vee \\overline r } \\right) \\wedge T \\vee r = \\left( {\\overline {\\left( {p \\vee q} \\right)} \\vee \\overline r } \\right) \\vee r = \\overline {\\left( {p \\vee q} \\right)} \\vee \\overline r \\vee r = \\overline {\\left( {p \\vee q} \\right)} \\vee T = T$\n\nQ. 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Эйнар Кваран (; , Вадланес — , Рейкьявик) — исландский писатель, драматург и поэт конца XIX — первой половины XX века.
Имя
После рождения он получил имя Эйнар и отчество (патроним) Хьорлейфссон. В 1913 году альтинг, по инициативе комитета по именам, членом которого был Эйнар Хьорлейфссон, принял закон (впоследствии отменённый), разрешающий исландцам брать имена и фамилии древнего происхождения. Пользуясь возможностью в 1916 году Эйнар взял себе фамилию Кваран, использовав эпоним из древней исландской саги о людях из Лососьей долины. Таким образом, полное имя Эйнара — Эйнар Хьйорлейфссон Кваран (), и он в своё время являлся одним из немногих исландцев имеющим и патроним, и фамилию. Устоявшимся в Исландии является употребление только имени и фамилии писателя — Эйнар Кваран (), иногда в сочетании с сокращением отчества — Эйнар Х. Кваран ().
Биография
Эйнар Кваран родился 6 декабря 1859 года в небольшой деревне Вадланес на востоке Исландии недалеко от Эйильсстадира в семье преподобного Хьйёрлейфюра Эйнарссона и домохозяйки Гвюдлёйг Эйоульфсдоуттир. Детство Эйнара прошло в различных селениях на севере возле Скага-фьорда. В 1877 году Эйнар поступил, а в 1881 году окончил колледж в Рейкьявике, известный как Латинская школа.
В 1882 году поступил на экономический факультет Копенгагенского университета, где вместе с тремя другими студентами-исландцами издавал исландский литературный журнал Verðandi, который отстаивал идеи реализма и разрыва с прошлым восхищением сагами.
В 1885 году Эйнар эмигрировал в Канаду, где он жил в центре исландской культуры — в Новой Исландии в Виннипеге и помог основать два исландскоязычных еженедельных изданий — «Heimskringla» и «Lögberg».
По возвращении в Исландию в 1895 году Эйнар стал журналистом и редактором в Рейкьявике и Акюрейри; участвовал в борьбе за исландскую независимость и писал об образовании и театре. Он был соредактором «Ísafold», тогда ведущей газеты Исландии, и редактором «Fjallkonan». С 1892 по 1895 год и в 1908—1909 годах редактировал «Skírnir» — журнал Исландского литературного общества.
Проработав 19 лет в журналистике в Канаде и Исландии, Эйнар в 1906 году решил полностью посвятить себя литературной работе и правительство Исландии предоставило ему стипендию, чтобы он мог полностью посвятить себя писательской деятельности. Начиная с 1906 года он опубликовал пять романов, две пьесы и какое-то время руководил театральной труппой Рейкьявика.
Эйнар был дважды женат. Его первая жена, Матильда Петерсен, была датчанкой; она умерла в Канаде, и их двое детей умерли в младенчестве. В 1888 году он женился на Гислине Гисладоуттир; у них было пятеро детей, один из которых — старший сын Сигюрдюр, умер от туберкулеза, когда ему было 15 лет.
Творчество
Эйнар очень рано заинтересовался книгами. По рассказам его родных, он начал сочинять стихи и рассказы ещё в раннем возрасте. Когда ему было двенадцать лет, он сжег целое собрание написанных им рассказов. Его интерес к литературе возрастает в годы учёбы в латинской гимназии в Рейкьявике, где он пишет стихи, пьесы и рассказы. Его лучшие работы этих лет, без сомнения, — это его два рассказа «Orgelið» () и «Hvorn eiðinn á ég að rjúfa?» (). Рассказы были напечатаны и получили неоднозначные отзывы, в частности их сочли несколько революционными, а автора посчитали аморальным.
Эйнар написал множество рассказов, романов, пьес и сборник стихов. Он был приверженцем чистоты и красоты языка, писал очень хорошо и стилистически красиво. Его революционной для исландской литературы работой стал рассказ «Vonir» (), который он написал в 1890 году, находясь в Канаде и повествующий об эмигрантском опыте.
Эйнар также был выдающимся спиритуалистом, автором первой положительной оценки спиритизма на исландском языке, а также соучредителем и президентом экспериментального общества, в результате которого было создано Исландское общество психических исследований (), в котором он был первый президент. Он сыграл важную роль в расследовании и популяризации многих исландских медиумов, особенно Индриди Индридасона и Хафстейна Бьёрднссона. В поздних литературных произведениях Эйнара значительное место занимал спиритизм, особенно в романе «Sögur Rannveigar» (), написанном в 1919—1922 годах, и христианский гуманизм. По мнению Стейнгримюра Торстейнссона, своим творчеством Эйнар оказал влияние на исландскую культуру и мировоззрение, в частности, сделав исландцев менее ортодоксальными и менее суровыми в воспитании своих детей.
В 1920-х годах ходили слухи, что Эйнара считают лауреатом Нобелевской премии по литературе.
Исландский историк литературы и литературный критик Сигюрдюр Нордаль пренебрежительно отозвался об Эйнаре как о чрезмерно сосредоточенном на всепрощении и, следовательно, терпимом к тем вещам, против которых писателю как-раз и следует следует скорее протестовать, чем прощать. По мнению Сигюрдюра Эйнару следовало бы писать в духе исландского национализма и современных ему интерпретаций Ницше, считая кровную месть лучшей этической моделью. В 1930-х годах лауреат Нобелевской премии Халльдор Лакснесс еще более резко критиковал Эйнара за его увлечение спиритизмом.
Примечания
Писатели Исландии
Поэты Исландии
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Q: PostgreSQL substitute case sensitive variables in a command Hi I have some variables in a plpgsql script. It's all case sensitive.
schema := 'myschema';
table_name := 'MyTable'
column_name := 'MyColumn'
I need to put those variables in a simple statement:
select max(column_name) from schema.table_name.
I feel like I am battling the case sensitivity with PostgreSQL. I basically need to recreate this:
select max("MyColumn") from myschema."MyTable";
For the life of me I can't get this work and am clearly too dumb for PostgreSQL. Trying with EXECUTE(), EXECUTE() w/ FORMAT(), quote_indent() etc etc. Any thoughts?
A: Got it with this gem
execute format('select max(%I) from %s.%I', column_name, schema_name, table_name);
Depressing that that took hours of my life...
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The world constantly throws up new challenges about what it means to be Christian and to live a distinctively Christian lifestyle.
The priest, broadcaster, writer and ethicist Samuel Wells considers some of the biggest contemporary political, social and moral challenges and grapples with them in the light of Christian hope and wisdom.
Under three headings - Engaging the World, Being Human, and Facing Mortality - he probes a wide range of issues including the rise of religious extremism, migration, ecology, social media, sexual identities, inequality, obesity, life stages from childhood to old age, dementia, facing death and much more.
This striking and profoundly wise book sets out to shape a theological imagination and fluency that is grounded in the reality of being human in a suffering world and yet open to transformation by the life and wisdom of God.
How Then Shall We Live? by Samuel Wells was published by Canterbury Press in June 2016 and is our 23940th best seller. The ISBN for How Then Shall We Live? is 9781848258624.
Reviews of How Then Shall We Live?
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Got a question? No problem! Just click here to ask us about How Then Shall We Live?.
Details for How Then Shall We Live?
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WEP calls for the London Mayor to join forces over domestic violence
Newsdesk 23rd August 2016 23rd August 2016 Politics
Read time:1 minute, 53 seconds
Sophie Walker invites London Mayor Sadiq Khan to work with her on tackling rising rates of domestic violence in the capital.
A release of new statistics from the Metropolitan Police show a rise of 8% in reported incidents of domestic violence in London.
The Women's Equality Party leader, Sophie Walker, responded to the news: "During my campaign to be London's first female Mayor earlier this year, I set out clear plans to tackle the horrifying prevalence of domestic violence in our city.
"It is simply terrible these figures are rising and, what's more, that less than a third of the incidents investigated by the Met Police resulted in further action.
"We have plans to tackle domestic violence both at the point at which it is reported and before this through preventative teaching in schools, rehabilitation of offenders, and specialist support for every survivor. I invite Sadiq Khan, who has styled himself as London's first 'feminist Mayor', to take on these plans and to show himself to be a man of action as well as words."
Walker explained that rising rates of violence against women and girls often occur at times of crisis, putting the figures in the context of current political, economic and international uncertainties and claimed that the party could bring support to City Hall.
"Understanding rising domestic violence rates is one thing, but the central issue for us is combatting them. I was the only London Mayoral candidate to set out a clear set of policies that prioritised this issue. The Women's Equality Party is dedicated to ending violence against women in all its forms."
WEP's policies for London include better police enforcement of Domestic Violence Protection Notices/Orders (DVPO), so that perpetrators and not victims are removed from their homes, and ring-fenced stable and sustainable funding for specialist support services, that are for and led by women, including BAME women and disabled women.
WEP would also work with local authority services to establish a solid pan-London system to help women fleeing domestic and sexual abuse find refuge and secure accommodation, as well as ring-fence a proportion of London's housing investment to build refuges those trapped in dangerous situations because they can't afford to escape.
Tagged : politicsSadiq KhanSophie WalkerWomen's Equality Party
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section*{Abstract}
{\bf
We propose using smeared boundary states $e^{-\tau H}|\cal B\rangle$ as variational approximations to the ground state of a conformal field theory deformed by relevant bulk operators. This is motivated by recent studies of quantum quenches in CFTs and of the entanglement spectrum in massive theories. It gives a simple criterion for choosing which boundary state should correspond to which combination of bulk operators, and leads to a rudimentary phase diagram of the theory in the vicinity of the RG fixed point corresponding to the CFT, as well as rigorous upper bounds on the universal amplitude of the free energy. In the case of the 2d minimal models explicit formulae are available. As a side result we show that the matrix elements of bulk operators between smeared Ishibashi states are simply given by the fusion rules of the CFT.
}
\vspace{10pt}
\noindent\rule{\textwidth}{1pt}
\tableofcontents\thispagestyle{fancy}
\noindent\rule{\textwidth}{1pt}
\vspace{10pt}
.
\section{Introduction}\label{sec1}
Conformal field theories (CFTs) are supposed to correspond to the non-trivial renormalization group (RG) fixed points of relativistic quantum field theories (QFTs). Such theories typically contain a number of scaling operators of dimension $\Delta<d$ (where $d$ is the space-time dimension), which, if added to the action, are relevant and drive the theory to what is, generically, a trivial fixed point. The points along this trajectory then correspond to a massive QFT. In general there is a multiplicity of such basins of attraction of the RG flows, but enumerating them and determining which combinations of relevant operators lead to which basins, and therefore to what kind of massive QFT, in general requires non-perturbative methods.
This problem is equivalent to mapping out the phase diagram in the vicinity of the critical point corresponding to the CFT.
Another way of characterizing these massive theories is through the analysis of the possible boundary states of the CFT. Imagine the scenario in which the relevant operators are switched on in only a half-space, say $x_0<0$. This will then appear as some boundary condition on the CFT in $x_0>0$. However the boundary conditions themselves undergo RG flows, with fixed points corresponding to so-called conformal boundary conditions. Therefore on scales $\sim M^{-1}$, where $M$ is the mass scale of the perturbed theory, the correlations near the boundary should be those of a conformal boundary condition, deformed by \em irrelevant \em boundary operators.
A similar question is raised through recent work on the spectrum of the entanglement hamiltonian in massive QFTs \cite{LR}. If the theory is defined in ${\bf R}^D$ and is in its ground state, and we study the entanglement between the degrees of freedom in the half-space $A: x_1>0$ and its complement, then the entanglement, or modular, hamiltonian $K_A=-(1/2\pi)\log\rho_A$, where $\rho_A$ is the reduced density matrix of $A$, takes the form \cite{HSK1,HSK2}
\be
K_A=\int_{x_1>0}x_1T_{00}(x)d^Dx\,,
\ee
which is nothing but the generator of rotations in euclidean space, or of boosts in lorentzian signature.
In 1+1 dimensions we may consider a conformal transformation $z=x_1+ix_0=\epsilon e^w$ which sends the euclidean $z$-plane, punctured at the origin by a disc of radius $\epsilon$ representing the UV cutoff, to an semi-infinite cylinder of circumference $2\pi$. $K_A$ is then simply the generator of translations around this cylinder. However, if the QFT corresponds to a perturbed CFT, it is not conformally invariant, but rather the couplings transform as
\be
\lambda\to\lambda\,\epsilon^{2-\Delta}e^{(2-\Delta){\rm Re}\,w}\,,
\ee
where $\Delta<2$ is the scaling dimension of the perturbing operator. Thus the dimensionless coupling
$g=\lambda\epsilon^{2-\Delta}$ is effectively switched on over a length scale $O(1)$ near ${\rm Re}\,w\sim\log(1/g) $. If we are interested the low-lying spectrum of $K_A$, corresponding to the R\'enyi entropies ${\rm Tr}\,\rho_A^n$ with $n\gg1$, the effective circumference of the cylinder is $2\pi n$ and we are then in a similar situation to the above, where the massive theory for ${\rm Re}\,w>\log(1/g)$ acts as an effective boundary condition on the CFT in ${\rm Re}\,w<\log(1/g)$. As concluded in \cite{LR}, the low-lying spectrum of $K_A$ should therefore be that of the (boundary) CFT, with an appropriate boundary condition depending on the bulk perturbation.
The same question arises in the context of quantum quenches \cite{CCQQ}. In this case we are interested in the real time evolution of an initial state $|\Psi_0\rangle$ under a hamiltonian $H$ of which it is not an eigenstate. An example is the case where $H=H_{CFT}$ and $|\Psi_0\rangle$ is the ground state of the massive perturbed CFT. This is a difficult problem, and in \cite{CCQQ,JCQQ} the step was taken of replacing this ground state by a conformal boundary state perturbed by irrelevant operators.\footnote{In \cite{CCQQ} only the smeared states of the form (\ref{smeared}) were considered, which happen to lead to subsystem thermalization, while in \cite{JCQQ} it was argued that more general states should lead to a generalized Gibbs ensemble.} This allows the explicit computation of the imaginary time evolution and the continuation to real time, which would be very difficult for the exact ground state of the massive theory.
Thus an important problem in all these cases is to determine to which conformal boundary condition a particular combination of bulk operators should correspond. For simple examples this is apparent by physical inspection. For example, the CFT corresponding to the critical Ising model has two relevant operators, coupling to the magnetic field $h$ and the deviation $t$ of the reduced temperature from its critical value. There are three stable RG fixed points at
$(h=0,t\to+\infty)$ and $h\to\pm\infty$, respectively the sinks for the disordered and the two ordered phases, and corresponding to the three conformal boundary conditions when the Ising spins are respectively free and fixed, either up or down.
One way to make this identification is to think of the boundary condition as defining a state $|\cal B\rangle$ when the theory is quantized on a time slice $x_0=$ constant. In that language we may regard the perturbed CFT as described by a hamiltonian operator
\be
\hat H=\hat H_{CFT}+\sum_j\lambda_j\int\hat\Phi_j(x)d^{d-1}x\,,
\ee
where the $\{\hat\Phi_j\}$ are relevant operators. We then ask which $|\cal B\rangle$, suitably deformed by boundary irrelevant operators, is closest in some sense to the ground state of $\hat H$ at strong coupling.
Conformal boundary states by themselves contain no scale, and therefore cannot be good candidates for the ground state of $\hat H$. Indeed, they must have infinite energy compared to this state. In known examples in 2d (and, for example, for free theories in higher dimensions) they are also non-normalizable. We must therefore deform them by irrelevant boundary operators in order to give them a scale. The simplest such operator is the stress tensor $\hat T_{00}$, which has scaling dimension $d$ and therefore boundary RG eigenvalue $(d-1)-d=-1$. Since its space integral is the CFT hamiltonian, including only this operator is tantamount to considering boundary states `smeared' by evolution in imaginary time:
\be\label{smeared}
e^{-\tau \hat H_{CFT}}|\cal B\rangle\,,
\ee
where $\tau>0$ is parameter with the dimensions of length. Such states have finite energy and correlation length $\propto\tau^{-1}$, and also finite norm.
Such smeared boundary states may be thought of as a continuum version of matrix product states (MPS). Indeed, a lattice discretization of the euclidean path integral, illustrated in Fig.~\ref{MPS}, suggests that such states correspond to matrices with internal dimension $\sim N^{\tau/\delta\tau}$, where $N$ is the number of states on each lattice edge and $\delta\tau$ is the time step. However, unlike discrete MPS states, the smeared boundary state (\ref{smeared}) automatically has the correct short-distance behavior of the CFT.
\begin{figure}[h]\label{MPS}
\centering
\includegraphics[width=0.9\textwidth]{smeared}
\caption{Path integral for smeared boundary state (left) and its lattice discretization (right) as a matrix product state. On the right, each vertical column of lattice sites represents a matrix. The horizontal lines represent contractions between these in the internal space, and the vertical dangling bonds label the physical degrees of freedom.}
\end{figure}
From this point of view it is therefore natural to regard
(\ref{smeared}) as a variational \em ansatz\em, with $\tau$ and the choice of boundary state $|\cal B\rangle$ as variational parameters.
In this paper we explore this idea further and show that this program can be carried through explicitly for the $A_m$ (diagonal) series of unitary minimal 2d CFTs. It should be extendable to the other non-diagonal minimal models, and in principle to other rational 2d CFTs, and indeed to higher dimensional theories if enough information is available about the CFT.
More specifically, given a set of physical conformal boundary states $\{|a\rangle\}$, (whose definition is recalled in Sec.~\ref{sec2}) we take as a variational ground state
\be
|\{\alpha_a\},\{\tau_a\}\rangle=\sum_a\alpha_a\,e^{-\tau_a\hat H_{CFT}}|a\rangle\,,
\ee
and compute the variational energy per unit volume
\be
\lim_{L\to\infty}\frac1{L^D}\frac{\langle\{\alpha_a\},\{\tau_a\}|\hat H_{CFT}+\sum_j\lambda_j\int\hat\Phi_j(x)d^{d-1}x|\{\alpha_a\},\{\tau_a\}\rangle}{\langle\{\alpha_a\},\{\tau_a\}|\{\alpha_a\},\{\tau_a\}\rangle}\,,
\ee
minimizing this with respect to the $\{\alpha_a\}$ and $\{\tau_a\}$.
An important general consequence of the analysis is that, in the limit $L\to\infty$, the minimizing states are always purely physical, that is all but one $\{\alpha_a\}$ vanishes. This is because both $H_{CFT}$ and the perturbing operators turn out to be diagonal in this basis. This is reassuring, as in principle the minimizers could be non-physical linear combinations of these, for example the Ishibashi states in 1+1 dimensions.
Specializing now to the case of 1+1 dimensions,
for the minimal models, the precise values of these diagonal matrix elements are related to the elements of the modular $S$-matrix of the CFT, and, with these in hand, it is straightforward, for a fixed set of couplings $\{\lambda_j\}$ to determine which values of $\tau_a$ and $a$ minimize the variational energy, and thus to map out a rudimentary phase diagram of the theory in the vicinity of the CFT.
It then turns out that although this approach yields correct results in some aspects, for example in determining which combination of bulk couplings $\{\lambda_j\}$ best matches a given boundary state $a$, it is not capable of reproducing some of the finer details of the phase boundaries between different states $a$. In this approximation these are always first-order, and `massless' RG flows to other non-trivial CFTs are
not properly accounted for. This can be seen as a limitation of the particular trial state, which could be remedied by including other operators acting on the boundary state, but at the cost of the loss of analytic tractability.
However, an amusing side result of the analysis is that matrix elements of primary bulk operators $\hat\Phi_j$ between Ishibashi states $\langle\langle i|$, $|k\rangle\rangle$ (which are boundary states within a single Virasoro module) are simply proportional to the fusion rule coefficients:
\be
\langle\langle i|e^{-\tau H}\,\hat\Phi_j\,e^{-\tau H}|k\rangle\rangle\propto N_{ijk}\,.
\ee
This is a consequence of the Verlinde formula \cite{Verlinde}, and to our knowledge has not been previously observed. Although this matrix element should be proportional to the OPE coefficient $c_{ijk}$ which governs the matrix element $\langle i|\hat\Phi_j|k\rangle$ between highest weight states, it is rather surprising that the contributions of all the descendent states should conspire to give the integer-valued fusion rule coefficient.
The outline of this paper is as follows. In Sec.~\ref{sec2} we set up the formalism and prove some general results. In Sec.~\ref{sec3} we apply this to the case of the diagonal minimal models, with the $A_3$ and $A_4$ cases as specific examples, and finally in Sec.~\ref{sec4} give a summary and some further remarks.
After this paper was completed, I was made aware of Ref.~\cite{Kon}, in which similar ideas are explored. However that paper is based on comparing ratios of overlaps between different boundary states and numerical approximations to the exact ground state of the deformed theory, rather than the variational method adopted here. The overlap method is shown to work well for the case of the perturbed Ising model, but is computationally more intensive.
\section{General formalism.}\label{sec2}
As described in the Introduction, we consider a $d$ $(=D+1)$-dimensional CFT perturbed by its bulk primary operators $\{\Phi_j\}$ with coupling constants
$\{\lambda_j\}$, so the hamiltonian is:
\be\label{Hp}
\hat H=\hat H_{CFT}+\sum_j\lambda_j\int\hat\Phi_j(x)d^{D}x\,.
\ee
The theory is quantized on a spatial torus of volume $L^{D}$, where $L$ is much larger than any other scale in the theory.
We assume for simplicity that the $\{\Phi_j\}$ are all scalars and that they have their CFT normalization
\be\label{norm}
\langle\hat\Phi_j(x)\hat\Phi_j(0)\rangle_{CFT}=|x|^{-2\Delta_j}\,,
\ee
where $\Delta_j$ is the scaling dimension of $\Phi_j$.
Although we are interested in relevant perturbations with $\Delta_j<d$, these will in general lead to a finite number of primitive UV divergences up to some finite order in the couplings (as for a super-renormalizable deformation of a free theory), in particular in the ground state energy which we are trying to approximate. These divergences may be subtracted by adding a finite number of counter-terms to $\hat H$ determined the OPEs of the $\{\Phi_j\}$. We assume this has been done.
For example, if $2\Delta_j\geq d$ there is a UV divergence in the ground state energy at $O(\lambda_j^2)$. This is subtracted by a term in $\hat H$ proportional to the unit operator. This does not affect the variational procedure in general.
The case $2\Delta_j=d$ is special and leads to a logarithmic anomaly in the energy. This will be discussed for the 2d Ising model in Sec.~\ref{seclog}.
Conformal boundary states $|\cal B\rangle$ are defined by the condition
\be\label{T0k}
\hat T_{0k}(x)|{\cal B}\rangle=0\,,\quad(k=1,\ldots,D)\,,
\ee
where $\hat T_{ij}$ is the energy-momentum tensor of the CFT. That is, they are annihilated by the momentum density operator, and so are invariant under local time reparametrizations. (For boundaries with a space-like normal there is no energy flow across the boundary.)
Although in higher dimensions these states, and their classification, are poorly understood except for free or weakly coupled CFTs, in 2d much more is known \cite{JC89,PZ}. The Hilbert space is acted on by two copies $({\cal V}\otimes\overline{\cal V})$ of the Virasoro algebra, generated by
\be
\hat L_n=\frac L{2\pi}\int e^{2\pi nix/L}\,\hat T(x)dx\,,\quad\hat{\overline L}_n=\frac L{2\pi}\int e^{-2\pi nix/L}\,\hat{\overline T}(x)dx\,,
\ee
where, as usual, $T\equiv T_{zz}=-T_{00}+T_{11}-2iT_{01}$ and
$\overline T\equiv T_{\bar z\bar z}=-T_{00}+T_{01}+2iT_{01}$, in euclidean signature.
It is spanned by states $|i,N\rangle\otimes\overline{|i',N'\rangle}$, where $N$ labels the states of a module of $\cal V$ with highest weight state labelled by $i$, and similarly for $\overline{\cal V}$. For CFTs with central charge $c\geq1$, this is a Virasoro module, while for the minimal models with $c<1$ it is a Kac module with the null states projected out.
The condition (\ref{T0k}) then corresponds to
\be
\big(\hat T(x)-\hat{\overline T}(x)\big)|{\cal B}\rangle=0\,.
\ee
In terms of the Virasoro generators this becomes
\be
\big(\hat L_n-\hat{\overline L}_{-n}\big)|{\cal B}\rangle=0\,,
\ee
whose solution is the span of the Ishibashi states
\be\label{Ishi}
|i\rangle\rangle=\sum_N|i,N\rangle\otimes\overline{|i,N\rangle}\,,
\ee
where the sum is over all the orthonormalized states in the module.
However, the Ishibashi states are not physical, in the sense that, when they are chosen as boundary states on the opposite edges $x_0=\pm\tau$ of an annulus, the partition function $Z={\rm Tr}\,e^{-L\hat H'}$ evaluated in terms of the generator $\hat H'$ of translations around the annulus does not have the form of a sum over eigenstates with non-negative integer coefficients, as it must if periodic spatial boundary conditions are imposed. For the diagonal minimal $A_m$ models, the physical states which do have this property are linear combinations of the Ishibashi states
\be\label{phys}
|a\rangle=\sum_j\frac{S^i_a}{(S^i_0)^{1/2}}\,|i\rangle\rangle\,,
\ee
where $S^i_k$ is the matrix by which the Virasoro characters transform under modular transformations.
The multiplicities of the eigenstates $j$ of $\hat H'$ which do propagate are then given by the fusion rule coefficients $N^j_{ab}$. In particular the vacuum state propagates only if $a=b$, that is $N^0_{ab}=\delta_{ab}$.
While similar results are available for the non-diagonal minimal models, wider results for general CFTs are not available, which is why we mainly restrict to the $A_m$ models in explicit calculations.
In higher dimensions, the boundary states satisfying (\ref{T0k}) also form a linear space, and we assume that the physical states may be identified analogously. Consider the partition function in the slab ${\mathbb T}^D\times\{-\tau,\tau\}$ (where ${\mathbb T}^D$ is a $D$-dimensional torus of volume $L^D$) with boundary states $|a\rangle$, $|b\rangle$ at $x_0=\pm\tau$:
\be
Z_{ab}=\langle b|e^{-2\tau\hat H_{CFT}}|a\rangle\,,
\ee
(where $\hat H_{CFT}$ is the generator of translations in $x_0$)
and, similarly to the 2d case, demand that when evaluated by quantizing in one of the spatial directions, it has the form of a trace over intermediate states whose energies all scale like $\tau^{-1}$. However this is difficult to implement since this spectrum on the torus is not related to the conformal spectrum for $d>2$.
In fact, we shall need only a weaker condition: that physical states $\{a,b,\ldots\}$ should satisfy
\be\label{gap}
Z_{ab}/(Z_{aa}Z_{bb})^{1/2}=O\big(e^{-{\rm const.}(L/2\tau)^D}\big)\qquad\mbox{for $L/\tau\to\infty$}\,.
\ee
This may be understood as follows: when the boundary conditions are the same, we expect that
\be
Z_{aa}\sim e^{\sigma_a(L/2\tau)^D}\,,
\ee
where the exponent is (minus) the Casimir energy of a system between two identical plates. In this geometry this is always attractive, thus $\sigma_a>0$. It must scale as $L^D$, and, since the boundary conditions and the bulk theory are scale-invariant, also as $\tau_a^{-D}$. The quantity $-\sigma_a L^{D-1}/(2\tau)^D$ is the ground state energy of the generator of translations around one of the spatial cycles of the torus. On the other hand the exponent on the right hand side of (\ref{gap}) is the gap to the lowest-energy state in the sector with $ab$ boundary conditions. It is the interfacial energy from the point of view of $d$-dimensional classical statistical mechanics.
Thus the physical boundary states may in principle be determined by diagonalizing the partition functions in the slab in the limit $L\gg\tau$. We assume that this has been done.
As discussed in the introduction, we use as variational states for the ground state of the perturbed hamiltonian (\ref{Hp}), the ansatz
\be
|\{\alpha_a\},\{\tau_a\}\rangle=\sum_a\alpha_a\,e^{-\tau_a\hat H_{CFT}}|a\rangle\,.
\ee
We first discuss the inner product of these states
\be
\langle a|e^{-\tau_aH_{CFT}}e^{-\tau_bH_{CFT}}|b\rangle\,.
\ee
This is the partition function $Z_{ab}$ in slab of width $\tau_a+\tau_b$ with boundary conditions $a,b$ on opposite faces. As discussed above, for physical boundary states in the limit $L\gg\tau_a+\tau_b$
\be\label{Zab}
Z_{ab}\sim\delta_{ab}\, e^{\sigma_a(L/(2\tau_a))^D}\,.
\ee
The matrix elements of the unperturbed hamiltonian $\hat H_{CFT}$ are, for the same reason, diagonal in this basis as long as
$L\gg\tau_{a,b}$, and may be found by differentiating (\ref{Zab})
\be
\langle a|\hat H_{CFT}\,e^{-2\tau_a\hat H_{CFT}}|a\rangle\sim\delta_{ab}\frac{D\sigma_aL^D}{(2\tau_a)^{D+1}} e^{\sigma_a(L/(2\tau_a))^D}\,.
\ee
Finally we need the matrix elements of the perturbation
\be
\langle a|e^{-\tau_a\hat H_{CFT}}\,\hat\Phi_j(x)\,e^{-\tau_b\hat H_{CFT}}|b\rangle\,,
\ee
which is a one-point function in the slab. Once again, if we evaluate this by inserting a complete set of eigenstates of a generator of translations around the torus, this is dominated by its ground state if $L\gg\tau_{a,b}$, but this contributes only if $a=b$. So the perturbation is also diagonal in the basis of physical states (but not in the Ishibashi basis: see Sec.~\ref{sec33}).
When $a=b$ the one-point functions in the mid-plane of the slab have the form
\be
\langle\Phi_j(x)\rangle=\frac{A_a^j}{(2\tau_a)^{\Delta_j}}\,,
\ee
where the amplitudes $A_a^j$ are universal given the normalization (\ref{norm}) of the operator.
Since the perturbed hamiltonian is diagonal in the physical basis of variational states, the problem becomes much simpler:
for each $a$ we should minimize the variational energy per unit volume
\be\label{Ea}
E_a=\frac{D\sigma_a}{(2\tau_a)^{D+1}}+\sum_j\lambda_j\frac{A_a^j}{(2\tau_a)^{\Delta_j}}\,,
\ee
with respect to $\tau_a$, and then choose the $a$ which gives the absolute minimum.
Note that having found this minimum $a$ for a particular set of couplings $\{\lambda_j\}$, since $E_a$ transforms multiplicatively under
\be
\lambda_j\to e^{(D+1-\Delta_j)\ell}\lambda_j\,,\quad \tau_a\to e^{-\ell}\tau_a\,,\quad
E_a\to e^{(D+1)\ell}E_a\,,
\ee
the absolute minimum will occur for the same value of $a$ along an RG trajectory. This is reassuring, since each point on the trajectory should be described by the same massive QFT up to a rescaling of the mass, which is proportional to $1/\tau_a^{\rm min}$.
Since the $\{\Phi_j\}$ are relevant,
$\Delta_j<D+1$, so that the behavior of $E_a$ as $\tau_a\to0$ (but still $\gg\epsilon$) is dominated by the first term and is positive if $\sigma_a>0$
(which corresponds to the physically reasonable case of an attractive Casimir force.) As $\tau_a\to\infty$ it approaches zero, dominated by the term with smallest $\Delta_j$ and $\lambda_j\not=0$. If the sign is negative this implies that $E_a$ has a negative minimum at some finite value of $\tau_a$. At least for the 2d minimal models we can show that there is always some $a$ for which $\lambda_jA_a^j<0$, so this minimum always exists.
\subsection{Trace of the energy-momentum tensor.}
We may infer a general result about the trace $\langle\Theta\rangle=\langle T^i_i\rangle$ of the energy-momentum tensor in the perturbed theory as approximated by this method. For given set of relevant perturbations $\{\lambda_j\}$ this is given by the response of the action to a scale transformation
\be
\Theta(x)=-\sum_j(D+1-\Delta_j)\lambda_j\Phi_j(x)\,.
\ee
Differentiating (\ref{Ea}) we see that at the minimum
\be
\frac{(D+1)D\sigma_a}{(2\tau_a)^{D+1}}+\sum_j\Delta_j\lambda_j\langle\Phi_j\rangle=0\,,
\ee
and so
\be
E=-(D+1)^{-1}\sum_j\Delta_j\lambda_j\langle\Phi_j\rangle+\sum_j\lambda_j\langle\Phi_j\rangle=-(D+1)^{-1}\langle\Theta\rangle\,.
\ee
Once again, this is reassuring, as we expect that in the ground state of a relativistic theory $\langle T_{00}\rangle=-\langle T_{kk}\rangle$ for $k\not=0$, and so
\be
\langle\Theta\rangle =-\langle T_{00}\rangle+\sum_{k=1}^D\langle T_{kk}\rangle=-(D+1)\langle T_{00}\rangle=-(D+1)E\,.
\ee
The variational method therefore gives a lower bound on $\langle\Theta\rangle$.
\section{2d minimal models}\label{sec3}
We now specialize to the case of the 2d $A_m$ minimal models.
In 2d, the Casimir amplitude $\sigma_a=\pi c/24$, independent of $a$, where $c$ is the central charge.
When $a=b$ the expectation values of the one-point functions in a long strip of width $2\tau_a$ may be found by a conformal mapping from the half-plane to have the form
\be\label{strip}
\langle\Phi_j(x,\tau)\rangle_{\text{strip}}
=\frac{\widetilde A_a^j}{\left((2\tau_a/\pi)\sin(\pi\tau/2\tau_a)\right)^{\Delta_j}}\,,
\ee
where the amplitude governs the behavior of the one-point function in the upper half-plane $y>0$ with boundary condition $a$ on the real axis:
\be
\langle\Phi_j(y)\rangle_{\text{half-plane}}=\frac{\widetilde A_a^j}{y^{\Delta_j}}\,.
\ee
In (\ref{strip}) we should set $\tau=\tau_a$, whence we read off that $A_a^j=\pi^{\Delta_j}\widetilde A_a^j$.
If the operator $\Phi_j$ has its standard CFT normalization (\ref{norm}), the amplitudes $\widetilde A_a^j$ are universal. In \cite{CL} they were computed in terms of the overlap between the boundary state $|a\rangle$ and the highest weight state $|j\rangle$ corresponding to the primary operator $\Phi_j$:
\be
\widetilde A^j_a=\frac{\langle j|a\rangle}{\langle 0|a\rangle}\,.
\ee
This follows by conformally mapping the upper half plane to a semi-infinite cylinder $x>0$ with a boundary condition $a$ at $x=0$, and comparing the result for $x\to\infty$ with the result of inserting a complete set of eigenstates of the generator of translations along the cylinder.
For the $A_m$ minimal models, inserting the expression (\ref{phys}) for $|a\rangle$ we then find \cite{CL}
\be\label{At}
\widetilde A^j_a=\frac{S_a^j}{S_a^0}\left(\frac{S_0^0}{S_0^j}\right)^{1/2}\,.
\ee
To summarize, the variational energy (\ref{Ea}) in this case is given by
\be
E_a=\frac{\pi c}{24(2\tau_a)^2}+\sum_j\lambda_j\frac{S_a^j}{S_a^0}\left(\frac{S_0^0}{S_0^j}\right)^{1/2}\frac{\pi^{\Delta_j}}{(2\tau_a)^{\Delta_j}}\,.
\ee
It is useful to rescale the couplings by positive constants $\tilde\lambda_j=\pi^{\Delta_j}(S_0^0/S^j_0)^{1/2}\lambda_j$
so that this simplifies to
\be
E_a=\frac{\pi c}{24(2\tau_a)^2}+\sum_{j\not=0}\frac{S_a^j}{S_a^0}\frac{\tilde\lambda_j}{(2\tau_a)^{\Delta_j}}\,.
\ee
Note that that sum over $j$ excludes $j=0$ which corresponds to adding the unit operator and therefore a constant shift in the energy.
There are two general statements which follow from the fact that $S$ is a symmetric orthogonal matrix, and that the elements $S_0^j$ are all positive.
First, since all its rows are orthogonal and non-zero it follows that, for $j\not=0$, some of the elements $S_a^j$ are positive and some negative. Therefore, if ${j^*}$ corresponds to the smallest value of $\Delta_j$ such that $\lambda_j\not=0$, and therefore dominates the behavior of $E_a$ as $\tau_a\to\infty$, no matter what the sign of $\lambda_{j^*}$ we may always find at least one $a$ such that
$\lambda_{j^*}S_a^j<0$, and so $E_a$ approaches zero from below. Since $E_a\to+\infty$ as $\tau_a\to0$, this implies that, for these $a$, $E_a$ has a negative minimum at finite $\tau_a$, corresponding to a finite correlation length. This rules out the possibility
that this variational ansatz can describe massless flows to another non-trivial CFT.
Second, we may ask whether there is a combination of couplings $\{\lambda_j\}$ which will lead to a prescribed $b$ as overall minimum. The answer is affirmative. For, suppose we choose
\be\label{combo}
\tilde\lambda_j=-g(2\mu)^{\Delta_j-2}\,S^j_b\,,
\ee
where $g$ is a positive constant and $\mu$ is some fixed scale $>\epsilon$. Then the second term in (\ref{Ea}) is, when $\tau_a=\mu$,
\be
-\frac g{S^0_a\mu^2}\sum_{j\not=0}S^j_aS^j_b=-\frac g{S^0_a\mu^2}\left(\delta_{ab}-S_a^0S_b^0\right)\,.
\ee
Since $0<S^0_a<1$, this is $<0$ if $a=b$ and $>0$ otherwise. Thus, at this scale, the boundary state $b$ will correspond to the lowest trial energy\footnote{This does not rule out the possibility that some other $E_a$ might come lower than this at some other scale, but in practice this does not seem to happen.}. Note that (\ref{combo}) implies including some irrelevant couplings in the mix of deformations.
Further results depend on the detailed form of the modular $S$-matrix for the $A_m$ models.
In particular, we may ask what happens if a single $\lambda_j$ is non-zero.
Depending on whether $\lambda_j>0$ or $<0$, we have to determine which value of $a$ minimizes (maximizes) the ratio $S^j_a/S^0_a$.
Label the positions of the bulk operators in the Kac table by $j=(r,s)$, with $1\leq r\leq m-1$ and $1\leq s\leq m$, and $(r,s)$ identified with $(m-r,m+1-s)$. The label $j=0$ corresponds to $(r,s)=(1,1)$). Similarly label the boundary states $a$ by $(\alpha,\beta)$.
The $A_m$ minimal series of CFTs is conjectured to be the scaling limit of the critical lattice RSOS $A_m$ models \cite{RSOS1,RSOS2}. These models are defined on a square lattice. At each node $R$ there is an integer-valued height variable
$h(R)$ satisfying $1\leq h(R)\leq m$, with the RSOS constraint that $|h(R)-h(R')|=1$ if $R$ and $R'$ are nearest neighbors. The heights may be thought of as living at the nodes of the Dynkin diagram $A_m$, so each configuration is a many-to-one embedding of the diagram into the square lattice. The critical Boltzmann weights of the lattice model are specified in terms of the elements $s_h^0(m)$ of the Perron-Frobenius eigenvector corresponding to the largest eigenvalue of the adjacency matrix of $A_m$. The general eigenvector has the form
\be
s_h^j(m)\propto\sin\frac{\pi jh}{m+1}\,.
\ee
The microscopic interpretation of the conformal boundary states (\ref{phys}) for these models has been given in
\cite{SB,PB}.
The simplest boundary states are when the boundary lies at 45$^{\circ}$ to the principal lattice vectors, and the heights on the boundary are all fixed to the same particular value $h$, say. These have been identified with the conformal boundary conditions with the Kac labels $(\alpha,\beta)=(1,h)$. The second simplest type of microscopic boundary condition is when the boundary heights are fixed to $h$ and on the neighboring diagonal they are fixed to $h+1$. These have been identified with $(\alpha,\beta)=(h,1)$. In \cite{PB} a complete set of microscopic boundary conditions was identified for each Kac label $(\alpha,\beta)$ but these become increasingly complicated. In general the microscopic boundary states corresponding to labels near the center of the Kac table are increasingly disordered.
The ratios of elements of the modular $S$-matrix for the diagonal $A_m$ models are
\be\label{ratio}
\frac{S^j_a}{S^0_a}=\frac{S^{r,s}_{\alpha,\beta}}{S^{1,1}_{\alpha,\beta}}
=(-1)^{(r+s)(\alpha+\beta)}\,\frac{\sin\frac{\pi r\alpha}m}{\sin\frac{\pi \alpha}m}
\,\frac{\sin\frac{\pi s\beta}{m+1}}{\sin\frac{\pi \beta}{m+1}}=(-1)^{(r+s)(\alpha+\beta)}\,\frac{s_\alpha^r(m-1)}{s_\alpha^1(m-1)}\frac{s_\beta^s(m)}{s_\beta^1(m)}\,.
\ee
Locating the global maximum and minimum of this expression for general $(r,s)$ is simplified by the fact that, for fixed $(r,s)$, it factorizes into expressions depending only on $\alpha$ and $\beta$ respectively. Thus we can restrict to the four possible products of the maximum and minimum of each factor, and compare these values.
In each factor the numerator is an oscillating function which is modulated by the positive denominator, which itself has minima
at $\alpha=1,m-1$ (and $\beta=1,m$), which for the lattice $A_m$ models correspond to the most ordered states.
The most relevant bulk operator corresponds to $(r,s)=(2,2)$, when (\ref{ratio}) becomes
\be
4\cos\frac{\pi\alpha}m\cos\frac{\pi\beta}{m+1}\,.
\ee
The extrema of each factor are at $\alpha=1, m-1$ and $\beta=1,m$. Thus for $\lambda_{2,2}>0$ the minimum energy corresponds to $(\alpha,\beta)=(1,m)=(m-1,1)$, and, for $\lambda_{2,2}<0$, $(\alpha,\beta)=(1,1)=(m-1,m)$.
These correspond to the most ordered states, at the ends of the Dynkin diagram. This is to be expected as, in the Landau-Ginzburg correspondence, $\Phi_{2,2}$ is the most relevant Z$_2$ symmetry breaking operator.
Similarly, the most relevant Z$_2$ even operator is $\Phi_{3,3}$, when (\ref{ratio}) becomes
\be
(2\cos\frac{2\pi\alpha}m+1)(2\cos\frac{2\pi\beta}{m+1}+1)\,.
\ee
If $m$ is even, the first factor varies between $2\cos\frac{2\pi}m+1$ at $\alpha=1,m-1$, and $-1$ at $\alpha=\frac12m$,
and the second factor varies between $2\cos\frac{2\pi}{m+1}+1$ at $\beta=1,m$, and $2\cos\frac{\pi m}{m+1}+1$ at $\beta=\frac12m, \frac12m+1$. Thus for $\lambda_{3,3}<0$ there are degenerate minima for $(\alpha,\beta)=(1,1)=(m-1,m)$ and $(\alpha,\beta)=(1,m)=(m-1,1)$ (the most ordered states, which break the Z$_2$ symmetry.).
On the other hand for $\lambda_{3,3}>0$ we need to compare the quantities
\be
(-1)(2\cos\frac{2\pi}{m+1}+1)\,, \quad (2\cos\frac{\pi m}{m+1}+1)(2\cos\frac{2\pi}m+1)\,.
\ee
Numerically, the first is more negative, so the minimum energy in this case corresponds to $\alpha=\frac12m$, $\beta=1,m$. These are Z$_2$-symmetric states.
For odd $m$ the same story holds, with $\alpha$ and $\beta$ interchanged.
Another interesting special case is $\Phi_{1,3}$. This is a perturbation which, with the correct sign, is supposed to flow to the $A_{m-1}$ minimal CFT, and for the other sign to a state with large degeneracy \cite{RSOS1}. As we have seen, such massless flows cannot be accounted for within this set of trial states. In this case (\ref{ratio}) simplifies to
\be
2\cos\frac{2\pi\beta}{m+1}+1\,,
\ee
independent of $\alpha$. Depending on the sign of the coupling, this picks out the boundary states either with $\beta=1,m$ or with $\beta\approx\frac12m$. In both cases, however, there is an $(m-1)$-fold degeneracy of candidate ground states. This reflects a flow towards a true first-order transition, as expected for one sign of the coupling \cite{RSOS1}, or the best attempt of this approximation to reproduce the critical point of the $A_{m-1}$ model, as expected for the other sign. This is an important check on the effectiveness of our approach.
These somewhat cryptic general remarks are best illustrated with some simple examples.
\subsection{The Ising model.}
This corresponds to $A_3$. The perturbed hamiltonian is
\be\label{HIsing}
\widehat H=\widehat H_{CFT}+t\int\hat\varepsilon dx+h\int\hat\sigma dx\,,
\ee
where $\varepsilon=\Phi_{2,1}=\Phi_{1,3}$ and $\sigma=\Phi_{1,2}=\Phi_{2,2}$ are the energy density and magnetization operators respectively.
In this case, the bulk operators are $\{\Phi_j\}=(1,\epsilon,\sigma)$, and the boundary states in the same labeling are $(+,-,f)$, corresponding to fixed$(+)$, fixed$(-)$ and free boundary conditions on the Ising spins. The $S$-matrix in this ordering of the basis is
\be
S=\left(\begin{array}{ccc}\ffrac12&\ffrac12&\ffrac1{\sqrt2}\\
\ffrac12&\ffrac12&-\ffrac1{\sqrt2}\\
\ffrac1{\sqrt2}&-\ffrac1{\sqrt2}&0\end{array}\right)\,.
\ee
After rescaling the couplings as above, we find that
\begin{eqnarray}
E_+&=&\frac\pi{48(2\tau_+)^2}+\frac t{2\tau_+}+\sqrt2\frac h{(2\tau_+)^{1/8}}\,,\label{Eaa}\\
E_-&=&\frac\pi{48(2\tau_-)^2}+\frac t{2\tau_-}-\sqrt2\frac h{(2\tau_-)^{1/8}}\,,\label{E2a}\\
E_f&=&\frac\pi{48(2\tau_f)^2}-\frac t{2\tau_f}\,.\label{Ef}
\end{eqnarray}
For $h=0$, $t>0$, corresponding to the disordered state, it is clear that the minimizer is $E_f$. For the opposite sign of $t$ with $h>0$, the minimizer is $E_-$, corresponding to negative magnetization (recall the definition of the sign of the couplings in (\ref{HIsing})), and vice versa. As $h\to0$ from either side with $t<0$, we remain in one or the other of these states, corresponding to spontaneous symmetry breaking.
However, for $t>0$ and $0<h\ll t^{15/8}$ there is a problem. The minimum of $E_-$ is found by balancing the last two terms in (\ref{E2a}), and therefore occurs when $\tau_-=O((t/h)^{7/8})$ at a value
$E_-=-O(h^{8/7}/t^{1/7})$. On the other hand the minimum of $E_f=-O(t^2)$ is much lower in this limit. This would suggest, incorrectly, that the magnetization is zero in the ground state. As we increase the ratio
$h/t^{15/8}$, eventually these levels cross, but there is no reason for $\tau_-$ and $\tau_f$ to be equal at this point.
This appears to be an inherent problem of using a variational ansatz which is not sufficiently complex. It could presumably be overcome by using a trial state of the form
\be
e^{-\tau\hat H_{CFT}}\,e^{-h_s\int\hat\sigma_s dx}\,|f\rangle\,,
\ee
where $\hat\sigma_s$ is the boundary magnetization coupling to a boundary magnetic field $h_s$, at the cost of loss of analytic tractability.
\subsubsection{Logarithmic anomaly.}\label{seclog}
When $h=0$ it follows from (\ref{Eaa},\ref{E2a},\ref{Ef}) that the minimum energy scales like $t^2$. Yet it has been known since Onsager that the correct behavior is $t^2\log t$. The origin of this logarithmic anomaly is a cancellation between the scaling term $t^{2/(2-\Delta_\varepsilon)}$ and the analytic background $\propto t^2$, both of which occur with amplitudes which diverge as $\Delta_\varepsilon\to1$. This may be accounted for within the variational approach by adding a counter-term as before, proportional to the space-time integral of the 2-point function, which will now also have a logarithmic dependence on the IR cutoff $\tau$. Thus, for example, (\ref{Ef}) becomes
\be
E_f=\frac\pi{48(2\tau_f)^2}-\frac t{2\tau_f}-At^2\log(\tau_f/\epsilon)\,,
\ee
where $\epsilon$ is the short-distance cutoff and $A$ is a (calculable) $O(1)$ constant. The minimum still occurs at
$\tau_f\sim t^{-1}$, but the last term now contributes the desired logarithm at the minimum.
\subsection{The tricritical Ising model.}
This corresponds to the $A_4$ lattice model with heights
$h(R)\in\{1,2,3,4\}$. The RSOS condition means that $h$ is even on even sites odd $s$ on odd sites, or vice versa.
In the Landau-Ginzburg picture it corresponds to a scalar field $\phi$ with a $\phi^6$ interaction, and a Z$_2$ symmetry under $\phi\to-\phi$.
Note that in the lattice model this Z$_2$ symmetry is implemented by reflecting the Dynkin diagram \em and \em a sublattice shift. The Kac table with bulk operators labelled by Landau-Ginzburg is shown in Fig.~2.
Note that odd $r$ is Z$_2$ odd and vice versa.
However another model in the same universality class is the spin-1 (Blume-Capel) Ising model, which may be thought of as an Ising model with vacancies.
\begin{figure}[h]\label{LG}
\centering
\includegraphics[width=0.25\textwidth]{A4LG}
\caption{Landau-Ginzburg assignment of bulk operators in the $A_4$ Kac table.}
\end{figure}
\begin{figure}[h]\label{A4}
\centering
\includegraphics[width=0.4\textwidth]{A4bcv2}
\caption{Correspondence between boundary conditions in lattice models and Kac labels of conformal boundary states: in the $A_4$ model according to Ref.~\cite{PB} (upper labels), and in the Blume-Capel model, according to Ref.~\cite{Affleck} (lower labels).}
\end{figure}
The usually accepted phase diagram and RG flows of the tricritical Ising model near the tricritical fixed point are quite complex. (See for example Fig.~4.2 of \cite{JCbook}.) In the Z$_2$-even sector, turning on the most relevant operator $\Phi_{3,3}\sim\phi^2$ gives flows either to the high-temperature disordered phase, or to the 2 coexisting low-temperature ordered phases. Turning on the $\Phi_{1,3}\sim\phi^4$ operator gives flows either to a first-order transition between these ordered phases and
a disordered phase with vacancies, or to the $A_3$ Ising fixed point.
As for the Ising model, turning on the $\Phi_{2,2}\sim\phi$ operator leads to broken-symmetry phases. However, at low temperatures there may be coexistence between two such phases with different densities of vacancies. These persist to finite temperature, giving `wings' in the phase diagram which end in lines of Ising-like transitions. These lines meet in the tricritical point and correspond to flows generated by the non-leading but relevant Z$_2$-odd operator $\Phi_{2,1}\sim\phi^3$.
According to Behrend and Pearce \cite{PB}, the labelling of the boundary states in the $A_4$ lattice model is as shown in Fig.~3. On the same diagram we have indicated their interpretation in the Blume-Capel model, due to Chim \cite{Chim} and Affleck \cite{Affleck}, which is perhaps more intuitive. Here $(\pm)$ label totally ordered states, $(0)$ is a vacancy-rich state, and $(0\pm)$ are partially ordered states. $(d)$ is a multicritical point separating these in the boundary RG flows \cite{Affleck}.
Note that the $\alpha=2$ states are Z$_2$ even while the Z$_2$ symmetry interchanges $\alpha=1$ and $\alpha=3$ (keeping $\beta$ the same.)
Let us see how well the variational approach reproduces this picture. According to the earlier analysis, turning on
the $\Phi_{2,2}$ operator corresponds to the boundary states at the corners of the Kac table in Fig.~\ref{A4}. These are the most ordered states.
Again, turning on the $\Phi_{3,3}$ perturbation corresponds to the boundary states
which extremize $\big(2\cos(\pi\alpha/2)+1\big)\big(2\cos(2\pi\beta/5)+1\big)$. This gives $\beta=1,4$, and, depending on the sign of the coupling, either $\alpha=2$ or $\alpha=1,4$. These correspond to the disordered and ordered Ising-like phases, respectively, as expected.
Turning on $\Phi_{1,3}$ corresponds to maximizing only the second factor $\big(2\cos(2\pi\beta/5)+1\big)$, and so, depending on the sign of the coupling gives either $\beta=1,4$, corresponding to coexistence between these Ising-like phases (instead of a second order critical point as it should), or $\beta=2,3$ coexistence between partially ordered phases and a disordered, vacancy-rich phase.
Turning on $\Phi_{2,1}$, on the other hand, corresponds to extremizing
$(-1)^{\alpha+\beta}\cos(\pi\alpha/4)$. For one sign of the coupling we get coexistence between the strongly ordered phase $(-)=(1,2,1,2)$ and the partially ordered phase $(0+)=(3,2/4,3,2/4)$, and for the other sign we get coexistence between their Z$_2$ partners. This is once again in general agreement with the wings of the phase diagram, except that the approximation suggests a first-order rather than an Ising-like continuous transition.
We conclude that for this model the boundary states roughly reproduce the expected RG flows when a single relevant operator is turned on, with the exception that flows to non-trivial CFTs are approximated by first-order rather than continuous transitions.
\subsection{Matrix elements between Ishibashi states and the fusion rules.}\label{sec33}
As an aside, we mention a curiosity which follows from the result (\ref{At}) for the matrix element of a bulk primary field between physical states in the limit $L\to\infty$:
\be
\langle a|e^{-\tau\hat H}\hat\Phi_je^{-\tau\hat H}|b\rangle=\delta_{ab}\left(\frac\pi{2\tau}\right)^{\Delta_j}\,
\frac{S_a^j}{S_a^0}\left(\frac{S_0^0}{S_0^j}\right)^{1/2}
\ee
and the definition of these states in terms of the Ishibashi states (\ref{phys}), which, on inverting becomes:
\be
|i\rangle\rangle=\sum_aS^i_a(S^i_0)^{1/2}\,|a\rangle\,.
\ee
Hence
\be
\langle\langle i|e^{-\tau\hat H}\hat\Phi_je^{-\tau\hat H}|k\rangle\rangle=
\left(\frac\pi{2\tau}\right)^{\Delta_j}\left(\frac{S_0^0}{S_0^j}\right)^{1/2}(S^i_0)^{1/2}(S^k_0)^{1/2}
\,\sum_a\frac{S_a^iS_a^jS_a^k}{S^0_a}\,.
\ee
We recognize the sum over $a$ as the Verlinde formula \cite{Verlinde} for the fusion rule coefficient $N_{ijk}$. Taking into account the normalization of the states, we have
\be\label{fusion}
\frac{\langle\langle i|e^{-\tau\hat H}\hat\Phi_je^{-\tau\hat H}|k\rangle\rangle}
{\big(\langle\langle i|e^{-2\tau\hat H}|i\rangle\rangle
\langle\langle k|e^{-2\tau\hat H}|k\rangle\rangle\big)^{1/2}}
=\left(\frac\pi{2\tau}\right)^{\Delta_j}\left(\frac{S_0^0}{S_0^j}\right)^{1/2}\,\,N_{ijk}\,.
\ee
Note that the first factor could be absorbed into a redefinition of the normalization of $\hat\Phi_j$.
This result is somewhat surprising, and, to our knowledge, has not been noticed before. If we insert the definition (\ref{Ishi}) of the Ishibashi states into the numerator, the leading term for $\tau\gg L$ is proportional to the OPE coefficient $c_{ijk}$, which certainly vanishes whenever $N_{ijk}$ does. The contributions of all the descendent states are all proportional to $c_{ijk}$, with coefficients which could, in principle, be computed from the Virasoro algebra. However, it is remarkable that they all conspire to sum to the integer-valued fusion rule coefficient $N_{ijk}$.
If there were an independent way of establishing (\ref{fusion}) this would give an alternative derivation of the Verlinde formula. It would be interesting to see whether this result extends to non-diagonal minimal models and to other rational CFTs. Although it has been derived here only for the Virasoro minimal models, there seems to be no obstacle in principle to its generalization to other rational CFTs, and it then suggests that the 1-point functions of bulk fields between suitable boundary states are determined by purely topological data of the fusion algebra. It also gives a possible way to \em define \em (at least ratios of) the fusion rules for non-rational CFTs.
\section{Conclusion.}\label{sec4}
We have proposed using smeared boundary states as trial variational states for massive deformations of CFTs. This is motivated by the uses of these states in quantum quenches and entanglement studies. In the case of the 2d minimal models we can perform explicit calculations which show this method gives a qualitative picture of the phase diagram in the vicinity of the CFT. Its main failing is that it cannot correctly predict a flow to a non-trivial CFT. In this case it appears to suggest phase coexistence rather than a continuous transition. In addition, the boundaries between different states corresponding to different renormalization group sinks are always first-order transitions. This is a necessary consequence of the variational method.
However the method, by its nature, always gives the correct scaling of the energy with the coupling constants. From a numerical point of view it cannot be competitive with earlier methods such as the truncated conformal space approach \cite{TCFT1,TCFT2}, but it is much simpler and moreover gives new insight into the physical relationship between conformal boundary states and ground states of gapped theories. Since it gives a bound on the universal term in the free energy, it would be interesting to make a detailed comparison with exact results available for integrable perturbations \cite{Fateev}.
\section*{Acknowledgements}
The author thanks V.~Pasquier, H.~Saleur and G.~Vidal for helpful discussions, and A.~Konechny for pointing out Ref.~\cite{Kon}.
\paragraph{Funding information.}
This work was supported in part by funds from the Simons Foundation, and by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic
Development and by the Province of Ontario through the Ministry of Research and Innovation.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Skulls of the Shogun Is The Wu-Tang Advance Wars We Didn't Know We Needed
Stephen Totilo
Filed to: Wait for thisFiled to: Wait for this
Wait for this
Jake kazdal
Haunted temple studios
It helps if you can hear the soundtrack. It helps if you loved Advance Wars. It might even help if you liked listening to the Wu-Tang Clan or watched Kill Bill a decade ago and developed a love for an old martial arts movie like Shogun Assassin.
These things will help, but if you don't know — or even if you do — allow me to translate Skulls of the Shogun for you.
What we've got is a a video game about a dead shogun who doesn't want to wait 500 years to get his turn at rest in the afterlife. No, he's going to fight his way through his fellow dead warriors to find peace, using the kind of aggressive problem-solving that makes video games action as undiplomatic as it is interesting.
The game, Skulls of the Shogun, is a thematic descendant of samurai movies filtered through the better martial arts hip-hop style of Wu-Tang producer the RZA (listen to the video here; that's in-game music. In terms of how it plays, Skulls is the child of Final Fantasy Tactics, X-Com, Advance Wars, Valkyria Chronicles and, of course, the game that spawned all of them: chess. This is a turn-based strategy game with the Shogun as the prize piece, the piece whose defeat renders a match immediately lost but who, along with his fighting allies, has abilities that make him a unique combatant on the board.
I saw Skulls of the Shogun last month at PAX East in Boston, where its lead creator, Jake Kazdal of Haunted Temple Studios, took me through the action. Then he e-mailed me a build of the unfinished game so I could help fill the maw in my life that's opened during the current hiatus of Advance Wars. Valkyria Chronicles hasn't done that; it doesn't play fast enough for me. In all these games, you bring a small army to battle, preparing to face off against an enemy squad, all of these units confined to a map marked with varied terrain that might affect play by, say, providing better or worse cover.
The best player lets their shogun wait to join the battle, then makes him eat three fallen skulls, producing one hellraiser of unit.
In each of these games, you play your turn, moving your units by some defined amount, letting any of them who can attack do so, and then waiting through the enemy's turns, watching their maneuvers and hoping that you marshaled the troops correctly. Play it well and there will be no surprises during the enemy turns. You'll know your gains and losses in your mind before you execute them on the map. These games are constrained by numbers, by the stats that define how far each unit can move and what its chances will be to win if it should attack or be attacked.
Advance Wars and Final Fantasy Tactics, the great turn-based strategy series from Nintendo and Square Enix, are set on grids, the latter getting so specific that it requires you to choose which way your unit will face at the end of fulfilling its round of orders. The pace of play in these games can be methodical, though accelerated if the game's battle animations are turned off (you want this... imagine having the option to either watch a chess knight joust against a chess pawn Every Time a pawn is getting killed or just seeing the knight knock the pawn piece off the board). Valkyria Chronicles, Sega's version of this, can play more slowly, due to much more graphical pomp and circumstance, though it hides some of the off-putting math by letting characters move across the map in any natural direction, instead of channeling them from grid square to grid square.
Skulls of the Shogun is a hybrid of the aforementioned games, moving with the gridless fluidity of a Valkyria Chronicles game and at the pace of an animations-off Advance Wars. For that achievement of subgenre excellence, it is welcome.
In the early levels of Skulls that I played, I commanded the Shogun and his men in his first few battles of the afterlife. He moves from map node to map node, facing several skirmishes in each mission. He fights, in these early levels, beside a couple of infantry samurai, a couple of archers and some cavalry. Befitting the genre, the units have different strengths and weaknesses. Each can move in any direction you send them, but the circle that defines the limit of where the cavalry can move in a single turn is the biggest. Those guys can chew up terrain. Those mounted fighters, however, are not the potent combatants that the infantry are. But they are better in close quarters fighting than the hopeless archers who are only a menace from afar. All of the units are weaker than the shogun, who grows in power the longer you wait to unleash him.
The numbers in Skulls of the Shogun are hidden or made pretty. Movement stats and attack ranges are displayed as concentric circles around characters. A unit's health is represented by the degree to which the flag he carries into battle is tattered. The odds of a successful attack are presented as color-coded red outlines around enemies whose vulnerability is certain and orange outlines for those who might be missed. You can read the status of every unit on the map with ease, recognizing that characters with weapons drawn still have an opportunity to attack and that those with weapons raised are within range to respond to an attack with their own counter-attack. (The game's chief inelegance is its still in-the-works controls, which don't yet allow as gracious and efficient character selection with an Xbox 360 controller as one would hope.)
Kazdal was surprised how slowly I played it. I was paying his game a compliment.
Into the old strategy formula there's an infusion of surprises. Units can form blocking walls by lining up next to each other, producing a glow around the clustered units that indicates that anyone behind them is safe. Units can knock other units back with the force of an attack, pushing an enemy either out of range or even off a cliff to sudden death. Units can haunt rice paddies to gain money for buying extra units or haunt temples to produce magic-wielding monks who heal friends or hurl fire. And then there are the skulls, which are dropped when a unit is defeated and can be eaten, powering the feeding unit up. The best player lets their shogun wait to join the battle, then makes him eat three fallen skulls, producing one hellraiser of unit who can rip the enemy apart. Ideally the enemy player — computer-controlled or one of three competing human players — hasn't prepped their shogun the same way.
When Kazdal showed me the game in Boston he was surprised how slowly I played it. I was paying his game a compliment, as I'm precious with the turns I take in Advance Wars or its Nintendo cousin Fire Emblem. I savor games like these and lament their lack of prominence, even wishing against harsh reality that one would have appeared from Nintendo during the launch of their latest portable gaming system. I can wait a little longer on Nintendo, Sega and the rest of the strategy-makers because I'm optimistic that Skulls of the Shogun can satisfy my fix.
The 3DS's Submarine Side-Scroller Teases Us With A Pseudo-Submarine Advance Wars
The 3DS launch game Steel Diver was first shown last June, back when it seemed like a simple, solid …
The game isn't done, but it's close. Kazdal is targeting a late summer or early fall release on Xbox Live Arcade, with a Steam and Windows Phone 7 version also around that time.
I'm eager for Skulls of the Shogun. If you're not, at least listen to the music (by Sam Bird and DJ Makyo). We don't just have a contender for a strong new strategy game; we've got one of 2011's best soundtracks in the making.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,221
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using namespace swift;
using namespace swift::syntax;
void UnknownSyntax::validate() const {
assert(Data->Raw->isUnknown());
}
#pragma mark - unknown-syntax API
size_t UnknownSyntax::getNumChildren() const {
size_t NonTokenChildren = 0;
for (auto Child : getRaw()->Layout) {
if (!Child->isToken()) {
++NonTokenChildren;
}
}
return NonTokenChildren;
}
Syntax UnknownSyntax::getChild(const size_t N) const {
// The actual index of the Nth non-token child.
size_t ActualIndex = 0;
// The number of non-token children we've seen.
size_t NumNonTokenSeen = 0;
for (auto Child : getRaw()->Layout) {
// If we see a child that's not a token, count it.
if (!Child->isToken()) {
++NumNonTokenSeen;
}
// If the number of children we've seen indexes the same (count - 1) as
// the number we're looking for, then we're done.
if (NumNonTokenSeen == N + 1) { break; }
// Otherwise increment the actual index and keep searching.
++ActualIndex;
}
return Syntax { Root, Data->getChild(ActualIndex).get() };
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 952
|
Visa and the U.S. Small Business Administration awarded $60K in cash prizes at their joint hackathon
The two organizations hosted their second annual hackathon over the weekend, this time focused on developers building app-based solutions that will help small businesses prepare and recover from major natural disasters.
By Michelai Graham / staff
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SnapShot received $25, 000 for their work at the Visa and SBA annual hackathon.
Photo by Joy Asico, AP Images for Visa.
Updated on 05/07/2019 at 9:42 a.m.
For a second year in a row, Visa and the U.S. Small Business Administration teamed up to host a hackathon to kick off #SmallBusinessWeek, taking place May 5-11.
The event took place on Sunday, May 5, at the Inclusive Innovation Incubator in Washington, D.C. Entrepreneurs, designers and developers from all over the U.S. began their hacking on Friday, May 3, and concluded for presentations on Sunday morning. Of the teams formed, 27 presented their final solutions to a panel of judges that included this reporter (me), Visa VP of Global Small Business Nate Smith, Three Brothers Bakery President Janice Jucker and SBA CIO Maria Roat.
The apps focused on solutions that will help small businesses prepare and recover from major natural disasters like hurricanes, wildfires and earthquakes.
David Simon, Visa's global head of small and medium enterprises, and Chris Pilkerton, Small Business Association's acting administrator, announced the final awards.
Here are this year's winners:
First place receiving $25,000: SnapShot, an app-based solution using bookkeeping APIs to help small businesses catalog their assets before a disaster, and then uses AI-based damage detection software to assess the value of damages through mobile photos.
Second place receiving $15,000: Disaster Recovery Score, this app solution digitally consolidates forms and disaster resources when filing for loans and claims. The app provided a disaster readiness score to determine a small businesses's ability to recover from a natural disaster.
Third place receiving $10,000: Route7, an app using Amazon virtual assistant Alexa's voice assistance and a visual recognition software to speed up the pre and post-inventory process.
Visa API Challenge Winner receiving $5,000: Disaster Recovery Score
Authorize.net API Winner receiving $5,000: SBAssist, a web platform that maps small businesses affected by federally and state declared disasters, giving users the ability to contribute to those businesses directly and receive social badges.
Congrats to the 2019 #SmallBusinessWeek Hackathon winners Team Snapshot! Their app detects damage to inventory/equipment in photos so #smallbusiness can file insurance claims bypassing days or weeks for an insurance adjuster to assess damage manually. @VisaDeveloper @Visa pic.twitter.com/3EAu1ems9F
— SBA (@SBAgov) May 5, 2019
Simon told Technical.ly that Visa intends to continue this partnership with SBA to host the hackathon annually. He said the hackathon grew from the first event last year with 12 teams, to the 27 who presented this year.
"I think we learn something each time," Simon said. "I think the most important thing to do is listen. It's about what we learn and how we apply that. We also learned that the format needs to continue to evolve."
With this year's hackathon, Simon said the topic was more specific, compared to last year's hackathon where teams didn't have a specific target. The disaster recovery topic stemmed from Jucker and her experience bringing her company back from a natural disaster after her restaurant Three Brothers Bakery in Houston was flooded during Hurricane Harvey. Jucker told us that it took 17 days to reopen the bakery and at that time, they only had a shelf of baked goods available.
Ashwin Kumar and Sarah Han, both West Coast residents, told Technical.ly that they formed their team a few weeks prior to the hackathon and traveled to the District to ultimately win the grand prize with SnapShot. Han works in Los Angeles as a software engineer on web development, and Kumar works as a product manager in San Francisco at an AI startup. Both are developers with a few years of experience participating in hackathons.
"That's how I learned how to code, just basically going to hackathons and doing it through that," Kumar said. "I win sometimes, and I don't win sometimes, but I think either way it's just a great way to kind of develop new technologies, look at a problem that maybe we don't generally think about in our own lives, and have a challenge and constraints and be like alright, for one weekend, we're going to focus on this."
Kumar and Han said they see themselves working together to flush out SnapShot's business model to scale the company beyond this hackathon experience. Further, Simon said Visa and SBA will continue to work with the winning companies to help nurture their ideas further.
Companies: Small Business Administration
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 88
|
\section{Introduction}
In the last several years much has been said about the domain
walls in various supersymmetric field theories in four dimensions
\cite{1}. The existence of the BPS saturated domain walls is in
one-to-one
correspondence with the central extension of ${\cal N} =1$
superalgebra, with the
central charge $Z_{\alpha\beta}$ lying
in the
representation $\{0,1\}$ or $\{1,0\}$ of the Lorentz
group (for brevity we will
refer to such charges as the $(1,0)$ charges).
In the non-Abelian gauge theories the $(1,0)$ central charge
emerges as a quantum anomaly
in the superalgebra \cite{two} -- \cite{four}.
The possibility of the
existence of the tensorial central charges in ${\cal N} =1$
superalgebras was noted in the
brane context in Ref. \cite{deAzcarraga:1989gm}.
The general theory of
the
central charges in ${\cal N} =1$
superalgebras was revisited recently \cite{FP}.
In this paper we will discuss, in various theories, the
central extensions of ${\cal N}
=1$ superalgebras with the central charge $Z_{\alpha\dot\beta}$
lying
in the
representation $\{1/2,1/2\}$ of the Lorentz
group (to be referred to as the (1/2,1/2) charges).
Such central charges are related to BPS objects with the axial
geometry,
in particular,
the saturated strings. The fact that
they exist is very well known in the
context of
supersymmetric QED (SQED) with the Fayet-Iliopoulos term,
see Ref. \cite{2,2prim} and
especially Ref. \cite{3}, specifically devoted
to this
issue. In Ref. \cite{3} it is shown, in particular,
that if the spontaneous breaking of U(1) is due to the superpotential
(the so-called $F$ model), then the Abrikosov strings cannot be
saturated. At the same time, if the spontaneous breaking of U(1) is
due
to the Fayet-Iliopoulos term (the so-called $D$ model, with the
vanishing superpotential) then the Abrikosov string is saturated, one
half of supersymmetry is conserved, and the string tension is given
by the value of the central charge.
\footnote{The statements above refer to ${\cal N}=1$ theories.
In certain ${\cal N}=2$ extensions of QED
one finds BPS saturated strings
without the Fayet-Iliopoulos term. See Secs. 4 and 9.3.}
Another physically interesting
example where the (1/2,1/2) charges
play a role is the wall junction.
The fact that generalized Wess-Zumino (GWZ) models
with a global symmetry of the U(1) or $Z_N$
type may contain BPS wall junctions
was noted in Ref. \cite{AT}.
The interest to the wall junctions preserving one quarter of the
original supersymmetry was revived recently
after the publications~\cite{wjone,3dwj}, discussing such
junctions
in some GWZ models.
In this work we calculate the central
extension of the ${\cal N}=1$ superalgebra
of the $Z_{\alpha\dot\beta}$ type for a
generic gauge theory, with or without
matter. As will be seen,
a spatial integral of a
full spatial derivative of the
appropriate structure does indeed emerge.
It will be explained how the mass of the saturated solitons with the
axial geometry depends on the combination of
the $(1,0)$ and $(1/2,1/2)$ central charges. For the solitons that are
pure BPS strings
(i.e. they posses axial geometry, {\em and} their energy density is
completely localized near some axis) only the $(1/2,1/2)$ charge
can contribute.
We found that in the Wess-Zumino models, as well as in
the gauge theories with matter, the expression for
this central charge {\em per se} contains certain terms with
coefficients
which are ambiguous. Of critical importance
is the ambiguity in the coefficient of the squark term.
Using this ambiguity, we
will prove that in {\em weak coupling}
the only ${\cal N}=1$
gauge model admitting the BPS strings is SQED with the
Fayet-Iliopoulos term. We then present some speculative ideas as to
the possibility of the BPS strings in the non-Abelian models in
{\em strong coupling}. For the objects of the type of the wall
junctions
the ambiguity mentioned above conspires with a related ambiguity
in the $(1,0)$ central charge, so that the resulting energy
of the wall junction configuration is unambiguous.
\section{Generalities}
Let $Q_\alpha\, , \bar Q_{\dot\alpha}$ be supercharges of
the ${\cal N} =1$ four-dimensional field theory under consideration.
The central charge relevant to strings, $Z_{\alpha\dot\alpha}$,
appears in the anticommutator
\begin{eqnarray}
\{Q_\alpha\, , \bar Q_{\dot\alpha}\}
&\!\! = &\!\!2 P_{\alpha\dot\alpha}
+2 Z_{\alpha\dot\alpha} \nonumber\\[0.2cm]
&\!\!\equiv&\!\!
2\left\{ P_\mu + \int\, {\rm d}^3 x\, \varepsilon_{0\mu\nu\chi}\,
\partial^\nu a^\chi \right\}
\left(\sigma^\mu \right)_{\alpha\dot\alpha}\, ,
\label{bsa}
\end{eqnarray}
where $P_\mu$ is the momentum operator, and $a^\nu$ is an
axial vector specific to the theory under consideration.
It must be built of
dynamical fields of the theory.
In other words, the $(1/2,1/2)$ central charge is
\begin{equation}} \def\eeq{\end{equation}
Z_\mu = \int\, {\rm d}^3 x\, \varepsilon_{0\mu\nu\chi}
\, \partial^\nu a^\chi\,.
\label{bsaa}
\eeq
The corresponding tensor current
$$
j_{\rho\mu} = \varepsilon_{\rho\mu\nu\chi}
\, \partial^\nu a^\chi
$$
is obviously conserved nondynamically, irrespective of the
concrete form of the axial current $ a^\chi$.
Assume that the string is
aligned along the vector $n_\mu$ (it is normalized by the condition
$n_\mu n^\mu =-1$), and $L$ is the length of the string ($L$ is
assumed
to tend to infinity).
Then the second term in Eq. (\ref{bsa}) can be always represented as
\begin{equation}} \def\eeq{\end{equation}
Z_\mu = \int\, {\rm d}^3 x\, \varepsilon_{0\mu\nu\chi}
\, \partial^\nu a^\chi = T L \, n_\mu\, ,
\label{bsaaprim}
\eeq
where $T$ is a parameter of dimension mass squared.
The
direction
of $n_\mu $ can always be chosen in such a way as to make
$T$ in Eq. (\ref{bsaa}) positive. We will always assume $T
>0$.
In the rest frame of the string lying along the $z$ direction
(i.e. ${\bf n} = \{0,0,1\}$,
or $ n_\mu = \{0,0,0,-1\}$) the superalgebra
(\ref{bsa})
takes the form
\begin{equation}
\{Q_\alpha\, , \bar Q_{\dot\alpha}\} = 2\, \left[ \begin{array}{cc}
M-T L & 0\\ 0 & M+ T L
\end{array} \right]_{\alpha\dot\alpha}\, ,
\label{ququ}
\end{equation}
where $M$ is the total mass of the string.
For the saturated strings
\begin{equation}} \def\eeq{\end{equation}
M = T L\, ,
\eeq
i.e. the mass of the string coincides with the central charge
appearing
in the ${\cal N} =1$ superalgebra (\ref{bsa}). The parameter
$T$ is then identified with the string tension. If the
state of the BPS string is denoted $|\, {\rm str}\rangle$, then
\begin{equation}} \def\eeq{\end{equation}
Q_1|\, {\rm str}\rangle = \bar Q_{\dot 1}|\, {\rm str}\rangle= 0\, .
\eeq
In other words, $Q_1$ and $\bar Q_{\dot 1}$ annihilate the string --
this half of supersymmetry is conserved in the saturated string
background. The action of $Q_2$ and $\bar Q_{\dot 2}$ on $|\, {\rm
str}\rangle$ produces the fermion zero modes.
Any four-dimensional ${\cal N}=1$
theory can be dimensionally reduced
to two dimensions, where it becomes ${\cal N}=2$ theory.
If the latter has topologically stable instantons,
elevating the theory back to four dimensions gives us strings.
Classical descriptions are totally
equivalent. Distinctions occur at the
level of quantum corrections, which are to be treated differently
in two- and four-dimensional theories. The topological charge of the
two-dimensional theory is related to the central charge of the
centrally
extended algebra (\ref{bsa}). This simple observation
allows one to use a wealth of information regarding various
two-dimensional models
in analysis of saturated strings
in four dimensions at the classical level.
For the solitons of the wall junction type, which preserve
a quarter of the original supersymmetry
(more generally, for the BPS solitons
with the axial geometry), it is necessary to consider, simultaneously,
the $(1,0)$ charge, which appears in the commutator
\begin{equation}
\{ Q_\alpha Q_\beta\} = -4 i \left(\vec\sigma\right)_{\alpha\beta}\,
\int\, {\rm d}^3 x\,\vec\nabla\, \bar \Sigma\, ,
\label{moycc}
\end{equation}
where $\bar\Sigma$ is a scalar operator built of the dynamical fields
of
the
theory, and
\begin{equation}
\left(\vec\sigma\right)_{\alpha\beta}=\{-\tau_3\,,\, i\,,\,
\tau_1\}_{\alpha\beta}\,.
\end{equation}
For the BPS strings the $(1,0)$ charge must vanish; however,
for the wall junctions and other axial geometry BPS solitons
both the $(1,0)$ and $(1/2,1/2)$ charges do not vanish (see Sec. 3).
In this case the general structure of the supercharge
anticommutators is as follows
\begin{equation}} \def\eeq{\end{equation}
\frac{1}{2L}\{{\cal Q}\, {\cal Q}\}\to
\begin{array}{c|c|c|c|c}
~~~ & ~\bar Q_{\dot 1}~ & ~\bar Q_{\dot 2}~ & ~ Q_{ 1}~ & ~Q_{ 2}~
\\[0.4cm]\hline
\vspace*{-0.2cm}
Q_{ 1}~ & \frac{M}{L} + \oint a_k {\rm d} x_k & ~~0~~ &
-2i\oint {\rm d}n_k S_k & ~~0~~\\[0.2cm]
Q_{ 2}~ & ~~0~~ & \frac{M}{L} - \oint a_k {\rm d} x_k & ~~0~~ &
~~0~~\\[0.2cm]
\bar Q_{\dot 1}~ & 2i\oint {\rm d}n_kS_k & ~~0~~ & \frac{M}{L} +
\oint a_k {\rm d} x_k & ~~0~~\\[0.2cm]
\bar Q_{\dot 2} & ~~0~~ & ~~0~~ & ~~0~~ & \frac{M}{L} - \oint a_k
{\rm
d} x_k
\end{array}
\eeq
where the integrals above are taken in the plane perpendicular to
the axis of the soliton
(i.e. in the $x,y$ plane), along a closed path of
radius $R$
(it is assumed that $R\to\infty$),
$dn_k$ is the element of the length of the curve, see Fig. 1 ($ d\vec
n$ is
perpendicular
to $ d\vec x$), and, finally,
\begin{equation}} \def\eeq{\end{equation}
\{ S_1\,,\, S_2\} = \{ {\rm Re}\Sigma \,,\, {\rm
Im}\Sigma\}\, ,
\eeq
so that
\begin{equation}} \def\eeq{\end{equation}
\oint a_k {\rm d} x_k = \int {\rm d}^2 x (\partial_x a_y - \partial_y
a_x
)
=\int {\rm d}^2 x \left[-i\partial_\zeta (a_x +ia_y) +
i\partial_{\bar\zeta}
(a_x -ia_y)\right]\, ,
\eeq
\begin{equation}} \def\eeq{\end{equation}
\oint {\rm d}n_k S_k = \int {\rm d}^2 x\left[\partial_\zeta \Sigma
+\partial_{\bar\zeta}\bar\Sigma \right]\, ,
\eeq
and the complex coordinates $\zeta, \bar\zeta$ are introduced below
in Eq. (\ref{zetabar}).
The BPS bound on the
soliton mass is
obtained from the requirement of vanishing of the determinant of
the
above matrix, which implies
\begin{equation}} \def\eeq{\end{equation}
\frac{M}{L} = -\oint a_k {\rm d} x_k + 2\oint {\rm d}n_k S_k
\, .
\label{ccbpss}
\eeq
For saturated objects
the master equation (\ref{ccbpss})
expresses the tensions in terms of two contour integrals
over the large circle.
\begin{figure}
\epsfysize=6cm
\centerline{\epsfbox{bpscharge.eps}}
\caption{The integration contour in the $x,\, y$ plane. The soliton
axis
(the closed circle) lies perpendicular to this plane.}
\end{figure}
\section{Generalized Wess--Zumino Models}
In this section, as a warm up exercise,
we will discuss the GWZ models which give rise to the
BPS solitons with the axial geometry, and derive the $(1/2,1/2)$
central charge in these models. The full expression for the
$(1,0)$ central charge was found previously \cite{four}.
The Lagrangian has the form
\begin{equation}
{\cal L} =
\frac{1}{4} \sum_i \int \! {\rm d}^2\theta {\rm d}^2\bar\theta\,
\bar \Phi_i \Phi_i +\left\{\frac{1}{2}\int \! {\rm d}^2\theta
{\cal W} (\Phi_i ) + \mbox{H.c.}
\right\}
\, ,
\label{GWZlagr}
\end{equation}
where $\Phi_i$ is the set of the chiral fields, and the superpotential
${\cal W}$ is an analytic function of the fields $\Phi_i$.
The original (renormalizable) Wess--Zumino model
implies that ${\cal W}$ is a cubic polynomial in $\Phi_i$.
We shall not limit ourselves to this assumption, keeping in mind
that
GWZ models with more contrived
superpotentials can appear as low-energy limits of some
renormalizable microscopic field theories. The case
of more general K\"ahler potential will
be considered later.
The equations of the BPS saturation
for the solitons with the axial geometry
in this model were first derived~\footnote{
See Sec. III.D of Ref.~\cite{four} entitled, rather awkwardly,
``BPS-saturated strings." In fact, the
authors meant BPS solitons with the
axial
geometry.}
in Ref.~\cite{four}; they have the form
\begin{equation}
\frac{\partial \phi_i }{\partial\zeta} = \frac{1}{2}\frac{\partial
\bar{\cal
W}}{\partial
\bar\phi_i}\, ,
\label{Aspenone}
\end{equation}
where
\begin{equation}
\zeta = x+i y\,,\quad \frac{\partial }{\partial\zeta}=\frac{1}{2}
\left( \frac{\partial }{\partial x}- i\frac{\partial }{\partial y} \right)\,
.
\label{zetabar}
\end{equation}
The soliton axis is assumed to lie along the $z$ axis,
while the soliton profile depends on $x,y$. Note that it is {\em not}
assumed that the solution of Eq. (\ref{Aspenone})
is analytic in $\zeta$ (in fact, one can prove that
it must depend on both $\zeta$ and $\bar\zeta$ in the general
case).
A constant phase, which could have appeared on the right-hand side
of
Eq. (\ref{Aspenone}), is absorbed in $\zeta$.
Given the solution of Eq. (\ref{Aspenone}), one gets two constraints
determining the parameter of the residual (conserved)
supersymmetry,
\begin{equation}
\left( 1 + \tau_3\right) \varepsilon = 0\, ,\quad
\frac{-i}{2} \left( 1 - \tau_3\right) \varepsilon = \bar\varepsilon
\, ,
\label{prs}
\end{equation}
where the spinorial indices of $\varepsilon,\bar\varepsilon$
are suppressed (both are assumed to be the upper indices),
and we follow the notations and conventions collected in
\cite{four}. The first constraint implies that $\varepsilon$ has only
the
lower
component, which reduces the number of supersymmetries from
four to
two;
the second constraint further reduces the number of the
residual supersymmetries to one.
In order to calculate the $(1,0)$ and $(1/2,1/2)$
central charges one needs the expression for the supercharges. In
fact,
since we focus on full derivatives, we need to know the
supercurrent
$J^\mu_\alpha =(1/2)\left(\bar\sigma\right)^{\dot\beta\beta}
J_{\alpha\beta\dot\beta}$, rather than the
supercharges {\em per se}. The corresponding expression is well
known
(see e.g. Ref. \cite{four}),
\begin{eqnarray}
J_{\alpha\beta\dot\beta} &\!\!=&\!\! 2\sqrt{2}\sum
\left[
\left({\partial
}_{\alpha\dot\beta}\bar\phi
\right)\psi_\beta -i\epsilon_{\beta\alpha} F\bar\psi_{\dot\beta}
\right]
\nonumber\\[0.2cm]
&\!\!- &\!\! \frac{\sqrt{2}}{3}\sum \left[
\partial_{\alpha\dot\beta}(\psi_\beta \bar\phi)
+\partial_{\beta\dot\beta}(\psi_\alpha \bar\phi)
-3\epsilon_{\beta\alpha}\partial^\gamma_{\dot\beta}(\psi_\gamma
\bar\phi)
\right]
\, .
\label{scwz}
\end{eqnarray}
The supercharge $Q_\alpha$ is defined as
\begin{equation}} \def\eeq{\end{equation}
Q_\alpha =\int d^3 x J_\alpha^0\, , \qquad
J^\mu_\alpha = \frac{1}{2} \left(
\bar\sigma^\mu\right)^{\beta\dot\beta}
J_{\alpha\beta\dot\beta}\, .
\label{oprsz}
\eeq
The term in the second line in Eq. (\ref{scwz}) is conserved by itself.
Moreover,
in the supercharge it is represented as an integral over the full
derivative.
Below we will discuss the impact of deleting this term.
We will keep it, however, for the time being,
since we want to use the supercurrent which enters in one
supermultiplet
with the geometric $R$ current \cite{scsm} (sometimes called the
$R_0$ current). The $R_0$ current is conserved in conformal
theories.
It is not difficult to find the full derivative terms in $\{Q_\alpha
\bar Q_{\dot\beta}\}$ by computing the canonic commutators of the
fields at the
tree level [the $(1/2,1/2)$ central charge appears already at the tree
level].
The task is facilitated if one observes that in order to get the
$(1/2,1/2)$ central
charge it is sufficient to keep only the terms of the mixed symmetry
in
$\{\bar Q_{\dot\alpha}\,
J_{\alpha\beta\dot\beta}\}$, namely, symmetric in $\alpha ,\beta$
and
antisymmetric in $\dot\alpha ,\dot\beta$ or {\em vice versa}.
The result of this calculation reduces to Eq. (\ref{bsaa})
with
\begin{equation}
a^\mu = \frac{1}{4} \, a^\mu_{(\psi )}- \frac{1}{6} a^\mu_{(\phi )}\, ,
\end{equation}
where $a^\mu_{(\psi )}$ and $ a^\mu_{(\phi )}$ are the fermion and
boson axial
currents, respectively,
\begin{equation}
a^\mu_{(\psi )} = -\sum \psi \sigma^\mu \bar \psi\, ,
\qquad
a^\mu_{(\phi )}= - i\sum \phi
\stackrel{\leftrightarrow}{\partial}^{\,\mu} \bar\phi \, .
\end{equation}
The expression for the $(1,0)$ central charge in the GWZ model
found previously \cite{four} at the tree level takes the form of Eq.
(\ref{moycc})
with
\begin{equation}} \def\eeq{\end{equation}
\bar\Sigma = \bar{\cal W} - \frac{1}{3}\sum\, \bar\Phi\,
\frac{\partial
{\cal W}}{\partial
\bar\Phi}\,.
\eeq
One can check that {\em only} the {\em combined} contribution
of the central charges above
correctly reproduces the mass of the BPS solitons
with the axial geometry, e.g. the wall junctions.
Indeed, Eq. (\ref{ccbpss}) implies that in the model at hand
\footnote{The term $-(1/3)\partial_k\partial_k (\bar\phi\phi)$ is
irrelevant both
for strings and wall junctions, since it vanishes
in the both cases. It contributes, however, in the energy of the
axial geometry solitons of the type discussed in \cite{four}. This
term
occurs in
passing from the canonic energy-momentum tensor
$$
\theta_{\mu\nu}^{\rm canonic} = \partial_\mu\bar\phi
\partial_\nu\phi +
\partial_\nu\bar\phi \partial_\mu\phi +{\rm fermions}
- g_{\mu\nu}{\cal L}
$$
to the one which is traceless in the conformal limit
$$
\theta_{\mu\nu}^{\rm traceless} = \theta_{\mu\nu}^{\rm canonic} +
\frac{1}{3}\,\left( g_{\mu\nu} \partial^\alpha\partial_\alpha
-\partial_\mu\partial_\nu
\right)\bar\phi\phi\,.
$$
}
\begin{eqnarray}
\frac{M}{L}&\!\!=&\!\! \int {\rm d}^2 x\left[\partial_k\bar\phi
\partial_k\phi
+\left|\frac{\partial W}{\partial\phi}\right|^2
-\frac{1}{3}\partial_k\partial_k (\bar\phi\phi)
\right]\nonumber\\[0.2cm]
&\!\!=&\!\! -2\left( 1 -\frac{2}{3}\right)\int {\rm d}^2 x\left[
\partial_\zeta\phi \partial_{\bar\zeta}\bar\phi
-\partial_\zeta\bar\phi
\partial_{\bar\zeta}\phi
\right]
\nonumber\\[0.2cm]
&\!\!+&\!\! 2\int {\rm d}^2 x\left[\partial_\zeta\left({\cal W}
-\frac{1}{3}\phi
\frac{\partial {\cal W}}{\partial\phi}\right)
+\partial_{\bar\zeta}\left(\bar{\cal W} -
\frac{1}{3}\bar\phi
\frac{\partial \bar{\cal W}}{\partial\bar\phi}\right)
\right] \, .
\label{smccp}
\end{eqnarray}
On the other hand, for the BPS-saturated solution one can write
\begin{eqnarray}
0&\!\!=&\!\! \int {\rm d}^2 x\left[2\partial_\zeta\phi-
\frac{\partial
\bar{\cal
W}}{\partial\bar\phi}\right]
\left[2\partial_{\bar\zeta}\bar\phi- \frac{\partial {\cal
W}}{\partial\phi}\right]
\nonumber\\[0.2cm]
&\!\!=&\!\! \int {\rm d}^2 x\left[\partial_k\bar\phi \partial_k\phi
+\left|\frac{\partial {\cal W}}{\partial\phi}\right|^2
\right] \nonumber\\[0.2cm]
&\!\!+&\!\! 2\int {\rm d}^2 x\left[
\partial_\zeta\phi \partial_{\bar\zeta}\bar\phi
-\partial_\zeta\bar\phi
\partial_{\bar\zeta}\phi
\right]
- 2\int {\rm d}^2 x\left[\partial_\zeta{\cal W}
+\partial_{\bar\zeta}\bar{\cal W}
\right] \, ,
\label{smbpssp}
\end{eqnarray}
or
\begin{equation}} \def\eeq{\end{equation}
\frac{M}{L} = -2\int {\rm d}^2 x\left[
\partial_\zeta\phi \partial_{\bar\zeta}\bar\phi
-\partial_\zeta\bar\phi
\partial_{\bar\zeta}\phi
\right]
+ 2\int {\rm d}^2 x\left[\partial_\zeta{\cal W}
+\partial_{\bar\zeta}\bar{\cal W}
\right] \, .
\label{smbpss}
\eeq
At first sight it might seem
that Eqs. (\ref{smccp}) and (\ref{smbpss}) contradict each other,
since the axial current contribution to the soliton mass in these two
expressions
(corresponding to the $(1/2,1/2)$ central charge)
has different coefficients (cf. $-2 +(4/3)$ in the first case and $-2$
in the
second).
Upon inspection one sees that Eq. (\ref{smccp}) has a different
expression for the
$(1,0)$ central charge too. The difference is
$$
-\frac{2}{3}\int {\rm d}^2 x\left[\partial_\zeta\left(\phi
\frac{\partial {\cal W}}{\partial\phi}\right)
+\partial_{\bar\zeta}\left(
\bar\phi
\frac{\partial \bar{\cal W}}{\partial\bar\phi}\right)
\right] \, .
$$
For the BPS saturated solitons satisfying Eq. (\ref{Aspenone})
it is easy to show that
\begin{eqnarray}
& &-\frac{2}{3}\int {\rm d}^2 x\left[\partial_\zeta\left(\phi
\frac{\partial {\cal W}}{\partial\phi}\right)
+\partial_{\bar\zeta}\left(
\bar\phi
\frac{\partial \bar{\cal W}}{\partial\bar\phi}\right)
\right]\nonumber\\[0.2cm]
&\!\!=&\!\!
-\frac{4}{3}\int {\rm d}^2 x\left[
\partial_\zeta\phi \partial_{\bar\zeta}\bar\phi
-\partial_\zeta\bar\phi
\partial_{\bar\zeta}\phi
\right] -\frac{1}{3} \int {\rm d}^2 x \partial^\alpha\partial_\alpha
\bar\phi\phi\,.
\label{vsprav}
\end{eqnarray}
This relation immediately implies the coincidence of the
soliton masses ensuing from Eqs. (\ref{smccp}) and (\ref{smbpss}),
respectively.
In fact, the superficial difference between them
is due to the ambiguity in the choice of the
supercurrent (the terms with the full derivatives in Eq. (\ref{scwz}))
and the corresponding ambiguity in the energy-momentum tensor.
Equation (\ref{smccp}) is derived on the basis
of the supercurrent and the energy-momentum tensor
with the properties
$\varepsilon^{\alpha\beta}J_{\alpha\beta\dot\beta}=0,\,\,\,
\theta^\mu_\mu =0 $
in the conformal limit. Passing to the minimal supercurrent
and the canonic energy-momentum tensor
one drops all terms containing the factor $1/3$ in Eq. (\ref{smccp})
and recovers Eq. (\ref{smbpss}). The mass of the soliton stays intact
due to a reshuffling of contributions due to $(1/2,1/2)$ and $(1,0)$
charges.
To illustrate the point let us consider, for instance, a $Z_N$
model suggested in Ref. \cite{GabDv},
with the
superpotential
\begin{equation}} \def\eeq{\end{equation}
{\cal W} =N \left\{ \Phi -
\frac{N}{N+1}\left(\frac{\Phi}{N}\right)^{N+1}\right\} \, ,
\eeq
where $\Phi$ is a chiral superfield. The model
obviously possesses a $Z_N$ symmetry, the vacuum manifold
corresponds
to $N$ points,
\begin{equation}} \def\eeq{\end{equation}
\phi_k = N \exp\left(\frac{2\pi i k}{N}\right)\,,\qquad k= 0,1,2, ...,N-
1\, ,
\eeq
while the vacuum value of the superpotential is
\begin{equation}} \def\eeq{\end{equation}
{\cal W} (\phi_k) = N^2 \exp\left(\frac{2\pi i k}{N}\right)\,,\qquad
N\to\infty\, .
\eeq
The solution of the BPS saturation equation for an isolated wall
exists, it was discussed in \cite{GabDv}. (Here and below $N$ will be
assumed large,
and only leading terms in $N$ will be kept.)
The tension of the minimal wall connecting the neighboring vacua is
\begin{equation}} \def\eeq{\end{equation}
T= 2|\Delta {\cal W}| = 4\pi N\, .
\eeq
Consider the BPS wall junctions of the type depicted in Fig. 2.
\begin{figure}
\epsfysize=6cm
\centerline{\epsfbox{bpscharge2.eps}}
\caption{The domain wall junction in the theory with $Z_N$
symmetry. The
``hub" is denoted by the closed circle.}
\end{figure}
Assuming that there is a solution of Eq. (\ref{Aspenone}),
to the leading order in $N$ one can write (at $|\zeta |\to \infty$)
\begin{equation}} \def\eeq{\end{equation}
\phi = N e^{i\alpha (\gamma )}\, , \quad \alpha (0) = 0\,,\quad
\alpha
(2\pi ) = 2\pi\,,
\eeq
which entails, in turn,
\begin{equation}} \def\eeq{\end{equation}
\oint_{|x|=R\to\infty}\, a_k {\rm d}x_k =\frac{N^2}{3}\left[ \alpha
(2\pi ) -
\alpha (0 )\right] = \frac{2\pi}{3}N^2\, .
\eeq
We also observe that
\begin{eqnarray}
2\oint {\rm d}n_k w_k &\!\!=&\!\! 2 N^2 R\int {\rm d}\gamma
\cos(\alpha -\gamma )
=4\pi N^2 R\, ,\nonumber\\[0.2cm]
\{w_1, \,w_2\} &\!\!=&\!\! \{{\rm Re }{\cal W}\, , \, {\rm Im }{\cal
W}\}\,.
\label{okot}
\end{eqnarray}
which is exactly the mass of $N$ isolated walls inside the contour.
Furthermore,
\begin{equation}} \def\eeq{\end{equation}
2\oint {\rm d}n_k S_k
=4\pi N^2 R -\frac{4\pi}{3}N^2\, .
\label{okotp}
\eeq
The total mass of the junction configuration comes out the same from
both
expressions, Eqs. (\ref{smccp}) and (\ref{smbpss}),
\begin{equation}} \def\eeq{\end{equation}
\frac{M}{L} = 4\pi N^2 R - 2\pi N^2\, ,
\eeq
(see also \cite{GabShi}).
The first term can be interpreted as the mass of the ``spokes"
joined at the origin,
while the second as that of the
``hub".
Let us remark that the stringy (``hub") contribution to the
total mass equals to twice the area of the contour
on the $\phi$ plane covered by the solution. Since
we consider the junction with $N$ ``minimal" domain walls
connecting the neighboring vacua, the contour is closed.
The closeness is nothing but the equilibrium
condition at the junction line.
Summarizing, we observe an ambiguity in the
$(1/2,1/2)$ central charge.
This ambiguity is due to the
fact that both, the supercurrent and the energy-momentum tensor,
are not uniquely determined. Both admit certain full
derivative terms which are conserved by themselves
and, therefore, do not affect the supercharges
and the energy-momentum four-vector.
They do affect the expressions for the central charges, however.
For the soliton solutions of the wall junction
type the ambiguity in the
$(1/2,1/2)$ central charge combines with another ambiguity,
in the $(1,0)$ central charge, to produce an
unambiguous expression for the soliton mass. As we will see shortly,
the same ambiguity (and a similar conspiracy) takes place in
the gauge theories
with matter.
Practically, it is more convenient to
work with the minimal supercurrents
(and the canonic energy-momentum tensor).
Then, one omits the second line in Eq.
(\ref{scwz}). The expression for $a^\mu$
in the $(1/2,1/2)$ central charge then becomes
\begin{equation}} \def\eeq{\end{equation}
a^\mu = \frac{1}{4} \, a^\mu_{(\psi )}- \frac{1}{2} a^\mu_{(\phi )}\, ,
\qquad
a^\mu_{(\phi )}= - i\sum \phi
\stackrel{\leftrightarrow}{\partial}^{\,\mu} \bar\phi \, ,
\eeq
while $\bar\Sigma$ in the $(1,0)$ central charge becomes
$$\bar\Sigma = \bar{\cal W}\,.$$
\section{SQED with the Fayet-Iliopoulos term}
The simplest theory (and the only one in the class
${\cal N}=1$, see
below) where saturated strings exist in the weak coupling
regime
is supersymmetric electrodynamics (SUSY QED, or SQED), with the
Fayet-Iliopoulos (FI) term.
In the superfield notation the Lagrangian of the
model has the form
\begin{equation}
{\cal L } = \left\{ \frac{1}{8\, e^2}\int\!{\rm d}^2\theta \, W^2 + {\rm
H.c.}\right\} +
\frac{1}{4}\int \!{\rm d}^4\theta \left(\bar{S}e^V S + \bar{T}e^{-V} T
\right)
- \frac{\xi }{4} \int\! {\rm d}^2\theta {\rm d}^2\bar \theta
\,V(\! x,\theta , \bar\theta ) \, ,
\label{sqed}
\end{equation}
where $e$ is the electric charge,
$S$ and $T$ are two chiral superfields with the electric charges
$+1$ and $-1$, respectively, $\xi$ is the coefficient
of the Fayet-Iliopoulos term. The
model with one chiral superfield is internally anomalous.
Topologically
stable solutions in this model and its modifications were considered
more than once in the past
\cite{2,2prim,3}. We combine various elements scattered in the
literature,
with a special emphasis on the algebraic aspect. Supersymmetry of
this
model is minimal, ${\cal N} =1$.
If $\xi\neq 0$, the vacuum state corresponds to the spontaneous
breaking of U(1). The spectrum of the model is that of a massive
vector
supermultiplet (one massive vector field, one real scalar and one
Dirac
fermion, all of one and the same mass),
plus a massless modulus (one chiral superfield)
parametrized by the product $ST$,
\begin{equation}} \def\eeq{\end{equation}
\Phi = 2ST\, .
\eeq
The vacuum valley is represented by the one-dimensional complex
manifold with the K\"ahler function
\begin{equation}} \def\eeq{\end{equation}
K(\Phi\, , \bar\Phi ) = \sqrt{\xi^2 +\Phi \bar\Phi}\, .
\eeq
In a generic point
nonsingular Abrikosov strings do {\em not} exist \cite{2prim}.
There is one special point, however, $\Phi = 0$, where
the theory supports the saturated string.
In components the Lagrangian of SQED (\ref{sqed}) has the form
(in the Wess-Zumino gauge)
\begin{eqnarray}
{\cal L } &\!\!=&\!\!-\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu}
+\left({\cal D}_\mu\phi\right)^\dagger {\cal D}^\mu\phi
+\left({\cal D}_\mu\chi\right)^\dagger {\cal D}^\mu\chi
-\frac{e^2}{2}\left( \phi^\dagger \phi -
\chi^\dagger\chi -\xi\right)^2\, ,
\nonumber\\[0.2cm]
&\!\!+&\!\!\mbox{fermions}
\label{lqedc}
\end{eqnarray}
where $\phi$ and $\chi$ are the lowest components
of the superfields $S$ and $T$, respectively, with the
electric charges $\pm 1$, e.g.
\begin{equation}} \def\eeq{\end{equation}
{\cal D}_\mu\phi =\partial_\mu \phi - i A_\mu\phi\, , \qquad
[{\cal D}_\mu\, , {\cal D}_\nu ]\phi = -i F_{\mu\nu }\phi\, .
\eeq
Without loss of generality we can assume that $\xi >0 $.
For the static field configurations, assuming in addition that all
fields
depend only on $x$ and $y$ and $A_0=A_3 = 0$, one gets the energy
functional in the form
\begin{eqnarray}
{\cal E} &\!\!=&\!\!\int{\rm d}x{\rm d}y
\left\{ \frac{1}{2e^2}F_{12}^2
+\sum_{i=1,2}\left({\cal D}_i\phi\right)^\dagger{\cal D}_i\phi +
\frac{e^2}{2}\left(\phi^\dagger\phi
-\xi
\right)^2\right\}\nonumber\\[0.2cm]
&\!\!\equiv &\!\! \int{\rm d}x{\rm d}y
\left\{ \left| \frac{1}{\sqrt{2}e} F_{12} +\frac{e}{\sqrt{2}}
\left(\phi^\dagger\phi -\xi\right)
\right|^2\right.
\nonumber\\[0.2cm]
&\!\!+&\!\!
\left. \left[ \left(
{\cal D}_1+ i{\cal D}_2\right)\phi\right]^\dagger\,
\left(
{\cal D}_1+ i{\cal D}_2\right)\phi \right\}+ {\cal Q}\, ,
\label{sqedef}
\end{eqnarray}
where ${\cal Q}$ is the surface (topological) term,
\begin{equation}} \def\eeq{\end{equation}
{\cal Q} = \int{\rm d}x{\rm d}y
\left\{ \xi F_{12} - \frac{i}{2} \partial_i (\phi^\dagger
\stackrel{\leftrightarrow}{\cal D}_j\phi )
\varepsilon^{ij}\right\}\, , \qquad i,j = 1,2\, .
\label{tt}
\eeq
We will discuss the value of the surface term later.
The saturation equations are
\begin{eqnarray}
F_{12} &\!\!=&\!\! -e^2
\left(\phi^\dagger\phi -\xi\right)\, ,\nonumber\\[0.2cm]
\left(
{\cal D}_1+ i{\cal D}_2\right)\phi &\!\!=&\!\! 0\, .
\label{sateq}
\end{eqnarray}
The {\em Ansatz} which goes through these equations is
\begin{eqnarray}
\phi &\!\!=&\!\! \sqrt{\xi}\eta e^{i\alpha}
\, ,\nonumber\\[0.2cm]
A_i &\!\!=&\!\! a\, \frac{\partial\alpha}{\partial x^i}\,, \quad
i=1,2\, ,
\label{agt}
\end{eqnarray}
where
\begin{equation}} \def\eeq{\end{equation}
\alpha = \mbox{Arg}\, \zeta\, , \qquad \zeta = x+i y\,,
\eeq
and $\eta,\,\, a$ are some functions depending on $r$.
This must be supplemented by the standard boundary conditions,
namely
\begin{equation}} \def\eeq{\end{equation}
\eta (r)\, ,\,\, a(r) \longrightarrow
\left\{ \begin{array}{c}
0\quad\mbox{at}\quad r\to 0\\ 1\quad\mbox{at}\quad r\to \infty
\end{array} \right.\, .
\eeq
For the given {\em Ansatz} the saturation equations (\ref{sateq})
degenerate into a system of first-order equations
\begin{eqnarray}
a ' &\!\!=&\!\! e^2\xi r (\eta^2 -1)
\, ,\nonumber\\[0.2cm]
\eta ' &\!\!=&\!\! -\frac{\eta(1-a)}{r} \, ,
\label{sateqa}
\end{eqnarray}
where the prime denotes differentiation over $r$.
Its solution is well known.
It is instructive to compare the topological term in Eq. (\ref{tt})
with the central charge of the superalgebra. To derive the central
charge one needs the
expression for the supercurrent in SQED, which takes the form (in
the
spinorial notation)
\begin{eqnarray}
J_{\alpha\beta\dot\beta} &\!\!=&\!\! \frac{2}{e^2}
\left(iF_{\beta\alpha} \bar\lambda_{\dot\beta}
+ \epsilon_{\beta\alpha} D\bar\lambda_{\dot\beta}
\right) + 2\sqrt{2}\sum\left({\cal D}_{\alpha\dot\beta}\phi^\dagger
\right)\psi_\beta
\nonumber\\[0.2cm]
&\!\!- &\!\! \frac{\sqrt{2}}{3}\sum \left[
\partial_{\alpha\dot\beta}(\psi_\beta \phi^\dagger)
+\partial_{\beta\dot\beta}(\psi_\alpha \phi^\dagger)
-3\epsilon_{\beta\alpha}\partial^\gamma_{\dot\beta}(\psi_\gamma
\phi^\dagger)
\right]
\, .
\label{scsqed}
\end{eqnarray}
Above it is assumed that there is no superpotential.
The second line may or may not be added, at will.
(The second line in Eq. (\ref{scsqed}) is conserved by itself;
in the supercharge it presents a full spatial derivative, hence, its
contribution
vanishes.)
The sum runs over various matter supermultiplets, in particular, $S$
and $T$ in
the case at hand.
To find the central charge one must compute the anticommutator
$\{ Q_\alpha \, , \bar J_{\dot\beta\dot\gamma\delta}\}$. Moreover,
we decompose the anticommutator above with respect to irreducible
representations of the Lorentz group, by singling out the
symmetric and
antisymmetric combinations of the dotted and undotted indices.
The one which is symmetric with respect to both pairs,
$(\alpha , \delta )$ and $(\dot\beta\dot\gamma )$, is the Lorentz
spin 2 (the energy-momentum tensor), which contributes to
$P_{\alpha\dot\alpha}$, rather than to the central charge.
The combination which is antisymmetric with respect to both pairs,
$(\alpha , \delta )$ and $(\dot\beta\dot\gamma )$ is Lorentz singlet,
it represents the trace terms in the energy-momentum tensor.
To single out the central charge we must isolate the terms of the
mixed
symmetry, i.e. symmetric with respect to $(\alpha , \delta )$ and
antisymmetric with respect to $(\dot\beta\dot\gamma )$,
and {\em vice versa}.
Keeping in mind this remark, and using the canonic commutation
relations and equations of motion for the $D$ field we get
an expression similar to that in the Wess-Zumino model,
plus an extra contribution due to the $D$ term,
\begin{equation}} \def\eeq{\end{equation}
\{ Q_\alpha \bar Q_{\dot\alpha}\} = i \xi \int d^3 x
\left[
F_{\beta\alpha} \varepsilon_{\beta\dot \alpha}
- \bar F_{\dot\gamma\dot\alpha}
\varepsilon_{\dot\gamma\alpha}
\right]\, .
\eeq
This implies
\begin{equation}} \def\eeq{\end{equation}
Z_\mu = \int {\rm d}^3 x\, \varepsilon_{0\mu\nu\rho}
\left(\xi\, \partial^\nu A^\rho -\sum \frac{i}{2}\partial^\nu
(\bar\phi
\stackrel{\leftrightarrow}{\cal D}^{\,\rho} \phi )
+ \frac{1}{4} \partial^\nu R^\rho
+\frac{1}{4}\partial^\nu a_{(\psi)}^\rho \right)\, ,
\label{tfit}
\eeq
where $R^\rho$ is the photino current, while $a_{(\psi)}^\rho$
is that of the electrons,
\begin{equation}} \def\eeq{\end{equation}
R^\rho = -\frac{1}{e^2}\lambda\sigma^\rho\bar\lambda\,,\qquad
a^\mu_{(\psi )} = -\sum \psi \sigma^\mu \bar \psi\, .
\eeq
Note that the coefficient of the $\bar\phi
\stackrel{\leftrightarrow}{\cal D}^{\,\rho} \phi$
(i.e. the selectron axial current) term is ambiguous --
it depends on whether the second line in Eq. (\ref{scsqed}) is
included
in the definition of the supercurrent. The result quoted above refers
to the minimal supercurrent, with the second line in Eq.
(\ref{scsqed})
discarded. Since the $(1,0)$
central charge is irrelevant for the string solution,
this ambiguity alone shows that the $\bar\phi
\stackrel{\leftrightarrow}{\cal D}^{\,\rho} \phi$
term cannot contribute to the central charge under consideration.
It is certainly the case, since ${\cal D}^{\,\rho} \phi$
falls off sufficiently fast at $r\to\infty$
(where $r$ is the distance to the string axis) for the string solution.
At the same time, the photon four-potential $A^\rho$
falls off slowly, as $1/r$.
Thus, the $(1/2,1/2)$ central charge is saturated
by the $\xi$ term exclusively. The latter is unambiguously
fixed in Eq. (\ref{tfit}), i.e. it does not depend on the full derivative
terms
in the supercurrent. The $(1/2,1/2)$ central charge is
obviously proportional to
$\xi$ and to the magnetic flux of the string,
\begin{equation}} \def\eeq{\end{equation}
\frac{M}{L}= \xi{\cal F}\,,
\eeq
where
\begin{equation}} \def\eeq{\end{equation}
{\cal F} =\int {\rm d}x {\rm d}y F_{12} =\oint A_k {\rm d } x_k\,.
\eeq
Note that the very same saturation equations (\ref{sateq})
are obtained in ${\cal N}=2$ SQED
with the vanishing Fayet-Iliopoulos term
and linear superpotential, see Sec. 9.3.
\section{The K\"ahler Sigma Models}
In this section we present some arguments concerning strings
in the four-dimensional $\sigma$ models on the K\"ahler manifolds.
The two-dimensional reductions of these models
are well studied, in the Euclidean formulation they admit instantons,
which are the solutions of the first order self-duality equations.
In the supersymmetric version the self-duality equations
in two dimensions
are reinterpreted as the BPS equations in higher-dimensional
theories (e.g. \cite{dopone}). It is obvious that
the instantons of the two-dimensional models are the BPS strings
in four dimensions. Thus, the four-dimensional
$\sigma$ models on the K\"ahler
manifolds do have the BPS strings at
the quasiclassical level, at weak coupling.
Keeping in mind the assertion we are going to prove
later (Sec. 8) we discuss where the K\"ahler sigma models
stand compared to other models.
Let us start with the $CP_1$ model. In a sense, this model can be
obtained as a limiting case of
SQED with a somewhat different matter content
compared to that of Sec. 4 (see, for instance,
\cite{wittenn=2}). Indeed, assume that the matter superfields
$S$ and $T$ have
both charges $+1$, rather than $\pm 1$. As a quantum theory, it is
anomalous,
but for the time being we limit ourselves to the classical
consideration. The limit to be taken is $e^2\to\infty$.
Let us have a closer look at Eq. (\ref{lqedc}), with the sign of the
charge
of the $\chi$ field reversed [correspondingly,
the $D$ term takes the form $D =e^2( \phi^\dagger \phi +
\chi^\dagger\chi -\xi)$].
In this limit the photon mass tends to infinity, the photon becomes
nondynamical and can be eliminated. It drags with itself
two real scalar degrees of freedom. The remaining two scalar degrees
of
freedom are massless. Their interaction reduces to
the sigma model on a sphere. This is most easily seen from Eq.
(\ref{lqedc}).
In the limit $e^2\to\infty$ the $D$ term must vanish, which implies
that $ \phi^\dagger \phi +
\chi^\dagger\chi =\xi$. In fact, the gauge freedom allows one to
identically eliminate
one out of four degrees of freedom residing in $\phi,\,\, \chi$.
The remaining three are subject to the constraint, telling
us that the radius of the sphere is $\xi$.
Thus, the SQED with the Fayet-Iliopoulos term, in the limit
$e^2\to\infty$, gives rise to the model with the
action
\begin{equation}} \def\eeq{\end{equation}
S = \frac{1}{2g^2} \int {\rm d}^4 x {\rm d}^2\theta
{\rm d}^2\bar\theta\, \ln \left (1+\bar\Phi\Phi \right)
\eeq
where $\Phi$ is a chiral superfield,
\begin{equation}
{\Phi ({x}_L,\theta )} = \phi ({x}_L) + \sqrt{2}\theta^\alpha
\psi_\alpha ({
x}_L) + \theta^2 F({x}_L)\, .
\label{chsup}
\end{equation}
The coupling constant $2/g^2$ has the dimension of mass squared
and is equal to $\xi$.
The string tension will be proportional to $2/g^2=\xi$.
The metric of the sphere in the target space $G$ in this case is
\begin{equation}} \def\eeq{\end{equation}
G = \frac{2}{g^2} \frac{1}{(1+\bar\Phi\Phi )^2 }\, .
\eeq
The energy functional for the stringy solution takes the
form which looks exactly as
the action in the Euclidean two-dimensional sigma
model whose world volume is transverse
to the string. It is easy to rewrite it in terms
of the topological charge plus a positive definite contribution,
\begin{eqnarray}
\frac{{\cal E}}{L} = \int {\rm d}^2x
\left\{\frac{8}{g^2}\left|\frac{\partial_{\bar\zeta}\phi}{1+\bar\phi
\phi}
\right|^2 + \frac{1}{g^2}\varepsilon_{\mu\nu}\partial_\mu
\left(\frac{\bar\phi\, i\stackrel{\leftrightarrow}{\partial_\nu}
\phi}{1+\bar\phi
\phi}
\right)
\right\}\, ,
\label{rttc}
\end{eqnarray}
where the second term, the integral over the full derivative,
presents the topological charge
and the integral runs in the plane transverse
to the string.
Instantons saturate the topological charge; since
$\pi_2 (S_2) = Z$, the saturated solutions are labeled by an
integer
$n$, equal to the topological charge. The surface term contribution in
Eq.
(\ref{rttc}) is thus proportional to $g^{-2} n = \xi n $.
In four dimensions the instantons
present the BPS saturated strings. These strings are rather peculiar.
Since the two-dimensional theory is classically (super)conformally
invariant, the two-dimensional instantons can have any size
(correspondingly,
the cross section of the string in four-dimensional theory
can be arbitrary). The larger is the
transverse size of the string the
smaller is the energy density in the string. However, the
string tension remains constant
proportional to $g^{-2} =\xi$.
This is the limiting profile of the Abrikosov string in SQED
with the Fayet-Iliopoulos term
-- the profile it acquires when the vector field mass tends to infinity
while the remaining degrees of freedom of the
matter fields remain massless.
For our purposes it is important to interpret
the surface term contribution in the
string tension in terms of the
$(1/2,1/2)$ central charge
of the four-dimensional SQED. Upon inspecting Eq. (\ref{tfit})
we conclude that this contribution comes from the
first term in Eq. (\ref{tfit}). The field $A_\mu$ is not dynamical in
the
limit under consideration, and is expressible in terms of
the residual scalars. Since our consideration
is quasiclassical, it is not surprising that the current of the matter
fermions does not contribute. The second term in Eq. (\ref{tfit})
does not contribute either -- as was discussed,
its coefficient is ambiguous.
The O(3) (or $CP_1$) model belongs to a
more general class of $CP_N$ models.
The latter can be derived as the low-energy limit of SQED
with the FI term and with $N+1$ chiral
matter superfields (all of them have charge $+1$),
in the limit $e^2\to\infty$. One can eliminate the nondynamical
$A_\mu$ field, much in the same way as in $CP_1$,
arriving in this way at a nonlinear sigma model.
One has to introduce complex coordinates
$w_{i}^j={\phi_i}/{\phi_j}$ where $i\neq j$ which
can be considered as the scalar components of the chiral
superfields $\Phi_{i}^{j}$. The action can be written
in terms of $\Phi_{i}^{j}$ as follows
\begin{equation}} \def\eeq{\end{equation}
S = \frac{1}{2g^2} \int {\rm d}^4 x {\rm d}^2\theta
{\rm d}^2\bar\theta\, \ln \left (1+\sum_{i,j}\bar\Phi_{i}^{j}
\Phi_{i}^{j}
\right)\,.
\eeq
The identification $\xi={1}/{g^2}$ is transparent
since both parameters determine the size of
the target manifold in two formulations.
The general expression for the central charge is \cite{3dwj}
\begin{equation}} \def\eeq{\end{equation}
Z= \int {\rm d}^2x
\left\{ \partial_\zeta \left(
K_{\phi}\partial_{\bar{\zeta}}\phi-
K_{\bar{\phi}}\partial_{\bar{\zeta}}\bar{\phi}\right)+
\partial_{\bar{\zeta}}(K_{\bar{\phi}}\partial_{\zeta}
\bar{\phi}-K_{\phi}\partial_{\zeta}\phi)
\right\}\,,
\eeq
where the complex variable $\zeta$ is defined in Eq. (\ref{zetabar})
the subscripts $\phi,\,\,\bar\phi$ denote the $\phi,\,\,\bar\phi$
partial derivatives of the K\"ahler metric.
More generally, we expect similar strings
for all toric varieties which can be presented
as low energy limits of gauged linear sigma model.
In Sec. 9 we shall encounter one more example of the
K\"ahler sigma model coupled to the
Abelian gauge field -- the low-energy effective action
for ${\cal N}=2$ SUSY Yang-Mills theory in four dimensions.
\section{Supersymmetric gluodynamics}
To begin with, consider the simplest non-Abelian gauge model, SUSY
gluodynamics. The Lagrangian is
\begin{equation}
{\cal L} = \frac{1}{4g^2} \int\!{\rm d}^2\theta \,\mbox{Tr}\, W^2 +
\,
\mbox{H.c.}
\, ,
\label{SFYML}
\end{equation}
where
$ W = W^aT^a$, and $T^a$ are the generators of the gauge group $G$
in
the
fundamental representation. Although the
gauge group $G$ can be arbitrary, for definiteness
we limit ourselves to SU($N$). In components
\begin{equation}
{\cal L} = \frac{1}{g^2} \left\{ -\frac 14
G_{\mu\nu}^aG^{a\mu\nu} +
i\lambda^{a \alpha}
{\cal D}_{\alpha\dot\beta}\bar\lambda^{a\dot\beta}
\right\} \, .
\label{SUSYML}
\end{equation}
There is a supermultiplet of the classically conserved currents
(for a recent review see e.g. \cite{SVr}),
\begin{eqnarray}
{\cal J}_{\alpha\dot\alpha} &\!\!=&\!\! -\frac{4}{g^2}\,\mbox{Tr}
\left[e^V
W_\alpha e^{-V}\bar W_{\dot \alpha}\right]\nonumber\\[0.2cm]
&\!\!=&\!\! R_{\alpha\dot\alpha} -
\left\{ i\theta^{\beta} J_{\beta\alpha\dot\alpha}
+
\mbox{H.c.} \right\} -
2\, \theta^{\beta}\bar{\theta}^{\dot\beta} \,
J_{\alpha\dot\alpha\beta\dot\beta} +\dots\;,
\label{decom2}
\end{eqnarray}
where $R_{\alpha\dot\alpha} $
is the chiral current,
$J_{\beta\alpha\dot\alpha} $ is the
supercurrent, and ${J_{\alpha\dot\alpha\beta\dot\beta}}$ is a
combination of the energy-momentum tensor
$\vartheta_{\alpha\dot\alpha\beta\dot\beta}=
(\sigma^\mu)_{\alpha\dot\alpha}(\sigma^\nu)_{\beta\dot\beta}\,
\vartheta_{\mu\nu}$
and a full derivative appearing in the central charge, namely,
\begin{eqnarray}
R_{\alpha\dot\alpha} &\!\!=&\!\!
-
\frac{4}{g^2}\,\mbox{Tr}\lambda_\alpha\bar\lambda_{\dot\alpha}\,
,
\nonumber\\[0.2cm]
J_{\beta\alpha\dot\alpha}&\!\!=&\!\!
(\sigma^\mu)_{\alpha\dot\alpha}
\,
J_{\mu\,,\beta}=\frac {4i}{g^2}\,\mbox{Tr}\, G_{\alpha\beta}\,\bar
\lambda_{\dot \alpha}\,,\nonumber\\[0.2cm]
J_{\alpha\dot\alpha\beta\dot\beta}&\!\!=&\!\!
\vartheta_{\alpha\dot\alpha\beta\dot\beta}
-\frac{i}{4}\varepsilon_{\alpha\beta}\partial_{\gamma\{\dot\beta}
\,
R^\gamma_{\dot\alpha\}}+
\frac{i}{4}\varepsilon_{\dot\alpha\dot\beta}\partial_{\dot\gamma
\{\beta}
\,
R^{\dot\gamma}_{\alpha\}}
\,,\nonumber\\[0.2cm]
\vartheta_{\alpha\dot\alpha\beta\dot\beta}
&\!\!=&\!\!
\frac{2}{g^2}\,\mbox{Tr}\left[i\,\lambda_{\{\alpha
}{\cal
D}_{\beta\}\dot\beta} \bar \lambda_{\dot \alpha}
-i\,\left(
{\cal D}_{\beta\{\dot\beta}\lambda_\alpha \right) \,\bar
\lambda_{\dot
\alpha\}}
+ G_{\alpha\beta}
\bar G_{\dot\alpha\dot\beta}\right]\,.
\label{arcr}
\end{eqnarray}
The symmetrization over $\alpha,\beta$ or $\dot \alpha,\dot\beta$
is marked by the braces. In fact, since the chiral current is
classically conserved
(so far we disregard anomalies), symmetrization in the third line is
superfluous: the corresponding expressions are automatically
symmetric.
To obtain the expression on the right-hand side from
$\mbox{Tr}
\left[e^V
W_\alpha e^{-V}\bar W_{\dot \alpha}\right]$
we observe
that the expression for $J_{\alpha\dot\alpha\beta\dot\beta}$
has mixed symmetry: the part symmetric in $\{\alpha \, ,\beta\}$
{\em and} $\{\dot\alpha \, ,\dot\beta\}$ is the $(1,1)$ Lorentz
tensor, it represents
the (traceless) energy-momentum
tensor. The remainder, i.e.
the part symmetric in $\{\alpha \, ,\beta\}$
and antisymmetric in $\{\dot\alpha \, ,\dot\beta\}$ or
{\em vice versa}, is the
$(0,1)+ (1,0)$ Lorentz tensor.
The part antisymmetric in both $\{\alpha \, ,\beta\}$
{\em and} $\{\dot\alpha \, ,\dot\beta\}$ is $(0,0)$.
It represents the traces which vanish in the classical approximation.
It is quite obvious that the only part relevant for the central charge
is $(0,1)+ (1,0)$ piece in $J_{\alpha\dot\alpha\beta\dot\beta}$.
This means, in particular, that the inclusion of the traces
will have no impact on the central charge.
It is easy to see that
\begin{equation}} \def\eeq{\end{equation}
\left\{ Q_\gamma\, , \bar J_{\dot\beta\dot\alpha\alpha}\right\}
=2 J_{\alpha\dot\alpha\gamma\dot\beta}\, .
\label{safd}
\eeq
Combining this equation with the third line in Eq. (\ref{arcr}) we
conclude that the centrally
extended algebra is given by Eq. (\ref{bsa})
with
\begin{equation}} \def\eeq{\end{equation}
a^\nu = \frac{1}{4} \, R^\nu\, .
\label{okon}
\eeq
Unlike the central extension relevant for the domain walls, which
appears \cite{two} -- \cite{four} as a quantum anomaly,
in the problem at hand the algebra gets a full-derivative term at the
tree level. The presence of the anomaly manifests itself
through the fact that the energy-momentum tensor ceases to be
traceless, and $\partial_\nu R^\nu$ no more vanishes. On general
grounds it is clear, however, that Eqs. (\ref{bsa}), (\ref{okon})
stay intact.
The occurrence of a full-derivative term in the algebra
presents a precondition for a nontrivial central extension.
Whether or not this term actually vanishes is a dynamical issue
which depends on the presence of the string-like solitons.
These may be strings, or domain-wall junctions, as in
Ref. \cite{3dwj}. SUSY gluodynamics
is a strongly coupled theory;
therefore, one cannot use
quasiclassical considerations to search/analyze
solitons. The hope is that there is a dual description
in terms of effective degrees of freedom, for which quasiclassical
analysis may be relevant. Within this dual description
the second term in Eq. (\ref{bsa}) is mapped onto some relevant
operator of the effective theory.
It is clear that the second term in Eq. (\ref{bsa})
is the necessary but not sufficient condition for the
existence of the saturated strings.
If it were absent, there would be no
hope.
\section{Generic Non-Abelian Model with Matter}
The $(1/2,1/2)$ central charge
in the generic non-Abelian theory is obtained by combining
the expressions we have derived in the previous sections.
The operator $a_\mu$ in Eq. (\ref{bsa}) receives contributions from
the gluino
term, as in Sec. 6, which is unambiguous,
and the contributions from matter
(both, the scalar and spinor components of matter enter),
as in the generalized Wess-Zumino model (Sec. 3), whose coefficients
are not fixed -- they depend on how one defines
the supercurrent in those terms that are total derivatives.
This ambiguity derives its origin from that in the definition of the
supercurrents,
\begin{eqnarray}
J_{\alpha\beta\dot\beta} &\!\!=&\!\! \frac{2}{g^2}
\left(iG^a_{\beta\alpha} \bar\lambda^a_{\dot\beta}
+ \epsilon_{\beta\alpha} D^a\bar\lambda^a_{\dot\beta}
\right) + 2\sqrt{2}\sum\left[
\left({\cal D}_{\alpha\dot\beta}\phi^\dagger
\right)\psi_\beta -i\epsilon_{\beta\alpha}
F\bar\psi_{\dot\beta}\right]
\nonumber\\[0.2cm]
&\!\!- &\!\! \frac{\sqrt{2}}{3}\sum \left[
\partial_{\alpha\dot\beta}(\psi_\beta \phi^\dagger)
+\partial_{\beta\dot\beta}(\psi_\alpha \phi^\dagger)
-3\epsilon_{\beta\alpha}\partial^\gamma_{\dot\beta}(\psi_\gamma
\phi^\dagger)
\right]
\, ,
\label{scsqcd}
\end{eqnarray}
where the sum runs over all matter supermultiplets, $D^a$ and $F$
are the corresponding $D$ and $F$ terms.
The second line is conserved by itself, nondynamically;
the spatial integral of the time-like component
reduces to the integral over the total derivative for the second line.
Therefore, it may or may not be included
in the definition of the supercurrents.
This is the supersymmetric analog of the ambiguity
in the energy-momentum tensor in nonsupersymmetric theories
with the scalar fields. The ambiguity in the choice
of $J_{\alpha\beta\dot\beta} $ leads, with necessity,
to the fact that the coefficients of the matter terms in $a^\mu$ in Eq.
(\ref{bsa}), namely, $a^\mu_{(\psi )}$ and $a^\mu_{(\phi )}$,
are not uniquely fixed.
Due to this ambiguity, the matter component of
$a^\mu$ cannot contribute to $Z$ for strings (it could contribute,
though,
for the wall junctions and other similar object with the axial
geometry).
\section{Strings
Cannot be Saturated in ${\cal N}=1$ Non-Abelian Gauge Theories in
Weak Coupling}
Here we will prove that
in the absence of the U(1) factors, even if the theory
under consideration does support string-like solitons
in the quasiclassical consideration (some examples are discussed
e.g. in Ref. \cite{Strass}), the central charge vanishes with necessity.
Therefore, these strings {\em cannot be saturated}.
In weak coupling (i.e. for the string solitons
in the quasiclassical treatment)
the $(1/2,1/2)$ central charge must be saturated
by the term with the {\em bosonic}
axial current. (We remind that the FI term is absent).
As was explained, the coefficient of this term is
not unambiguous -- it depends on the definition of the
supercurrent (e.g. minimal {\em versus} conformal).
Since we are interested in the
string solitons, rather than the wall junctions,
this ambiguity cannot be canceled by that in the $(1,0)$
central charge, since the latter must identically vanish.
This is dictated by the Lorentz symmetry arguments.
This means that the $(1/2,1/2)$
central charge must vanish identically.
The consideration above
shows that if the BPS objects with the axial geometry
exist in the quasiclassical limit (in non-Abelian
gauge theories), the stringy core must be
accompanied by objects with the $(1,0)$ charges.
In four dimensions
domain walls do the job. Within the brane picture
it is possible to consider four-dimensional
theories as that on the brane embedded in $M$ theory.
For instance, the expected domain wall junctions
in ${\cal N}=1$ Yang-Mills theory -- the
gauge analog of the junctions in the GWZ models --
can be identified as a
junction of M5 branes, so that the definition
of the current for the theory on M5
removes any ambiguity.
\section{Strings in the Seiberg-Witten ${\cal N }=2$ Model }
Here we will speculate on possible BPS strings at strong coupling.
As we already know, such strings do not appear in weak coupling.
The $(1/2,1/2)$ central charge
(appearing in the anticommutator $\{ Q,\, \bar Q\}$)
is not holomorphic -- it need not depend holomorphically
on the chiral parameters, in contradistinction with the $(1,0)$ charge.
This means, that even if both the weak and strong coupling regimes
are
attainable in one and the same theory,
generally speaking, nothing
can be said regarding the BPS strings in the strong coupling regime
from the behavior at weak coupling.
\subsection{Strings in Pure ${\cal N }=2$ Yang-Mills Theory}
Turn now to discussion of the ${\cal N }=2$ Yang-Mills theory
without matter hypermultiplets. The
exact solution for the low-energy effective
action, as well as the exact spectrum of the BPS particles,
are known \cite{ws}. Now
we address the issue of possible stringy central
charges, besides the standard ones,
saturated by particles \cite{wo}.
>From the discussion above we saw that it can be
attributed only to the gluino axial current since
there is no FI term in the model. Let us
restrict ourselves to SU(2) gauge
group.
The key features of the Seiberg-Witten
solution can be summarized as
follows.
The vacuum manifold develops the Coulomb branch which is
parametrized by the global coordinate, the order parameter
$u=\langle {\rm tr}\phi^2\rangle$. At low energies
the effective theory becomes Abelian and
is described by a single holomorphic function -- prepotential
$\cal {F}$ which determines the effective coupling constant
of the theory $\tau={\partial^2 \cal{F}}/{\partial a^2}$,
as well as the K\"ahler metric on the Coulomb
branch of the moduli space, which appears to be a
one-dimensional special K\"ahler
manifold. The K\"ahler potential
can be found from the prepotential as follows
\begin{equation}} \def\eeq{\end{equation}
K(a,\bar a)={\rm Im} a_{D}\bar a
\eeq
where $a$ is the vacuum value of the third component of the
scalar field and
$a_{D}={\partial \cal{F}}/{\partial a}$. The variable $a$ can be
expressed in terms of variable $u$ as follows:
\begin{equation}} \def\eeq{\end{equation}
a(u)=\int_{-\sqrt{u+\Lambda^2}}^{-\sqrt{u-\Lambda^2}}
\frac{x^2 dx}{\pi\sqrt{(x^2-u)^2-\Lambda^4}}\, .
\eeq
Unlike the variable $u$, the variable $a$ cannot be considered as a
global
coordinate on the moduli space since
the K\"ahler metric ${\rm Im}\tau(a)$ has zeros
(here $\tau$ is the complexified
coupling constant). Therefore,
to analyze the complex plane of $a$, an explicit expression for $a(u)$
is
needed. A direct inspection shows that the region of small $a$
is essentially removed from the
complex plane so that $|a(u)|> {\rm const}\, \Lambda$.
The lower bound on $a$ can be seen also geometrically
if we recall that it is just the mass of the $W$ boson, which can be
represented in the theory on D3 probe
as the pronged string connecting the
probe and the split O7 orientifold \cite{probe}.
It is clear that the
minimal mass of the $W$ boson geometrically is the distance
between
the
7-branes on the $u$ plane; it is, thus,
proportional to $\Lambda$. Therefore,
we see that $\pi_1$ of the scalar field manifold is nontrivial --
topologically stable objects with the axial geometry are expected,
provided $a$
winds around the ``forbidden" region.
Whether these objects are strings (i.e. have finite energy per unit
length)
depends on dynamics, on how fast the volume energy density
dies off as we go away from the axis in the perpendicular
direction. The convergence could be ensured by the
appropriate form of the K\"ahler metric, as in the sigma models.
It is quite obvious, that in this case
the string tension
\begin{equation}} \def\eeq{\end{equation}
T={\rm const}\, \Lambda^2\, .
\eeq
The existence of
such the stable objects with the axial geometry would be a purely
strong coupling effect since at the classical level the
point $ a=0$ is attainable, and, correspondingly,
$\pi_1$ is trivial. If the strings do exist, they may be BPS-saturated
provided the term due to the gluino current $R^\mu$ in the central
charge is
nonvanisihing.
To this end the gluino current must fall off at large distances
$r$ from the axis as $1/r$. Finiteness of the string tension
would imply then that effective degrees of freedom coupled to
$R^\mu$
form a U(1) gauge interaction. If the string tension is finite and
the gluino current falls off at large distances
$r$ from the axis faster than $1/r$, the string is tensionless.
\subsection{Strings in ${\cal N}$=2 SQCD}
Adding the matter hypermultiplets to the model discussed in Sec. 9.1
we get ${\cal N}$=2 SQCD.
Since
there is no restoration of the SU(2) gauge symmetry
at the generic point at the Coulomb branch, the
``forbidden" region on the complex $a$ plane exists in the
theory with the fundamental matter too. The
BPS strings may appear on the Coulomb
branch, with the tension saturated
by the $R$ current of gluinos. The
tension now depends on the masses of the
fundamental matter and can be determined, in principle,
from the explicit expression for $a(u, \Lambda, m)$.
Moreover, the Higgs branch (parametrized by
the vacuum expectation values of the fundamentals
$\langle Q\rangle , \,\, \langle\tilde Q\rangle $) is possible, and
the question of the BPS strings on the Higgs
branch can be addressed. We recall that geometrically
the Higgs branch is the hyper-K\"ahler manifold \cite{aps}
(for a review see \cite{ap}) whose metric can be determined
classically.
It is not renormalized by quantum corrections. Actually,
the Higgs branch for SU$(N_c)$ theory with $N_f$
flavors is the cotangent bundle of the Grassmannian
$T^{*}{\rm Gr}_{N_c,N_f}$, with the antisymmetric $N_c$\,-form.
The metric on this manifold can be found from the K\"ahler potential
\begin{equation}} \def\eeq{\end{equation}
K(Q,\tilde Q)={\rm Tr}\sqrt{k^2+MM^{\dagger}}\, ,
\eeq
where $k$ is a solution of the equation
\begin{equation}} \def\eeq{\end{equation}
{\rm det}\, \left( k 1_{N_f} +\sqrt {k^2 1_{N_f}+MM^{\dagger}}
\right) = {\rm det}\,(QQ^{\dagger})\,,
\eeq
and $M=Q\tilde Q$ is the meson matrix.
Since $\pi _{2}({\rm Gr}_{n,k})\neq 0$, instantons
in the two-dimensional sigma model on $T^{*}{\rm Gr}_{n,k}$
are possible. The arguments presented in Sec. 5 suggest that these
instantons can be interpreted as strings on the Higgs branch.
It would be interesting to understand whether a version of the
string on the Higgs branch recently found in \cite{yung}
can be BPS saturated.
The existence of the BPS string on the Higgs branch was
recently conjectured within the brane approach \cite{oz}.
This string was expected to be tensionless at the
root of the Higgs branch, which qualitatively
agrees with the discussion above.
\subsection{Softly broken ${\cal N}$=2 theory
(strings in ${\cal N}=2$ SQED)}
If the softly broken ${\cal N}=2$ Yang-Mills theory
is considered near the monopole or dyon singularities
the effective low-energy theory which ensues is
${\cal N}=2$ dual SQED. This is the famous Seiberg-Witten
result. A small mass term of the chiral superfields
of the original ${\cal N}$=2 non-Abelian theory
is translated
in a small perturbation of the superpotential for the
matter fields in SQED.
If the monopole (or dyon) superfields
are denoted as $M,\, \tilde M$
the
superpotential in the low-energy SQED
can be written as
\begin{equation}} \def\eeq{\end{equation}
{\cal W}=\mu\, u(a_{D}) +\tilde{M}a_{D}M\, ,
\label{ddd}
\eeq
where $a_D$ is a chiral superfield which is
the ${\cal N}=2$ superpartner of the (dual) vector superfield.
The second term in Eq. (\ref{ddd})
is fixed by ${\cal N}=2$ supersymmetry.
The parameter
$\mu$ in the first term is small. Generically
$\mu\, u(a_{D}) $ breaks ${\cal N}=2$ supersymmetry
down to ${\cal N}=1$. However, in the linear approximation,
when
\begin{equation}} \def\eeq{\end{equation}
{\cal W}=\mu a_{D} \Lambda +\tilde{M}a_{D}M\, ,
\label{fff}
\eeq
${\cal N}=2$ is unbroken.
Let us forget about the origin
of ${\cal N}=2$ SQED and discuss this U(1) theory
with the superpotential (\ref{fff}) {\em per se}.
Minimization of the potential stemming from
(\ref{fff}) yields
the monopole condensation.
The Abrikosov strings obviously do exist.
Their tension is
proportional
to $\mu$.
They were discussed in the literature previously~\cite{DS,HSZ}.
The classical equations for the string reduce
to Eq. (\ref{sateq}). \footnote{It should be taken
into account that on the solution
$ | M| = |\tilde M|$.}
Thus, the string is
saturated. The question is how this could happen
given that the $(1/2, 1/2)$ central charge must vanish
in the absence of the FI term.
The central charge in the anticommutator
$\{Q_\alpha \bar Q_{\dot\alpha} \}$
is indeed zero. One should not forget however,
that SQED with the superpotential (\ref{fff})
is an ${\cal N}=2$ theory --
there exist two supercharges $Q, \, Q '$
of the type $(1/2,0)$
and two supercharges $\bar Q, \,\bar Q '$
of the type $(0,1/2)$. Therefore, one should look
for the central extension in the anticommutator
of the general form
$\{ {\cal Q}_\alpha \bar{\cal Q}_{\dot\alpha} \}$
where ${\cal Q}$ is a linear combination
of $Q$ and $Q '$. A nonvanishing cenral term
of this type does exist.
If we now return to the
original non-Abelian ${\cal N}$=2 theory,
we conclude that at
small $\mu$ the string is (approximately) BPS saturated.
It becomes exactly saturated in the limit
$\mu\to 0\,,\quad \Lambda\to\infty$
with $\mu\Lambda$ fixed~\cite{DS,HSZ}.
The saturation is approximate, rather than exact, since
higher order terms in $\mu$ (non-linear in $a_D$ terms in the
superpotential
of the low-energy U(1) theory)
break ${\cal N}=2$
and
return us back to the ${\cal N}=1$ theory.
In ${\cal N}=1$ the extra supercharges $Q'
, \, \bar Q '$ disappear, while the
central charge in the anticommutator
$\{Q_\alpha \bar Q_{\dot\alpha} \}$
vanishes.
\section{The Brane Picture: How It Corresponds to Field Theory}
\subsection{The Fayet-Iliopoulos string as a membrane}
With the brane picture in mind, we
can look for the brane configuration corresponding to
the BPS strings discussed above. The interpretation of the
strings
whose tension is proportional to the four-dimensional
FI terms is rather simple.
Let us consider the brane configuration relevant for the Abelian
${\cal N}=2$
Yang-Mills
theory in the IIA picture. It consists of the pair of the parallel
NS5 branes with the worldvolumes $(x^0, x^1, x^2, x^3, x^4, x^5)$,
plus
a single D4 brane with the worldvolume $(x^0, x^1, x^2, x^3, x^6)$.
The gauge theory is defined on the worldvolume of the D4 brane,
and the
distance between the NS5 branes along the $x^6$ direction
plays the role of the
inverse coupling in the Abelian theory.
Since the four-dimensional FI terms have the meaning of the relative
distance between the
NS branes in $(x^7, x^8, x^9)$ \cite{GK},
the
``FI strings" are nothing but the D2 branes stretched between the NS
branes in some of $(x^7, x^8, x^9)$ directions. The rest of their
worldvolume
coordinates coincide with the D4 ones.
This picture gets slightly modified if one considers
the Abelian ${\cal N}=1$ theory. According to
the well-known procedure (see, for instance, \cite{GK}), one has then
to rotate one of the NS5 branes, which now has
$(x^0, x^1, x^2, x^3, x^8, x^9)$ as the
worldvolume. The Fayet-Iliopoulos term
has now the meaning of the displacement of the NS5 branes along
$x^7$.
The D2 brane stretched between the NS5 branes with the
worldvolume
$(x^0, x^1, x^7)$
plays the role of the BPS string.
Let us note that the FI string can be elevated smoothly in the $M$
theory.
Indeed, the NS5 branes and the D4 brane can be identified with the
single
M5 brane in the $M$ theory. The FI string
can be considered as an M2 brane
stretched between two components of the M5 branes.
The tension of the
FI string is proportional to the length of the M2
brane along the $x^7$ direction
and, therefore, proportional to the value of the FI parameter $\xi$, in
full
agreement with the field theory expectations. Recently a
similar picture for the FI strings was discussed in \cite{oz}.
\subsection{On (conjectured) strong coupling BPS strings
via
\\ branes}
The BPS saturated objects with the axial geometry were
discussed in the brane picture previously. For instance,
the domain wall junction in ${\cal N}=1$ supersymmetric
gluodynamics
which is expected to saturate both, the $(1/2, 1/2)$ and $(1,0)$
central charges, occurs as the intersection
of the M5 branes, since
the domain walls were identified as the M5
branes wrapped on 4-manifold in $M$ theory \cite{wittenmqcd}.
Here we would like to add a few remarks on a
possible interpretation of the strong coupling
BPS strings in the brane picture.
Previous attempts to recognize BPS tensionless
string in four dimensions, apparently seen within the brane
approach \cite{HK}, were based on the intersection
of the M5 branes
or the M2 brane stretched between two M5 branes.
The BPS string on the Coulomb branch discussed in Sec.~9
is nothing but a wrapped M5 brane, since its
tension is proportional to the area on the region on the Coulomb
branch.
However, the explicit geometry of
intersection of the M5 branes yielding saturation
of the $(1/2,1/2)$ central charge is still to be clarified.
Another possible approach to the brane interpretation of
the BPS strings in the Yang-Mills theories
follows from the correspondence between the Yang-Mills theories
in four dimensions and two-dimensional sigma models.
It was recently recognized
\cite{hh,d,dht} that there is a close relation
between the two-dimensional $CP_N$ models (which have ${\cal
N}=2$)
and the Yang-Mills theories in four dimensions, with ${\cal N}=1$
or ${\cal N}=2$,
with or without fundamental matter. In the latter case
the correspondence relies on the coincidence
of the spectra of the BPS domain walls in four
dimensions and BPS solitons in two.
A more direct relation connects the ${\cal N}=2$ theory with $N_f$
flavors
at the root of the baryonic branch of the
moduli space with the $CP_{2N_c-N_f-1}$ model
\cite{d,dht}. The translation dictionary between the two
models looks as follows: the complex coupling
in four dimensions corresponds to a complex parameter
combining the two-dimensional
FI term with the $\theta$ term; twisted mass terms
in $d=2$ correspond to the coordinates on
the Coulomb branch in the $d=4$ theory; finally,
the Riemann surfaces providing the BPS
spectra in both theories are the same.
The correspondence above has a rather simple
explanation in the brane description of both theories. It appears that
the brane configurations for both theories are actually the same.
The $d=4$ theory is defined on the worldvolume
of the D4 branes stretched between a pair
of the NS5 branes. The coupling constant is just the distance between
the NS5 branes. In $M$ theory all branes above are elevated
to a pair of the M5 branes,
one of which is flat and the second is wrapped around the Riemann
surface. The configuration is described by the holomorphic
embedding into four-dimensional space
\begin{equation}} \def\eeq{\end{equation}
(t-\Lambda^N)
\left\{ t\Lambda^{N-N_f}\prod ^{N-N_f} (v-\tilde{m_i}) -
\prod^N (v-m_i)\right\}
=0\,,
\eeq
where the first factor represents the flat brane, while the second
the curved one, and $m_i$'s correspond to the masses
of the fundamental hypermultiplets.
Let us add a D2 brane and consider the Abelian
gauge theory on its worldvolume. If the D2 brane is stretched
between the same
NS5 branes we arrive at the $CP_N$ model in $d=2$ where it has
the extended supersymmetry,
${\cal N}=2$. This explains
the coincidence between the complexified coupling constant in
the four-dimensional theory and the
FI term in the two-dimensional theory. Therefore, the picture can
be
apparently
interpreted as follows: the $d=2$ sigma model,
with the twisted masses added,
is the theory on the brane which is the
probe for the ${\cal N}=2$ low-energy theory in four dimensions.
In \cite{d,dht} it was shown that the spectrum
of the BPS particles in ${\cal N}=2$ theory
at the root of the baryonic branch exactly coincides with
the spectrum of BPS dyonic
kinks in the corresponding $CP_N$ model.
Moreover, the brane identification shows that the hypermultiplets
in $d=4$ and $d=2$ arise essentially in the same way. Therefore, we
can
use the relation between the models in a two-fold way. The existence
of instantons in the $CP_N$ model implies that one can expect
BPS saturated strings at the root of the baryonic branch.
In the opposite direction, the $(1/2,1/2)$ central charge in
four dimensions can be mapped into the central
charge of the ${\cal N}=2$ two-dimensional theory.
Since in the formulation of the
sigma model with the nondynamical vector field,
the gauge 2-potential plays the role of the current
$a^\mu$ in Eq. (\ref{bsa}), the central charge is actually mapped
onto the Chern number $\int Adx$. Certainly, these issues
need further clarification. We hope to discuss them
elsewhere.
\section{Conclusions and Discussion}
In this paper we elaborated the generic structure of
of the central charges in supersymmetric gauge theories in
four dimensions. The central finding is that the $(1/2,1/2)$
charge is ambiguous in the part related to the matter fields,
due to possible total derivative terms in the supercurrents.
The part related to the gauge fields (including gaugino)
is unambiguous. That is why in the weak coupling regime
the only model admitting BPS strings is SQED with the
Fayet-Iliopoulos term. In the non-Abelian theories
the
Fayet-Iliopoulos term is forbidden; hence, at weak coupling there can
be no
BPS strings. Even if some strings exist, they are nonsaturated
with necessity.
These assertions are proven at the theorem level.
The ambiguity we found does not preclude from existence
other BPS saturated objects with the axial geometry, i.e. the wall
junctions.
The ambiguity in the $(1/2,1/2)$ charge is combined with that
in the $(1,0)$ charge to produce a well-defined answer
for the tension of the walls and the ``hub" in the middle.
We presented some examples.
The strong coupling regime is a different story.
Since the analyticity argument does not apply
to the $(1/2,1/2)$ charge, the existence/non-existence of the
BPS strings should be discussed separately
at weak and strong couplings -- the lessons we learn at weak
coupling say nothing about possible scenarios at strong coupling.
We speculated on different cases when the
BPS-saturated objects with axial geometry may appear in the strong
coupling regime. We argued that saturation
of the $(1/2,1/2)$ charge at strong coupling can be attributed
to the M5 brane intersection.
(The Fayet-Iliopoulos BPS strings come from the M2 branes.)
The BPS-saturated strings may be expected in
the ${\cal N}=2$ Yang-Mills theories
on the Coulomb branch.
In the ${\cal N}=1$ gauge theories
an obvious candidate for the BPS saturation is the domain wall
junction. One cannot assert at the moment with absolute certainty
that the strong coupling junction
exists in ${\cal N}=1$ supersymmetric
gluodynamics, but the $M$ theory arguments
suggest that such junctions do exist.
Additional support in favor of this conclusion is
provided by field-theoretic models considered in
\cite{GabShi}.
A comment is in order regarding the situation in supergravity
coupled to the Yang-Mills theory. Upon inspecting the (1/2, 1/2)
central
charge one finds the term $H=dB-K$
in the anticommutator $\{Q,\,\bar Q\}$, where $B$ is the
two-form field and $K=AdA-\frac{2}{3}A^3$
is the dual of the Chern-Simons current
in the Yang-Mills theory. Therefore, we see that the axial
current of gluons enters into
the central charge, if gravity degrees of freedom
are taken into account. We plan to discuss this point in more detail
elsewhere.
Since the ${\cal N}=2$ Yang-Mills
theory enjoys duality, one can pose a question of
the duality partner of the BPS string.
Four-dimensional BPS strings can be viewed as objects
dual (in the Dirac sense) to localized objects.
Indeed, since in
$d$ space-time dimensions the $ p$ brane is dual to a $(d-p-4)$
brane,
the Dirac quantization condition amounts
to the observation that the $(d-p-4)$ brane
is weakly coupled if the $p$ brane
is strongly coupled and {\em vice versa}. Therefore, one can expect
that within the framework of duality
the strongly coupled BPS string
has something to do with the instantons at weak coupling.
To make a conjecture regarding the central charge dualizing the
stringy one,
let us observe that there is a contribution in the
central charge for $\{Q,\bar Q\}$ in six dimensions,
saturated by instantonic strings. In five dimensions the
instanton presents a particle with mass ${1}/{g^2}$,
saturating \cite{seiberg} the central
charge $\int d^4 x \tilde{F}F$. In four dimensions we can expect a
remnant of this central charge resulting from
dimensional reduction.
\vspace{1cm}
{\bf Acknowledgments}
\vspace{0.2cm}
\noindent
We would like to thank M. Strassler,
A. Vainshtein and A. Yung for
useful discussions.
A part of this work was done while one of the authors (M.S.) was
visiting
the
Aspen Center for Physics, within the framework of the program
{\em Phenomenology of Superparticles and Superbranes}.
A.G. thanks TPI at the Minnesota University for the kind
hospitality.
This work was supported in part by DOE under the grant number
DE-FG02-94ER408. The work of A.G. was also supported by the
INTAS
grant number
INTAS-97-0103.
|
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| 5,623
|
The 2014–15 season was Notts County Football Club's 126th year in the Football League and their 5th consecutive season in Football League One, the third division of the English League System.
Match Details
Pre-season
League One
League table
Matches
The fixtures for the 2014–15 season were announced on 18 June 2014 at 9am.
FA Cup
The draw for the first round of the FA Cup was made on 27 October 2014.
League Cup
The draw for the first round of the 2014–15 Football League Cup was made on 17 June 2014 at 10:00. Notts County were drawn away to Sheffield Wednesday and were eliminated from the competition after suffering a 3–0 defeat.
Football League Trophy
Transfers
In
Out
Loans In
Loans Out
References
Notts County F.C. seasons
Notts County
|
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The World's Longest Running Talent Showcase
CSN Story
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About Nashville CSN:
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{"url":"https:\/\/www.physicsforums.com\/threads\/turning-a-vector-into-a-vector-function-of-time.789926\/","text":"# Turning a vector into a vector function of time\n\n## Homework Statement\n\n[\/B]\nA Velocity vector: V = (12,4)\n\nwrite the vector as a vector function of Displacement.\n\n2. The attempt at a solution\n\nI integrated the components of the Vector and got the function S(t) = (S(12t), S(4t))\n\nI this correct at all?\n\nRay Vickson\nHomework Helper\nDearly Missed\n\n## Homework Statement\n\n[\/B]\nA Velocity vector: V = (12,4)\n\nwrite the vector as a vector function of Displacement.\n\n2. The attempt at a solution\n\nI integrated the components of the Vector and got the function S(t) = (S(12t), S(4t))\n\nI this correct at all?\n\nThis cannot be answered because the notation S(12t), etc., is undefined.\n\nwhy?\n\nRay Vickson\nHomework Helper\nDearly Missed\nwhy?\n\nWhy what?\n(1) Why the notation is undefined? The answer is: because you have not defined it.\n(2) Why I can't answer the question? Because---as I have already said---I have no idea what you mean by what you wrote.\n\nMark44\nMentor\nI integrated the components of the Vector and got the function S(t) = (S(12t), S(4t))\nDo you mean S(t) = <12t, 4t>?\nIf so, don't forget you need the constant of integration.\n\nRay, I asked why because I literally have no idea what I'm talking about.\n\nMark, from V = 12x + 4y, am i allowed to turn it into that function that you wrote.\n\nmfb\nMentor\n\n## Homework Statement\n\n[\/B]\nA Velocity vector: V = (12,4)\n\nwrite the vector as a vector function of Displacement.\nIs this the exact and full problem statement? It sounds a bit odd.\n\nmfb\nMentor\nI don't think it makes sense.\n\nRay Vickson\nHomework Helper\nDearly Missed\nRay, I asked why because I literally have no idea what I'm talking about.\n\nMark, from V = 12x + 4y, am i allowed to turn it into that function that you wrote.\n\nYou say you literally have no idea what you are talking about. So, let's look at that.\n\nIf you are told a function f(t) for x-displacement, say x = f(t), can you figure out from that how to obtain velocity? Can you go backwards: given a velocity function g(t), so that v = g(t), can you figure out from that how to get displacement x = f(t)? [In a nutshell, that is what you are being asked to do in this problem.]\n\nIf the answers to both (or even one) of these questions is NO, you need to upgrade your background. You can Google \u201cdisplacement and velocity\u201d to find numerous articles on these issues.\n\nWait a minute.\n\nI can breakup the vector into:\n\nVx = 12\n\nand\n\nVy = 4\n\nthese can easy be described as a function. a constant function of velocity. I can find the integral of the function V(t) = 12, and turn it into S(t) = 12t\n\nand furthermore, from S(x) = x, which is a function of displacement, I CAN find the function for instantaneous velocity by simply finding the derivative. The answer is, it has no velocity as the function is constant.\n\nMark44\nMentor\nMark, from V = 12x + 4y, am i allowed to turn it into that function that you wrote.\nYou could have V(t) = 12i + 4j, which would be the same as V(t) = <12, 4>. i and j are the unit vectors in the direction of the positive x- and y-axes.\nWait a minute.\n\nI can breakup the vector into:\n\nVx = 12\n\nand\n\nVy = 4\n\nthese can easy be described as a function. a constant function of velocity. I can find the integral of the function V(t) = 12, and turn it into S(t) = 12t\nIs this a different example? Above you have Vx = 12, and now you have V(t) = 12. This would be incorrect if V(t) = <12, 4>.\n\nIf you really mean Vx(t), the x-component of velocity, then Sx(t) = 12t + C1. As I said before, you have to add the constant of integration.\n\nand furthermore, from S(x) = x, which is a function of displacement, I CAN find the function for instantaneous velocity by simply finding the derivative. The answer is, it has no velocity as the function is constant.\nThis makes no sense. If you're talking about displacement and velocity, the independent variable should be t, not x. Having said that, if S(x) = x, then S'(x) = 1 represents the \"velocity\" here, a constant velocity, which is not the same as \"no velocity.\"\n\nI get what your saying. S(x) = x was meant to be a position function. The point is i wanted to turn the velocity vector into a function. that's all. I didn't know if that's even possible.\n\nMark44\nMentor\nI get what your saying. S(x) = x was meant to be a position function. The point is i wanted to turn the velocity vector into a function. that's all. I didn't know if that's even possible.\nV(t) = <12, 4> is a vector-valued function, albeit one that produces a vector constant. Or in slightly different form, V(t) = 12i + 4j.","date":"2021-04-15 10:15:58","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8104618787765503, \"perplexity\": 656.4566194171647}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038084765.46\/warc\/CC-MAIN-20210415095505-20210415125505-00225.warc.gz\"}"}
| null | null |
[Search] [txt|pdfized|bibtex] [Tracker] [WG] [Email] [Diff1] [Diff2] [Nits]
Versions: 00 01 02 03 04 05 06 07 08 09 10 11 12
13 14 15 16 17 18 19 20 21 22 23 rfc6412
Expires in: June 2006
Scott Poretsky
Reef Point Systems
Brent Imhoff
Terminology for Benchmarking
IGP Data Plane Route Convergence
<draft-ietf-bmwg-igp-dataplane-conv-term-09.txt>
Intellectual Property Rights (IPR) statement:
other groups may also distribute working documents as
Internet-Drafts.
This document describes the terminology for benchmarking IGP
Route Convergence as described in Applicability document [1] and
Methodology document [2]. The methodology and terminology are to
be used for benchmarking Convergence Time and can be applied to
any link-state IGP such as ISIS [3] and OSPF [4]. The data plane
is measured to obtain the convergence benchmarking metrics
described in [2].
Poretsky, Imhoff [Page 1]
INTERNET-DRAFT Benchmarking Terminology for January 2006
1. Introduction .................................................2
2. Existing definitions .........................................3
3. Term definitions..............................................3
3.1 Convergence Event.........................................3
3.2 Route Convergence.........................................4
3.3 Network Convergence.......................................4
3.4 Full Convergence..........................................5
3.5 Convergence Packet Loss...................................5
3.6 Convergence Event Instant.................................6
3.7 Convergence Recovery Instant..............................6
3.8 Rate-Derived Convergence Time.............................7
3.9 Convergence Event Transition..............................7
3.10 Convergence Recovery Transition..........................8
3.11 Loss-Derived Convergence Time............................8
3.12 Sustained Forwarding Convergence Time....................9
3.13 Restoration Convergence Time.............................9
3.14 Packet Sampling Interval.................................10
3.15 Local Interface..........................................11
3.16 Neighbor Interface.......................................11
3.17 Remote Interface.........................................11
3.18 Preferred Egress Interface...............................12
3.19 Next-Best Egress Interface...............................12
3.20 Stale Forwarding.........................................13
3.21 Nested Convergence Events................................13
4. IANA Considerations...........................................13
5. Security Considerations.......................................14
6. Normative References..........................................14
7. Author's Address..............................................14
This draft describes the terminology for benchmarking IGP Route
Convergence. The motivation and applicability for this
benchmarking is provided in [1]. The methodology to be used for
this benchmarking is described in [2]. The methodology and
terminology to be used for benchmarking Route Convergence can be
applied to any link-state IGP such as ISIS [3] and OSPF [4]. The
data plane is measured to obtain black-box (externally observable)
convergence benchmarking metrics. The purpose of this document is
to introduce new terms required to complete execution of the IGP
Route Convergence Methodology [2]. These terms apply to IPv4 and
IPv6 traffic as well as IPv4 and IPv6 IGPs.
An example of Route Convergence as observed and measured from the
data plane is shown in Figure 1. The graph in Figure 1 shows
Throughput versus Time. Time 0 on the X-axis is on the far
right of the graph. The components of the graph and metrics are
defined in the Term Definitions section.
Convergence Convergence
Recovery Event
Instant Instant Time = 0sec
Maximum ^ ^ ^
Throughput--> ------\ Packet /---------------
\ Loss /<----Convergence
Convergence------->\ / Event Transition
Recovery Transition \ /
\_____/<------Maximum Packet Loss
X-axis = Time
Y-axis = Throughput
Figure 1. Convergence Graph
2. Existing definitions
document are to be interpreted as described in BCP 14, RFC 2119.
RFC 2119 defines the use of these key words to help make the
intent of standards track documents as clear as possible. While this
document uses these keywords, this document is not a standards track
document. The term Throughput is defined in RFC 2544.
3. Term Definitions
3.1 Convergence Event
The occurrence of a planned or unplanned action in the network
that results in a change in the egress interface of the Device
Under Test (DUT) for
routed packets.
Convergence Events include link loss, routing protocol session
loss, router failure, configuration change, and better next-hop
learned via a routing protocol.
Measurement Units:
Convergence Packet Loss
Convergence Event Instant
3.2 Route Convergence
Recovery from a Convergence Event indicated by the DUT
Throughput equal to the offered load.
Route Convergence is the action of all components of the router
being updated with the most recent route change(s) including the
Routing Information Base (RIB) and Forwaridng Information Base
(FIB), along with software and hardware tables. Route
Convergence can be observed externally by the rerouting of data
Traffic to a new egress interface.
Full Convergence
Convergence Event
3.3 Network Convergence
The completion of updating of all routing tables, including the
FIB, in all routers throughout the network.
Network Convergence is bounded by the sum of Route Convergence
for all routers in the network. Network Convergence can be
determined by recovery of the Throughput to equal the
offered load, with no Stale Forwarding, and no blenders[5][6].
Route Convergence
Stale Forwarding
3.4 Full Convergence
Route Convergence for an entire FIB.
When benchmarking convergence, it is useful to measure
the time to converge an entire FIB. For example,
a Convergence Event can be produced for an OSPF table of
5000 routes so that the time to converge routes 1 through
5000 is measured. Full Convergence is externally observable
from the data plane when the Throughput of the data
plane traffic on the Next-Best Egress Interface equals the
offered load.
Issues: None
3.5 Convergence Packet Loss
The amount of packet loss produced by a Convergence Event
until Route Convergence occurs.
Packet loss can be observed as a reduction of forwarded traffic
from the maximum Throughput. Convergence Packet Loss
includes packets that were lost and packets that were delayed
due to buffering. The maximum Convergence Packet Loss observed
in a Packet Sampling Interval may or may not reach 100% during
Route Convergence (see Figure 1).
number of packets
Rate-Derived Convergence Time
Loss-Derived Convergence Time
Packet Sampling Interval
INTERNET-DRAFT Benchmarking Terminology for January 2006
3.6 Convergence Event Instant
The time instant that a Convergence Event becomes observable in
the data plane.
Convergence Event Instant is observable from the data
plane as the precise time that the device under test begins
to exhibit packet loss.
hh:mm:ss:nnn, where 'nnn' is milliseconds
Convergence Recovery Instant
3.7 Convergence Recovery Instant
The time instant that Full Convergence is measured
and then maintained for an interval of duration equal to
the Sustained Forwarding Convergence Time
Convergence Recovery Instant is measurable from the data
plane as the precise time that the device under test
achieves Full Convergence.
hh:mm:ss:uuu
Sustained Forwarding Convergence Time
3.8 Rate-Derived Convergence Time
The amount of time for Convergence Packet Loss to persist upon
occurrence of a Convergence Event until occurrence of Route
Convergence.
Rate-Derived Convergence Time can be measured as the time
difference from the Convergence Event Instant to the
Convergence Recovery Instant, as shown with Equation 1.
(eq 1) Rate-Derived Convergence Time =
Convergence Recovery Instant - Convergence Event Instant.
Rate-Derived Convergence Time should be measured at the maximum
Throughput. Failure to achieve Full Convergence results in
a Rate-Derived Convergence Time benchmark of infinity.
seconds/milliseconds
3.9 Convergence Event Transition
The characteristic of a router in which Throughput
gradually reduces to zero after a Convergence Event.
The Convergence Event Transition is best observed for
Full Convergence. The Convergence Event Transition may
not be linear.
Convergence Recovery Transition
3.10 Convergence Recovery Transition
gradually increases to equal the offered load.
The Convergence Recovery Transition is best observed for
Convergence Event Transition
3.11 Loss-Derived Convergence Time
The amount of time it takes for Route Convergence to
to be achieved as calculated from the Convergence Packet
Loss. Loss-Derived Convergence Time can be calculated
from Convergence Packet Loss that occurs due to a
Convergence Event and Route Convergence.as shown with
Equation 2.
(eq 2) Loss-Derived Convergence Time =
Convergence Packets Loss / Offered Load
NOTE: Units for this measurement are
packets / packets/second = seconds
Loss-Derived Convergence Time gives a better than
actual result when converging many routes simultaneously.
Rate-Derived Convergence Time takes the Convergence Recovery
Transition into account, but Loss-Derived Convergence Time
ignores the Route Convergence Recovery Transition because
it is obtained from the measured Convergence Packet Loss.
Ideally, the Convergence Event Transition and Convergence
Recovery Transition are instantaneous so that the
Rate-Derived Convergence Time = Loss-Derived Convergence Time.
However, router implementations are less than ideal.
For these reasons the preferred reporting benchmark for IGP
Route Convergence is the Rate-Derived Convergence Time.
Guidelines for reporting Loss-Derived Convergence Time are
provided in [2].
3.12 Sustained Forwarding Convergence Time
The amount of time for which Full Convergence is maintained
without additional packet loss.
The purpose of the Sustained Forwarding Convergence Time is to
produce Convergence benchmarks protected against fluctuation
in Throughput after Full Convergence is observed. The
Sustained Forwarding Convergence Time to be used is calculated
as shown in Equation 3.
(eq 3)
Sustained Forwarding Convergence Time = 5 packets/Offered Load
units are packets/pps = sec
for which at least one packet per route in the FIB for all
routes in the FIB MUST be offered to the DUT per second.
seconds or milliseconds
3.13 Restoration Convergence Time
The amount of time for the router under test to restore
traffic to the original outbound port after recovery from
a Convergence Event.
Restoration Convergence Time is the amount of time for routes
to converge to the original outbound port. This is achieved
by recovering from the Convergence Event, such as restoring
the failed link. Restoration Convergence Time is measured
using the Rate-Derived Convergence Time calculation technique,
as provided in Equation 1. It is possible to have the
Restoration Convergence Time differ from the Rate-Derived
Convergence Time.
3.14 Packet Sampling Interval
The interval at which the tester (test equipment) polls to make
measurements for arriving packet flows.
Metrics measured at the Packet Sampling Interval may include
Throughput and Convergence Packet Loss.
Packet Sampling Interval can influence the Convergence Graph.
This is particularly true when implementations achieve Full
Convergence in less than 1 second. The Convergence Event
Transition and Convergence Recovery Transition can become
exaggerated when the Packet Sampling Interval is too long.
This will produce a larger than actual Rate-Derived
Convergence Time. The recommended value for configuration
of the Packet Sampling Interval is provided in [2].
Poretsky, Imhoff [Page 10]
3.15 Local Interface
An interface on the DUT.
Neighbor Interface
Remote Interface
3.16 Neighbor Interface
The interface on the neighbor router or tester that is
directly linked to the DUT's Local Interface.
3.17 Remote Interface
An interface on a neighboring router that is not directly
connected to any interface on the DUT.
3.18 Preferred Egress Interface
The outbound interface from the DUT for traffic routed to the
preferred next-hop.
The Preferred Egress Interface is the egress interface prior
to a Convergence Event.
Next-Best Egress Interface
3.19 Next-Best Egress Interface
second-best next-hop. It is the same media type and link speed
as the Preferred Egress Interface
The Next-Best Egress Interface becomes the egress interface
after a Convergence Event.
Preferred Egress Interface
3.20 Stale Forwarding
Forwarding of traffic to route entries that no longer exist
or to route entries with next-hops that are no longer preferred.
Stale Forwarding can be caused by a Convergence Event and is
also known as a "black-hole" since it may produce packet loss.
Stale Forwarding exists until Network Convergence is achieved.
3.21 Nested Convergence Events
The occurence of Convergence Event while the route table
is converging from a prior Convergence Event.
The Convergence Events for a Nested Convergence Events
MUST occur with different neighbors. A common
observation from a Nested Convergence Event will be
the withdrawal of routes from one neighbor while the
routes of another neighbor are being installed.
This document requires no IANA considerations.
Documents of this type do not directly affect the security of
Internet or corporate networks as long as benchmarking
is not performed on devices or systems connected to production
6.1 Normative References
[1] Poretsky, S., "Benchmarking Applicability for IGP Data Plane
Route Convergence", draft-ietf-bmwg-igp-dataplane-conv-app-09,
work in progress, January 2006.
[2] Poretsky, S., "Benchmarking Methodology for IGP Data Plane
Route Convergence", draft-ietf-bmwg-igp-dataplane-conv-meth-09,
[3] Callon, R., "Use of OSI IS-IS for Routing in TCP/IP and Dual
Environments", RFC 1195, December 1990.
[4] Moy, J., "OSPF Version 2", RFC 2328, IETF, April 1998.
6.2 Informative References
[5] S. Casner, C. Alaettinoglu, and C. Kuan, "A Fine-Grained View
of High Performance Networking", NANOG 22, June 2001.
[6] L. Ciavattone, A. Morton, and G. Ramachandran, "Standardized
Active Measurements on a Tier 1 IP Backbone", IEEE
Communications Magazine, pp90-97, May 2003.
7. Author's Address
8 New England Executive Park
EMail: sporetsky@reefpoint.com
1194 North Mathilda Ave
Sunnyvale, CA 94089
EMail: bimhoff@planetspork.com
This document is subject to the rights, licenses and restrictions
contained in BCP 78, and except as set forth therein, the authors
retain all their rights.
this standard. Please address the information to the IETF at ietf-
ipr@ietf.org.
Poretsky, Imhoff [Page 15]
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Q: How to reduce the number of independent variables in mathematica I am not srue whether this is really a mathematical question, or actually a mathematica question. :D
suppose I have a matrix
{{4/13 + (9 w11)/13 + (6 w12)/13,
6/13 + (9 w21)/13 + (6 w22)/13}, {-(6/13) + (6 w11)/13 + (4 w12)/
13, -(9/13) + (6 w21)/13 + (4 w22)/13}}
with w11, w12, w21, w22 as free parameters.
And I know by visual inspection that 3*w11+2*w12 can be represented as one variable, and 3*w21+2*w22 can be represented as another. So essentially this matrix only has two independent variables. Given any matrix of this form, is there any method to automatically reduce the number of independent variables? I guess I am stuck at formulating this in a precise mathematical way.
Please share your thoughts. Many thanks.
Edit:
My question is really the following.
Given matrix like this
{{4/13 + (9 w11)/13 + (6 w12)/13,
6/13 + (9 w21)/13 + (6 w22)/13}, {-(6/13) + (6 w11)/13 + (4 w12)/
13, -(9/13) + (6 w21)/13 + (4 w22)/13}}
or involving some other symbolical constants
{{a+4/13 + (9 w11)/13 + (6 w12)/13,
6/13*c + (9 w21)/13 + (6 w22)/13}, {-(6/13)/d + (6 w11)/13 + (4 w12)/
13, -(9/13) + (6 w21)/13 + (4 w22)/13}}
I want to use mathematica to automatically identify the number n of independent variables (in this case is 2), and then name these independent varirables y1, y2, ..., yn, and then re-write the matrix in terms of y1, y2, ..., yn instead of w11, w12, w21, w22.
A: Starting with
mat = {{4/13 + (9 w11)/13 + (6 w12)/13,6/13 + (9 w21)/13 + (6 w22)/13},
{-(6/13) + (6 w11)/13 + (4 w12)/13, -(9/13) + (6 w21)/13 + (4 w22)/13}};
Form a second matrix, of indeterminates, same dimensions.
mat2 = Array[y, Dimensions[mat]];
Now consider the polynomial (actually linear) system formed by setting mat-mat2==0. We can eliminate the original variables and look for dependencies amongst the new ones. Could use Eliminate; I'll show with GroebnerBasis.
GroebnerBasis[Flatten[mat - mat2], Variables[mat2], Variables[mat]]
Out[59]= {-3 + 2 y[1, 2] - 3 y[2, 2], -2 + 2 y[1, 1] - 3 y[2, 1]}
So we get a pair of explicit relations between the original matrix elements.
---edit---
You can get expressions for the new variables that clearly indicates the dependency of two of them on the other two. To do this, form the Groebner basis and use it in polynomial reduction.
gb = GroebnerBasis[Flatten[mat - mat2], Variables[mat2], Variables[mat]];
vars = Flatten[mat2];
PolynomialReduce[vars, gb, vars][[All, 2]]
Out[278]= {1 + 3/2 y[2, 1], 3/2 + 3/2 y[2, 2], y[2, 1], y[2, 2]}
---end edit---
Daniel Lichtblau
Wolfram Research
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\section{Introduction}
In \cite{lind}, Lindestrauss and Weiss introduced the notion of metric mean dimension for any continuous map $\phi:X\rightarrow X$, where $X$ is a compact metric space with metric $d$. We will denote by $
\underline{\text{mdim}}_{\text{M}}(X,d,\phi)$ and $\overline{\text{mdim}}_{\text{M}}(X,d,\phi)$, respectively, the lower and upper metric mean dimension of $\phi:X\rightarrow X$. We have
\begin{equation}\label{ejfkfg}
\underline{\text{mdim}}_{\text{M}}(X,d,\phi)\leq \overline{\text{mdim}}_{\text{M}}(X,d,\phi).\end{equation}
\medskip
Denote by $h_{\text{top}}(\phi)$ the topological entropy of $\phi:X\rightarrow X $. We have if $\underline{\text{mdim}}_{\text{M}}(X,d,\phi)>0$, then $h_{\text{top}}(\phi)=\infty$.
Example \ref{equaltozero} proves there exist continuous maps $\phi:X\rightarrow X $ with infinite topological entropy and metric mean dimension equal to zero.
\medskip
Let $N$ be a compact Riemannian manifold with topological dimension $\dim(N)=n$. In \cite{Carvalho}, the authors proved if $n\geq 2$, then the set consisting of homeomorphisms on $N$ with upper metric mean dimension equal to $ n$ is residual in $ \text{Hom}(N)$. This fact is proved in \cite{JeoPMD} for $C^{0}(N)$ instead of $ \text{Hom}(N)$. On the other hand,
any Lipschitz continuous map defined on a finite dimensional compact metric space has finite entropy (see \cite{Katok}, Theorem 3.2.9), therefore, it has metric mean dimension equal to zero.
\medskip
For any $\alpha\in (0,1)$, we denote by $H^{\alpha}([0,1])$ the set consisting of $\alpha$-H\"older continuous maps on the interval $[0,1]$. Hazard in \cite{Hazard} proves there exist continuous maps on the interval with infinite entropy which are $\alpha$-H\"older for any $\alpha\in (0,1)$. However, the example showed by Hazard has zero metric mean dimension (see Example \ref{equaltozero}). In this work, we will show, for any $\alpha\in (0,1)$, there exists a $\phi\in H^{\alpha}([0,1])$ with $$
\underline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)= \overline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)=1-\alpha.$$
\medskip
In the next section, we will present the definition of metric mean dimension. In Section \ref{hoorseshoe}, we will show some results about the metric mean dimension for continuous maps with horseshoes and several examples. Although the inequality given in \eqref{ejfkfg} is clear, we do not know any reference in which is showed an explicit example of a continuous map on the interval such that the inequality is strict. We will construct this kind of examples in Section \ref{hoorseshoe}.
Furthermore, we prove for $a,b\in [0,1]$, with $a<b$, the set consisting of continuous maps $\phi:[0,1]\rightarrow [0,1]$ such that $$ \underline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\phi)=a\quad\text{and}\quad \overline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\phi)=b$$ is dense in
$C^{0}([0,1]).$ In Section \ref{section4} we show the existence of H\"older continuous maps with positive metric mean dimension. Finally, in the last section we will leave some conjectures that arise from this research.
\medskip
\section{Metric mean dimension for continuous maps}
Throughout this work, $X$ will be a compact metric space endowed with a metric $d$ and $\phi: X\rightarrow X$ a continuous map. For any
$n\in\mathbb{N}$, we define the metric $d_n:X\times X\to [0,\infty)$ by
$$
d_n(x,y)=\max\{d(x,y),d(\phi(x),\phi(y)),\dots,d(\phi^{n-1}(x),\phi^{n-1}(y))\}.
$$ \begin{definition} Fix $\varepsilon>0$.
\begin{itemize}\item We say that $A\subset X$ is an $(n,\phi,\varepsilon)$-\textit{separated} set
if $d_n(x,y)>\varepsilon$, for any two distinct points $x,y\in A$. We denote by $\emph{sep}(n,\phi,\varepsilon)$ the maximal cardinality of any $(n,\phi,\varepsilon)$-{separated}
subset of $X$.
\item We say that $E\subset X$ is an $(n,\phi,\varepsilon)$-\textit{spanning} set for $X$ if
for any $x\in X$ there exists $y\in E$ such that $d_n(x,y)<\varepsilon$. Let $\emph{span}(n,\phi,\varepsilon)$ be the minimum cardinality
of any $(n,\phi,\varepsilon)$-spanning subset of $X$. \end{itemize}\end{definition}
\begin{definition}
The \emph{topological entropy} of $(X,\phi,d)$ is defined by
\begin{equation*}\label{topent}h_{\emph{top}}(\phi)=\lim _{\varepsilon\to0} \emph{sep}(\phi,\varepsilon)=\lim_{\varepsilon\to0} \emph{span}(\phi,\varepsilon),
\end{equation*}
where $$\emph{sep}(\phi,\varepsilon)=\underset{n\to\infty}\limsup \frac{1}{n}\log \emph{sep}(n,\phi,\varepsilon)\quad\text{and}\quad\emph{span}(\phi,\varepsilon)=\underset{n\to\infty}\limsup \frac{1}{n}\log \emph{span}(n,\phi,\varepsilon).$$\end{definition}
\begin{definition}
We define the \emph{lower metric mean dimension} and the \emph{upper metric mean dimension} of $(X,d,\phi )$ by
\begin{equation*}\label{metric-mean}
\underline{\emph{mdim}}_{\emph{M}}(X,d,\phi)=\liminf_{\varepsilon\to0} \frac{\emph{sep}(\phi,\varepsilon)}{|\log \varepsilon|}\quad \text{ and }\quad\overline{\emph{mdim}}_{\emph{M}}(X,d,\phi)=\limsup_{\varepsilon\to0} \frac{\emph{sep}(\phi,\varepsilon)}{|\log \varepsilon|},
\end{equation*}
respectively. \end{definition}
We also have that
$$
\underline{\text{mdim}}_{\text{M}}(X,d,\phi )=\liminf_{\varepsilon\to0} \frac{\text{span}(\phi,\varepsilon)}{|\log \varepsilon|}
\quad\text{ and }\quad \overline{\text{mdim}}_{\text{M}}(X,d,\phi )=\limsup_{\varepsilon\to0} \frac{\text{span}(\phi,\varepsilon)}{|\log \varepsilon|}.
$$
\begin{remark}
Throughout the paper, we will omit the underline and the overline on the notations of the metric mean dimension when the result be valid for both cases.
\end{remark}
\begin{remark}\label{kjjj} Topological entropy does not depend on the metric $d$. However, the metric mean dimension depends on the metric (see \cite{lind}), therefore, it is not an invariant under topological conjugacy.
If $X=[0,1]$, we consider the metric $d(x,y)=|x-y|$, for every $x,y\in [0,1]$. We will denote this metric by $|\cdot|$.
\end{remark}
\begin{remark} For any continuous map $\phi:X\rightarrow X$, it is proved in \cite{VV} that
\begin{equation}\label{boundd}0\leq \overline{\emph{mdim}}_{\emph{M}}(X ,d,\phi) \leq \overline{\emph{dim}}_{\emph{B}} (X,d) \quad \text{ and }\quad 0\leq \underline{\emph{mdim}}_{\emph{M}}(X ,d,\phi) \leq \underline{\emph{dim}}_{\emph{B}} (X,d) ,\end{equation}
where $\underline{\emph{dim}}_{\emph{B}} (X,d) $ and $\overline{\emph{dim}}_{\emph{B}} (X,d) $ are respectively the lower and upper box dimension of $X$ with respect to $d$. \end{remark}
From the above remark we have if $N$ is an $n$-dimensional compact Riemannian manifold with Riemannian metric $d$ we have
\begin{equation*}\label{dbounddw} 0\leq {\text{mdim}}_{\text{M}}(N ,d,\phi) \leq n,\end{equation*}
for any continuous map $\phi:N\rightarrow N$. In particular, if $\phi: [0,1]\rightarrow [0,1]$ is a continuous map, we have
\begin{equation}\label{bounddw} 0\leq {\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\phi) \leq 1.\end{equation}
\section{Horseshoes and metric mean dimension}\label{hoorseshoe}
An $s$-\textit{horseshoe} for $\phi:[0,1]\rightarrow [0,1]$ is an interval $J=[a,b]\subseteq [0,1]$ which has a partition into $s$ subintervals $J_{1},\dots,J_{s}$, such that $J_{j}^{\circ}\cap J_{i}^{\circ}=\emptyset$ for $i\neq j$ and $J\subseteq \phi (\overline{J}_{i})$ for each $i=1,\dots, s$ (in Figure \ref{Ima222} we show a 3-horseshoe). The subintervals $J_{i}$ will be called \textit{legs} of the horseshoe $J$ and the length $|J|:=b-a$ is its \textit{size}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.3\textwidth]{Piernas.png}
\caption{$J$ is an $3$-horseshoe}
\label{Ima222}
\end{figure}
\medskip
Misiurewicz in \cite{Misiurewicz}, proved if $\phi:[0,1]\rightarrow [0,1]$ is a continuous map with $h_{\text{top}}(\phi)>0$, then there
exist sequences of positive integers $k_{n}$ and $s_{n}$ such that, for each $n$, $\phi^{k_{n}}$ has
an $s_{n}$- horseshoe and
$$ h_{\text{top}}(\phi)= \lim_{n\rightarrow \infty}\frac{1}{k_{n}}\log s_{n}. $$
For metric mean dimension, in \cite{VV}, Lemma 6, is proved that:
\begin{lemma}\label{lema1} Suppose that $I_{k}=[a_{k-1},a_{k}]\subseteq [0,1]$ is an $s_{k}$-horseshoe for $\phi:[0,1]\rightarrow [0,1]$ consisting of $s_{k}$ subintervals with the same length $I_{k}^{1},\dots, I_{k}^{s_{k}}$. Setting $\varepsilon_{k}=\frac{|I_{k}|}{s_{k}}$,
we have
$$ \emph{sep}( \phi , \varepsilon_{k})\geq \log(s_{k}/2). $$
\end{lemma}
The above lemma provides a lower bound for the \textit{upper} metric mean dimension of a continuous map $\phi:[0,1]\rightarrow [0,1]$ with a sequence of horseshoes, since, with the conditions presented, we can prove that \begin{equation}\label{feff}\overline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)\geq \limsup_{k\rightarrow \infty}\frac{\log s_{k}}{|\log\varepsilon_{k}|}= \underset{k\rightarrow \infty}{\limsup}\frac{ 1}{\left|1-\frac{\log |I_{k}|}{\log s_{k}}\right|}, \end{equation} as can be seen in the proof of Proposition 8 from \cite{VV}. In order to obtain the exact value of the (lower and upper) metric mean dimension of a continuous map on the interval, we must be carefully choosing both the number and the size of the legs of the horseshoes. Inspired by the examples presented in \cite{Kolyada}, in \cite{JeoPMD} was proved the next result, which, together with the Lemma \ref{lema1}, will give us examples of continuous maps on the interval with metric mean dimension equal to a fixed value (see Examples \ref{med1}, \ref{med12} and \ref{1example}).
\begin{theorem}\label{misiu}
Suppose for each $k\in\mathbb{N}$ there exists a $s_{k}$-horseshoe for $\phi\in C^{0}([0,1])$, $I_{k}=[a_{k-1},a_{k}]\subseteq [0,1]$, consisting of sub-intervals with the same length $I_{k}^{1}, I_{k}^{2},\dots, I_{k}^{s_{k}} $ and $[0,1]=\cup_{k=1}^{\infty}I_{k}$.
We can rearrange the intervals and suppose that $2\leq s_{k}\leq s_{k+1}$ for each $k$. If each $\phi|_{I_{k}^{i}}:I_{k}^{i}\rightarrow I_{k} $ is a bijective affine map for all $k$ and $i=1,\dots, s_{k}$, we have \begin{enumerate}[i.]
\item $ {\underline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\phi) \leq \underset{k\rightarrow \infty}{\liminf}\frac{1}{\left|1-\frac{\log |I_{k}|}{\log s_{k}}\right|}. $
\item If the limit $\underset{k\rightarrow \infty}{\lim}\frac{1}{\left|1-\frac{\log |I_{k}|}{\log s_{k}}\right|}$ exists, then $\overline{\emph{mdim}}_{\emph{M}}([0,1] ,|\cdot |,\phi) = \underset{k\rightarrow \infty}{\lim}\frac{1}{\left|1-\frac{\log |I_{k}|}{\log s_{k}}\right|}.$ \end{enumerate}
\end{theorem}
In Figure \ref{Ima22} we present the graph of a continuous map $\phi:[0,1]\rightarrow [0,1]$ such that each $I_{k}$ is an $3^{k}$-horseshoe for $\phi$. Note that in Theorem \ref{misiu},i. is presented an upper bound for the \textit{lower} metric mean dimension and in ii. a condition to obtain the exact value of the \textit{upper} metric mean dimension for a certain class of continuous maps on the interval.
\begin{figure}[ht]
\centering
\subfigure[Each $I_{k}$ is an $3^{k}$-horseshoe]{\includegraphics[width=0.34\textwidth]{Ima22.png} \label{Ima22}}\quad \subfigure[Each $I_{k}$ is an $3^{k}$-horseshoe]{\includegraphics[width=0.34\textwidth]{Ima223.png}\label{Ima223}}
\caption{$s$-horseshoes}
\end{figure}
In the next proposition we show a lower bound for the \textit{lower} metric mean dimension of a continuous map on the interval satisfying the conditions in Theorem \ref{misiu}.
\begin{proposition}\label{ddf}
If $\phi:[0,1]\rightarrow [0,1]$ satisfies the conditions in Theorem \ref{misiu}, we have \begin{equation}\label{fefeesc} {\underline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\phi) \geq \underset{k\rightarrow \infty}{\liminf}\frac{\frac{\log s_{k-1}}{\log s_{k}}}{1 - \frac{\log |I_{k}|}{\log s_{k}}} . \end{equation}
If $\underset{k\rightarrow \infty}{\liminf}\frac{\frac{\log s_{k-1}}{\log s_{k}}}{1 - \frac{\log |I_{k}|}{\log s_{k}}}= \underset{k\rightarrow \infty}{\liminf}\frac{1}{\left|1-\frac{\log |I_{k}|}{\log s_{k}}\right|},$ we have $$ {\underline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\phi) = \underset{k\rightarrow \infty}{\liminf}\frac{1}{\left|1 - \frac{\log |I_{k}|}{\log{ s_{k}}}\right|} . $$
\end{proposition}
\begin{proof}
Take any $\varepsilon\in (0,1)$. For any $k\geq 1$ set $ \varepsilon_k= \frac{|I_{k}|}{s_{k}} .$ There exists some $k\geq 1$ such that $\varepsilon \in [\varepsilon_{k}, \varepsilon_{k-1}]$. We have
\begin{equation*}\label{equ12ssswe} \text{sep}( n, \phi , \varepsilon)\geq \text{sep}( n, \phi , \varepsilon_{k-1})\geq \text{sep}( n, \phi |_{I_{k-1}}, \varepsilon_{k-1}), \quad\text{for any }n\geq 1, \end{equation*}
and thus $ \text{sep}( \phi , \varepsilon)\geq \text{sep}( \phi |_{I_{k-1}}, \varepsilon_{k-1}). $
Hence, from Lemma \ref{lema1}, it follows that
$
\text{sep}(\phi, \varepsilon)\geq \log \left(\frac{s_{k-1}}{2}\right) .$
Therefore, \begin{align*}\label{exxample12} {\underline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\phi) &= \liminf_{\varepsilon\rightarrow 0} \frac{\text{sep}(\phi ,\varepsilon )}{|\log {\varepsilon}|} \geq \liminf_{k\rightarrow \infty}\frac{ \log {s_{k-1}}}{|\log \varepsilon_{k}|}\\
&=\underset{k\rightarrow \infty}{\liminf}\frac{\log s_{k-1}}{\log s_{k} - \log |I_{k}|}=\underset{k\rightarrow \infty}{\liminf}\frac{\frac{\log s_{k-1}}{\log s_{k}}}{1 - \frac{\log |I_{k}|}{\log s_{k}}}.\end{align*}
The second part of the proposition follows from the above fact and Theorem \ref{misiu}.
\end{proof}
As we mentioned above, Lemma \ref{lema1} provides a lower bound for the upper metric mean dimension of a continuous map on the interval. For instances, in \cite{VV}, Proposition 8, is proved if $\varrho:[0,1]\rightarrow [0,1]$ satisfies the conditions in Lemma \ref{lema1}, with $s_{k}=k^{k}$ and $|I_{k}|=\frac{6}{\pi^{2} k^{2}}$ for any $k\in \mathbb{N}$, then \begin{equation}\label{mapavv}\overline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\varrho)=1.\end{equation} This fact is a consequence of \eqref{feff} and \eqref{bounddw}. Next, it follows from \eqref{fefeesc} that \begin{equation}\label{defeeee} {\underline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\varrho) \geq \underset{k\rightarrow \infty}{\liminf}\frac{\frac{\log s_{k-1}}{\log s_{k}}}{1 - \frac{\log |I_{k}|}{\log s_{k}}}=\underset{k\rightarrow \infty}{\liminf}\frac{\frac{\log (k-1)^{k-1}}{\log k^{k}}}{1 - \frac{\log |6/\pi^{2}k^{2}|}{\log k^{k}}}=1.\end{equation} Therefore, from \eqref{mapavv} and \eqref{defeeee} we have $$\overline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\varrho)=\underline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\varrho)=1.$$
We will obtain a continuous map $\varphi_{0,1}:[0,1]\rightarrow [0,1]$ such that $$ 0= \underline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\varphi_{0,1})< \overline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\varphi_{0,1})=1.$$
\begin{example}\label{med1} Set $a_{0}=0$ and $a_{n}= \sum_{i=1}^{n}\frac{6}{\pi^{2}i^{2}}$ for $n\geq 1$. Set $I_{n}:=[a_{n-1},a_{n}]$ for any $n\geq 1$. Let $\varphi_{0,1}\in C^{0}([0,1])$ be defined by \begin{equation*} \varphi_{0,1} (x) =\begin{cases}
T_{n^{n}}^{-1}\circ g^{n^{n}}\circ T_{n^{n}} & \text{if }x\in I_{n^{n}},\text{ for }n\geq 1\\
x & \text{ otherwise}
\end{cases},\end{equation*} where $T_{n}: I_{n}:=[a_{n},a_{n+1}] \rightarrow [0,1] $ is the unique increasing affine map from $I_{n}$ onto $[0,1]$ and $g(x)= |1-|3x-1||$ for any $x\in[0,1]$. Each $I_{n^{n}}$ is an $3^{n^{{n}}}$-horseshoe for $\varphi_{0,1}$ (see Figure \ref{Ima223}). It follows from Theorem \ref{misiu} that $$ {\overline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\varphi_{0,1}) = \underset{n\rightarrow \infty}{\lim}\frac{1}{\left|1-\frac{\log |I_{n^{n}}|}{\log 3^{n^{n}}}\right|} = \underset{n\rightarrow \infty}{\lim}\frac{1}{\left|1-\frac{\log \left|\frac{6}{\pi^{2}(n^{n})^{2}}\right|}{\log 3^{n^{n}}}\right|}=1. $$
Next, we prove that $$ {\underline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\varphi_{0,1}) = 0. $$
Firstly, given that $\varphi_{0,1}$ is the identity outside of $Y=\cup_{j=1}^{\infty}I_{j^{j}}$, we have
$$ {\underline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\varphi_{0,1})= {\underline{\emph{mdim}}_{\emph{M}}}(Y ,|\cdot |,\varphi_{0,1}|_{Y}). $$
Next, note that $\frac{\log 3^{(k-1)^{k-1}}}{\log 3^{k^{k}}}\rightarrow 0$ as $k\rightarrow \infty$. Hence, for any $\delta >0$ there exists $k_{0}\geq 1$ such that for any $k> k_{0}$ we have $\frac{\log 3^{(k-1)^{k-1}}}{\log 3^{k^{k}}}< \delta$. For any $k\geq k_{0}$, set $ \varepsilon_k= \frac{|I_{k^{k}}|}{3(3^{k^{k}})} =\frac{6}{3^{k^{k}+1}\pi^{2}k^{2k}} .$ It follows from Corollary 7.2, \cite{demelo}, page 165, for each $j=1, 2,\dots, {k-1}$, we have $$ \emph{span}(n,\varphi_{0,1} |_{I_{j^{j}}}, {\varepsilon_{k}})\leq \frac{(3^{j^{j}})^{n}}{\varepsilon_{k}}.$$ Hence, if $Y_{k}=\cup_{j=1}^{k-1}I_{j^{j}}$, for every $ n\geq1$ we have
\begin{align*} \emph{span}(n,\varphi_{0,1} |_{Y_{k}}, {\varepsilon_{k}}) & \leq \sum_{j=1}^{k-1} \frac{(3^{j^{j}})^{n}}{\varepsilon_{k}} \leq \sum_{j=1}^{k-1} \frac{(3^{(k-1)^{k-1}})^{n}}{\varepsilon_{k}} \leq (k-1) \frac{(3^{(k-1)^{k-1}})^{n}}{\varepsilon_{k}}. \end{align*}
Therefore,
\begin{align*} \limsup_{n\rightarrow \infty}\frac{\emph{span}(\varphi_{0,1} |_{Y_{k}} , {\varepsilon_{k}})}{n|\log {\varepsilon_{k}}|}&\leq \limsup_{n\rightarrow \infty}\frac{\log\left[(k-1) \frac{(3^{(k-1)^{k-1}})^{n}}{\varepsilon_{k}}\right]}{n[\log (3^{k^{k}+1}\pi^{2}k^{2k}/ 6)]} \leq \frac{\log 3^{(k-1)^{k-1}}}{\log 3^{k^{k}}}.
\end{align*}
This fact implies that for any $\delta >0$ we have $$ {\underline{\emph{mdim}}_{\emph{M}}}(Y ,|\cdot |,\varphi_{0,1}|_{Y})<\delta$$ and hence $$ {\underline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\varphi_{0,1})={\underline{\emph{mdim}}_{\emph{M}}}(Y ,|\cdot |,\varphi_{0,1}|_{Y})=0.$$
\end{example}
Fix $a\in [0,1]$ and let $\phi_{a}\in C^{0}([0,1])$ be such that $$ {\underline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\phi_{a})={\overline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\phi_{a})=a$$ (two different constructions of this kind of maps can be seen in \cite[Example 3.1]{JeoPMD} and \cite[Proposition 9.1]{Carvalho}). Set $ {\varphi}_{a,1}\in C^{0}([0,1])$ be defined by \begin{equation*} {\varphi}_{a,1} (x) =\begin{cases}
T_{1}^{-1}\circ \varphi_{0,1}\circ T_{1} & \text{if }x\in [0,\frac{1}{2}],\\
T_{2}^{-1}\circ \phi_{a}\circ T_{2} & \text{if }x\in [\frac{1}{2},1]
\end{cases},\end{equation*} where $T_{1}: [1,\frac{1}{2}] \rightarrow [0,1] $ and $T_{2}: [\frac{1}{2},1] \rightarrow [0,1] $ are, respectively, the unique increasing affine map from $[0,\frac{1}{2}]$ onto $[0,1]$ and from $[\frac{1}{2},1]$ onto $[0,1]$.
We have \begin{equation}\label{a1} {\underline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |, {\varphi}_{a,1})=a\quad\text{and}\quad {\overline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\varphi_{a,1})=1.\end{equation}
Next, for fixed $b \in (0,1)$, we present an example of a continuous map $\varphi_{0,b}\in C^{0}([0,1])$ such that \begin{equation*}\label{a1e2} {\underline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |, {\varphi}_{0,b})=0\quad\text{and}\quad {\overline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\varphi_{0,b})=b.\end{equation*}
\begin{example}\label{med12} Fix $r>0$ and set $ b=\frac{1}{r+1}$. Set $a_{0}=0$ and $a_{n}= \sum_{i=0}^{n-1}\frac{C}{3^{ir}}$ for $n\geq 1$, where $C=\frac{1}{\sum_{i=0}^{\infty}\frac{1}{3^{ir}}}= \frac{3^{r}-1}{3^{r}}$. Let $\varphi_{0,b}\in C^{0}([0,1])$ be defined by \begin{equation*} \varphi_{0,b} (x) =\begin{cases}
T_{n^{n}}^{-1}\circ g^{n^{n}}\circ T_{n^{n}} & \text{if }x\in I_{n^{n}},\text{ for }n\geq 1\\
x & \text{ otherwise}
\end{cases},\end{equation*} where $T_{n}: I_{n}:=[a_{n-1},a_{n}] \rightarrow [0,1] $ is the unique increasing affine map from $I_{n}$ onto $[0,1]$ and $g(x)= |1-|3x-1||$ for any $x\in[0,1]$. Each $I_{n^{n}}$ is a $3^{n^{{n}}}$-horseshoe for $\varphi_{0,b}$. It follows from Theorem \ref{misiu} that $$ {\overline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\varphi_{0,b}) = \underset{n\rightarrow \infty}{\lim}\frac{1}{\left|1-\frac{\log |I_{n^{n}}|}{\log 3^{n^{n}}}\right|} = \underset{n\rightarrow \infty}{\lim}\frac{1}{\left|1+\frac{\log (3^{n^{n}})^{r}}{\log 3^{n^{n}}}\right|}=\frac{1}{1+r}=b. $$
We can prove that $$ {\underline{\emph{mdim}}_{\emph{M}}}([0,1] ,|\cdot |,\varphi_{0,b}) = 0$$ analogously as in Example \ref{med1}.
\end{example}
For $a<b$, let $\phi_{a}\in C^{0}([0,1])$ be such that $$ {\underline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\phi_{a})={\overline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\phi_{a})=a.$$ Set $ {\varphi}_{a,b}\in C^{0}([0,1])$ be defined by \begin{equation*} {\varphi}_{a,b} (x) =\begin{cases}
T_{1}^{-1}\circ \varphi_{0,b}\circ T_{1} & \text{if }x\in [0,\frac{1}{2}],\\
T_{2}^{-1}\circ \phi_{a}\circ T_{2} & \text{if }x\in [\frac{1}{2},1]
\end{cases}.\end{equation*}
We have \begin{equation}\label{ab1} {\underline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |, {\varphi}_{a,b})=a\quad\text{and}\quad {\overline{\text{mdim}}_{\text{M}}}([0,1] ,|\cdot |,\varphi_{a,b})=b.\end{equation}
We will endow $C^{0}([0,1])$ with the metric
$$\hat{d}(\phi,\psi)=\max_{x\in [0,1]}|\phi(x)-\psi(x)|.
$$
In \cite{VV}, Proposition 9, is proved the set consisting of $\phi\in C^{0}([0,1])$ with $\overline{\text{mdim}}_{\text{M}}([0,1],|\cdot|,\phi)=1$ is dense in $ C^{0}([0,1])$. More generally, in \cite{Carvalho}, Theorem B, is proved for a fixed $a\in [0,1]$, the set consisting of continuous maps $\phi\in C^{0}([0,1])$ such that $$ \underline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\phi)= \overline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\phi)=a$$ is dense in
$C^{0}([0,1])$ (see also \cite{JeoPMD}, Theorem 4.1). Furthermore, we have:
\begin{theorem}
For $a,b\in [0,1]$, with $a<b$, the set $C_{a}^{b}([0,1])$ consisting of continuous maps $\phi:[0,1]\rightarrow [0,1]$ such that $$ \underline{\emph{mdim}}_{\emph{M}}([0,1] ,|\cdot |,\phi)=a\quad\text{and}\quad \overline{\emph{mdim}}_{\emph{M}}([0,1] ,|\cdot |,\phi)=b$$ is dense in
$C^{0}([0,1]).$
\end{theorem}
\begin{proof} From \eqref{a1} and \eqref{ab1} we have for any $a,b\in[0,1]$, with $a\leq b$, there exists $\varphi_{a,b}\in C_{a}^{b}([0,1])$.
Fix $\psi\in C^{0}([0,1])$ and take any $\varepsilon >0$. Given that $C^{1}([0,1])$ is dense in $C^{0}([0,1])$, we can assume that $\psi\in C^{1}([0,1])$ and therefore $$ \underline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\phi)= \overline{\text{mdim}}_{\text{M}}([0,1] ,|\cdot |,\phi)=0.$$
Let $p^{\ast}$ be a fixed point of $\psi$. Choose $\delta>0$ such that $|\psi(x)-\psi (p^{\ast})|<\varepsilon/2$ for any $x$ with $|x-p^{\ast}|<\delta$. Take $\varphi_{a,b}\in C_{a}^{b}([0,1])$. Set $J_{1}=[0,p^{\ast}]$, $J_{2}=[p^{\ast},p^{\ast}+\delta/2]$, $J_{3}=[p^{\ast}+\delta/2,p^{\ast}+\delta] $ and $J_{4}=[p^{\ast}+\delta,1]$. Take the continuous map $\psi_{a,b}:[0,1]\rightarrow [0,1]$ defined as $$
\psi_{a.b}(x)= \begin{cases}
\psi(x), & \text{ if }x\in J_{1}\cup J_{4}, \\
T_{2}^{-1}\varphi_{a,b}T_{2}(x), & \text{ if }x\in J_{2}, \\
T_{3}(x), & \text{ if }x\in J_{3},
\end{cases}
$$ where $T_{2}:J_{2}\rightarrow [0,1] $ is the affine map such that $T_{2}(p^{\ast})=0 $ and $T_{2}(p^{\ast}+\delta/2)=1 $, and $ T_{3}:J_{3}\rightarrow [ p^{\ast}+\delta/2, \psi(p^{\ast}+\delta)]$ is the affine map such that $ T_{3}(p^{\ast}+\delta/2)=p^{\ast}+\delta/2$ and $ T_{3} (p^{\ast}+\delta)=\psi(p^{\ast}+\delta)$. Note that $\hat{d}(\psi_{a,b},\psi)<\varepsilon.$ We have $$\underline{\text{mdim}}_\text{M}([0,1] ,|\cdot |,\psi_{a,b})
= \underline{\text{mdim}}_\text{M}(J_{2} ,|\cdot |,\varphi_{a,b})
=a$$
and $$ \overline{\text{mdim}}_\text{M}([0,1] ,|\cdot |,\psi_{a,b})
= \overline{\text{mdim}}_\text{M}(J_{2} ,|\cdot |,\varphi_{a,b})
=b,$$
which proves the theorem.
\end{proof}
\section{H\"older continuous maps with positive metric mean dimension}\label{section4}
We say that $\phi:[0,1]\rightarrow [0,1]$ is an $\alpha$-\textit{H\"older continuous map}, for $\alpha\in (0,1]$, if there exists $K>0$ such that
$$ \frac{|\phi(x)-\phi(y)|}{|x-y|^{\alpha}}\leq K\quad \text{for all }x, y\in [0,1],\text{ with }x\neq y. $$
If $\phi$ satisfies the above condition for $\alpha =1$, then $\phi $ is called a \textit{Lipschitz continuous map}.
\medskip
For $\alpha\in (0,1),$ $ H^{\alpha}([0,1])$ will denote the space of $\alpha$-H\"older continuous maps on $[0,1]$. $C^{1}([0,1])$ and $H^{1}([0,1])$ will denote respectively the space of $C^{1}$-maps and the space of Lipschitz continuous maps on $[0,1]$. We have $$C^{1}([0,1])\subset H^{1}([0,1])\subset H^{\beta}([0,1]) \subset H^{\alpha}([0,1]) \subset C^{0}([0,1]),\quad\text{ where }0<\alpha<\beta<1.$$
Next, suppose that $N$ is a compact Riemannian manifold with topological dimension equal to $n$ and Riemannian metric $d$. In \cite{JeoPMD}, Theorem 4.5, is proved the set consisting of continuous maps on $N$ with \textit{lower} and \textit{upper} metric mean dimension equal to a fixed $a\in [0,n]$ is dense in $C^{0}(N)$. Furthermore, in Theorem 4.6, is showed the set consisting of continuous maps on $N$ with \textit{upper} metric mean dimension equal to $n$ is residual in $C^{0}(N)$. If $n\geq 2$, in \cite{Carvalho}, Theorem A, is proved the set consisting of homeomorphisms on $N$ with \textit{upper} metric mean dimension equal to $n$ is residual in the set consisting of homeomorphisms on $N$. It is well known any homeomorphism on $[0,1]$ has zero entropy and therefore has zero metric mean dimension.
\medskip
Hence, it remains to show the existence of H\"older continuous maps on finite dimensional compact manifolds. In Theorem \ref{example-a} we will prove there exist H\"older continuous maps on the interval with positive metric mean dimension, with certain conditions of the H\"older exponent. In Conjectures A, B and C the authors leave three problems that can be objects of future studies.
\medskip
The next lemma, whose proof is straightforward and
left to the reader, will be useful in order to prove that some functions are H\"older continuous.
\begin{lemma}\label{hfnff} Let $\phi:[0,1]\rightarrow [0,1]$ be a continuous map such that $ \frac{|\phi(x)-\phi(y)|}{|\omega(x-y)|}\leq K$ for some $K>0$ and any $x\neq y\in [0,1]$, where $\omega(t)=-t\log(t)$ for any $ t>0$ and $\omega (t)=0$. Then $\phi$ is an $\alpha$-H\"older continuous map for any $\alpha\in (0,1)$.
\end{lemma}
\begin{definition} Let $\phi:[0,1]\rightarrow [0,1]$ be a continuous map. If there exists $K>0$ such that $ \frac{|\phi(x)-\phi(y)|}{|\omega(x-y)|}\leq K$ for any $x\neq y\in [0,1]$, we will say that $\phi$ has \textit{modulus of continuity} $\omega$.
\end{definition}
Next example, which was introduced in \cite{Hazard}, proves there exist continuous maps with infinite entropy and metric mean dimension equal to zero. That map is also $\alpha$-H\"older for any $\alpha\in (0,1)$. We will include the details of its construction for the sake of completeness.
\begin{example}\label{equaltozero} For $n\geq 1$, take $I_{n}=[2^{-n},2^{-n+1}]$. Note that $$|I_n|=2^{-n+1}-2^{-n}=2^{-n}(2-1)=2^{-n} \quad\text{for each }n.$$ Divide each interval $I_{n}$ into $2n+1$ sub-intervals with the same lenght, $I_{n}^{1}$, $\dots$, $I_{n}^{2n+1}$. For $k=1,3,\dots, 2n+1$, let $\phi|_{I_{n}^{k}} :I_{n} ^{k}\rightarrow I_{n} $ be the unique increasing affine map from $I_{n} ^{k}$ onto $I_{n}$ and for $k=2,4,\dots, 2{n}$, let $\phi|_{I_{n}^{k}} :I_{n} ^{k}\rightarrow I_{n} $ be the unique decreasing affine map from $I_{n} ^{k}$ onto $I_{n}$. Note that $\phi$ is a continuous map ($\phi(x)=x$ for any $x\in \partial I_{n}$) and each $I_{n}$ is a $(2n+1)$-horseshoe (see Figure \ref{Ima1}), therefore $h_{\emph{top}}(\phi)=\infty$. It follows from Theorem \ref{misiu} that $$\emph{mdim}_{\emph{M}}([0,1] ,| \cdot |,\phi)=0.$$ We will prove
$\phi$ is $\alpha$-H\"older for any $\alpha\in(0,1).$ We will consider the map $\omega $ defined in Lemma \ref{hfnff}. \begin{figure}[ht]
\centering
\includegraphics[width=0.28\textwidth]{Ima1.png}
\caption{Each $ I_{n}$ is a $(2n+1)$-horseshoe}
\label{Ima1}
\end{figure}
Let $x,y\in [0,1]$ be two distinct points. Then there exist $n\geq 1$ and $m\geq 1$ such that $x\in I_{n}$ and $y\in I_{m}$. Given that every $I_{k}$ is $\phi$-invariant, we have $\phi(x)\in I_{n}$ and $\phi(y)\in I_{m}$. We have the next cases:
\medskip
\noindent \textbf{Case} $x=0:$ We have $\phi(0)=0$ and hence $|\phi(0)-\phi(y)|=|\phi(y)|\leq 2^{-m+1}$. Furthermore, $$ 2^{-m}\leq|y|\leq2^{-m+1} \quad\text{ and thus }\quad \frac{1}{|y|}\leq 2^{m}\quad\text{and}\quad \frac{1}{\log\frac{1}{|y|}}\leq \frac{1}{\log 2^{m-1} } . $$ Hence, $$ \frac{ |\phi(y)|}{\omega(|y|)}<\frac{2^{-m+1}2^{m}}{\log 2^{m-1}} =\frac{2}{(m-1)\log 2}.$$
\noindent \textbf{Case} $n> m+1:$ In this case we have
$$ |\phi(x)-\phi(y)|\leq |2^{-n}-2^{-m+1}|< 2^{-m+1} \quad\text{and} \quad 2^{-m+1}>|x-y|>|2^{-n+1}-2^{-m}|>2^{-m} .$$
Therefore, $$ \frac{1}{|x-y|}<2^{m}\quad\text{and}\quad \frac{1}{\log\frac{1}{|x-y|}}< \frac{1}{\log 2^{m-1} } $$ and thus
$$ \frac{ |\phi(x)-\phi(y)|}{\omega(|x-y|)}<\frac{2^{-m+1}2^{m}}{\log 2^{m-1}} =\frac{2}{(m-1)\log 2}.$$
\noindent \textbf{Case} $n=m+1:$ Note that $I_m=[2^{-m}, 2^{-m+1}],$ $ I_{m+1}=[2^{-(m+1)}, 2^{-m}]$ and consider the sub-intervals \begin{align*}
I^1_{m}&=\biggl[2^{-m}, 2^{-m}+\frac{|I_{m}|}{2m+1}\biggr]=\biggl[2^{-m}, 2^{-m}+\frac{2^{-m}}{2m+1}\biggr]=\biggl[2^{-m}, \frac{2m+2}{2^m(2m+1)}\biggr]=[B_m, C_m]\subseteq I_{m}
\end{align*} and
\begin{align*}
I^{2m+1}_{m+1} &=\biggl[2^{-m}-\frac{1}{2^{m+1}(2m+3)},2^{-m}\biggr]=\biggl[\frac{4m+5}{2^{m+1}(2m+3)}, 2^{-m}\biggr]=[A_m, B_m]\subseteq I_{m+1}.
\end{align*}
For any $x\in I^{2m+1}_{m+1}$, we have
\begin{align*}
\phi (x)&= \frac{2^{-m}-2^{-(m+1)}}{2^{-m}-A_m}\ (x-2^{-m})+2^{-m}=\frac{2^{-m}-2^{-(m+1)}}{2^{-m}-\frac{4m+5}{2^{m+1}(2m+3)}}\ (x-2^{-m})+2^{-m}\\
&=(2m+3)(x-2^{-m})+2^{-m}.
\end{align*}
For any $x\in I^{1}_{m}$, we have
\begin{align*}
\phi (x)&= \frac{2^{-m+1}-2^{-m}}{C_m-2^{-m}} \ (x-2^{-m})+2^{-m}=(2m+1)(x-2^{-m})+2^{-m}.
\end{align*}
Hence, if $x\in I^{2m+1}_{m+1}$ and $y\in I^{1}_{m}$, we have \begin{align*}
|x-y|\leq C_m-A_m &=\frac{2m+2}{2^{m}(2m+1)}-\frac{4m+5}{2^{m+1}(2m+3)}\\
& =\frac{1}{2^m}\biggl[\frac{2m+1}{2m+1}+\frac{1}{2m+1}-\frac{2(2m+3)}{2(2m+3)}+\frac{1}{2(2m+3)}\biggr]\\
&=\frac{1}{2^m}\biggl[\frac{1}{2m+1}+\frac{1}{2(2m+3)}\biggr]\\
&\leq \frac{1}{2^m}\ \frac{2}{2m+1}=\frac{1}{2^{m-1}(2m+1)}
\end{align*} and, furthermore,
$$
|\phi(y)-\phi(x)|= (2m+1)(y-2^{-m})-(2m+3)(x-2^{-m})\leq (2m+3)(y-x).$$
Therefore,
\begin{align*}
\frac{|\phi(y)-\phi(x)|}{\omega(|x-y|)}& \leq \frac{(2m+3)(y-x)}{-|x-y| \log |x-y|}\leq \frac{2m+3}{\log (2^{m-1}(2m+1))}.
\end{align*}
Given that
$$\lim_{m\to \infty}\frac{2m+3}{(m-1)\log 2+ \log (2m+1) }=\frac{2}{\log 2},$$
we have
\begin{equation}\label{escfr}
\frac{|\phi(y)-\phi(x)|}{\omega(|x-y|)}\leq 3.
\end{equation}
If $x\in I^{k}_{m+1}$ and $y\in I^{j}_{m}$, then there are ${x}'\in I^{2m+1}_{m+1}$ and ${y}'\in I^{1}_{m}$ such that $\phi(x)= \phi({x}')$ and $\phi(y)= \phi({y}')$. We have
$$
|{x}'-{y}'|\leq |x-y|\leq C_m-A_m= \frac{1}{2^m}\ \frac{2}{2m+1}=\frac{1}{2^{m-1}(2m+1)}<e^{-1}, \quad\text{for all } \ m\geq 2.
$$ Since $\omega(t)$ is increasing on $[0, e^{-1}]$, it follows that
$ \omega (|{x}'-{y}'|)\leq \omega (|x-y|)$ and therefore, by \eqref{escfr}, we have
$$ \frac{|\phi(y)-\phi(x)|}{\omega(|y-x|)}\leq \frac{|\phi({y}')-\phi({x}')|}{\omega(|{y}'-{x}'|)}\leq 3.$$
Note also, if $m=1$, then $\phi|_{I_1\cup I_2}$ is a Lipschitz function. Hence, in particular, it has modulus of continuity $\omega $.
\medskip
\noindent \textbf{Case} $n=m$: In this case we have there exists $y^{\prime}$ in the same branch as $x$ such that $f(y^{\prime})=f(y)$ and $|x-y|\geq |x-y^{\prime}|$. Note that $$ |x-y^{\prime}|\leq \frac{|I_{n}|}{2n+1}=\frac{1}{2^{n}(2n+1)} \quad\text{and hence }\quad \frac{1}{\log(|x-y^{\prime}|^{-1})}\leq \frac{1}{\log(2^{n}(2n+1))} . $$ Therefore, \begin{align*}\frac{ |\phi(x)-\phi(y)|}{\omega(|x-y|)}& \leq \frac{ |\phi(x)-\phi(y^{\prime})|}{\omega(|x-y^{\prime}|)} = \frac{(2n+1)|x-y^{\prime}|}{|x-y^{\prime}|\log(|x-y^{\prime}|^{-1})} = \frac{2n+1}{\log(2^{n}(2n+1))}.
\end{align*}
In each case we have $\frac{ |\phi(x)-\phi(y)|}{\omega(|x-y|)}$ is bounded. Therefore, $\phi$ is an $\alpha$-H\"older continuous map for any $\alpha\in (0,1)$.
\end{example}
We recall the map $\varrho$ presented in \eqref{mapavv} has (lower and upper) metric mean dimension equal to 1. In that case, for any $k\geq 1$, $\varrho$ has a $k^{k}$-horseshoe with length equal to $ \frac{6}{\pi^{2}k^{2}}$. We can prove that $\varrho$ is not $\alpha$-H\"older for none $\alpha\in (0,1)$, because the number of legs of each horseshoe is so big, and therefore the slopes of the map on each subinterval increases very quickly. Next, we will present a path (depending of the length of the horseshoes) consisting of continuous maps such that each of them have less legs than $\varrho$, their metric mean dimension is equal to one, however, they are not $\alpha$-H\"older for none $\alpha\in (0,1)$.
\begin{example}\label{1example} Fix any $\beta\in (1,\infty)$ and take $g\in C^{0}([0,1])$, defined by $x\mapsto |1-|3x-1||$. Set $a_{n}=\sum_{k=1}^{n}\frac{C}{k^{\beta}}$ for $n\geq 1$, where $C=\frac{1}{\sum_{k=1}^{\infty} {k^{-\beta}}}$. For each $n\geq 2$, let
$T_{n}: I_{n}:=[a_{n-1},a_{n}] \rightarrow [0,1] $ be the unique increasing affine map from $I_{n}$ onto $[0,1]$.
Consider the map $\phi_{\beta}:[0,1]\rightarrow [0,1]$ defined by $$\phi_{\beta}|_{I_{n}}= T_{n}^{-1}\circ g^{{n}}\circ T_{n}\quad \text{ for each }n\geq 2.$$ Hence, each $I_{n}$, whose length is $\frac{C}{n^{\beta}}$, is a $3^{n}$-horseshoe for $\phi$. From Theorem \ref{misiu} and Proposition \ref{ddf} we have $$ \underline{\emph{mdim}}_{\emph{M}}([0,1] ,|\cdot |,\phi_{\beta})= \overline{\emph{mdim}}_{\emph{M}}([0,1] ,|\cdot |,\phi_{\beta})=1.$$
Hence, $\{\phi_{\beta}\}_{\beta\in(1,\infty)}$ is a path of continuous map with metric mean dimension equal to 1.
Next, we will prove that $\phi$ is not $\alpha$-H\"older for none $\alpha\in (0,1)$. Note that each $I_{n}$ can be divided into $3^{n}$ sub-intervals with the same length, $I_{n,1}, I_{n,2},\dots, I_{n,3^{{n}}}$ such that for any $j\in\{1,2\dots,3^{{n}}\}$ we have
$$\phi_{\beta}(x)=\begin{cases} 3^{{n}}(x-a_{n-1})-(j-1)(a_{n}-a_{n-1})+a_{n-1} & \text{ if }x\in I_{n, j} \text{ for } j \text{ odd }\\
-3^{{n}}(x-a_{n-1})+(j-1)(a_{n}-a_{n-1})+a_{n} & \text{ if }x\in I_{n, j} \text{ for } j \text{ even}.\end{cases}$$
Hence, for each $x,y\in I_{n,j}$, we have $$ |x-y|\leq {|I_{n,j}|}= \frac{|I_{n}|}{3^{{n}}}= \frac{C}{3^{{n}}[n^{\beta}]}\quad\text{and}\quad |\phi_{\beta}(x) -\phi_{\beta}(y)|=3^{{n}}|x-y|. $$
Therefore, if $x=a_{n-1}\in I_{n,1} $ and $y=a_{n-1}+\frac{a_{n}-a_{n-1}}{3^{{n}}} \in I_{n,1}$, we have
$$ \frac{|\phi_{\beta}(x)-\phi_{\beta}(y)|}{|x-y|^{\alpha}}= 3^{{n}}|x-y|^{1-\alpha}\leq \frac{3^{{n}}C^{1-\alpha}}{3^{(1-\alpha){n}}[n^{\beta}]^{(1-\alpha)}} = \frac{3^{\alpha {n}}C^{1-\alpha}}{[n^{\beta}]^{(1-\alpha)}} \rightarrow \infty $$ as $n\rightarrow \infty$. Thus, $\phi_{\beta}$ is not $\alpha$-H\"older.
\end{example}
In the above example, we can see for every ${\beta}$ and $\gamma$ in $(1,\infty)$, we have $\phi_{\beta}$ and $\phi_{\gamma}$ are topologically conjugate and they have the same metric mean dimension. As we mentioned in Remark
\ref{kjjj}, the metric mean dimension is not invariant under topological conjugacy. In \cite{JeoPMD}, Remark 3.2, is showed a path of continuous maps, $\{\phi_{\beta}\}_{\beta\in (0,\infty)}$, such that for every ${\beta}$ and $\gamma$ in $(0,\infty)$, $\phi_{\beta}$ and $\phi_{\gamma}$ are topologically conjugate, however, $$ \text{mdim}_{\text{M}}([0,1],|\cdot|,\phi_{\beta})\neq \text{mdim}_{\text{M}}([0,1],|\cdot|,\phi_{\gamma})\quad\text{if }\beta\neq \gamma. $$
\medskip
In the next theorem we prove the existence of H\"older continuous map on the interval which have positive metric mean dimension. Note for the map presented in Example \ref{equaltozero}, the increase of the number of legs, which is $2n+1$ for each $n\geq 1$, is very slow compared to the decrease in size of the horseshoes, which is $\frac{1}{2^{n}}$ for each $n\geq 1$. Therefore, that map has metric mean dimension equal to zero, from Theorem \ref{misiu} and Proposition \ref{ddf}. If we keep the same subintervals $I_{n}=[2^{-n},2^{-n+1}]$ and we wish to obtain a continuous map on $[0,1]$ with positive metric mean dimension, we must have $\underset{n\rightarrow \infty}{\limsup}\frac{\log 2^{n}}{\log s_{n}}<\infty$, where $s_{n}$ is the number of legs of the map on each $I_{n}$. This fact implies that $s_{n}$ must increase very quickly and therefore, the H\"older exponent of the map must be zero or very small. Hence, we must modify the size of the horseshoe in order to obtain a continuous map with positive metric mean dimension and a greater H\"older exponent: the number of legs must not increase so fast (otherwise the H\"older exponent decreases) and the size of the horseshoe must not decay too fast (otherwise the metric mean dimension decreases). In the next theorem, is presented an example where the size of the horseshoe is $\frac{C}{3^{nr}}$, for a fixed $r\in(0,\infty)$ and a constant $C>0$, and the number of legs is $3^{n}$, for any $n\in\mathbb{N}$. These sequences, $\frac{C}{3^{nr}}$ and $3^{n}$, are reasonably similar and we will prove that in this case the map has positive metric mean dimension and a considerable H\"older exponent.
\begin{theorem}\label{example-a} Fix $ a\in [0,1)$ and take $\alpha= 1-a$. There exists an $\alpha$-H\"older continuous map $\phi_{a}:[0,1]\rightarrow [0,1]$ such that $$ \underline{\emph{mdim}}_{\emph{M}}([0,1] ,| \cdot |,\phi_{a})=\overline{\emph{mdim}}_{\emph{M}}([0,1] ,| \cdot |,\phi_{a})=a. $$
\end{theorem}
\begin{proof} If $a=0$ we can take $\phi_{0}$ as the identity on $[0,1]$.
Fix $r>0$ and let $ a=\frac{1}{r+1}$. Set $a_{0}=0$ and $a_{n}= \sum_{i=0}^{n-1}\frac{C}{3^{ir}}$ for $n\geq 1$, where $C=\frac{1}{\sum_{i=0}^{\infty}\frac{1}{3^{ir}}}= \frac{3^{r}-1}{3^{r}}$. For each $n\geq 1$, set $I_{n}=[a_{n-1},a_{n}]$ and take
$T_{n}:I_{n}\rightarrow [0,1]$ and $g$ as in Example \ref{1example}. Set $\phi_{a}:[0,1]\rightarrow [0,1]$, given by $\phi_{\alpha}|_{I_{n}}= T_{n}^{-1}\circ g^{n}\circ T_{n}$ for any $n\geq 1$. It follows from Theorem \ref{misiu} that
$$ \underline{\text{mdim}}_{\text{M}}([0,1], | \cdot |,\phi_{a})= \overline{\text{mdim}}_{\text{M}}([0,1],| \cdot |,\phi_{a})=a .$$
We will prove that $\phi_{a}: [0,1] \to [0,1]$ is $\alpha$-H\"older with $\alpha = \frac{r}{r+1}=1-a.$
Let $x,y\in [0,1]$ be two distinct points. Thus, there exist $n\geq 1$ and $m\geq 1$ such that $x\in I_{n}$ and $y\in I_{m}$. Given that each $I_{k}$ is $\phi_{a}$-invariant, we have $\phi_{a}(x)\in I_{n}$ and $\phi_{a}(y)\in I_{m}$. We have the following cases:
\medskip
\noindent \textbf{Case} $x=0:$ Since $\phi_{a}(0)=0$, we have $$|\phi_{a}(0)-\phi_{a}(y)|=|\phi_{a}(y)|\leq \frac{1-3^{-mr}}{1-3^{-r}} .$$ Furthermore, $$ \frac{1-3^{(-m+1)r}}{1-3^{-r}} \leq|y|\leq \frac{1-3^{-mr}}{1-3^{-r}}\quad\text{ and thus }\quad \frac{1}{|y|}\leq \frac{1}{\frac{1-3^{(-m+1)r}}{1-3^{-r}}}\quad\text{and}\quad \frac{1}{\log\frac{1}{|y|}}\leq \frac{1}{\log \frac{1-3^{-mr}}{1-3^{-r}}}. $$ Hence, if $\omega $ is the map defined in Lemma \ref{hfnff}, we have $$ \frac{ |\phi(y)|}{\omega(|y|)}<\frac{\frac{1-3^{-mr}}{1-3^{-r}}}{\frac{1-3^{(-m+1)r}}{1-3^{-r}} \log \frac{1-3^{-mr}}{1-3^{-r}} } = \frac{1-3^{-mr}}{[1-3^{(-m+1)r}]\log \frac{1-3^{-mr}}{1-3^{-r}} } ,$$
which converges to $\frac{1}{\log \frac{1}{ 1-3^{-r}} }$. This fact implies that $\frac{ |\phi(y)|}{\omega(|y|)}$ is bounded.
\medskip
\noindent \textbf{Case} $n=m+k$, \textbf{with} $k>1$ \textbf{for a fixed} $m$: we know that
\[
|\phi_{a}(x)-\phi_{a}(y)|\leq \frac{3^{(1-m)r}-3^{-nr}}{1-3^{-r}}=\frac{3^{(n-m+1)r}-1}{3^{nr}(1-3^{-r})}\quad\text{ and }\quad
\frac{1}{|x-y|}\leq \frac{(1-3^{-r})3^{nr}}{3^{(n-m)r}-1},\]
hence
\begin{equation}\label{fbbg}
\frac{|\phi_{a}(x)-\phi_{a}(y)|}{|x-y|^{\alpha}}\leq \frac{3^{nr\alpha} (1-3^{-r})^{\alpha-1} [3^{(n-m+1)r}-1]}{3^{nr}[3^{(n-m)r}-1]^{\alpha}} ={\frac{3^{nr(\alpha-1)} (1-3^{-r})^{\alpha-1} [3^{(k+1)r}-1]}{(3^{kr}-1)^{\alpha}}}.
\end{equation}
\medskip \noindent Note that, if the sequence $\{k\}$ is bounded, then \eqref{fbbg} is convergent to $0$ as $n\to \infty$, since $\alpha-1<0$. Hence \eqref{fbbg} is bounded if $\{k\}$ is bounded. Therefore, we can assume that the sequence $\{k\}$ is unbounded. From \eqref{fbbg}, we write
\begin{align*}
\frac{|\phi_{a}(x)-\phi_{a}(y)|}{|x-y|^{\alpha}} &\leq 3^{nr(\alpha-1)} (1-3^{-r})^{\alpha-1}\ \frac{\frac{3^{(k+1)r}-1}{3^{kr\alpha}}}{\frac{(3^{kr}-1)^\alpha}{3^{kr\alpha}}}=3^{nr(\alpha-1)} (1-3^{-r})^{\alpha-1} \frac{(3^{kr+r-kr\alpha}-3^{-kr\alpha})}{\biggl(1-\frac{1}{3^{kr}}\biggr)^\alpha}.
\end{align*}
Since
$\underset{k\to \infty}{\lim}\biggl(1-\frac{1}{3^{kr}}\biggr)=1$ and $ \underset{k\to \infty}{\lim} 3^{-kr\alpha}=0,$
we have
\begin{align*}
\lim_{k\to \infty} 3^{nr(\alpha-1)} (1-3^{-r})^{\alpha-1} \ \frac{(3^{kr+r-kr\alpha}-3^{-kr\alpha})}{\biggl(1-\frac{1}{3^{kr}}\biggr)^\alpha}&=(1-3^{-r})^{\alpha-1} \lim_{k\to \infty} 3^{nr\alpha-nr+kr+r-kr\alpha}\\
&=(1-3^{-r})^{\alpha-1} \ 3^r\ \lim_{k\to \infty}3^{(n-k)r\alpha-(n-k)r}\\
&=(1-3^{-r})^{\alpha-1} \ 3^r\ \lim_{k\to \infty} 3^{(n-k)(\alpha-1)r}\\
&=(1-3^{-r})^{\alpha-1} \ 3^r\ \lim_{k\to \infty} 3^{m(\alpha-1)r}\\
&= (1-3^{-r})^{\alpha-1} 3^{m(\alpha-1)r+r}.
\end{align*}
Thus, in any case, $\frac{|\phi_{a}(x)-\phi_{a}(y)|}{|x-y|^{\alpha}}$ is bounded with $x\in I_n$, $y\in I_m$, $n>m+1$.
\medskip \noindent
\textbf{Case} $n=m+1$: we have $I_m=[a_{m-1}, a_m] ,$ $I_{m+1}=[a_m, a_{m+1}],$
$$I_m=I^{1}_m\cup \cdots \cup I^{3^m}_{m}; \ \ I_{m+1}=I^{1}_{m+1}\cup \cdots \cup I^{3^{m+1}}_{m+1} \ \ \text{and} \ \ I_m\cap I_{m+1}=\{a_{m}\},$$ where $|I_{m}^{j}|=\frac{|I_{m}|}{3^{m}}$ and $|I_{m+1}^{j}|=\frac{|I_{m+1}|}{3^{m+1}}$.
Suppose that $x\in I^{1}_{m+1}$ and $y\in I^{3^m}_{m}$.
Hence \begin{equation}\label{bcmvns}
|x-y|\leq a_m+\frac{|I_{m+1}|}{3^{m+1}}-\biggl(a_m-\frac{|I_{m}|}{3^{m}}\biggr)=\frac{|I_{m+1}|}{3^{m+1}}+\frac{|I_{m}|}{3^{m}}=\frac{C(1+3^{r+1})}{3^{m(r+1) +1}}.
\end{equation}
It is easy to check that
$$\phi _{a}(y)=3^m(y-a_m)+a_m\quad\text{and}\quad \phi _{a} (x)=3^{m+1}(x-a_m)+a_m.$$
Hence,
\begin{align*}
|\phi_{a}(x)-\phi_{a}(y)|&=|3^{m+1}(x-a_m)-3^{m}(y-a_m)|=3^{m+1}x-3^{m}y-(3^{m+1}-3^m)a_m\\
(y\leq a_{m} )\quad & \leq 3^{m+1}x-3^{m}y-(3^{m+1}-3^m)y= 3^{m+1}(x-y).
\end{align*}
Therefore, from \eqref{bcmvns} we have
\begin{align*}
\frac{|\phi_{a}(x)-\phi_{a}(y)|}{|x-y|^{\alpha}}&\leq 3^{m+1} |x-y|^{1-\alpha}\leq 3^{m+1}\biggl(\frac{C(1+3^{r+1})}{3^{m(r+1) +1}} \biggr)^{1-\alpha}= \frac{3^{m+1}(C(1+3^{r+1}))^{1-\alpha}}{3^{(m(r+1)+1)(1-\alpha)}}\\
&= 3^{m+1 +(m(r+1)+1)(\alpha-1)}(C(1+3^{r+1}))^{1-\alpha}=3^{m(\alpha(r+1)-r)}3^{\alpha}(C(1+3^{r+1}))^{1-\alpha},
\end{align*}
which is bounded if and only if
$\alpha(r+1)-r\leq 0,$ that is, if $\alpha\leq \frac{r}{1+r}.$
If $x\in I^{j}_{m+1}$ and $y\in I^{i}_{m}$ for some $j$ and $i$, then there are ${x}'\in I^{1}_{m+1}$ and ${y}'\in I^{3^m}_{m}$ such that $ \phi_a(x)= \phi_a({x}') $ and $ \phi(y)= \phi({y}').$ Moreover, $|x-y|\geq |{x}'-{y}'|$, which provides
\begin{align*}
| \phi_a(x)-\phi(y)| &=| \phi_a({x}')- \phi_a({y}')|\leq 3^{m(\alpha(r+1)-r)}3^{\alpha}(C(1+3^{r+1}))^{1-\alpha}|{x}'-{y}'|^{\alpha} \\
&\leq 3^{m(\alpha(r+1)-r)}3^{\alpha}(C(1+3^{r+1}))^{1-\alpha}|{x}-{y}|^{\alpha}.
\end{align*}
\noindent \textbf{Case} $n=m$: we have $\phi (x)=3^n(x-a_n)+a_n$ for any $x\in I^{3n}_{n}$. Hence, if $ x, y\in I^{3n}_{n}$, then
$
|\phi_{a}(x)-\phi_{a}(y)| =3^n |x-y|$ and therefore
\begin{align*}
\frac{|\phi_{a}(x)-\phi_{a}(y)|}{|x-y|^{\alpha}}&= 3^n |x-y|^{1-\alpha}\leq 3^n\ |I^{3n}_{n}|^{1-\alpha}=3^n\biggl(\frac{|I_n|}{3^n}\biggr)^{1-\alpha}=3^n \biggl(\frac{C}{3^{(n-1)r}3^n}\biggr)^{1-\alpha}\\
&=C^{1-\alpha}3^{n-(1-\alpha)(nr+n)}=C^{1-\alpha}3^{r-\alpha r} 3^{n( \alpha -r+\alpha r)},
\end{align*}
which is bounded if and only if
$ \alpha -r+\alpha r\leq 0$, that is, if and only if $ \alpha \leq \frac{r}{1+r} .$
We can conclude that $\phi_a$ is an $(1-a)$-H\"older continuous map. \end{proof}
Let $X$ be a compact metric space with metric $d$. If $\phi_{1},\dots,\phi_{n}\in C^{0}(X)$ and $$\overline{\text{mdim}}_{\text{M}}(X ,d,\phi_{k})=\underline{\text{mdim}}_{\text{M}}(X ,d,\phi_{k})=a_{k},\quad\text{for any }k=1,\dots,n,$$
we have \begin{equation*}\label{jme} \underline{\text{mdim}}_{\text{M}}(X\times \cdots\times X ,d^{n},\phi_{1}\times\cdots \times \phi_{n}) = \overline{\text{mdim}}_{\text{M}}(X\times \cdots\times X ,d^{n},\phi_{1}\times\cdots \times \phi_{n})= \sum_{k=1}^{n}a_{k},\end{equation*}
where \begin{equation*}\label{bnm}d^{n}((x_{1},\dots,x_{n}),(y_{1},\dots,y_{n}))=d(x_{1},y_{1}) +\cdots+ d (x_{n},y_{n}),\quad \text{ for }x_{1},\dots,x_{n},y_{1},\dots,y_{n}\in X.\end{equation*} (see \cite{JeoPMD}, Theorem 3.13, v). Hence, it follows from Theorem \ref{example-a} that:
\begin{corollary}
Fix $b\in [0,n]$ and $\alpha ={1-\frac{b}{n}}$. There exists $\psi_{b}\in H^{\alpha} ([0,1]^{n})$ such that $$\overline{\emph{mdim}}_{\emph{M}}([0,1]^{n} ,d^{n},\psi_{b})=\underline{\emph{mdim}}_{\emph{M}}([0,1]^{n} ,d^{n},\psi_{b})=b.$$
\end{corollary}
\begin{proof}
Take $\phi_{a}:[0,1]\rightarrow [0,1]$ defined in the proof of Theorem \ref{example-a}, where $a=\frac{b}{n}$, and set $\psi_{b}=\phi_{a}\times \cdots \times \phi_{a}:[0,1]^{n}\rightarrow[0,1]^{n}$. We have $$ \underline{\text{mdim}}_{\text{M}}([0,1]^{n} ,d^{n},\psi_{b}) = \overline{\text{mdim}}_{\text{M}}([0,1]^{n} ,d^{n},\psi_{b})= na=b.$$
Given that each $\phi_{a}$ is an $(1-\frac{b}{n})$-H\"older continuous map, we have $\psi_{b}$ is an $(1-\frac{b}{n})$-H\"older continuous map.
\end{proof}
\section{Some conjectures about this research}
Note in Theorem \ref{example-a} there exists a relationship between the H\"older exponent $\alpha$ and the metric mean dimension of a continuous map on the interval. We will present the next conjectures about this relationship which can be the subject of future research.
\begin{conjecturea}\label{conj1} If $\phi:[0,1]\rightarrow [0,1]$ is an $\alpha$-H\"older continuous map, then $$\emph{mdim}_{\emph{M}}([0,1],|\cdot |,\phi) \leq 1-\alpha.$$\end{conjecturea}
Note in Theorem \ref{example-a} we prove there exists an $\alpha$-H\"older continuous map $\phi:[0,1]\rightarrow [0,1]$ with $\text{mdim}_{\text{M}}([0,1],|\cdot|,\phi)= 1-\alpha $, where $\alpha\in (0,1)$. From \eqref{boundd} we have does not exist any continuous map on the interval with metric mean dimension biggest to 1.
\begin{conjectureb} There is no any $\alpha$-H\"older continuous map $\phi:[0,1]\rightarrow [0,1]$, with $\alpha>0$ and $$\emph{mdim}_{\emph{M}}([0,1],|\cdot |,\phi) = 1.$$\end{conjectureb}
Note if Conjecture A is true, then Conjecture B is a consequence of Conjecture A.
\begin{conjecturec}If $\phi:[0,1]\rightarrow [0,1]$ is an $\alpha$-H\"older continuous map for any $\alpha\in (0,1)$, then $$\emph{mdim}_{\emph{M}}([0,1],|\cdot |,\phi) = 0.$$\end{conjecturec}
Note if Conjecture A is true, then Conjecture C is a consequence of Conjecture A.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,478
|
A 40. Le Mans-i 24 órás versenyt 1972. június 10. és június 11. között rendezték meg.
Végeredmény
Nem értékelhető
Nem ért célba
Megjegyzések
Pole Pozíció - #14 Equipe Matra-Simca Shell - 3:42.02
Leggyorsabb kör - #8 Ecurie Bonnier Switzerland - 3:46.90
Táv - 4691.343 km
Átlag sebesség - 195.472 km/h
Források
https://web.archive.org/web/20090801025233/http://www.experiencelemans.com/en-us/dept_165.html
Le Mans-i 24 órás versenyek
Lemans
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,983
|
Q: Majority filter (minimum mapping unit) in ArcGIS Desktop with larger window size? I want to run the majority filter in ArcGIS Desktop with a window size of 4 or 5 i.e. greater than the default 3.
Is it possible, or is there some other command (or tool) which might help me achieve that?
A: Try using the Focal Statistics tool with a Majority rule. It allows you to set a variety of window shapes, sizes, etc.
A: A Focal Majority function does a very poor job at establishing a MMU. I would recommend using a sieve approach. This will provide an exact defined MMU.
I believe that GDAL has a sieve model and it is also available in our Gradient Metrics ArcGIS Toolbox. It is an easy procedure to implement. The ArcGIS steps for using sieve to establish a minimal mapping unit of 10 cells are, more or less, as follows:
MinCells = 10
tmp1 = RegionGroup(InRaster, "EIGHT", "WITHIN", "ADD_LINK", "")
query = "VALUE > " + minCells
tmp2 = ExtractByAttributes(tmp1, query)
outraster = Nibble(InRaster, tmp2)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 872
|
var socketio = require('socket.io');
module.exports = Sockets;
function Sockets(server) {
this.clients = {};
this.socketsOfClients = {};
this.server = server;
}
Sockets.prototype.init = function () {
var self = this;
this.io = socketio.listen(this.server);
this.io.sockets.on('connection', function (socket) {
socket.on('set username', function (userName) {
// Is this an existing user name?
if (self.clients[userName] === undefined) {
// Does not exist ... so, proceed
self.clients[userName] = socket.id;
self.socketsOfClients[socket.id] = userName;
self.userNameAvailable(socket.id, userName);
self.userJoined(userName);
} else
if (self.clients[userName] === socket.id) {
// Ignore for now
} else {
self.userNameAlreadyInUse(socket.id, userName);
}
});
socket.on('message', function (msg) {
var srcUser;
if (msg.inferSrcUser) {
// Infer user name based on the socket id
srcUser = self.socketsOfClients[socket.id];
} else {
srcUser = msg.source;
}
if (msg.target === "All") {
// broadcast
self.io.sockets.emit('message', {
"source": srcUser,
"message": msg.message,
"target": msg.target
});
} else {
// Look up the socket id
self.io.sockets.sockets[self.clients[msg.target]].emit('message', {
"source": srcUser,
"message": msg.message,
"target": msg.target
});
}
});
socket.on('disconnect', function () {
var uName = self.socketsOfClients[socket.id];
delete self.socketsOfClients[socket.id];
delete self.clients[uName];
// relay this message to all the clients
self.userLeft(uName);
});
});
};
Sockets.prototype.userJoined = function (uName) {
var self = this;
Object.keys(this.socketsOfClients).forEach(function (sId) {
self.io.sockets.sockets[sId].emit('userJoined', {
"userName": uName
});
});
}
Sockets.prototype.userLeft = function (uName) {
this.io.sockets.emit('userLeft', {
"userName": uName
});
}
Sockets.prototype.userNameAvailable = function (sId, uName) {
var self = this;
setTimeout(function () {
console.log('Sending welcome msg to ' + uName + ' at ' + sId);
self.io.sockets.sockets[sId].emit('welcome', {
"userName": uName,
"currentUsers": JSON.stringify(Object.keys(self.clients))
});
}, 500);
}
Sockets.prototype.userNameAlreadyInUse = function (sId, uName) {
var self = this;
setTimeout(function () {
self.io.sockets.sockets[sId].emit('error', {
"userNameInUse": true
});
}, 500);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,014
|
Britons confused by new technology
By Anna Lagerkvist (TechRadar ) 2007-01-31T00:00:00.3Z MP3 players
One in three still can't set the VCR
Over 60 per cent only ever use four features on their mobile phone
Many Britons think new technology and gadgets are too complicated, a new PayPal survey has shown. Over half of those questioned (53 per cent) said they don't bother with new technology, as techie jargon and new gadgets confuse them.
Even though video players are almost a thing of the past, one in three respondents admitted to still having trouble setting them. The situation with digital video recorders was even worse, with 77 per cent saying they don't know how.
Feature-heavy mobile phones are another tech blackspot. Over 60 per cent said they only use four features on their phones, and two in five didn't even know if their mobile phone had a camera.
Neil Edwards from PayPal said: "It's a worrying sign for Britain that so many of us are baffled and therefore turned off by technology. There's no hiding from technology, so burying your head in the sand won't make it go away. We all must embrace technology or risk becoming the tech illiterates of the world."
Of the 1,000 or so people surveyed, 70 per cent used a computer regularly, 75 per cent owned a mobile phone, and 77 per cent had a DVD player or recorder. A quarter (27 per cent) owned a portable music player.
However, these figures are higher for younger age groups. Of 16- to 24-year-olds, 93 per cent regularly use a computer, and 70 per cent use a digital music player player. Among 25- to34-year-olds, 36 per cent have an MP3 player, whereas only 11 per cent of over 45s do.
Check your technical knowledge on www.whatisyourtq.com .
See more MP3 players news
Best wireless earbuds: the best Bluetooth earbuds and earphones in 2019
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,107
|
package com.tle.web.qti.viewer.questions.renderer.interaction.unsupported;
import com.google.inject.assistedinject.Assisted;
import com.google.inject.assistedinject.AssistedInject;
import com.tle.web.qti.viewer.QtiViewerContext;
import com.tle.web.qti.viewer.questions.renderer.QtiNodeRenderer;
import com.tle.web.qti.viewer.questions.renderer.unsupported.UnsupportedQuestionException;
import com.tle.web.sections.render.SectionRenderable;
import uk.ac.ed.ph.jqtiplus.node.item.interaction.AssociateInteraction;
/** NOT SUPPORTED */
public class AssociateInteractionRenderer extends QtiNodeRenderer {
@AssistedInject
public AssociateInteractionRenderer(
@Assisted AssociateInteraction model, @Assisted QtiViewerContext context) {
super(model, context);
}
@Override
protected SectionRenderable createTopRenderable() {
throw new UnsupportedQuestionException("associateInteraction");
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,492
|
Q: How to load image if the image is available on AWS? I upload image from vue, to server. The server will return the image url, but it's will process the image first, and then upload it to AWS. And the process may takes about 1 - 3 second untill the image fully uploaded to AWS
The problem is my vue already get the image url before the image fully upload, and try to load it in <img>, and an error happen, 404 not found. Because it's not fully uploaded to AWS.
Question:
How to automatically load an image as soon as the image available on AWS using best practice / clean way?
For now, I use looping, and use @error, and check every 1 second. But I think it's not clean
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 450
|
Q: Alternative to DNS for internal servers I am setting up my first network at work. All the servers are running CentOS 6.3 and have statically assigned IP addresses.
For example:
server1 192.168.0.101
server2 192.168.0.102
I want to be able to type something at the command line like
$> ssh myuser@server1
The quickest solution is to edit the hosts file and after some research the preferred way seems to be running a DNS server.
I think both solutions mean that I have to manually edit files but I would like my internal servers to auto-discover over UDP.
I could write something that runs a UDP service and updates the hosts file with new servers as they come online / get removed, but it seems like such a simple thing that it must exist already and I don't want to reinvent (this potentially dangerous) wheel, does anyone know of software that already does this?
A: You could use ZeroConf: http://en.wikipedia.org/wiki/Multicast_DNS
Check avahi out: https://help.ubuntu.com/community/HowToZeroconf (ubuntu howto, should be easily adaptable to CentOS)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,619
|
\section{Introduction}
Despite of the increasing success of QCD in describing a large variety of phenomena, both in the perturbative
as well as in the non-perturbative regimes, some fundamental questions remain unsolved. Prominent examples are the
very nature and detailed properties of the strongly coupled quark gluon plasma which is the conjectured state of QCD
matter at temperatures comparable and larger than the QCD energy scale $\Lambda_{\mathrm{QCD}}$.
Furthermore, the nature and properties of the chiral and deconfinement phase transition as well as the position
of a conjectured critical point (CP) in the QCD phase diagram are among the still challenging issues \cite{Friman:2011zz}.
To answer these questions experimentally, a number of large scale experiments is currently running
(ALICE, ATLAS and CMS at the LHC, and STAR and PHENIX at RHIC),
planned (MPD at NICA) or under construction (CBM and HADES at FAIR).
On the theory side, lattice QCD yields a smooth
crossover from the hadronic phase to the quark-gluon phase for small chemical potential at temperatures of about
$150\MeV$. At sufficiently large net baryon density the crossover may turn into a first order phase transition
at a CP.
At non-zero net densities (\ie non-zero quark chemical potential) there is no first-principle approach to the phase diagram.
Therefore, one has to rely on effective models or on
truncation schemes. Nevertheless, many of these approaches seem to point to a first order phase transition
connected to the spontaneous breaking of chiral symmetry at densities a few times the nuclear density.
The end point of this transition line has interesting properties on its own.
From macroscopic examples, the phenomenon of critical opalescence, \ie the diverging
scattering strength of transparent media in the vicinity of a critical point, has been known for a long time
\cite{Smoluchowski:1908}. Quite common also is the phenomenon of critical slowing down, \ie the diverging relaxation
time at criticality \cite{Hohenberg:1977ym}. These two examples as well as most of the special properties of critical points
have their reason in the diverging correlation length making the system scale free.
To understand the mass generation connected to chiral symmetry breaking, several effective models have been constructed
with the Nambu-Jona-Lasinio (NJL) model \cites{Klevansky:1992qe,Asakawa:1989bq} and the
linear sigma model (L$\sigma$M) \cites{Bochkarev:1995gi, Jungnickel:1995fp} being the most prominent ones.
Our present investigation is motivated by the question whether penetrating probes reflect directly the phase structure
of strongly interacting matter. We focus here on real photons and select the $\LSM$ to mimic the above
anticipated phase structure. The $\LSM$ contains quark and meson (pion and sigma) fields as basic degrees of freedom, where the fluctuations
of the latter ones are accounted for in linear approximation, as in \cites{Mocsy:2004ab,Bowman:2008kc,Ferroni:2010ct} and
the photon field is minimally coupled to the strongly interacting components of the $\LSM$.
There is a large difference in the time scales concerning the strong and the electromagnetic interactions, respectively.
This makes possible separating the two interactions involved. The strong interaction is responsible for the relaxation
towards a local thermal equilibrium as well as to the mass generation via the spontaneous breaking of chiral symmetry.
The electromagnetic interaction with a perturbative radiation field contributes little to this,
because its effects are $ \Ord{\alpha_{\mathrm{em}}/\alpha_s }$
suppressed. Therefore we might calculate the thermodynamics without regarding electromagnetism and use the thermodynamic
properties as well as the effective masses of the dressed quarks and mesons later on in the photon emission calculations.
\section{Thermodynamics of the L$\sigma$M with linearized meson fluctuations}
The $\LSM$ is a widely used effective model of QCD and has been applied often
for studying various aspects of thermodynamics of strongly interacting matter.
It was suggested by Gell-Mann and Levy in 1960 \cite{GellMann:1960np} for
studying chiral symmetry breaking. In absence of an explicit symmetry breaking term, the model has a
\mbox{$SU(2)\times SU(2)\simeq O(4)$} symmetry and therefore belonging to the same
universality class as $N_f=2$ QCD in the chiral limit \cite{Pisarski:1983ms}. This symmetry present
at high temperatures is spontaneously broken to a residual $SU(2)$ symmetry with the three pseudoscalar $\pi$ mesons
being the Goldstone modes. Breaking chiral symmetry explicitly, the pions acquire non-zero masses. Besides these
satisfying properties
there is a close connection to the non-linear $\sigma$ model, which in turn is equivalent to leading order
chiral effective field theory of QCD. Compared to the NJL model the $\LSM$ has the advantage of including the mesons
directly as dynamic field quanta, making it easier to address their properties.
The $\LSM$ Lagrangian reads
\begin{eqnarray}
\LLSM &=& \bar\psi(i\gamma^\nu\partial_\nu - g(\sigma + i\gamma^5\vec\tau\vec \pi))\psi
+ \frac12\partial_\rho \sigma \partial^\rho \sigma + \frac12\partial_\kappa\vec\pi\partial^\kappa \vec\pi
+ \frac{\lambda}{4}(\sigma^2 + \vec\pi^2 - v^2)^2 - H \sigma,
\end{eqnarray}
where the Dirac field $\psi$ describes a doublet of quarks,
$\sigma$ corresponds to an iso-scalar and Lorentz-scalar field,
and $\vec{\pi}$ describes an iso-vector and Lorentz-pseudoscalar field, the latter ones conveniently interpreted as the
$\sigma$ and $\pi$ mesons.
From the Lagrangian the thermodynamic
potential $\Omega$ is constructed via the path integral of the exponential of the Euclidean action
and evaluated following the procedure described
in \cites{Mocsy:2004ab,Bowman:2008kc,Ferroni:2010ct} for including linearized fluctuations.
First, one integrates over the fermionic fields $\psi$ and $\bar\psi$.
The remaining path integral corresponds to a purely mesonic theory with a complicated interaction potential, which
is approximated by a quadratic one to account for small fluctuations.
The parameters of this quadratic potential are identified with the masses and thermodynamic averages of the
meson fields. This leads to self consistency relations for the masses.
The parameters are fixed by the following requirements:
The mass of the pions is set to $138\MeV$ in vacuum
($T=\mu=0$) and the sigma meson mass to $700\MeV$. The effective quark mass in the vacuum
is fixed to one third of the nucleon mass $m_{\mathrm{eff}}^0 = g v = 312\MeV$, and the parameter $v$ is
identified with the pion decay constant in vacuum, $v=92.4\MeV$.
With these parameters one obtains the results depicted in Fig.~\ref{fig_Thermodyn}.
\begin{figure}[htp]
\centering
\subfigure{\includegraphics[width = 0.47\textwidth,clip=true,trim=7mm 12mm 30mm 25mm]
{Mesonmassen_pioncolor_LinFluk.eps}
\label{subfig_m_pi_LF}
\put(-191, 30){\fcolorbox{black}{white}{(a)}}
\subfigure{\includegraphics[width = 0.47\textwidth,clip=true,trim=7mm 12mm 30mm 25mm]
{Mesonmassen_sigmacolor_LinFluk.eps}
\label{subfig_m_si_LF}
\put(-191, 30){\fcolorbox{black}{white}{(b)}}}\\
\subfigure{\includegraphics[width = 0.47\textwidth,clip=true,trim=7mm 12mm 30mm 25mm]
{Quarkmasse_LinFluk.eps}
\label{subfig_m_q_LF}
\put(-191, 30){\fcolorbox{black}{white}{(c)}}
\subfigure{\includegraphics[width = 0.47\textwidth,clip=true,trim=7mm 12mm 30mm 25mm]
{Suszeptibitaet_mumu_color_LinFluk.eps}
\label{subfig_chi_LF}
\put(-191, 30){\fcolorbox{black}{white}{(d)}}
}
\caption{Contour plots of dynamically generated masses for the pion (a) and sigma (b) mesons as well as
the quarks (c) (in MeV).
The quark number susceptibility (d), normalized to the susceptibility of a ideal massless Fermi-gas,
is increased around the CP (white circle).
The solid white curve denotes the coexistence curve for the 1st order phase transition
and the dashed line estimates the pseudocritical temperature. The later one is defined as the
the position of the extremal normalized heat capacity $\bar c = -c_0^{-1}\partial^2_T \Omega(\mu,T)$
with $-c_0$ being the second derivative w.r.t. temperature of the grand canonical potential $\Omega_0$
of an ultrarelativistic free Fermi gas.
}
\label{fig_Thermodyn}
\end{figure}
Figs.~\ref{subfig_m_pi_LF}-\ref{subfig_m_q_LF} show contour plots of the masses over the phase diagram.
One notes that the pion mass (Fig.~\ref{subfig_m_pi_LF}) increases with temperature and chemical potential
with the strongest
change at the phase boundary. The sigma meson mass (Fig.~\ref{subfig_m_si_LF}) on
the other hand exhibits a valley of low mass values around
the phase boundary and with a global minimum at the critical point. The quark mass plotted in Fig.~\ref{subfig_m_q_LF}
drops from its vacuum value to about $30\MeV$. The most drastic change, again, is at the phase boundary,
signaling that the mechanism for mass generation is indeed the spontaneous breaking
of chiral symmetry within the $\LSM$. Because the chiral symmetry is also explicitly
broken by a nonzero $H$ in the Lagrangian, the quark mass does not drop to zero, but stays finite in the high
temperature phase.
Comparing the meson masses (\cf Figs.~\ref{subfig_m_pi_LF} and \ref{subfig_m_si_LF}), one realizes that they are
degenerate above the 1st order phase transition curve and the crossover region,
respectively, but very different below. This behavior of the mass difference of these chiral partners is another
sign of the chiral symmetry breaking and restoration.
For quantifying the size of the critical region the quark number susceptibility
\mbox{$\chi:=-\partial^2\Omega/\partial \mu^2$}
is chosen, since the susceptibility scales with the correlation length whose divergence causes many of the special features
of a CP. In Fig.~\ref{subfig_chi_LF}, $\chi$ is normalized to the susceptibility $\chi_0$
of a massless ideal fermion gas to scale out trivial contributions.
\section{Photon emission rates within the L$\sigma$M}
For calculating photon emission rates, the $\LSM$ Lagrangian is extended by an electromagnetic
sector coupled minimally (\cf \cite{Mizher:2010zb}) to the strongly interacting part.
\begin{eqnarray}
\Lag_{\gamma\mathrm{L}\sigma\mathrm{M}}
&=& \LLSM + \Lag_\gamma + \Lag_{\mathrm{int}},\\
\Lag_{\mathrm{int}} &=& -eQ_f\bar\psi \slashed A \psi
+ \frac12 e^2 \pi^+\pi^-A^\nu A_\nu
+ \frac12 e A_\nu(\pi^-\partial^\nu\pi^+ + \pi^+\partial^\nu\pi^-),
\end{eqnarray}
where $\Lag_\gamma = -\frac14 F^{\mu\nu}F_{\mu\nu}$ is the free photon Lagrangian and $A^\mu$ denotes the photon field.
Photon emission rates are, in a kinetic theory approach, convolutions of squared matrix elements $|M|^2$ and
phase space distribution functions $f_\pm$, the latter ones explicitly depending on $T$ and $\mu$.
Superimposed are implicit $T$ and $\mu$ dependencies from the effective masses of the involved fields, as displayed in
\mbox{Figs.~\ref{subfig_m_pi_LF}-\ref{subfig_m_q_LF}}. Given the marked variations of these masses
one can expect an pronounced impact on the emission rates
\begin{eqnarray}
\omega \frac{d^7 N}{dx^4dk^3} = \frac{\mathcal{N}}{(2\pi)^5} \int \frac{dp^3}{2p^0}\int \frac{dq^3}{2q^0}\int \frac{dz^3}{2z^0}
|\mathcal{M}|^2 f_\pm(p^0)f_\pm(q^0)(1\mp f_\pm(z^0))\delta^{(4)}(p+q-z-k).\label{rate_allg}
\end{eqnarray}
Owing to the weakness of the electromagnetic interaction we restrict the calculations to
first order in the electromagnetic coupling. Since we expect to have captured the dominant part of the strong interaction
in the calculation of the thermodynamic potential and the effective masses, the residual interaction
is expected to be relatively weak.
Therefore we restrict our calculation to 1st order processes in the quark-meson coupling.
Within this approximation the contributing processes are the tree-level processes in the $s$, $t$ and $u$ channels.
In \eqref{rate_allg}, four of the nine integrations can be carried out exactly applying the delta distribution.
Another (angular) integration drops out by symmetry reasons, so one is left with four integrals, which have to be
executed numerically resulting in the rates depicted in Fig.~\ref{fig_rates}.
\begin{figure}[hbt]
\centering
\subfigure{\includegraphics[width=0.47\textwidth,clip=true,trim=7mm 12mm 23mm 25mm]{rate_qq_gp_omega=0010_color.eps}
\put(-60, 77){\fcolorbox{black}{white}{(a)}}
\label{subfig_rate_qq_gp_omega=0010}}\hfill
\subfigure{\includegraphics[width=0.47\textwidth,clip=true,trim=7mm 12mm 23mm 25mm]{rate_qp_gq_omega=0010_color.eps}
\put(-60, 77){\fcolorbox{black}{white}{(b)}}
\label{subfig_rate_qp_gq_omega=0010}}\\
\subfigure{\includegraphics[width=0.47\textwidth,clip=true,trim=7mm 12mm 23mm 25mm]{rate_qq_gs_omega=0010_color.eps}
\put(-60, 77){\fcolorbox{black}{white}{(c)}}
\label{subfig_rate_qq_gs_omega=0010}}\hfill
\subfigure{\includegraphics[width=0.47\textwidth,clip=true,trim=7mm 12mm 23mm 25mm]{rate_qs_gq_omega=0010_color.eps}
\put(-60, 77){\fcolorbox{black}{white}{(d)}}
\label{subfig_rate_qs_gq_omega=0010}}
\caption{Contour plots of the photon emissivity $\omega\frac{d^7N}{d^4x d^3k}$ in units of $\MeV^2$ as functions of
$T$ and $\mu$ at photon energies $\omega = 10\MeV$ for the processes
\mbox{$\overline{\psi} \psi\rightarrow \pi\gamma$ (a)},
\mbox{$\psi\pi\rightarrow \psi\gamma$ (b)},
\mbox{$\overline{\psi} \psi\rightarrow \sigma\gamma$ (c)}
and \mbox{$\psi\sigma\rightarrow \psi\gamma$ (d)}.
Phase contour definitions as in Fig.~\ref{fig_Thermodyn}}
\label{fig_rates}
\end{figure}
When photon energies $\omega$ are much larger than the respective masses, it is not expected
to see much of the details of the phase structure. Contrary, at lower energies there are huge differences in available
phase space and matrix elements squared leading to pronounced patterns which reflect phase diagram features, in particular
the effective masses. For this reason $\omega=10\MeV$ is chosen.
Figure~\ref{fig_rates} shows contour plots of the photon rates for the different contributing processes over the phase diagram.
In Figs.~\ref{subfig_rate_qp_gq_omega=0010} and \ref{subfig_rate_qs_gq_omega=0010}
we see an enhancement in the
crossover region and in Fig.~\ref{subfig_rate_qs_gq_omega=0010} a global maximum in the critical region.
In Figs.~\ref{subfig_rate_qq_gp_omega=0010} and ~\ref{subfig_rate_qq_gs_omega=0010}
one notices
large rates in the chirally restored phase and much less photon emission in the chirally broken phase, which in case of
\ref{subfig_rate_qq_gs_omega=0010} is superimposed by an island of enhanced rates for $T\sim100\MeV$ and $\mu\lesssim200\MeV$.
Figures \ref{subfig_rate_qq_gp_omega=0010} and \ref{subfig_rate_qp_gq_omega=0010} show photon rates from processes involving
pions. Pions exhibit a large mass difference between the two
phases, but contrary to the sigma meson whose mass has a global minimum at the CP
the pion mass does not show special features at this point. This leads to
a large difference in the emissivity between the phases but no features characteristic for the CP itself.
For the pion-involving Compton process (Fig.~\ref{subfig_rate_qp_gq_omega=0010}) there is an enhancement in the
crossover region. This is probably due to a combination of phase space
effects and the (comparatively) large probability for the internally propagating pion to get on-shell.
A better channel for obtaining signatures of a CP are sigma involving processes. This is expected, since the sigma meson
is precisely the mode getting massless at the CP making long range interactions possible
and thus driving the critical processes. Unfortunately, the inclusion
of linearized fluctuations increases the sigma mass, so it is not clear whether the endpoint of the 1st order phase transition
shows correctly the critical behavior. But linearizing fluctuations anyhow restricts to small fluctuations making it not adequate
very near the CP. Nevertheless the sigma mass drops to small values in the critical
region, which has a notable effect on the corresponding processes, \eg the excess of the photon rate in the critical region
in Fig.~\ref{subfig_rate_qs_gq_omega=0010}.
There is a large difference in the rates for the processes
under consideration, even between the corresponding Compton (Figs.~\ref{fig_rates}(b) and \ref{fig_rates}(d)) and annihilation
(Figs.~\ref{fig_rates}(a) and ~\ref{fig_rates}(c)) processes. These can be understood in terms
of available phase space in combination with thermal suppression. Within a Boltzmann approximation two of the remaining integrals
in \eqref{rate_allg} can be solved to obtain
\begin{eqnarray}
\omega \frac{d^7 N}{dx^4dk^3}
&\stackrel{\omega\ll m_i}{\sim}&\int \limits_{s_0}\frac{ds}{s-z^2}\int dt |M(s,t)|^2 \exp\{-(s-z^2)/(4\omega T)\}.
\end{eqnarray}
The difference between the minimal kinematically allowed value of the center of mass energy
$\sqrt{s_0} = \max\{m_1+m_2, m_3\}$
for the different processes, together with a small value of $\omega$
leads to the huge thermal suppression at small $T$ seen in Figs.~\ref{fig_rates} (a) and (d).
\section{Summary}
Focusing on soft-photon emission rates we demonstrate that some features of the phase diagram provided by the
linear sigma model are nicely mapped out. Being aware of some limitations, such as the restriction to linearized
fluctuations (\cf \cite{Tripolt:2013jra} for a proper account of fluctuations) and the need to implement more complete
rates in a model of space-time evolution of the matter, we hope that improved calculations can provide useful complementary
information on strongly interacting matter produced in the course of relativistic heavy-ion collisions at various energies,
system sizes and centralities.
\begin{bibdiv}
\begin{biblist}
\bib{Friman:2011zz}{book}{
author = {Friman, Bengt},
author = {Hohne, Claudia},
author = {Knoll, Jorn},
author = {Leupold, Stefan},
author = {Randrup, Jorgen},
author = {Rapp, Ralf},
author = {Senger, Peter},
editor = {Friman B et al},
title = {The CBM Physics Book: Compressed Baryonic Matter in
Laboratory Experiments},
publisher = {Springer},
address = {Berlin},
series = {Lect.~Notes~Phys.},
volume = {814},
pages = {1},
doi = {10.1007/978-3-642-13293-3},
year = {2011},
}
\bib{Smoluchowski:1908}{article}{
author = {von Smoluchowski M},
title = {Molekular-kinetische Theorie der Opaleszenz von Gasen im kritischen Zustande,
sowie einiger verwandter Erscheinungen},
journal = {Ann. Phys.},
volume = {330},
pages = {205},
doi = {10.1002/andp.19083300203},
year = {1908},
}
\bib{Hohenberg:1977ym}{article}{
author = {Hohenberg P C},
author = {Halperin B I},
title = {Theory of Dynamic Critical Phenomena},
journal = {Rev. Mod. Phys.},
volume = {49},
pages = {435},
doi = {10.1103/RevModPhys.49.435},
year = {1977},
}
\bib{Klevansky:1992qe}{article}{
author = {Klevansky S P},
title = {The Nambu-Jona-Lasinio model of quantum chromodynamics},
journal = {Rev. Mod. Phys.},
volume = {64},
pages = {649},
doi = {10.1103/RevModPhys.64.649},
year = {1992},
}
\bib{Asakawa:1989bq}{article}{
author = {Asakawa M},
author = {Yazaki K},
title = {Chiral Restoration at Finite Density and Temperature},
journal = {Nucl. Phys.},
volume = {A504},
pages = {668},
doi = {10.1016/0375-9474(89)90002-X},
year = {1989},
}
\bib{Bochkarev:1995gi}{article}{
author = {Bochkarev A},
author = {Kapusta J I},
title = {Chiral symmetry at finite temperature: Linear versus
nonlinear sigma models},
journal = {Phys. Rev.},
volume = {D54},
pages = {4066},
doi = {10.1103/PhysRevD.54.4066},
year = {1996},
eprint = {arXiv:hep-ph/9602405},
}
\bib{Jungnickel:1995fp}{article}{
author = {Jungnickel D U },
author = {Wetterich C},
title = {Effective action for the chiral quark-meson model},
journal = {Phys. Rev.},
volume = {D53},
pages = {5142},
doi = {10.1103/PhysRevD.53.5142},
year = {1996},
eprint = {arXiv:hep-ph/9505267},
}
\bib{Mocsy:2004ab}{article}{
author = {Mocsy A},
author = {Mishustin I N },
author = {Ellis P J},
title = {Role of fluctuations in the linear sigma model with
quarks},
journal = {Phys. Rev.},
volume = {C70},
pages = {015204},
doi = {10.1103/PhysRevC.70.015204},
year = {2004},
eprint = {arXiv:nucl-th/0402070},
}
\bib{Bowman:2008kc}{article}{
author = {Bowman E S},
author = {Kapusta J I},
title = {Critical Points in the Linear Sigma Model with Quarks},
journal = {Phys. Rev.},
volume = {C79},
pages = {015202},
doi = {10.1103/PhysRevC.79.015202},
year = {2009},
eprint = {arXiv:nucl-th/0810.0042},
}
\bib{Ferroni:2010ct}{article}{
author = {Ferroni L},
author = {Koch V},
author = {Pinto M B},
title = {Multiple Critical Points in Effective Quark Models},
journal = {Phys. Rev.},
volume = {C82},
pages = {055205},
doi = {10.1103/PhysRevC.82.055205},
year = {2010},
eprint = {arXiv:nucl-th/1007.4721},
}
\bib{GellMann:1960np}{article}{
author = {Gell-Mann M},
author = {Levy M},
title = {The axial vector current in beta decay},
journal = {Nuovo Cim.},
volume = {16},
pages = {705},
doi = {10.1007/BF02859738},
year = {1960},
}
\bib{Pisarski:1983ms}{article}{
author = {Pisarski R D},
author = {Wilczek F},
title = {Remarks on the Chiral Phase Transition in
Chromodynamics},
journal = {Phys. Rev.},
volume = {D29},
pages = {338},
doi = {10.1103/PhysRevD.29.338},
year = {1984},
}
\bib{Mizher:2010zb}{article}{
author = {Mizher A J},
author = {Chernodub M N},
author = {Fraga E S},
title = {Phase diagram of hot QCD in an external magnetic field:
possible splitting of deconfinement and chiral
transitions},
journal = {Phys. Rev.},
volume = {D82},
pages = {105016},
doi = {10.1103/PhysRevD.82.105016},
year = {2010},
eprint = {arXiv:hep-ph/1004.2712},
}
\bib{Tripolt:2013jra}{article}{
author = {Tripolt R-A},
author = {Strodthoff N},
author = {von Smekal L},
author = {Wambach J},
title = {Spectral Functions for the Quark-Meson Model Phase
Diagram from the Functional Renormalization Group},
journal = {Phys. Rev.},
volume = {D89},
pages = {034010},
doi = {10.1103/PhysRevD.89.034010},
year = {2014},
eprint = {arXiv:hep-ph/1311.0630},
}
\end{biblist}
\end{bibdiv}
\end{document}
|
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| 3,944
|
Tukwila businesses press city for relocation help after police station plans
Posted on Jan 8, 2018 by Goorish Wibneh
The city of Tukwila has selected this location for its voter-approved justice center — a police station and courthouse. (Image via the city of Tukwila.)
Tukwila businesses facing displacement because of the city's police station and city courthouse plans say the city needs to increase the help it's offering them — or some businesses owners will go under.
"I hate to say it but they'll go out of business without relocation [benefits]," said Simon Castle, paint shop manager of Heiser Body Co., one of the dozens of businesses at South 150th Street and Military Road that face displacement for a planned police station and municipal courthouse.
The city of Tukwila plans to acquire the property for the justice center via eminent domain, a process that allows the city to compel the landowners to sell their properties to the city at fair market value.
The process does not require compensation for business owners who lease the properties. Most of the affected Tukwila business owners — many of whom are immigrant entrepreneurs who have spent years in their locations — are renters.
Some of those businesses have organized to push the city to increase the relocation assistance it is offering. They met for the first time last month.
"The intent of the meeting was to figure a way that we can petition for relocation expenses," Castle said. "Since the city has indicated there will be no or only minimal relocation costs paid to the displaced businesses, we were advised that coming together as a group would be a more effective measure than going at it alone."
The group includes a 6-member "multicultural coalition" committee, a majority of which are people of color and a woman.
"The meeting went well… we did organize a small committee to represent the group," said Bayview Motor Club owner Tofeek Mauda.
Property value and rents are rising in Tukwila. Many small businesses will struggle to survive unless they get the city to fully reimburse displacement costs such as moving costs, downtime expense, finding new property and increased rent, Castle said.
Castle says the body shop employs 40 union-affiliated painters and that it would cost the business $2 million to move. He said in the current economic climate, it seems unlikely that they would be able to find another appropriate location within the Tukwila city limits.
"Without these expenses reimbursed, there are going to be many unique businesses that will have to close up due to the fact they can't afford to move, or can't afford to pay higher rents in this economic climate. As such, we have had to form this coalition to fight for the rights of all the business owners involved so that we can preserve our businesses that we have all worked so hard to build," Castle said.
The group has hired Pacific Public Affairs, a public relations firm, and at least two businesses are privately retaining law firms whose representatives attended the meeting for informational exchange. Castle said he hopes attorneys are not needed for the coalition and that Tukwila treats the small business fairly.
The group plans to submit a petition the mayor's office and city council members at the first Tukwila council meeting in January. There'll be two new council members, one of whom is an immigrant.
Castle said a coalition of business owners can mobilize their collective resources.
Despite public outcry during a public hearing in November, Tukwila City Council members unanimously voted in November in favor of allowing the mayor's office to use condemnation proceedings to acquire the land needed for a voter-approved a $77 million public safety bond measure that passed last year.
Castle said he and other business owners are still upset that they received two weeks notice for that meeting.
"As a business owner, I am disappointed that we, like every other business involved, received only two weeks notice prior to the 'rubber stamp' vote at the City Council meeting."
Other businesses owners have hired their own attorneys to push the city to help them with relocation expenses. Attorney Kinnon Williams, who represents Riverton Heights Grocery, also criticized Tukwila's notification process.
Williams said other public entities, including the city of Seattle and Sound Transit, have a practice of notifying businesses and property owners months — not weeks — in advance of a possible public project that might be built on their locations.
"Sound Transit, they're sending out letters years in advance, even when they don't know exactly which properties will be affected," Williams said. "The thinking is, 'How are we going to do this with the least amount of harm?' That really wasn't done in Tukwila."
But Tukwila officials say they are offering relocation assistance to the businesses — in the form of advice and other help. And officials are working on a formula to provide some monetary assistance, based on the business owners' needs, said Derek Speck, the city's Economic Development administrator.
"Currently, we offer listings of available properties for sale and lease," he said. "We also make introductions to specific property owners in the area who might have available sites."
Speck said the city is sympathetic to the business owners' plights and is doing what it can.
"They are going through a very significant disruption that affects their owners, their employees, and their customers," Speck said. "We will do the best with the resources we have… Although we are not required by law to provide assistance to the businesses, we want to do what we can afford."
Speck said the city is also trying to find a new location for the smaller affected businesses.
"We are researching if there is a way to partner with Forterra and Abu Bakr Islamic Center on their project to purchase and convert an old motel into commercial spaces for small businesses so that the affected businesses can move as a group into that location," Speck said.
Williams criticized the city's plans, saying that the state and federal government already have a standard, which includes paying for all of the business' moving expenses.
"It's about equity and it's about fairness," he said.
While Castle and other business owners plan to go before the city council this month to argue for increased help, Castle remains frustrated with the process so far.
"I am thoroughly disillusioned with the city of Tukwila," he said. "The fact that they can state that they are not paying relocation costs, despite the fact that it's the right thing to do and despite other city councils in the King County area that have done so."
Additional reporting by Venice Buhain.
Tags: gentrification, Tukwila, Tukwila immigrant businesses fight displacement.
Goorish Wibneh
I am a "community journalist". I write about education and politics. My other interests include critical thinking, technology and international affairs. I have also discovered my latent interest in art, lately. I am Aquarius but I don't believe in psychoanalysis or personality tests.
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|
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| 1,835
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Johan Helmich Roman (ur. 26 października 1694 w Sztokholmie, zm. 20 listopada 1758 w Haraldsmåla koło Kalmaru) – szwedzki kompozytor późnego baroku. Członek Królewskiej Szwedzkiej Akademii Nauk.
Życiorys
W Londynie uczyli go Johann Christoph Pepusch i Attilio Ariosti. Romana nazywano ojcem muzyki szwedzkiej lub "szwedzkim Händlem", zarówno z powodu jego wybitnych umiejętności kompozytorskim, jak i inspirowania się muzyką Händla. Roman był tak jak Händel zwolennikiem polifonii włoskiej, takiej jakiej hołdowali Domenico Scarlatti i Giovanni Battista Pergolesi.
Najsłynniejszym dziełem Romana jest Drottningholmsmusik z 1744, 24-częściowa suita, skomponowana z okazji ślubu szwedzkiego następcy tronu Adolfa Fryderyka z siostrą Fryderyka II Pruskiego, Ludwiką Ulryką (Lovisa Ulrika).
Roman był mistrzem wielu gatunków muzyki instrumentalnej i wirtuozem skrzypiec, dobrym oboistą i dyrygentem. Organizował w Sztokholmie pierwsze publiczne koncerty. W ostatnich latach życia podupadł na zdrowiu (m.in. miał problemy ze słuchem) i wyjechał na południe Szwecji. Zmarł na raka.
Profesor Ingmar Bengtsson skatalogował dzieła Romana używając numerów BeRI.
Dzieła
Muzyka świąteczna
Die "Golovin-Musik", BeRI 1
Die "Drottningholmsmusik", BeRI 2
Suite ur Drottningholmsmusiken (1744)
13 kantat koronacyjnych i świątecznych
Utwory orkiestrowe i instrumentalne
23 symfonie
6 uwertur
5 suit orkiestrowych
2 Concerti Grossi
5 koncertów skrzypcowych
l concerto per oboe d'amore, BeRI 53
17 Trio-Sonaten
XII Sonate a flauto traverso, violone e cembalo (tr. 1727, poświęconych Ulryce Eleonorze gewidmet.
1 Sonate für Flöte und Cembalo
Utwory na klawesyn i fortepian (u.a. 12 Suiten und 12 Sonaten)
Assaggio à violino solo, BeRI 301, tr. 1740)
Violinenduos
Utwory wokalne
Szwedzka Msza (für Solisten, Chor und Orchester)
Kantaty (Dixit, Jubilate, Oh Gott, wir loben Dich)
Hymny (min. Beati omnes)
Psalmy Dawidowe
ok. 80 sakralnych pieśni w języku szwedzkim i łacińskim.
Przypisy
Linki zewnętrzne
Johan Helmich Roman, Petrucci Music Library
Szwedzcy kompozytorzy baroku
Członkowie Królewskiej Szwedzkiej Akademii Nauk
Urodzeni w 1694
Zmarli w 1758
Ludzie urodzeni w Sztokholmie
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\section{\label{S1:Intro}Introduction}
Complex mixtures of solute and solvent molecules are widespread, encompassing subjects ranging from physics and chemistry to materials science and even biology. These materials organise on a mesoscopic length scale, which lies between the smaller microscopic and larger macroscopic length scales and are inherently soft \cite{nagel2017experimental,van2018grand,evans2019simple}. This softness arises from relatively weak interactions ($\sim k_B T$) between molecular constituents and as such thermal fluctuations play a major role in deciding both their structural and dynamical behaviour. Therefore both entropic and enthalpic effects are important in determining their phase behaviour. The solute molecules can attract or repel each other and their relative strengths can be manipulated by changing the temperature or composition, which results in a series of different ordered self-assembled structures.
On going from spherically symmetric to anisotropic molecules an even richer phase behaviour \cite{de1993physics} is observed, not only controlled by entropy and enthalpy but also directional interactions between the anisotropic components. The simplest phase behaviour arises in polymer solutions, where the mixed state is stabilised by the entropy of mixing at higher temperatures. Upon lowering the temperature the enthalpic effects take over and below the bulk melting temperature $T_{c}$, it is energetically more favourable for the system to phase separate and exist as a mixture of polymer rich and solvent rich regions. In the reverse scenario, cooling polymer melts results in the appearance of a semi-crystalline polymeric glass phase, in which polymer chains are packed parallel to each other forming lamellar regions which coexist with amorphous regions with an imperfect packing \cite{de1993physics,evans2019simple,cingil2017illuminating,iwaura2006molecular,nagel2017experimental,van2018grand}.
Polymer dispersed liquid crystals (PDLCs) are one such example and are an important class of materials with applications ranging from novel bulk phenomena in electro-optic devices \cite{bronnikov2013polymer} to very rich and unique surface phenomena like tunable surface roughness \cite{liu2015reverse} and electric field driven meso-patterning on soft surfaces \cite{roy2019electrodynamic,dhara2018transition}. These soft materials can be termed multi-responsive as they can be controlled by electro-magnetic fields, the presence of interfaces or substrates and temperature or concentration-gradients \textit{etc}. While there is a lot of literature available on the synthesis and application of these novel materials, a fundamental understanding of the thermodynamics and kinetics of phase transformations in these complex mixtures is still missing.
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{Figure1-01.png}
\caption{\label{fig:figure1} A typical ``teapot'' phase diagram for a mixture of longer flexible polymers and shorter rod-like smectic-A mesogens, reproduced from \cite{kyu1996phase}, where $\phi$ indicates the LC volume fraction and $T$ temperature. The inset panels illustrate the phases that can exist in each of the respective regions demarcated by the (blue) phase boundaries; the flexible polymers and rod-like mesogens are represented by (purple) dots and (yellow) oblate spheres respectively. The four different phases include: mesogen-poor liquid, mesogen-rich liquid, nematic and smectic as denoted by $L_{1}$, $L_{2}$, $N_{1}$ and $S_{1}$ respectively. Dashed lines mark the triple points, $T_{c}$ the (continuous) transition from single-phase to two-phase liquid, $T_{SN}$ the (first-order) transition from smectic to nematic and $T_{NI}$ the (first-order) transition from nematic to isotropic liquid. Parameters used: $T_{NI}=333K$, $\alpha=0.851$, $r_{2}/r_{1}=2.25$ and $\chi(T)=-1+772/T$.}
\label{fig:fig_1}
\end{figure}
Interestingly, a rich phase diagram is also observed for complex mixtures with only repulsive interactions. The fundamental reason behind this phase behaviour was identified fairly early, in mixtures of rod-like hard particles and non-adsorbing polymers, from density functional theory (DFT) calculations \cite{lekkerkerker1992phase,lekkerkerker1994phase}. It is due to an effective depletion attraction between the rod-like particles when they are at small separation and thus, even in systems with only excluded volume interactions, one observes three distinct phases: of which two are isotropic ($L_{1}+L_{2}$) (one polymer-rich and the other mesogen-rich) and one mesogen-rich nematic $N_{1}$. Simple mean-field models of mixtures of polymers and liquid crystals has, however, considered both entropic and enthalpic effects \cite{mcmillan1971simple,chiu1998phase,matsuyama2002non}. The phase boundary between the one-phase and the two-phase regions of these mixtures in the temperature-order parameter plane (shown in Figure \ref{fig:fig_1} by the blue line) is commonly referred to as having a ``teapot" topology and is characterised by a number of special points. The primary order parameter, $\phi$, is the difference between the local densities of the two components, the polymers and the liquid crystals. The ``top" of the teapot is the critical point and its "lid" is coexistence of the polymer-rich and the mesogen-rich isotropic phases (see Figure \ref{fig:fig_1}). In this region,
the order parameter $\phi$, which is of the Ising universality class, grows continuously from zero as one cools the system below $T_c$. At order parameter values close to unity, the system consists primarily of the mesogens. For a purely mesogenic phase ($\phi = 1$), as one cools the system the nematic order parameter discontinuously jumps to a non-zero value, at the isotropic-nematic transition temperature, $T_{NI}$ which forms one ``spout" of the teapot. At this point the rotational invariance of the configurations are broken and the mesogens spontaneously order along a common director. This order parameter belongs to a different universality and the phases formed by the mixture of polymers and liquid crystals allow a novel interplay between order parameters of different symmetries which affects both the thermodynamics and kinetics of these complex mixtures. In some situations, cooling the system further results in the sudden appearance of a non-zero smectic order at $T_{SN}$, a thermodynamic state characterised by broken orientational symmetry and a one dimensional positional ordering. The phases coexisting in this region are pictorially shown in Figure \ref{fig:fig_1}.
The relative positions of these special points in the temperature-composition plane, which again are functions of the strengths of the microscopic interactions, shape the phase diagram. The four different phases appearing here are : mesogen-poor liquid, mesogen-rich liquid, nematic and smectic as denoted by $L_{1}$, $L_{2}$, $N_{1}$ and $S_{1}$ respectively and their coexistence regions are also shown in Figure \ref{fig:fig_1}. For a more detailed discussion on how the shape of the phase boundaries is affected by the parameter values please refer to Section 4 of Appendix C.
Recently, the nematic ordering of semi-flexible macromolecules, in implicit solvents, have been studied in the limit where the contour length, $L$, is much greater than its persistence length $l_{p}$ using large-scale molecular dynamics simulations. Owing to large director fluctuations, the effective tube radius within which each macro-molecule is confined is much greater than what should be expected from the length scale arising from average density \cite{egorov2016anomalous}. These director fluctuations modify the phase diagram one computes from density functional theories. In material systems both entropic and enthalpic interactions decide the phase behaviour of complex mixtures and an interplay between nematic order and phase separation has been recently studied for polymeric chains in implicit solvents of varying quality \cite{midya2019phase}. The stiffer chains showed a single transition from isotropic to nematic, while the softer chains also exhibited a demixing between isotropic fluids, one polymer-rich and the other mesogen-rich \cite{midya2019phase}.
Phase diagrams are central to the understanding of material properties as the regions of thermodynamic stability of materials are encoded in them. Calculating phase diagrams from molecular simulations however is a task which is far from trivial \cite{frenkel2001understanding}. One of the most prominent methods is the Gibbs ensemble technique \cite{panagiotopoulos1987direct} which is used for computing phase diagrams of liquid-vapour systems and for fluid mixtures, with the method of thermodynamic integration being another \cite{grochola2004constrained}. A number of recent publications have introduced a powerful method for estimating the whole phase diagram from a single molecular dynamics simulation by leveraging the multithermal-multibaric ensemble \cite{valsson2014variational,piaggi2019calculation,piaggi2019multithermal}.
In this work, we develop a multi-scale simulation methodology to map out the phase diagram of a binary mixture of rod-like mesogens in an explicit solvent of oligomers via CGMD simulations and qualitatively match it to its theoretical counterpart from mean-field theory \cite{chiu1998phase}. By scanning the
temperature-composition space via multiple CGMD simulations and by monitoring the resulting order parameter distributions we infer about the boundary between the locally stable and unstable regions. This maps out the phase boundaries and by appropriately tuning parameters appearing the mean-field theory we obtain a phase diagram which is qualitatively similar to the CGMD phase diagram. In principle, this method can be applied to a host of soft matter systems involving ordering fields competing
with phase separation like associating fluids like gels, gel-nematic mixtures, nematic-nematic mixtures etc. The remaining paper is organised into the following sections : Section II discusses our simulation methodologies, section III discusses the results of the molecular dynamics simulations, the global order parameters, the mean-field phase diagram is discussed next, along with how the phase boundaries inferred from the analyses of the partial free-energies obtained from the CGMD trajectories qualitatively agrees with the mean-field phase diagram upon a reasonable choice of parameter values. In Section IV, we conclude.
\section{\label{S2:Method}The Methodology}
\subsection{Mapping Phase Boundaries}
Our method, developed to extract phase boundaries from MD simulation trajectories, proceeds as follows. Simulations of a binary system, in this case a mixture of rod-like mesogens and oligomers, are performed for a series of initial starting compositions $\phi_{0}$, at high temperature and quenched to carefully chosen points in the $T-\phi$ plane. Details of the simulation results and coarse-grained model, including the parameters used, can be found in Section \ref{S3:Results} and Appendix \ref{Appendix:CGMD_Model} respectively. For a given volume fraction of the LC component $\phi_{0}$, the system will phase separate depending on its location in the underlying free energy landscape. In order to probe the topology of the underlying landscape, a new procedure has been devised.
From the resulting simulation library an estimate of the correlation length $\xi$, is first made, for a set of independent quenches at a given point and used to inform a specially devised binning procedure. Each trajectory is then binned into cubes such that the composition of the cubes may be evaluated, using a suitably defined continuum order parameter, to produce a histogram of the continuum order parameter distribution, $P(\phi; \phi_{0})$. The extracted distributions are then inverted to reveal a partial free energy $f(\phi; \phi_{0})=-k_{B}T\log P(\phi; \phi_{0})$ containing several minima, which consequently expose the topology of the free energy landscape and the approximate location of the phase boundaries.
The first step of our numerical recipe is to determine the correlation length at each point under consideration in the $T-\phi$ plane. This is achieved by coarse-graining the order parameter field and effectively reducing it to a spin-1/2 Ising-like configuration. Each of the instantaneous simulation snapshots are binned into cubes of size $\approx (2\sigma)^{3}$, $\sigma$ is defined in Appendix \ref{Appendix:CGMD_Model} and the number of monomers of each species $n_{A}$ and $n_{B}$ inside are totalled. A state $\Psi=\pm1$ is then assigned to each cell such that $\Psi=1$ if $n_{A}>n_{B}$ and $\Psi=-1$ otherwise. The spatial correlation function is then calculated,
\begin{eqnarray}
C(r_{ij})=(\Psi_{i} - \langle\Psi\rangle)(\Psi_{j} - \langle\Psi\rangle)
\label{e:correlation}
\end{eqnarray}
where $r_{ij}$ is the radial distance between the respective cubes and the angle brackets indicate averaging over a suitable long time period, in this case the last 20ns of all independent quenches are used. Figure \ref{fig:figure2} (a) depicts typical correlation functions calculated from MD simulations as quenched from $T^{*}=10.5$ to $T^{*}=5.1$ for all compositions considered in this work. The zero-crossing point for each composition indicates the correlation length $\xi$ as indicated explicitly for the $\phi_{0}=0.5$ composition in the figure. The correlation length for each of the simulations considered then serves as a customised estimate for the bin size used in the subsequent binning procedure to determine the continuum order parameter distribution $P(\phi; \phi_{0})$.
\begin{figure}[htb]
\includegraphics[]{Figure2-01.png}
\caption{\label{fig:figure2} Correlation functions, local nematic order parameter and continuum order parameter distributions from MD simulations at $T^{*}=5.1$. (a) Correlation function used to estimate the correlation length and bin size. The zoomed inset shows the zero-crossing points more clearly for all compositions and the correlation length $\xi$, for the $\phi_{0}=0.5$ composition, is indicated by the arrows as an example. (b) Local $P_{2}$ order as a function of the cutoff distance, the HWHM is indicated and used as the cutoff distance $r_{c}$ when assigning $P_{2}$ values to each rod.(c) The probability distribution as a function of the density, the cartoon panels indicate the mesogen-rich and mesogen-poor regions where the flexible polymers and rod-like mesogens are represented by (purple) dots and (yellow) oblate spheres respectively.}
\end{figure}
In the second step, the order parameter distribution is determined by re-binning the simulation cell into cubes with dimensions $\approx\xi^{3}$, this is illustrated in Figure \ref{fig:figure7} (a). The number of monomers of each species inside each bin are counted and a value assigned, using the order parameter of an arbitrary bin $i$, which is defined as
\begin{eqnarray}
\phi_{i}=\frac{1}{2}\Bigg( \frac{n_{A}^{i}-n_{B}^{i}}{n_{A}^{i}+n_{B}^{i}} + 1\Bigg)
\end{eqnarray}
In this case however, the continuum order parameter $\phi_{i}$, is bounded between zero and unity and is the continuum definition of the order parameter $\Psi$, defined above. The probability distribution $P(\phi;\phi_{0})$ may be found by averaging this process over the last 20ns of each independent quench and producing a histogram of the bin values. Figure \ref{fig:figure2} (c) shows typical probability distributions which reveal a distinct splitting of the simulation cell to its bracketing densities, revealing a series of mesogen-rich and mesogen-poor regions as indicated by the inset cartoon panels.
In the final step the order parameter distributions are inverted to reveal the topology of the free energy landscape though a partial free energy $f(\phi;\phi_{0})=-k_{B}T\log P(\phi;\phi_{0})$ at each composition, $\phi_{0}$. Figure \ref{fig:figure3} (e) shows an example inversion from MD simulations from the different compositions at $T^{*}=5.1$. A series of minima are present indicating the system is splitting to lower its free energy. In Appendix \ref{Appendix:Methods_Rationalise} this is rationalised using a conserved order parameter dynamics model which links the starting composition to the nature of the probability distribution of the order parameter at long times, which in turn is linked to the topology of the underlying free energy landscape. This leads to an important rule which may be used to understand the resulting partial free energy profiles. Simulations that converge onto their starting compositions $\phi_{0}$ with a single minimum are initiated from region of positive curvature, or $f^{\prime \prime} (\phi;\phi_{0}) > 0$ and those with that split into two or more successive minima are initiated from a region of negative curvature $f^{\prime \prime} (\phi;\phi_{0}) < 0$ and spontaneously phase separate. In Section \ref{S3:Results_MFTvsMD} this rule is employed to map out the approximate location of the phase boundaries from our CGMD simulations.
\subsection{Characterising Phase Boundaries}
The second half of our numerical recipe is concerned with identifying which liquid or liquid crystalline phase each of the respective minimums, in the partial free energy profiles, correspond to. Depending on whether or not the system splits or converges onto its equilibrium composition a local or global approach must be used respectively, in order to determine the extent of orientational ordering of the rod-like mesogens. In the scenario where the system splits between different ordered and disordered phases, a global approach cannot be used to determine the type of LC phase since it would mask the locally ordered nematic regions. Instead the local nematic order $P_{2}(r)$ of each rod-like molecule is probed as a function of the cutoff distance $r_{c}$,\cite{mukherjee2012derivation,cinacchi2009diffusivity},
\begin{eqnarray}
\scalebox{1.05}[1]{$P_{2}(r)=\Bigg\langle \frac{\Sigma^{N-1}_{i=1}\Sigma^{N}_{j=i+1}\delta(r-|\mathbf{r}_{j}-\mathbf{r}_{i}|)P_{2}(\hat{\textbf{u}_{i}}(\mathbf{r}_{i})\cdot\hat{\textbf{u}_{j}}(\mathbf{r}_{j}))}{\Sigma^{N-1}_{i=1}\Sigma^{N}_{j=i+1}\delta(r-|\mathbf{r}_{j}-\mathbf{r}_{i}|)} \Bigg\rangle$}
\end{eqnarray}
where $P_{2}$ is the second Legendre polynomial and $\hat{\textbf{u}_{i}}(\mathbf{r}_{i})$ the unit vector associated with the largest eigenvalue of the inertia tensor of particle $i$, with its centre of mass located at $\mathbf{r}_{i}$, see Appendix \ref{S2:Intro_Order} for methodological details. The angular brackets indicate statistical averaging over the last 20ns of each independent quench. Figure \ref{fig:figure2} (b) shows a series of $P_{2}$ curves as a function of the cutoff distance $r_{c}$ for each of the compositions considered at $T^{*}=5.1$. As a convention the half width half maximum (HWHM) is then used as the cutoff $r_{c}$, to assign a $P_{2}$ value to each rod-like mesogen in the system. For reference $P_{2}\sim1$ indicates perfect orientational order of the rods and $P_{2}\sim0$ a completely random orientation as illustrated in Figure \ref{fig:figure8}.
In order to then isolate the extent of nematic ordering within each of the distinct splitting regions, the $P_{2}$ ordering of the molecules is then coupled with the order parameter distributions $P(\phi;\phi_{0})$. This is achieved by isolating the bins at different points along the $P(\phi;\phi_{0})$ histogram and then averaging the local $P_{2}$ values of the molecules inside. In Figure \ref{fig:figure3} (f) points at different intervals along the $f(\phi;\phi_{0})$ profiles have been coloured from blue (isotropic) to red (anisotropic) according to their local $P_{2}$ values and consequently numerous different liquid and liquid crystalline phases are revealed. We note that it is possible for a bin to have a non-zero continuum order parameter $\phi_{0}$ and return a null local nematic order parameter $P_{2}$ value. In this situation there are no rods in the system with a centre of mass (COM) that lie inside the bin and thus the $P_{2}$ values cannot be averaged. The continuum order parameter $\phi_{i}$ however counts beads of each type ($A$ or $B$) inside the bins and is not concerned with full molecules. Therefore those points with non-zero $\phi_{i}$ and null $P_{2}$ are drawn as empty circles within the partial free energy profiles.
On the other hand, when the system converges to its equilibrium starting composition and there is not splitting, a global approach may be used to identify the structure of different LC phases using a suitably defined order parameter. The isotropic and nematic phases can be characterised by defining the usual tensor $Q$
\begin{eqnarray}
Q \equiv \frac{1}{2N}\sum^{N}_{i=1}(3\hat{\textbf{u}_{i}}\otimes\hat{\textbf{u}_{i}} - \textbf{1})
\end{eqnarray}
where $\otimes$ is the dyadic product and $\textbf{1}$ is a unit tensor and the summation is taken over all the rod-like mesogens. The unit vector $\hat{\textbf{u}_{i}}$ points along the backbone of the rod like mesogens and is defined as the vector spanning the first and last beads $x^{(i)}_{1}-x^{(i)}_{N_A}$ for an arbitrary molecule $i$. The global nematic order parameter $S$ corresponds to the largest eigenvalue of the tensor $Q$, such that $S\approx0$ in the I phase and $S\approx1$ in the nematic phase ($N_{1}$) where molecules are aligned parallel to the nematic director $\hat{\textbf{n}}$. The eigenvector associated with the largest eigenvalue is the global nematic order parameter $S$ and therefore contains information about the orientational ordering of molecules.
In order to probe the long ranged positional ordering in the smectic-A phase ($S_{1}$) and the distributions of the centre of mass of the rod-like mesogens along $\hat{\textbf{n}}$, the smectic order parameter must be introduced. It is given by the leading coefficient of the Fourier transform of the local density $\rho(\textbf{r}_{i}\cdot\hat{\textbf{n}})$.
\begin{eqnarray}
\Lambda \equiv \frac{1}{N}\Bigg\langle\Bigg|\sum^{N}_{i=1}\exp\Bigg[\frac{2\pi i (\textbf{r}_{i}\cdot\hat{\textbf{n}})}{d}\Bigg]\Bigg|\Bigg\rangle
\end{eqnarray}
where $d$ represents the spacing between layers of rod-like molecules in a perfect Sm-A phase. This is predetermined to be $8.2\sigma$ from the density waves discussed in Figure \ref{fig:figure3} (d) for the $\phi_{0}=0.9$ composition. In a pure system of rod-like molecules, i.e. $\phi_{0}=1$, one might reasonably expect a perfect Sm-A phase to form, such that $d\approx k$ and $\Lambda=1$ but in the systems considered here this is rarely the case due to thermal fluctuations and the long run-times required to achieve perfect ordering. It is clear that any non-zero value indicates some degree of smectic ordering as evidenced by the density waves and snapshots, $\Lambda=0.1$ is therefore taken as a reasonable cutoff. By combining our new method, with the local and global approaches discussed here, it becomes possible to both map out the phase boundaries and characterise them. This is demonstrated in Section \ref{S3:Results_MFTvsMD} where the phase diagram is built from our CGMD simulations.
\section{Results \& Discussion}
\subsection{\label{S3:Results}Molecular Dynamics Simulations}
\begin{figure}[h]
\includegraphics[width=\columnwidth]{Figure3-01.pdf}
\caption{\label{fig:figure3} (a) and (b) global nematic $P_{2}$ and smectic order $\Lambda$ parameters for each composition from MD simulation vs temperature. Both order parameters are calculated by averaging over the last 20ns of 5 independent quenches at each temperature. (c) and (d) density waves in the smectic phase for $\phi_{0}=0.75$ and $\phi_{0}=0.9$ compositions obtained by binning along the nematic director and averaging waves over the last 5ns of a single quench. The solid and dashed lines indicate the densities of the rod-like mesogens and flexible polymers respectively.}
\end{figure}
Before analysing splitting and phase separation we first discuss the global observations from our CGMD simulations. Five separate initial LC volume fractions were considered in this study: $\phi_{0}=0.1,0.25,0.5,0.75$ and $0.9$ where the freely flexibly polymer and semi-flexible rod-like mesogen species had fixed lengths $N_{A}=4$ and $N_{B}=8$ bead units respectively. The global nematic and smectic order parameters, for all the compositions considered in this work, are shown in Figures \ref{fig:figure3} (a) and (b) respectively and have been averaged using the last 20ns of 5 independent quenches. It is apparent, from Figure \ref{fig:figure3} (a), that the nematic-liquid transition temperature $T_{NL}$, increases monotonically with an increasing LC volume fraction, towards the bulk nematic-isotropic transition temperature $T_{NI}$. This corresponds to the drop in the $\phi_{0}=1$ composition (black line), for a pure system of rod-like mesogens at around $T^{*}=10.5$, which was estimated by melting the pure system of rods. The $\phi_{0}=1$ composition exhibits LC phases on its own, at the lowest temperatures $T^{*}<5.5$ the $S_{1}$ phase appears first in which layers of aligned rods stack on top of one-another with well defined spacing, where $\Lambda \approx 1$. This gradually decreases until $T^{*}\approx5.5$ at which point long-range positional order is lost ($\Lambda\approx0$) and the $N_{1}$ phase appears where the rods retain rotational order, $S\approx0.8$. As the temperature is raised the nematic order continues to decrease towards a value of $S\approx0.75$ until $T_{NI}=10.5$ at which point all rotational order is lost and the system is completely isotropic. This behaviour has been observed in a number of similar studies of rod-like mesogens \cite{milchev2019smectic} and is not unexpected. Aside from the pure system, the remaining compositions with a flexible polymeric component, were studied upon quenching the system from the isotropic phase at $T^{*}=10.5$ ($T_{NI}$) to ensure that no rotational ordering of the mesogens remains in any of the compositions studied for $\phi_{0}\leq0.9$.
For $T^{*}\geq6.0$ the compositions with a large volume fraction of mesogens $0.75\leq\phi_{0}\leq1$ are clearly nematic ($N_{1}$) with $S\approx0.8$ and $\Lambda\approx0$. Compositions with $\phi_{0}<0.75$ are completely isotropic ($L_{1}$) with $S\approx0$. As temperature is further decreased to $T^{*}=5.6$ compositions with $\phi_{0}\geq0.5$ show a non-zero $S$ and $\Lambda$ indicating the presence of both rotationally ordered and positionally ordered regions. We speculate that $T^{*}=5.6$ is close to the point where $S_{1}$, $N_{1}$ and $L_{1}$ phases may coexist \cite{mukherjee2020wetting}. Even though the smectic ordering retains only a small non-zero value ($\Lambda\approx 0.25$) for the $\phi_{0}=0.9$ composition, it is clear from Figure \ref{fig:figure3} (d) that there is preferential ordering of the rod-like mesogens into bands, with the flexible polymers filling the interstitial regions indicated by the solid and dashed lines respectively. Those compositions with compositions lying between $0.5\leq\phi_{0}\leq0.75$ also show a non-zero $\Lambda$ indicating small $S_{1}$ domains may exist. All other compositions $\phi\leq0.25$ are completely isotropic at this temperature.
In the region where $4.2\leq T^{*} \leq 4.6$ the smectic ordering $\Lambda$ for compositions $0.5\leq\phi_{0}\leq1$ gradually increases accompanied by an increased $S$ indicating a more ordered and micro-phase separated $S_{1}$ phase appears. This is no more apparent than in Fig \ref{fig:figure3} (c) and (d) where the mesogen-rich regions contain no flexible polymers in comparison to higher temperatures $T^{*}>4.6$ as well as a reduction in the number of mesogens in the polymer-rich regions. Importantly for the $\phi_{0}=0.9$ composition, the system fully adopts the $S_{1}$ phase as seen in Figure \ref{fig:figure3} (d) whereas the $\phi_{0}=0.75$ composition always contains regions with what appears to be some splitting with a low density liquid phase. This is shown most clearly at $T^{*}=4.2$ in Figure \ref{fig:figure3} (c) where the system is split between the $S_{1}$ phase and with 3 clearly defined peaks in one half of the simulation cell, with the other side containing a small number of rod-like mesogens dispersed in the flexible polymers. This should feature prominently in the MD phase diagram; the absence of the $N_{1}$ phase would also suggest that $T^{*}=4.6$ is below the triple point where only $S_{1}$ and $L_{1}$ phases may coexist. Similar self-organisation is also evident in experimental systems of
binary mixtures of long and short PDMS molecules, where they phase-segregate into alternate layers of long and short smectic phase owing to entropic stabilisation \cite{okoshi2010alternating,kato2019smectic}. Here, we observe similar micro phase-separated phases owing to combined effects of entropy and enthalpy.
\subsection{\label{S3:Results_MFTvsMD}Phase Diagrams}
\begin{figure}[htb]
\includegraphics[width=\columnwidth]{Figure4-01.png}
\caption{\label{fig:figure4} Constructing partial free energy landscapes from MD simulations at $T^{*}=5.1$. (a-e) Snapshots taken from MD simulations as the LC component is increased, the flexible polymers and semi-flexible rod-like mesogens are coloured purple and yellow respectively to enhance their orientational alignment and in panels (d) and (e) half the rods have been removed to reveal the banding of purple polymers in the smectic phase, produced using OVITO \cite{stukowski2009visualization}. (e) Free energy profiles as inverted from the probability distributions in Figure \ref{fig:figure2} (c), the approximate locations of the phase boundaries are indicated by dashed (grey) lines. The points along the histogram have been coloured continuously, according the local nematic order parameter $P_{2}$ of the rods, as indicated by the colourbar on the RHS, from $P_{2}=0$ (blue) to $P_{2}=0.6$ (red). In this way it is possible to distinguish between isotropic and anisotropic minima in the free energy landscape.}
\end{figure}
For reference a brief recap of the theoretical model for predicting phase diagrams of mixtures of polymers and smectic liquid crystals is presented and the key parameters which govern the phase behaviour are described. More details about the model and its development can be found in refs \cite{mcmillan1971simple,kyu1996phase,chiu1998phase} and Appendix \ref{Appendix:MFT}. The free energy of a mixture of a polymer and a smectic liquid crystal $f=f_{iso}+f_{aniso}$ comprises of two parts, an isotropic part describing the thermodynamics of isotropic liquids $f_{iso}$ and an anisotropic part which accounts for the ordering of the liquid crystals $f_{aniso}$. Flory-Huggins theory, \cite{flory1953principles} is used to describe the former for a liquid crystal - polymer mixture such that
\begin{eqnarray}
\scalebox{0.95}[1]{$f_{iso}(\phi,T) = \frac{\phi}{r_{1}}\ln\phi + \frac{1-\phi}{r_{2}}\ln(1-\phi) + \chi(T) \phi(1-\phi)$}
\label{e:fh}
\end{eqnarray}
where $r_{1}$ is the length of the rod-like mesogens, $r_{2}$ is the length of the polymer and $\phi$ is the volume fraction of the LC component. The F-H interaction parameter $\chi(T)$ is a quantity accounting for the enthalpic interactions and it is generally described by an inverse temperature relationship of the form $\chi=A+\frac{B}{T}$, where $A$ and $B$ are material specific parameters. The anisotropic portion, which couples the LC composition into the free energy is given by
\begin{eqnarray}
f_{aniso}(\phi,T,m_{n},m_{s}) = -\Sigma(m_{n},m_{s}) \phi \\*
\nonumber
- \frac{1}{2}\nu(T)(s(m_{n})^{2}+\alpha \kappa(m_{s})^{2})\phi^{2}
\label{e:msm}
\end{eqnarray}
where $\Sigma$ represents the decrease in entropy as the rod-like polymers align (Equation \ref{SI_MFT_ENTROPY}), $s$ is the nematic order parameter (Equation \ref{SI_MFT_NEMATIC}) and $\kappa$ is the smectic order parameter (Equation \ref{SI_MFT_SMECTIC}). All of these quantities are functions of the dimensionless nematic and smectic mean-field terms $m_{n}$ and $m_{s}$. The nematic coupling term $\nu(T)$ is a temperature dependent term which depends on the nematic-isotropic transition temperature $T_{NI}$, such that $\nu(T)=4.541 T/T_{NI}$, note the pre-factor is a universal quantity \cite{kyu1996phase,chiu1998phase}. The smectic interaction coupling $\alpha$, is a dimensionless quantity as defined in Equation \ref{SI_MFT_ALPHA} and is kept fixed. By minimising the free energy functional with respect to the order parameters ($\frac{\partial f_{aniso}}{\partial s}=0$ and $\frac{\partial f_{aniso}}{\partial \kappa}=0$), the resulting expressions (Equations \ref{SI_SELFCON_1} and \ref{SI_SELFCON_2}) can be evaluated numerically using the procedure outlined in Appendix \ref{Appendix:MFT}. The renormalised free energy landscape (obtained after re-substituting the minimised values of the nematic and the smectic order parameters back into the full free energy expression) is then determined at a given temperature (see Figure \ref{fig:figure9}) and the phase diagram can be mapped out as illustrated in Figure \ref{fig:figure5} (a). This is discussed in conjunction with the phase diagram extracted from our CGMD simulations.
\begin{figure*}[htb]
\includegraphics[width=\textwidth]{Figure5-01.png}
\caption{\label{fig:figure5} Phase diagram of a binary polymer-smectic-liquid crystal mixture. (a) Phase diagram as calculated from the mean-field theory with parameters as indicated in the figure in the long rod regime, see Appendix \ref{Appendix:MFT} for a detailed methodology of the numerical procedure. (b) Phase diagram as extracted from CGMD simulations, point types correspond to different phases: $\blacksquare$-Liquid, $\CIRCLE$-Nematic and $\blacktriangle$-Smectic. Each point is coloured according to the local nematic order $P_{2}(\phi)$, with the corresponding value in the colourbar (rhs). (c-d) Effective free energy profiles $f(\phi;\phi_{0})$ extracted from CGMD simulations at all compositions considered for $T^{*}=5.6$, $6.0$ and $6.5$ where the points are coloured according to their local $P_{2}$ values according to the colour scale in panel (b). Shown alongside are the corresponding snapshots for each of the compositions considered at each temperature, the flexible polymers and semi-flexible rod-like mesogens are coloured purple and yellow respectively.}
\end{figure*}
Figure \ref{fig:figure5} (b) shows the binary phase diagram for the semi-flexible rod-like mesogens with stiffness constant $k_{bend}=50$ and $N_{A}=8$ and fully-flexible polymers with $N_{B}=4$ as extracted from our CGMD simulations. This has been reconstructed using the procedure outlined in Section \ref{S2:Method} such that the local minimums of $f(\phi;\phi_{0})$, that result from the splitting of the effective free energies or order parameter distributions, are taken to define the phase boundaries. For $T^{*}<5.1$ the system shows a pure liquid phase $L_{2}$ and pure smectic-A phase $S_{1}$, at very low and very high mesogen concentrations respectively. This is evidenced by the value of the local $P_{2}$ order parameter in both regions as highlighted by the colouring of the points in Figure \ref{fig:figure4} (f) and a non-zero value of the global smectic ordering $\Lambda$ in Figure \ref{fig:figure3} (b). Between these two regions lies a large $L_{2}+S_{1}$ coexistence region spanning the intermediate concentrations.
At $T^{*}=5.1$, $L_{2}$ moves inwards to higher concentrations ($\phi\approx0.25$) and $S_{1}$ moves similarly to lower concentrations ($\phi\approx0.9$) and another highly ordered nematic phase appears $N_{1}$, this demarcates the $L_{2}+N_{1}$ and $N_{1}+S_{1}$ coexistence regions. The effective free energy profiles at $T^{*}=5.1$ are shown in Figure \ref{fig:figure4} (f) where the $\phi_{0}=0.5$ simulation (cyan line) shows the 3 distinct regions corresponding to the $L_{2}$, $N_{1}$ and $S_{1}$ phase boundaries. This is further confirmed by the local $P_{2}$ ordering in all 3 regions with $L_{2}$ phase corresponding to $P_{2}\approx 0$ and the $N_{1}$ and $S_{1}$ phases reaching $P_{2}\approx 0.5$ showing that they are highly ordered. In order to distinguish these two ordered minima we consult Figure \ref{fig:figure3} (a) and (b) and note that at high concentrations $\phi_{0}>0.5$ which show this splitting have both $\Lambda \neq 0$ and $S \neq 0$ indicating the presence of nematic and smectic phases. However this is not enough to identify which minimum corresponds to the smectic phase since a measure of the local smectic ordering or indeed its distribution is not possible to calculate, unlike the local nematic ordering ($P_{2}$), see Figure \ref{fig:figure4} (f). We therefore examine the simulation snapshots in Figure \ref{fig:figure2} taken at $T^{*}=5.1$ which shows the $\phi_{0}=0.5$ concentration (Figure \ref{fig:figure2} (c)) is predominantly split between $L_{1}$ at low concentrations and $N_{1}$ at high concentrations, whereas the snapshots for the $\phi_{0}>0.5$ concentrations (Figure \ref{fig:figure2} (d-e)) are predominantly $S_{1}$.
As temperature is further increased to $T^{*}=5.6$ both $L_{2}$ and $N_{1}$ appear to move inward
with only the highest composition $\phi_{0}=0.9$ having any smectic ordering with $\Lambda>0$, see Figure \ref{fig:figure3}(b). At $T^{*}=6.0$ all positional ordering is lost and the $S_{1}$ phase is replaced by the $N_{1}$ phase, this is the first spout of the double ``teapot'' topology indicated by $T_{SN}$ in figure \ref{fig:figure5} (b). The $L_{2}$ region moves to even higher $\phi$ values leaving a narrow $L_{1}+N_{1}$ coexistence region as evidenced by the splitting of the $\phi_{0}>0.75$ simulation in Fig \ref{fig:figure5}(d). This region narrows further at $T^{*}=6.5$ in Figure \ref{fig:figure5}(e) after which the minimums appear to merge with no apparent splitting at higher temperatures. However this does not mean that the coexistence region is lost but instead that it has narrowed sufficiently such that the compositions at which simulations have been performed do not fall inside this narrow coexistence region. We speculate that this region could be isolated by considering compositions in the region $0.75\leq\phi_{0}\leq0.9$, it is sufficient however to simply connect these minimums to the nematic-isotropic transition temperature $T_{NI}$ determined by melting the pure system ($\phi_{0}=1$). This give the second spout of the double ``teapot'' topology indicated by $T_{NI}$ in figure \ref{fig:figure5} (b).
From the MFT we find a similar picture to that extracted from the CGMD simulations, this is shown in Figure \ref{fig:figure5} (a) where two crucial changes have been made to the parameter values originally presented in \cite{kyu1996phase}, see Figure \ref{fig:figure1} or \ref{fig:figure10} (a) for the original phase diagram (for a detailed discussion on how the shape of the phase-boundary is affected by the parameter values refer to Section 4 of Appendix C). Firstly the polymer:mesogen length ratio $r_{2}/r_{1}=2$ has been inverted such that the mesogens are twice as long as the polymers to match our CGMD simulations. This has the effect of pushing the original $L_{1}+L_{2}$ coexistence region to lower concentrations as well as suppressing the temperature at which these two phases merge $T_{c}$, see Figure \ref{fig:figure10} (a) and (c). Secondly the nematic-isotropic transition temperature $T_{NI}$ has been raised from 333K to 400K, motivated by the high stiffness of our mesogens in our CG simulation model, which raises the temperature of the melt. This has the effect of pushing the double chimney-like topology upwards in the phase diagram to higher temperatures. As a direct consequence the $L_{1}+L_{2}$ coexistence region then becomes buried inside the phase diagram and is replaced by the $L_{1}+S_{1}$ region which also moves outward such that the pure $L_{2}$ and $S_{1}$ phases are forced to extremely low and high concentrations respectively due to an increased $T_{NI}$. Thus we observe $L_{2}+S_{1}$, $N_{1}+L_{2}$ and $N_{1}+S_{1}$ regions as well as pure $N_{1}$ and $S_{1}$ regions but no $L_{1}+L_{2}$ region. This bears a similar resemblance to the phase diagram obtained from our model CGMD simulations, possessing identical qualitative features.
\section{\label{S4:Conc}Conclusions}
A new methodology has been developed to probe the topology of free energy landscapes, from CGMD simulations of binary mixtures, by manipulating continuum order parameter distributions. Using our method we have shown how the approximate locations of the phase boundaries (spinodals) can be extracted and then characterized by analysing global nematic and smectic order parameters, local nematic order parameter distribution and simulation snapshots. The resulting phase diagram was then compared with Maier-Saupe type mean-field theory using comparable parameters to our MD simulations. Both diagrams possess an identical double chimney-like topology, even with modest computational resources, demonstrating the power of this method.
The accuracy of our method has a strong dependence on the shape of the phase diagram, specifically the width of the region in $\phi$ space. If the region is sufficiently wide, it is more likely that one of the initial starting compositions $\phi_{0}$, from our MD simulations, will fall inside the region and splitting will be observed. For our regime, long rods and short polymers, the $S_{1}$, $N_{1}$, $L_{1}+N_{1}$ and $N_{1}+S_{1}$ regions appear at very high volume fractions of the LC component and are narrow. In addition as the temperature is raised these regions further narrow considerably which hinders the method at higher temperature. In the reverse scenario however, with short rods and long polymers, an additional $L_{1}+L_{2}$ region is present. This region would be more accurately probed by our method since the different phases are more well separated in $\phi$ space.
Additionally, a quantitative estimate of $\chi(T)$ from the temperature dependence of the location of the peaks of the order parameter distribution is a relatively simple exercise for an isotropic polymer mixture (see the isotropic $L_{1}+L_{2}$ coexistence region in the phase diagram of short rods and long polymers) described by a Flory-Huggins free energy. This estimate, however, gets more involved as one includes anisotropic phases in the description, especially for systems with long mesogens, as in the systems consided here. The anisotropic phases start appearing at even lower volume fractions and interfere with the Ising like critical point. Here one observes the effects of the interference of the discrete Ising symmetry associated with the $\phi$ order parameter and the continuous symmetry of the nematic and the smectic order parameters and this makes the quantitative estimate of $\chi(T)$ more difficult. This is an aspect associated with the exact matching of the phase diagram resulting from MD simulations to that from the MFT, which would be resolved in future.
A computational method like this is general enough to be applied to the sub-cellular environment where
semi-flexible bio-polymers undergo liquid-liquid phase separation and under specific physico-chemical conditions they can self-assemble into non-random filamentous structures with anisotropic interactions promoting nematic ordering. It is also known that mechanical strain may induce alignment of semi-flexible polymers. Thus a method like this becomes an important tool for estimating parameters ($\chi(T)$, $\nu(T)$, $T_{NI}$ etc.) for constructing phase diagrams which thus enables a realistic meso-scale description of specific bio-polymers which already accounts for the specific chemical details. This specific meso-scale model can also be used for non-equilibrium kinetic simulations where one can probe the important role of various metastable intermediates in these complex systems.
\begin{acknowledgments}
W.F., B.M. and B.C. acknowledge funding support from EPSRC via grant EP/P07864/1, and P\& G, Akzo-Nobel, and Mondelez Intl. Inc. This work used the Sheffield Advanced Research Computer (ShARC) and BESSEMER HPC Cluster.
\end{acknowledgments}
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Q: Does iOS MultipeerConnectivity provide routing layer? I am writing an app that is suppose to work without a connection to mobile carrier and without local WiFi. Each device will act as transmitter, receiver and router.
My main challenge so far is that I cannot figure out how exactly MultipeerConnectivity works as documentation on MC is really limited. Apple denied revealing technical specification of MC claiming it's a proprietary network stack, so I have to rely on network sniffers and reverse-engineering which is not the quickest way to figure out how MC works.
Suppose I have 100 devices forming a mesh network in such way that each device is within the range of at least one other device and at maximum three other devices.
Is there any way to send a message from node A to node B that is not within the range of node A without the need to broadcast the message to all other nodes? I mean that message should be properly routed through all other nodes.
Does MC include a routing layer too or I have to write it myself?
From what I can see ad hoc delay tolerant wireless networks is still a hot subject in research.
These slides on ad hoc delay tolerant wireless network shed more light on the subject as it was a few years ago. And also this paper. Has Apple progressed it much with MC?
I cannot really see any way to send a message between nodes not directly connected to each other without flooding.
Correct?
A: The MCSession Reference states that
Sessions currently support up to 8 peers, including the local peer.
Also, the overview you cited says
In the discovery phase, your app uses a browser object […] to browse for nearby peers[.]
Moreover, the documentation on managing peers manually suggests that all peers in a session must be connected with each other to have them in a session.
This is suggesting that the framework only covers the communication between nearby devices, as in 'reachable by bluetooth or WiFi'. Naturally, those devices do not need complex routing, as they do communicate with each other and the benefit of the framework is simple multicasting between nearby devices, from a programmers' point of view.
As far as your question goes, this is about it - trivially, since all peers an a MCSession have links to each other - there is no routing needed.
This does however, allow you to construct a routing layer pretty easy.
Given your scenario, there will be multiple MCSessions with devices being part of at least one. All devices that are part of more than one MCSession do become routers and interconnect the MCSessions with each other.
The rest of the task should be straight forward; defining a namespace for addressing devices and implementing a routing protocol of your choice.
The old days of the internet, with unstable dialup connections, might be a plus factor for you as the routing protocols in place are rather stable in regards of link loss.
Here are two good starting points for you to make your choice of better fit:
*
*Link state routing
*Distance vector routing
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\begin{document}
\begin{titlepage}
\begin{center}
\today \hfill LBL-32484 \\
\hfill UCB-PTH-92/21 \\
\hfill OHSTPY-HEP-T-92-015\\
\hfill Stanford ITP 922/92\\
\vskip .20in
{\large \bf Neutrino Masses and Mixing Angles in a
Predictive Theory of Fermion Masses }
\footnote{This work was supported in part by the Director, Office of
Energy Research, Office of High Energy and Nuclear Physics, Division of
High Energy Physics of the U.S. Department of Energy under Contracts
DE-AC03-76SF00098 and DOE-ER-01545-585 and in part by the National Science
Foundation under grants PHY90-21139 and PHY86-12280.}
\vskip .20in
Savas Dimopoulos\\[.20in]
{\em Department of Physics,
Stanford University,
Stanford, CA 94305}\\[.20in]
Lawrence J. Hall\\[.20in]
{\em Department of Physics and Lawrence Berkeley Laboratory,\\
University of California, Berkeley, California 94720}\\[.20in]
Stuart Raby\\[.20in]
{\em Department of Physics,
The Ohio State University,
Columbus, OH 43210}
\end{center}
\vskip .10in
\begin{abstract}
A framework for predicting charged fermion masses in supersymmetric grand
unified theories is extended to make predictions in the neutrino sector.
Eight new predictions are made: the two neutrino mass ratios and the three
mixing angles and three phases of the weak leptonic mixing matrix.
There are three versions of the theory which are relevant for
producing MSW neutrino oscillations in the sun. One of these is
prefered by the combined solar neutrino observations. Another will be probed
significantly by the searches for $\nu_\mu\nu_\tau$ oscillations at the
NOMAD, CHORUS and P803 experiments. In this second
version $\nu_\tau$ could be a significant component of
the dark matter in the universe.
\end{abstract}
\end{titlepage}
PAGE (PAGE 1) \renewcommand{\thepage}{\roman{page}} \setcounter{page}{2}
\mbox{ }
\vskip 1in
\begin{center}
{\bf Disclaimer}
\end{center}
\vskip .2in
\begin{scriptsize}
\begin{quotation}
This document was prepared as an account of work sponsored by the United
States Government. Neither the United States Government nor any agency
thereof, nor The Regents of the University of California, nor any of their
employees, makes any warranty, express or implied, or assumes any legal
liability or responsibility for the accuracy, completeness, or usefulness
of any information, apparatus, product, or process disclosed, or represents
that its use would not infringe privately owned rights. Reference herein
to any specific commercial products process, or service by its trade name,
trademark, manufacturer, or otherwise, does not necessarily constitute or
imply its endorsement, recommendation, or favoring by the United States
Government or any agency thereof, or The Regents of the University of
California. The views and opinions of authors expressed herein do not
necessarily state or reflect those of the United States Government or any
agency thereof of The Regents of the University of California and shall
not be used for advertising or product endorsement purposes.
\end{quotation}
\end{scriptsize}
\vskip 2in
\begin{center}
\begin{small}
{\it Lawrence Berkeley Laboratory is an equal opportunity employer.}
\end{small}
\end{center}
\newpage
\renewcommand{\thepage}{\arabic{page}}
\setcounter{page}{1}
%
The solar neutrino problem and the several existing and upcoming
neutrino experiments have caused interest in neutrino masses and
mixings to escalate in the last few years.
Theoretical advances on this subject are very difficult to come by since the
subject of neutrino masses is, in general, coupled to the problem
of quark and charged lepton masses on which very little progress
has been made.
Recently, we proposed a predictive framework, based on
supersymmetric grand unified theories (GUTs), in which the
14 parameters of the quark and charged lepton mass matrices,
plus the ratio of the Higgs vacuum expectation values (vevs), can
be obtained in terms of just 8 input parameters, thus
leading to 6 predictions \cite{dhr}.
The consequences of these predictions will be tested in planned experiments
\cite{barger}.
Our framework has
the virtue that it is a consequence of a large class of GUTs.
In this paper we wish to study a subset of models
within this class which are very predictive in the neutrino
sector. We will show how the addition of one more input parameter,
for a total of 9 inputs, allows us to account for 23 parameters,
resulting in 14 predictions. In particular we will predict the 2
neutrino mass ratios, 3 mixing angles and 3 phases of the lepton
sector.
We begin with a rapid overview of our framework. A key ingredient
is the Georgi-Jarlskog texture for fermion mass matrices at the
GUT scale \cite{georgi}:
$$M_D = \left( \begin{array}{ccc}
0 & F & 0\\
F & E & 0\\
0 & 0 & D
\end{array} \right){v \over \sqrt{2}}\cos\beta
\;\;\; \; \; \; \;
M_E = \left( \begin{array}{ccc}
0 & F & 0\\
F & -3E &0 \\
0 & 0 & D
\end{array} \right){v \over \sqrt{2}}\cos\beta $$
\\
$$M_U = \left( \begin{array}{ccc}
0 & C & 0\\
C & 0 & B\\
0 & B & A
\end{array} \right){v \over \sqrt{2}}\sin\beta \eqno (1)
$$
where tan $\beta = v_2/v_1$, is the ratio of electroweak breaking
vevs. The factor of 3 difference in the 22 elements of $M_D$ and
$M_E$ is of crucial importance. It arises naturally as a
consequence of the breaking of the Pati-Salam SU(4) via a vev
pointing parallel to the hypercharge generator
$$Y_{15} = \left( \begin{array}{cccc}
1 & & & \\
& 1 & & \\
& & 1 & \\
&&&-3
\end{array} \right) \eqno (2)
$$
Three examples where this happens
are when the 22 entries are generated by Higgs doublets which lie
in
(a) a $\overline{45}$ of SU(5),
(b) a $\overline{126}$ of SO(10),
(c) a higher dimension operator, for example 45 $\times$ 45 $\times$ 10 of
SO(10).
For simplicity and definiteness we will focus on case (b) in this
paper and thus study theories where the $M_D$ and $M_E$ arise from
the following matrix of Yukawa
couplings.\footnote{Many of our conclusions are
probably more general and valid even if fermion masses come from
higher dimension operators.}
$$
16 \left( \begin{array}{ccc}
0 & f\, 10^d & 0\\[5pt]
f\, 10^d & e\, \overline{126}^d & 0 \\[5pt]
0 & 0 & d\,10^d
\end{array} \right)16 \eqno (3)
$$
Here $d$, $e$ and $f$ are Yukawa couplings and $10^d$ and
$\overline{126}^d$ are scalar fields getting SU(2) breaking vevs
that contribute to $M_D$ and $M_E$.
The entries $A$, $B$, $C$ of the up matrix can each arise from couplings to 10
or $\overline{126}$ scalar mesons.\footnote{The 120 scalar meson
would lead to antisymmetric contributions to the mass matrix, and
is therefore not needed.} There may be several such scalar mesons,
distinguished by discrete symmetries necessary to ensure the
texture structure of eq.(1). Each $\overline{126}$ contains an
SU(5) preserving vev that can contribute to the Majorana mass
matrix $M_{NN}$ of the SU(5) singlet right handed neutrinos.
To proceed further and make predictions for neutrino masses, we
need to introduce some hypothesis that limits the number of
parameters in the theory. We already know that we will need to
introduce at least one additional parameter into the theory,
namely the overall scale of the right handed neutrino masses. To
maximize predictive power we will seek theories where this is the
{\em only} additional parameter. This has some implications for
both the Yukawa couplings and the vevs:
(I) There are no new Yukawas, i.e., the Yukawa couplings giving
rise to $M_{NN}$ are the same as those that give rise to $A,\; B,\;
C,\; D,\; E,\; F$.
(II) All the entries in the $M_{NN}$ matrix must be generated
from the vev of only {\em one} of the $\overline{126}$ multiplets.
(III) Each fermion mass matrix element is generated by the vev of
only {\em one} of the 10 or $\overline{126}$ multiplets.
It is now easy to see that these minimality hypotheses lead us to
a theory in which $M_U$ originates in the following Yukawa
couplings
$$16 \left( \begin{array}{ccc}
0 & c\overline{126}\,^{uN} & 0\\[5pt]
c\overline{126}\,^{uN} & 0 & bX^u\\[5pt]
0 & bX^u & a\overline{126}\,^{uN}
\end{array} \right)16 \eqno (4)$$
\noindent
where:
(a) $\overline{126}\,^{uN}$ gets an electroweak breaking vev
giving rise to Dirac masses for up quarks and neutrinos.
(b) $\overline{126}\,^{uN}$ also gets an SU(5) preserving vev
contributing to $M_{NN}$.
(c) $X^u$ can be either $10^u$ or $\overline{126}\,^u$. In
either case it only gets a vev in an electroweak breaking
direction and gives Dirac masses to up quarks and neutrinos.
It is easy to see that if we try to deviate away from eqs.\ 3 and
4 or from (a), (b), (c) we would either unnecessarily increase the
number of parameters, or we would end up with a light right-handed
neutrino that would become the Dirac partner to $\nu_e$, $\nu_\mu$
or $\nu_\tau$, with a mass which is much larger than present
laboratory limits. It is straightforward to find a set of
symmetries which guarantees the textures of equations (3) and (4).
Note that $X^u$ cannot be $\overline{126}\,^{uN}$; if it were,
wavefunction mixing would induce a non-zero 22 entry in the matrix.
The neutrino masses which follow from equation 4 are:
$$M_{\nu N} = \left( \begin{array}{ccc}
0 & -3C & 0\\
-3C & 0 & -3\kappa B\\
0 & -3\kappa B & -3A
\end{array} \right){v\over \sqrt{2}}\sin\beta$$\\
$$
M_{N N} = \left( \begin{array}{ccc}
0 & C & 0\\
C & 0 & 0\\
0 & 0 & A
\end{array} \right)V \eqno (5)$$
\\[-.2in]
\noindent
where $\kappa = 1$ if $X^u = \overline{126}\,^u$ (case I) and
$\kappa = - {1 \over 3}$ if
$X^u = 10^u$ (case II). $V$, the superheavy singlet vev, is the
one additional free parameter which occurs in the neutrino mass
matrix.
There are two important ingredients we left out of our discussion
so far:
1) All the quantities involved in the mass matrices are complex.
This appears to limit the predictive power; however all but one of
the phases can be eliminated by rephasing the fields.
2) The mass matrices that we measure at the weak scale are not
the same as those at the GUT scale (eqs.\ 1, 5); the two are
connected via the renormalization group [RG].
In our previous paper\cite{dhr}, we analytically solved the RG
equations for quark and charged lepton masses. We shall restrict
ourselves in this paper to the same approximations used there.
We use one loop RG equations, neglecting all Yukawa couplings
except for the top, $\lambda_t$. This is a good approximation only
for sufficiently small $\tan\beta$. The effects of large $\tan\beta$
will be studied in a forthcoming paper \cite{gregetal}.
The neutrino mass derives from effective dimension 5
operators, involving lepton doublets $L_i$ and Higgs doublet $H$,
of the form
$${{M^{ij}_{\nu\nu}} \over 2} L_i L_j \left({H\over v\sin\beta/\sqrt{2}}
\right)^2
$$
\noindent
where
$$M_{\nu\nu} = M_{\nu N}\, M^{-1}_{NN}\, M_{\nu N}^T$$
\noindent
$M_{\nu\nu}$ gets rescaled by an overall factor, as a result of
RG running from $M_G$ to $M_W$.
These ingredients result in the following mass matrices at the
weak scale:
$$
M_E = \eta_e \left( \begin{array}{ccc}
0 & Fe^{i\phi} & 0\\
Fe^{-i\phi} & -3E & 0 \\
0 & 0 & D
\end{array} \right) {v\over\sqrt{2}} \cos \beta
\eqno (6)
$$
$$
M_{\nu\nu} = \eta_\nu {9A v^2 \over 2V}\left( \begin{array}{ccc}
0 & C/A & 0\\
C/A & \kappa^2\, B^2/A^2 & 2\kappa\, B/A\\
0 & 2\kappa\, B/A & 1
\end{array} \right)\; \sin^2 \beta\,
$$
where $\eta_e$ and
$\eta_\nu$ take into account the RG scaling from $M_G$ to $M_W$.
Note that, apart from the overall scale, the neutrino mass matrix
depends only on $\kappa$, B/A, and C/A. As always A,B and C refer
to parameters renormalized at the GUT scale.
Harvey, Ramond and Reiss\cite{harvey} have discussed neutrino mass matrices
in SO(10) GUTs which incorporate the Georgi-Jarlskog ansatz. However, they
did not make any predictions for the neutrino masses. This is because,
even though our ansatz is a special limit of theirs, we rely on
the additional hypothesis that the vev of only one multiplet
contributes to any one matrix element (item III above). It is this
additional hypothesis which gives our ansatz its strong predictive
power with all neutrino masses and mixing angles determined, up to one overall
scale, in terms of parameters fixed in the charged fermion sector
of the theory.
The mass matrices in the lepton sector, Eq. 6, can be
diagonalized by bilinear transformations of the form:
\begin{eqnarray*}
M_E^{diag} & = & V^L_e\, M_E\, V_e^{R \dagger}\\[5pt]
M_{\nu\nu}^{diag} & = & V_{\nu}\, M_{\nu\nu}\, V_{\nu}^T
\end{eqnarray*}
\vspace{-.8in}
$$\eqno (7)$$
\\
\noindent
The leptonic CKM matrix is:
$$V' = V_{\nu}\; V_e^{L \dagger} \eqno (8)$$
\noindent
The natural parameterization for $V_e^L$, $V_{\nu}$ and $V'$ are:
$$V_e^L = \left( \begin{array}{rrr}
c_1^\prime & -s_1^\prime & 0\\[5pt]
s_1^\prime & c_1^\prime & 0\\[5pt]
0 & 0 & 1 \end{array} \right)
\left( \begin{array}{rrr}
1 & &\\[5pt]
&e^{i\phi} & \\[5pt]
& & e^{i\phi} \end{array} \right )$$
\\
$$V_{\nu} = \left( \begin{array}{rrr}
c_2^\prime & s_2^\prime &0\\[5pt]
-s_2^\prime & c_2^\prime & 0\\[5pt]
0 & 0 & 1 \end{array} \right)
\left( \begin{array}{rrr}
1 & 0 & 0\\[5pt]
0 & c_3^\prime & s_3^\prime\\[5pt]
0 & -s_3^\prime & c_3^\prime \end{array} \right) \eqno (9)$$
\\
$$V' = \left( \begin{array}{ccc}
c_1^\prime c_2^\prime - s_1^\prime s_2^\prime e^{-i\phi} &
s_1^\prime + c_1^\prime s_2^\prime e^{-i\phi} & s_2^\prime s_3^\prime\\[5pt]
-c_1^\prime s_2^\prime - s_1^\prime e^{-i\phi} &
-s_1^\prime s_2^\prime + c_1^\prime c_2^\prime c_3^\prime e^{-i\phi} &
s_3^\prime\\[5pt]
s_1^\prime s_3^\prime & -c_1^\prime s_3^\prime & c_3^\prime e^{i\phi}
\end{array} \right)$$
\noindent
The angles are given by:
\begin{eqnarray*}
s_1^\prime & =& - {\frac{F}{3E}}\\[5pt]
s_2^\prime & = & {C/A \over {3\kappa^2 \; B^2/A^2}} > 0\\[5pt]
s_3^\prime & = & -2\kappa \; B/A
\end{eqnarray*}
\vspace{-.9in}
$$\eqno (10)$$
\\[.1in]
and the CP violating angle $\phi$ is determined in the previous
paper [1].
In general $V'$ contains three independent phases.
However, for our case all three are related to the phase $\phi$
which is identical to that occurring in the quark sector, and is
determined to be cos $\phi = .38 {{+ \ .21}\atop{- \ .14}}$.
Diagonalization of the quark mass matrices leads to a KM matrix
$V$ which is the same function of angles $\theta_i, \phi$ as $V'$
is of $\theta'_i$ and $\phi$:
$V'(\theta'_i, \phi) = V(\theta_i, \phi )$.
The relations between the mixing angles in the quark and lepton
sectors just involve simple group theory numerical factors
$$
\eqalignno{
s'_1 &= - {1\over 3} \ s_1\cr
s'_2 &= {1\over 3\kappa^2} \ s_2\cr
s'_3 &= 2\kappa \eta_3 \ s_3&(11)\cr}
$$
except for $\eta_3 = \eta_c V^2_{cb} m_t/m_c$ which comes from the effect of
the large top Yukawa coupling on the RG scaling of $V_{cb} = s_3$.
While the mixing angle from the D/E sector is most accurately
determined by
$$
s'_1 =- {\sqrt{ {m_e\over m_\mu}}}\eqno(12)
$$
the angles in the $U/\nu$ sector must be determined from quark
physics via:
$$
\eqalignno{
s_2 &= \sqrt{{m_u\over m_c}}\cr
s_3 &= V_{cb}.&(13)
\cr}
$$
We now use the known numerical inputs from the $U/D/E$ sectors to
make precise numerical predictions of the neutrino masses.
The neutrino mass ratios are
$$
{m_{\nu_\tau}\over m_{\nu_\mu} } = {1 \over 3\kappa^2} \left( {B\over
A}\right)^{-2} = {1 \over 3\kappa^2} (\eta_3 V_{cb})^{-2} =
\cases{ 208 \pm 42 &(I)\cr
1870 \pm 370 &(II) \cr}\eqno(14)
$$
and
$$
{m_{\nu_\mu}\over m_{\nu_e}} = \left({C/A\over 3\kappa^2
B^2/A^2}\right)^{-2} = 9\kappa^4 \left( {m_u\over m_c}\right)^{-1} =
\cases{ 3100 \pm 1000 &(I)\cr 38 \pm 12 &(II)\cr}\eqno(15)
$$
where the ranges obtained correspond to ranges of the input parameters\\
$(V_{cb}, m_b, m_c, \alpha_s, m_u/m_d)$ which we have found to be consistent
with the quark mass and mixing predictions of our scheme [1]. We
cannot predict the overall mass scale. We find
$$ m_{\nu_\tau} = \eta_\nu {9A v^2 \over 2V}
\sin^2 \beta = .8 eV \left({m_t \over 170 GeV}\right)
\left({10^{14} GeV \over V}\right)
$$
where we have ignored electroweak contributions to $\eta_\nu$.
The most useful form for the prediction of the three mixing angles
is in terms of the predictions for neutrino oscillations from
flavor $i$ to flavor $j$. For sufficiently large $\Delta m_{ij}^2$
the oscillation probabilities can be approximated by the familiar
$2 \times 2$ case: $P_{ij} = \frac{1}{2} sin^2 2\theta_{ij}$.
We find
$$\theta_{\mu\tau}\simeq V'_{\nu_\mu\tau} = s'_3 ,$$
$$\theta_{e\mu} \simeq |V'_{\nu_e\mu}| = \left(\frac{m_e}{m_\mu} +
\frac{m_{\nu_e}}{m_{\nu_\mu}} - 2 \sqrt{\frac{m_e m_{\nu_e}}{m_\mu
m_{\nu_\mu}}} \cos \phi \right)^{1/2} $$ and
$$
\theta_{e\tau} \simeq V'_{\nu_\tau e}= s'_1s'_3 .
$$
Using Equations 11-13, we obtain
$$
sin^2 2\theta_{\mu\tau} = \cases{ (2.6 \pm 0.5) 10^{-2} &(I)\cr
(2.9 \pm 0.6)10^{-3} &(II)\cr}\eqno(16)
$$
$$
sin^2 2\theta_{e\mu} = \cases{ (1.7 \pm 0.2) 10^{-2} &(I)\cr
(9.0 \pm 4.3) 10^{-2} &(II)\cr}\eqno(17)
$$
$$
sin^2 2\theta_{e\tau} = \cases{(1.3 \pm 0.3) 10^{-6} &(I)\cr
(1.4 \pm 0.3)10^{-7} &(II).\cr}\eqno(18)
$$
The best hope for an experimental laboratory test of these numbers
is provided by $\nu_\mu \nu_\tau$ oscillations.
Present limits and the reaches of proposed experiments are shown
in the $\Delta m^2-sin^2 2\theta$ plot in Figure 1, together with
our two predictions as vertical lines.
In model I $\theta_{\mu\tau}$ is sufficiently large that the Fermilab E531
results imply that $m_{\nu_\tau} \leq 2.5$ eV. This
means that it is unlikely that planned neutrino oscillation
experiments\cite{p803,chorus,nomad} will be able to detect the neutrino
masses of this model. Given the upper bound on $m_{\nu_\tau}$ we
have, using Eq. 14, $m_{\nu_\mu} < 1.5 \times 10^{-2}$ eV or
$m^2_{\nu_\mu} < 2.3 \times 10^{-4}$ (eV)$^2$. Now given this upper
bound on $m_{\nu_\mu}$ and our value for $\theta_{e\mu}$, Eq. 17, (see
vertical line labelled I in figure 2) we find a possible resolution of the
Cl, Kamiokande and Gallium\cite{davis,kam,sage,gallex} solar neutrino
experiments by MSW oscillations, at the 90 \% confidence level. Our
value of $\theta_{e\mu}$ implies that, as the error bars on the Ga
experiments\cite{sage,gallex} are decreased, a low number of about
50$\pm10$ SNUs will result. To test this region of parameter space in the
lab would require a long baseline $\nu_\mu \nu_\tau$ oscillation search
with sensitivity to smaller mixing angles than the present proposals.
We note that the neutrinos in this solution are all too light to be a
significant component of the dark matter.
In model II $\theta_{\mu\tau}$ is just beyond the E531 limits. This is very
exciting because it means that the upcoming $\nu_\mu \nu_\tau$ oscillation
searches will probe a large range of $\Delta m^2$ in this
model\cite{p803,chorus,nomad}. In particular, if $\nu_\tau$ makes a
significant contribution to the dark matter in the universe, then O(50)
events will be seen and $\sin^2 2\theta_{\mu\tau}$ will be determined to
be within 15\% of 3.10$^{-3}$. We can still obtain an upper limit on
$m_{\nu_\tau}$ if we demand that $\nu_\tau$ doesn't
overclose the universe. We have $$m_{\nu_\tau} \leq 93 eV (\Omega_\nu
h^2) $$ with the Hubble constant $H_0 = 100 h km/s/Mpc$ and $ 1/2 \leq h
\leq 1$. For $h = 1/2$ and $\Omega_\nu = 1$ , we have $m_{\nu_\tau} < 23
$ eV. This implies, using Eq.14, $m_{\nu_\mu} < 1.5 \times 10^{-2}$ eV
or $m^2_{\nu_\mu} < 2.4 \times 10^{-4}$ (eV)$^2$.
Now consider possible MSW oscillations. We find two possible
solutions to the solar neutrino problem consistent with the ``cosmological"
upper bound on $m_{\nu_\mu}$ and our value for $\theta_{e\mu}$, Eq. 17,
(depicted as the vertical line labelled II in figure 2). In fact, if a
signal is seen in $\nu_\mu \nu_\tau$ oscillations at CERN or at Fermilab
which is consistent with our value of $\theta_{\mu\tau}$, then we predict
values of $\theta_{e\mu}$ and $m_{\nu_\mu}$ which are significant for
the Cl, Kamiokande and Gallium solar neutrino experiments. This is our
upper model II solution. In this case we find about $110$ SNUs in Ga.
Note, this region is not consistent with combined fits to the
present observations of solar
neutrinos at 90 \% CL\cite{davis,kam,sage,gallex}. Nevertheless, perhaps
it is too early to rule it out.\footnote{It might, in fact, be consistent
with the solar model with a 5\% higher core temperature\cite{bludman}.}
It is an interesting solution since, in addition to having a tau neutrino
with mass of order 20 eV and thus a significant component of the dark
matter\footnote{This could lead to a mixture of cold dark matter, some
type of neutralino, and hot dark matter which seems to be preferred by
recent COBE data\cite{cobe}}, it also provides a possible solution to
the supernova shock reheating problem. The values of $m_{\nu_\tau}$ and
$\theta_{e\tau}$ are just in the range required by Fuller et.
al.\cite{fuller} to allow for $\nu_\tau \nu_e$ oscillations in the
supernova. This results in higher $\nu_e$ energies behind the shock,
thus producing an increase in the heating rate.
If no signal is seen in $\nu_\mu \nu_\tau$ oscillations at CERN or
at Fermilab, then the neutrino mass limits are sufficiently suppressed
that there is no hope that $\nu_\mu \nu_e$ oscillations could be found
at subsequent experiments such as the long baseline proposal at
Fermilab (P822)\cite{p822}. Even if a signal were seen in $\nu_\mu
\nu_\tau$ oscillations at CERN, a subsequent signal could only be seen at
experiments such as proposed in Fermilab P822 if the $\nu_\tau$ mass were
above the cosmological limit of $\sim$ 23 eV; a limit which we would expect to
apply to $\nu_\tau$ in this theory.
A third possible MSW solution to the solar neutrino problem is seen in
figure 2 as the lower model II solution. It is not favored by GALLEX,
but is consistent with Chlorine and Kamiokande.
In conclusion, we have a very predictive model for neutrino masses and
mixing angles. In terms of just one arbitrary parameter, the overall scale
of the neutrino masses, we predict all 9 observable quantities of the
neutrino sector. In version I of our model, we find the value of
$\theta_{e \mu}$ and the allowed values for $m_\mu$ leads to $\nu_e
\nu_\mu$ resonant MSW neutrino oscillations which seems to be favored by
present experiments to solve the solar neutrino problem. In this case we
predict that with greater statistics the GALLEX and SAGE experiments will
settle on a result of around 50 SNUs. On the other hand, in version II of
our model a large region of parameter space will be probed by the NOMAD
and CHORUS experiments for $\nu_\mu \nu_\tau$ oscillations under
construction at CERN or by P803 proposed at Fermilab. For a sufficiently
large $\nu_\tau$ mass, including values for which the $\nu_\tau$
contributes a significant amount of dark matter to the universe, these
experiments will make a precision test of our theory, measuring sin$^2 2
\theta_{\mu\tau}$ to 15\% accuracy. This model is also relevant for solar
neutrino experiments, but is in a region of the $\Delta m^2 - \sin^2
2\theta_{e\mu}$ which is not consistent with the present data at 90\%
CL. Our three possible MSW solutions to the solar neutrino problem are
given in table I, along with some of their properties.
S.R. thanks the Aspen Center For Physics where part of this work was carried
out.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,110
|
package entidades;
/**
* Clase Nodo de Pila.
*/
public class NodoDePila {
/** Atributo x. */
private int x;
/** Atributo y. */
private int y;
/** Atributo ptrSiguiente. */
private NodoDePila ptrSiguiente;
/**
* Constructor de la clase Nodo de Pila.
*
* @param xValue
* valor de x donde esta el personaje
* @param yValue
* valor de y donde esta el personaje
*/
public NodoDePila(final int xValue, final int yValue) {
this.x = xValue;
this.y = yValue;
ptrSiguiente = null;
}
/**
* Pide el siguiente.
*
* @return devuelve un nodo de pila con el siguiente
*/
public final NodoDePila obtenerSiguiente() {
return ptrSiguiente;
}
/**
* Setea el siguiente.
*
* @param nodo
* nuevo nodo a setear
*/
public final void establecerSiguiente(final NodoDePila nodo) {
ptrSiguiente = nodo;
}
/**
* Pide el valor de X.
*
* @return devuelve el valor de X
*/
public final int obtenerX() {
return x;
}
/**
* Pide el valor de Y.
*
* @return devuelve el valor de Y
*/
public final int obtenerY() {
return y;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,798
|
Aura-Note is a simple note taking sample application that demonstrates many of the features and patterns used when building an Aura based application. The [Aura 101 tutorial](https://github.com/forcedotcom/aura-note/blob/master/Aura101.pdf) walks you through building a component and putting it in the app.
To find out more about Aura please see the [Aura Documentation](http://documentation.auraframework.org/auradocs) site or see the instructions at the end of this README for
accessing the documentation on your localhost after you build the project.
### Prerequisites
You need:
* Java Development Kit (JDK) 1.6
* Apache Maven 3
### Getting Started
1. Clone the repo:
`git clone https://github.com/forcedotcom/aura-note.git`
2. Change to the newly created aura-note folder:
`cd aura-note`
If you want a pre-populated sample not database just copy the file auranote.db.h2.db (in the same folder as this README) to your home folder
3. To start jetty on port 8080 run this:
`mvn jetty:run -Pdev`
4. To see the app go to:
`http://localhost:8080/auranote/notes.app`
5. To edit the app open:
`src/main/webapp/WEB-INF/components/auranote/notes/notes.app`
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,455
|
Q: Can't update array value in mysql I'm not sure what is the problem with my query, because I can't update my database with the latest value, but I can (print_r) the value.
$serial[$i]= $_POST['serial'][$i];
print_r($serial);
$a = array(1,2,3,4,5,6);
print_r( $a);
$i=0;
$i=0;
foreach($serial as $s => $m){
$sqlw = "update speciform set nam5 = '$m[$i]' where nn = '$a[$i]' AND
nam11= CURDATE()";
mysql_query($sqlw) or die(mysql_error());
$i++;
}
Below is my table for update in (HTML and PHP):
Update Table
when I echo the $serial, I get this:-
echo $serial
when I echo the $a, I get this:-echo $a
and my database is like this:-Database Table
Really need someone to help me because I'm currently blank with this problem.
A: $sqlw = "update speciform set nam5 = '$m[$i]' where nn = '$a[$i]' AND
nam11= CURDATE()";
Change this query
$sqlw = "update speciform set nam5 = '$m' where nn = '$a[$i]' AND
nam11= CURDATE()";
Because as you have taken foreach loop so array of single element is their only in $m.
A: foreach($serial as $s => $m) { ... } will loop through the array $serial as you expect.
If you write :
foreach($serial as $s => $m)
{
echo $m;
}
You'll notice that at each iteration, test1 test2 test3 and so on will be displayed.
Since $m is a string, $m[0] is the first character. Hence, $m[$i] will display the first character at first iteration, the second character at second iteration, and so on...
$sqlw = "UPDATE speciform SET nam5 = '" . $m . "' WHERE nn = '" . $a[$i] . "' AND nam11= CURDATE()";
This query is vulnerable to SQL injections but this is another topic. I strongly advice you to sanitize user inputs. Never trust a user.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,088
|
With over 20 years experience, Darren has a wealth of knowledge and is on hand to offer customer support, help and advice.
DA Components UK deliveries are all sent by Royal Mail First Class. Delivery options are available for overseas orders.
Browse through our list of favourite products and find just what you need.
We always have an offer in store. Why not checkout our blog page.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,623
|
{"url":"http:\/\/lib.physcon.ru\/doc?id=da77b037eb25","text":"# IPACS Electronic library\n\n## PERTURBATION ANALYSIS OF SIMPLE EIGENVALUES OF SINGULAR LINEAR SYSTEMS\n\nM. Isabel Garcia-Planas, Sonia Tarragona\nIn this work a study of\nthe behavior of a simple eigenvalue of singular linear system family $E(p)\\dot x=A(p)x+B(p)u$, $y=C(p)x$ smoothly dependent on a vector of real parameters $p=(p_{1},\\ldots ,p_{n})$ is presented.","date":"2018-05-27 01:12:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8472756147384644, \"perplexity\": 2765.5722388339086}, \"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-2018-22\/segments\/1526794867977.85\/warc\/CC-MAIN-20180527004958-20180527024958-00054.warc.gz\"}"}
| null | null |
{"url":"https:\/\/opentextbc.ca\/mathfortrades2\/chapter\/volume-of-a-sphere\/","text":"Volume\n\n# 16 Volume of a Sphere\n\nClick play on the following audio player to listen along as you read this section.\n\nTo the left we have a picture of our beautiful planet, Earth. Although the picture itself is two dimensional, we know that Earth is three dimensional. This implies that Earth has a volume. What is also true is that Earth is an example of a sphere.\n\nSo far we\u2019ve calculated the volume of cubes, rectangular tanks, and cylinders. Using that information, how do you think we would calculate the volume of a sphere? What variables do you think you would use? Take a few moments to think about it before we go through the explanation below.\n\nJust in case you were wondering, Earth has a volume of\u2026\n\n$\\Large 1,083,206,916,846 \\text{ cubic kilometres}$\n\nThere are two things to note here:\n\n1. The answer is once again in cubic units.\n2. That\u2019s a lot of volume.\n\nWe should start this explanation off by revisiting the formula for a circle. Remember that a circle had a radius, diameter, and circumference. Also remember that to find the formula for a circle, we could have used one of two formulas.\n\n$\\Large \\begin{array}{ll} \\text{Formula one:} & \\text{ area} = {\\text{d}}^{2} \\times 0.7854 \\times \\text{h} \\\\ \\text{Formula two:} & \\text{ area} = \\pi {\\text{r}}^{2} \\end{array}$\n\nWhen dealing with the formula for a sphere, we can use formula two as a starting point. The formula for a sphere has a similar feel to it but with a little twist. The similarities include having both pi and radius in the formula, but that is where the similarities end.\n\nHere is the formula:\n\n$\\Large \\text{volume} = \\dfrac{4}{3} \\pi {\\text{r}}^{3}$\n\nThe question becomes, \u201cWhere does the 4\/3 come from?\u201d Well the explanation for that is actually quite a long one, and I\u2019ll leave that for another day. Quickly stated, it comes from the fact that if you took two cones with similar measurements to the sphere, it would end up that the volume of those two cones would equal the volume of the sphere. Using a bit of mathematical wizardry the 4\/3 ends up being derived from this fact.\n\nThe pi in the formula is the constant that we use when finding the circumference of a circle, and the radius, as you might remember, is half the length of the diameter. The last thing is that the radius is cubed. This relates to the fact that in the end we are solving for volume, which has three dimensions.\n\nNow I didn\u2019t expect you to get the formula as it\u2019s quite a bizarre one, but understanding where it comes from helps to conceptualize the formula. As we talked about previously, the purpose of this exercise is to rely less on memorization and more on an understanding of how stuff is derived.\n\nExample\n\nFind the volume of a sphere with a radius of 7 inches.\n\nStep 1:\u00a0As usual write down the formula.\n\n$\\Large \\text{volume} = \\dfrac{4}{3} \\pi {\\text{r}}^{3}$\n\nStep 2: Plug in the variables and solve for volume\n\n$\\Large \\begin{array}{c}\\text{volume} = \\dfrac{4}{3} \\pi {\\text{r}}^{3} \\\\ \\text{volume} = \\dfrac{4}{3} \\pi {7}^{3} \\\\ \\text{volume} = \\dfrac{4}{3} \\times \\pi \\times 7 \\times 7 \\times 7 \\\\ \\text{volume}= 1436{\\text{ in}}^{3} \\end{array}$\n\nExamples\n\nLet\u2019s take a step up here and add a little twist. Find the volume of a sphere with a diameter of 24. Note that the diameter will go through the exact center of the sphere. Think of where you are standing right now, and drill a hole straight down through the Earth to the exact other side making sure to go through the dead centre of the Earth. (Disclaimer: I think there is an Arnold Schwarzenegger movie plot somewhere inside this question.) Anyway, back to the question.\n\nStep 1:\u00a0As usual, write down the formula.\n\n$\\Large \\text{volume} = \\dfrac{4}{3} \\pi {\\text{r}}^{3}$\n\nI\u2019m going to guess that by this time in the book, you\u2019ve started to see some patterns and have picked out the fact that we need to use the radius in the formula but we only have diameter. We\u2019ll have to work a little math magic, and pull the radius out of the diameter before we start.\n\n$\\Large \\begin{array}{c}\\text{diameter} = \\text{radius} \\times 2 \\\\ \\text{radius} = \\dfrac{\\text{diameter}}{2} \\\\ \\text{radius} = \\dfrac{24}{2} \\\\ \\text{radius} = 12 \\end{array}$\n\nNow we have what we need to work with.\n\nStep 2: Plug in the variables and solve for volume.\n\n$\\Large\\begin{array}{c} \\text{volume} = \\dfrac{4}{3} \\pi {\\text{r}}^{3} \\\\ \\text{volume} = \\dfrac{4}{3} \\pi {12}^{3} \\\\ \\text{volume} = \\dfrac{4}{3} \\times \\pi \\times 12 \\times 12 \\times 12 \\\\ \\text{volume}= 7234.6 {\\text{ in}}^{3} \\end{array}$\n\nHey! Should we throw in a bonus question here? Okay, let\u2019s do it. Change the volume from cubic inches into cubic feet.\n\nStep 1: Write down the formula.\n\n$\\Large 1 \\text{ cubic foot} = 1728 \\text{ cubic inches}$\n\nStep 2: Cross multiply\n\n$\\Large \\begin{array}{c}\\dfrac{1 {\\text{ ft}}^{3}}{{\\text{X ft}}^{3}} = \\dfrac{1728 {\\text{ in}}^{3}}{7234.6 {\\text{ in}}^{3}} \\\\ 1 \\times 7234.6 = \\text{X} \\times 1728 \\\\ \\text{X} = \\dfrac{7234.6}{1728} \\\\ \\text{X} = 4.19{\\text{ ft}}^{3} \\end{array}$\n\n# Practice Question\n\nTry a practice question yourself and check the video answer to see how you did.\n\nQuestion 1\n\nJosh and Jatinder are both fans of the NBA and in class they are discussing what the volume of a basketball might be. They\u2019ve figured out that the diameter of a basketball is 9.5 inches. Calculate the volume of the basketball.","date":"2022-08-14 01:29:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.8342140316963196, \"perplexity\": 335.8624691270741}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571989.67\/warc\/CC-MAIN-20220813232744-20220814022744-00751.warc.gz\"}"}
| null | null |
Q: Table view cell's label won't center align unless number of lines is 1 I have a UITableViewCell subclass alone in a xib. The cell has a UILabel subview in it. The text alignment in IB is set to center, but unless numberOfLines is set to 1, the label snaps over to left alignment (even though the "Alignment" in IB is set to center sill). I tried setting the textAlignment to NSTextAlignmentCenter in the cell's awakeFromNib, and also in the corresponding table view's cellForRowAtIndexPath:,
but still no luck, even though when I check the cell's textAlignment value at various breakpoints it's NSTextAlignmentCenter.
How can I center align multiple lines of text in a UILabel?
A: Of course, right after I post the question, I figure it out.
If the "Autoshrink" has "Tighten Letter Spacing" selected in IB, it snaps it to the left alignment. Uncheck this to have it stay centered.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,142
|
Q: Secure Null Explain I need help to fix the error for Secure Null in this part of my code in Dart
if (selectedDatum.isNotEmpty) {
time = selectedDatum.first.datum;
selectedDatum.forEach((SeriesDatum datumPair) {
measures[datumPair.series.displayName] = datumPair.datum;
});
}
And this part with error for int?
Series<double, int>(
id: 'Gasto',
colorFn: (_, __) => MaterialPalette.blue.shadeDefault,
domainFn: (value, index) => index,
measureFn: (value, _) => value,
data: data,
strokeWidthPxFn: (_, __) => 4,
)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,886
|
FROM debian:wheezy
ENV DEBIAN_FRONTEND noninteractive
# Set the locale
RUN \
apt-get update && apt-get install -y --no-install-recommends locales &&\
rm -rf /var/lib/apt/lists/* /var/cache/apt/archives/* /var/cache/debconf/*-old &&\
localedef -i en_US -c -f UTF-8 -A /usr/share/locale/locale.alias en_US.UTF-8
ENV LANG en_US.UTF-8
ENV LANGUAGE en_US:en
ENV LC_ALL en_US.UTF-8
# Define commonly used JAVA_HOME variable
ENV JAVA_HOME /usr/lib/jvm/java-8-oracle
# Install Deps
# - http://www.oracle.com/technetwork/java/javase/jre-8-readme-2095710.html
RUN \
echo deb http://ppa.launchpad.net/webupd8team/java/ubuntu trusty main | tee /etc/apt/sources.list.d/webupd8team-java.list &&\
apt-key adv --keyserver hkp://keyserver.ubuntu.com:80 --recv-keys EEA14886 &&\
apt-get update &&\
echo oracle-java8-installer shared/accepted-oracle-license-v1-1 select true | debconf-set-selections &&\
apt-get -y --no-install-recommends install oracle-java8-installer unzip procps &&\
rm -rf /var/lib/apt/lists/* /var/cache/apt/archives/* /var/cache/debconf/*-old /var/cache/oracle-jdk*-installer \
${JAVA_HOME}/src.zip \
${JAVA_HOME}/*-src.zip \
${JAVA_HOME}/db \
${JAVA_HOME}/lib/missioncontrol \
${JAVA_HOME}/lib/visualvm \
${JAVA_HOME}/lib/*javafx* \
${JAVA_HOME}/jre/lib/plugin.jar \
${JAVA_HOME}/jre/lib/ext/jfxrt.jar \
${JAVA_HOME}/jre/bin/javaws \
${JAVA_HOME}/jre/lib/javaws.jar \
${JAVA_HOME}/jre/lib/desktop \
${JAVA_HOME}/jre/plugin \
${JAVA_HOME}/jre/lib/deploy* \
${JAVA_HOME}/jre/lib/*javafx* \
${JAVA_HOME}/jre/lib/*jfx* \
${JAVA_HOME}/jre/lib/amd64/libdecora_sse.so \
${JAVA_HOME}/jre/lib/amd64/libprism_*.so \
${JAVA_HOME}/jre/lib/amd64/libfxplugins.so \
${JAVA_HOME}/jre/lib/amd64/libglass.so \
${JAVA_HOME}/jre/lib/amd64/libgstreamer-lite.so \
${JAVA_HOME}/jre/lib/amd64/libjavafx*.so \
${JAVA_HOME}/jre/lib/amd64/libjfx*.so
# Install SonarQube
ENV SONARQUBE_VERSION 4.5.6
ENV SONARQUBE_HOME /opt/sonarqube
WORKDIR /opt
RUN \
wget --no-check-certificate https://sonarsource.bintray.com/Distribution/sonarqube/sonarqube-${SONARQUBE_VERSION}.zip &&\
unzip sonarqube-${SONARQUBE_VERSION}.zip &&\
rm sonarqube-${SONARQUBE_VERSION}.zip &&\
ln -s sonarqube-${SONARQUBE_VERSION} ${SONARQUBE_HOME} &&\
# Remove unnecessary files
rm -r \
${SONARQUBE_HOME}/bin/linux-ppc-64 \
${SONARQUBE_HOME}/bin/linux-x86-32 \
${SONARQUBE_HOME}/bin/macosx-* \
${SONARQUBE_HOME}/bin/solaris-* \
${SONARQUBE_HOME}/bin/windows-*
# Upgrade SonarQube plugins
# - http://docs.sonarqube.org/display/PLUG/Plugin+Version+Matrix
# - http://www.sonarsource.com/category/plugins-news/
WORKDIR ${SONARQUBE_HOME}/extensions/plugins
RUN \
wget --no-check-certificate https://bintray.com/artifact/download/sonarsource/SonarQube/org/sonarsource/sonar-findbugs-plugin/sonar-findbugs-plugin/3.3/sonar-findbugs-plugin-3.3.jar &&\
wget --no-check-certificate https://sonarsource.bintray.com/Distribution/sonar-java-plugin/sonar-java-plugin-3.7.1.jar
# Add a directory to process setup scripts for the container
RUN mkdir /docker-entrypoint-init.d
COPY docker-entrypoint.sh /
# forward sonar logs to docker log collector
RUN ln -sf /dev/stdout ${SONARQUBE_HOME}/logs/sonar.log
EXPOSE 9000
ENTRYPOINT ["/docker-entrypoint.sh"]
CMD ["sonar"]
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,554
|
Is shouldice treatment a good one to repair my hernia?
what sort of a doctor do I need to find?
Help, my doctor was not very approachable! Dr Bilkey anyone?
Yay! First consulation appointment over, 2 more visits to go!
How do I best approach my family doctor?
WHY bother with a assessment?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,484
|
Q: Calling Webservice from Apex class, getting 302 response error I have the below code trying to make a HTTP callout from a APEX class. The response I get back is
System.HttpResponse[Status=Moved Temporarily, StatusCode=302]
My class is
public with sharing class soapServiceApexClass {
@future(callout=true)
@AuraEnabled
public static void createSOAPWebSevice(Integer x, Integer y){
partnerSoapSforceCom.Soap myPartnerSoap = new partnerSoapSforceCom.Soap();
partnerSoapSforceCom.LoginResult partnerLoginResult = myPartnerSoap.login('username', 'pwd');
system.debug('partnerLoginResult'+partnerLoginResult);
soapSforceComSchemasClassMywebservi.SessionHeader_element webserviceSessionHeader = new soapSforceComSchemasClassMywebservi.SessionHeader_element();
webserviceSessionHeader.sessionId = partnerLoginResult.sessionId;
soapSforceComSchemasClassMywebservi.MyWebService mySOAPWebservice = new soapSforceComSchemasClassMywebservi.MyWebService();
mySOAPWebservice.SessionHeader = webserviceSessionHeader;
Integer z = mySOAPWebservice.Mul(1,2);
system.debug ('z========='+z);
}
@AuraEnabled(continuation=true)
public static map<String, Object> sendRequest(integer x,integer y) {
map<String, Object> mapJsonData = new map<String, Object>();
Continuation con = new Continuation(40);
string strResponse;
con.continuationMethod='getResponse';
HttpRequest httpRequest = new HttpRequest();
httpRequest.setMethod('GET');
httpRequest.setEndpoint('https://datasirpicnat-dev-ed.lightning.force.com/');
//loc : https://datasirpicnat-dev-ed.my.salesforce.com/
try {
Http http = new Http();
HttpResponse httpResponse = http.send(httpRequest);
system.debug('res body=========='+httpResponse.getBody());
System.debug('httpResponse'+httpResponse);
boolean redirect = false;
if (httpResponse.getStatusCode() >=300 && httpResponse.getStatusCode() <= 307 && httpResponse.getStatusCode() != 306) {
do {
redirect = false;
String loc = httpResponse.getHeader('Location');
system.debug('loc =========='+loc);
if(loc == null) {
redirect = false;
continue;
}
httpRequest req = new HttpRequest();
req.setEndpoint(loc);
req.setMethod('GET');
HttpResponse res = http.send(req);
if(httpResponse.getStatusCode() != 500) {
if(res.getStatusCode() >=300 && res.getStatusCode() <= 307 && res.getStatusCode() != 306) {
redirect= true;
}
system.debug('Inside this if');
}
system.debug('body of the 302 res is'+ res.getBody());
if (res.getStatusCode() == 200 ) {
strResponse = res.getBody();
system.debug('strResponse====='+strResponse);
}
} while (redirect && Limits.getCallouts() != Limits.getLimitCallouts());
}
system.debug('================================================');
if (httpResponse.getStatusCode() == 200 ) {
strResponse = httpResponse.getBody();
system.debug('strResponse====='+strResponse);
}
else {
throw new CalloutException(httpResponse.getBody());
}
system.debug('after callouts ========= '+httpResponse.getBody());
}
catch(Exception ex) {
throw ex;
}
if(!String.isBlank(strResponse)) {
mapJsonData = (map<String, Object>)JSON.deserializeUntyped(strResponse);
System.debug('mapJsonData ===> '+mapJsonData);
}
return mapJsonData;
}
}
also tried to handle the statuscode 302 in my class
exception System.CalloutException and not displaying any message same status code remains as 302 in developer console
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,224
|
{"url":"https:\/\/physics.stackexchange.com\/questions\/255464\/can-we-see-the-past-view-of-our-galaxy-while-observing-in-the-sky-because-of-the","text":"# Can we see the past view of our galaxy while observing in the sky because of the homogeneous structure of our universe?\n\nAs a sense, we think that simple speed addition should work ($V_1+V_2$) for all references but we know today that universe does not match our this sense. We cannot exceed the speed of light in any references.\n\nThis example shows that we can fail if we use our senses to explain some concepts in universe. We feel a line should go endless without a loop to itself but every definition of big bang mentions that the whole universe is homogeneous and there is no big bang center. If so, I have some questions about our universe geometry:\n\n1. Can a line go to endless in the universe as we imagine if the distribution of matters in the universe is homogeneous after bing-bang ?\n2. Is there any border of observed universe if universe is expanding?\n3. If the universe is homogeneous and it is expanding as similar as a balloon surface like sphere, can we see the past view of our galaxy while observing in the sky?\n\nI will appreciate if someone explains more detail how the universe geometry structure is if we have a homogeneous universe after big bang.","date":"2019-12-08 02:29:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2518463134765625, \"perplexity\": 434.1618646925541}, \"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-2019-51\/segments\/1575540504338.31\/warc\/CC-MAIN-20191208021121-20191208045121-00332.warc.gz\"}"}
| null | null |
from itertools import repeat
def is_immutable(self):
raise TypeError('%r objects are immutable' % self.__class__.__name__)
class ImmutableDict(dict):
_hash_cache = None
@classmethod
def fromkeys(cls, keys, value=None):
return cls(zip(keys, repeat(value)))
def __reduce_ex__(self, protocol):
return type(self), (dict(self),)
def __hash__(self):
if self._hash_cache is not None:
return self._hash_cache
rv = self._hash_cache = hash(frozenset(self.items()))
return rv
def setdefault(self, key, default=None):
is_immutable(self)
def update(self, *args, **kwargs):
is_immutable(self)
def pop(self, key, default=None):
is_immutable(self)
def popitem(self):
is_immutable(self)
def __setitem__(self, key, value):
is_immutable(self)
def __delitem__(self, key):
is_immutable(self)
def clear(self):
is_immutable(self)
def __repr__(self):
return '%s(%s)' % (
self.__class__.__name__,
dict.__repr__(self),
)
def copy(self):
"""Return a shallow mutable copy of this object. Keep in mind that
the standard library's :func:`copy` function is a no-op for this class
like for any other python immutable type (eg: :class:`tuple`).
"""
return dict(self)
def __copy__(self):
return self
class ConstantsObject(ImmutableDict):
def __getattr__(self, name):
return self[name]
def __setattr__(self, name, value):
self[name] = value
def __dir__(self):
return self.keys()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,892
|
\section{Introduction}\label{sec:introduction}}
\else
\section{Introduction}
\label{sec:introduction}
\fi
\IEEEPARstart{R}{ecommender} systems (RS) have become extremely common in recent years, and are applied in a variety of domains, from virtual community web sites like movielens.org to electronic commerce companies like amazon.com. In spite of the widespread application of RS, one difficult and common problem is the cold-start problem, where no prior events, like ratings or clicks, are known for certain users or items. The user cold-start problem may lead to the loss of new users due to the low accuracy of recommendations in the early stage. The item cold-start problem may make the new item miss the opportunity to be recommended and remain ``cold'' all the time. In this paper, we focus on the item cold-start problem, where recommendations are required for items that no one has yet rated.
\begin{figure}[tb!]
\begin{center}
\includegraphics[scale=0.39]{takenDis.pdf}
\end{center}
\vspace*{-10pt}
\caption{Attribute information and users' rating distributions of films \emph{Taken} and \emph{Taken 3}. Genres of these two films are both \emph{Action} and \emph{Thriller}. Common scriptwriters and actors are connected by red lines. The overall ratings of \emph{Taken} are high with a mean equal to 8.0 and the mean rating is 6.3 for \emph{Taken 3}. Attributes are modeled as features in this paper, whose values are 1 if corresponding attributes exist or 0 if they do not exist. In this example, features include all genres, directors, scriptwriters and actors.
}
\label{fig:takenDis}
\vspace*{-10pt}
\end{figure}
Content information, such as item attributes, were exploited to address such issues in previous methods \cite{hong:2013co,gantner:2010learning,hauger:2008comparison}. However, items with similar attributes may be of different interest for the same user. As shown in Figure \ref{fig:takenDis} (data is collected from \emph{IMDB} \footnote{http://www.imdb.com/}), movie \emph{Taken} is favored by many people after release, with a mean rating equal to 8.0. When the follow-up \emph{Taken 3} was first released in 2014, it can be seen as a ``cold'' film. Since genres, screenwriters and many actors of these two films are the same, then if we exploit film attributes to perform hybrid recommendations, we may recommend this ``cold'' film to users who favored \emph{Taken} before. However, as can be seen from the figure, the peak of \emph{Taken 3}'s overall ratings moves down to rating 6, which means that many users might favor \emph{Taken} but would give low ratings to \emph{Taken 3}. One reason could be that, although \emph{Taken 3} and \emph{Taken} have many attributes in common, \emph{Taken 3} has a lower quality than \emph{Taken}, thus users who favored \emph{Taken} before may dislike \emph{Taken 3}, i.e. the recommendation of \emph{Taken 3} to users who favored \emph{Taken} before might not be accurate. Therefore, it is not a safe way to handle the cold-start issue based on film attributes only. A natural solution is to select a small set of users to watch this ``cold'' film first, whose feedback can give us more understanding of users' preferences on this ``cold'' film. Then we can perform more accurate recommendations. Interestingly, this is similar to the key idea of active learning in the machine learning literature \cite{rubens:2015active}.
Most works that apply active learning to recommender systems focus on the user cold-start problem \cite{elahi:2014active,rashid:2008learning,golbandi:2011adaptive}. New users' preferences are typically obtained by directly interviewing the new user about what his interest is, or asking him to rate several items from carefully constructed seed sets. Seed sets may be constructed based on popularity, contention and coverage \cite{rashid:2008learning}. Items in these constructed seed sets will be rated by every new user. However, the item cold-start problem is different because items cannot be interviewed and typically there are no users willing to rate every new item. Thus we need to construct different user sets to rate different new items, ensuring that users are not always selected for rating requests. In addition, the user set for each new item must be carefully constructed so that we can learn as much as possible about the new item given a limited number of rating requests. However, limited works have been conducted to address the item cold-start problem by active learning. \cite{aharon:2015excuseme,anava:2015budget} use the active learning idea but ignore items' attribute information. Meanwhile, they select users based on limited criteria. In fact, the new item's attributes give us some understanding of this item and can be exploited to improve our user selection strategy. For example, we tend to select users who favor attributes existing in the new item, since these users are more willing to give ratings.
In this paper, we propose a novel recommendation framework for the item cold-start problem, where items' attributes are exploited to improve active learning methods in recommender systems. The attribute-driven active learning scheme has following characteristics:
\begin{itemize}
\item Explicitly distinguishing 1) whether a user will rate the new item and 2) what rating the user will give to the new item. The former helps us to select users who are willing to give ratings to the new item (feedback ratings). The latter allows us to exploit the rating distribution to improve the selection strategy. For example, we expect to select users who give diverse ratings to generate unbiased predictions. This is easy to understand since if we select users who all give high ratings, then the trained prediction model will generate high biased ratings for all other users, though the other users may not favor the new item at all.
\item Personalized selection strategy to ensure fairness. We construct our selection strategy based on four criteria, of which two are personalized criteria. The personalized criteria ensure that for new items with different attributes, users selected by our method would be different. This can avoid selecting the same user to rate every new item, which will negatively influence the user experience. These criteria are uniformly modeled as an integer quadratic programming (IQP) problem, which can be efficiently solved by some relaxation.
\item Dynamic active learning budget. In previous active learning works \cite{huang:2007selectively,anava:2015budget}, the \emph{budget} of active learning (i.e. the number of users selected for rating requests) for a new item is fixed. However, in real-world applications, 1) some new items are under the attention of a small set of users (not popular), e.g. films with unpopular actors and directors, and 2) some would be obviously favored by almost all users (popular and not controversial), e.g. \emph{Harry Potter and the Deathly Hallows: Part 2} \footnote{http://www.imdb.com/title/tt1201607/}, while 3) others are popular but controversial, and the recommender is not sure about users' preferences on them, e.g. although \emph{Taken 3} is famous, the qualities of films previously acted by its main actors vary a lot, thus it is difficult to predict users' preferences on \emph{Taken 3}. It is the items in the third case that need more feedback ratings so as to be learned more about. In this paper, we are the first to propose a dynamic active learning budget so that the limited active learning resources will be properly distributed, which can improve the overall prediction accuracy.
\item Considering \emph{exploitation}, \emph{exploration} and their trade-off. Traditional active learning methods aim at maximizing the performance measured on unselected instances in the prediction phase \cite{chattopadhyay:2012batch,chakraborty:2015batchrank}, regardless of the cost in the active learning phase, since they assume the labeling cost for each instance is the same. However, in our active learning phase, we prefer a rating request for a user who is willing to rate the item rather than a user who is not, because the latter one will negatively influence the user experience. Our solutions are inspired by \cite{rubens:2007influence, rokach:2008pessimistic}, which try to maximize the sum of \emph{rewards} by balancing the trade-off of \emph{exploitation} and \emph{exploration}. The \emph{rewards} in our task contain two parts, i.e. the user experience in the active learning phase and the prediction phase, respectively. By exploiting ``existing knowledge'' (\emph{exploitation}) from the model trained in Figure 3 (b), we are able to select willing users to obtain good user experience in the active learning phase. For the user experience in the prediction phase, we want to learn as much ``new knowledge'' (\emph{exploration}) about unselected users' preferences as possible, so as to generate accurate rating predictions for them. Note that users selected that best satisfy \emph{exploitation} may not be the most helpful for \emph{exploration}. Therefore, the ``exploitation-exploration trade-off'' in our task lies in how we optimize our user selection strategy, in order to obtain relatively good user experience in both of the active learning phase and the prediction phase. Our method considers both of these two goals and can further balance their trade-off by adjusting the parameter setting.
\end{itemize}
\section{Related Work}
\subsection{The Item Cold-start Problem}
To address the item cold-start problem, a common solution is to perform hybrid recommendations by combining content information and collaborative filtering \cite{agarwal:2009regression,gunawardana:2008tied,park:2009pairwise,nasery:2016recommendations}. A regression-based latent factor model is proposed in \cite{agarwal:2009regression} to address both cold and warm item recommendations in the presence of items' features. Items' latent factors are obtained by low-rank matrix decomposition. \cite{park:2009pairwise} solve a convex optimization problem, instead of the matrix decomposition, to improve this work. Another approach based on Boltzmann machines is proposed in \cite{gunawardana:2008tied, gunawardana:2009unified} to solve the item cold-start problem, which also combines content and collaborative information. LCE \cite{saveski:2014item} exploits the manifold structure of the data to improve the performance of hybrid recommendations. Other works are under a different setting where few ratings of new items exist, but no items' attribute information is known. \cite{aharon:2012dynamic, aizenberg:2012build} use a linear combination of raters' latent factors weighted by their ratings to estimate new items' latent factors.
\subsection{Active Learning in Recommender Systems}
Most active learning methods in recommender systems focus on the user cold-start problem, where they select items to be rated by newly-signed users \cite{rubens:2015active,elahi:2016survey}. We briefly introduce these methods since most of them can also be adapted to our new item task. The Popularity strategy \cite{golbandi:2010bootstrapping,golbandi:2011adaptive} and the Coverage strategy \cite{golbandi:2010bootstrapping} are two representative attention-based methods, where the former one selects items that have been frequently rated by users and the latter one selects items that have been highly co-rated with other items. Uncertainty reduction methods aim at reducing the uncertainty of rating estimates \cite{golbandi:2010bootstrapping,rubens:2007influence,rokach:2008pessimistic}, model parameters \cite{hofmann:2003collaborative, jin:2004bayesian} and decision boundaries \cite{danziger:2007choosing}. Error reduction methods try to reduce the prediction error on the testing set by either 1) optimizing the performance measure (e.g. minimizing \emph{RMSE}) on the training set \cite{golbandi:2010bootstrapping,golbandi:2011adaptive}, or 2) directly controlling the factors that influence the prediction error on the testing set \cite{rubens:2009output, settles:2008multiple}. \cite{harpale:2008personalized} uses some initial ratings to perform personalized active learning in a non-attribute context. There are also combined strategies \cite{rubens:2007influence,mello:2010active,elahi:2012adapting} considering several objectives at the same time. When applied to our new item task, some of these works require a few initial ratings on new items, which are not available in our task. The other works do not need initial ratings, but perform active learning regardless of the content information. However, the new item's content information gives us some understanding of the new item and we can exploit it to better perform active learning. In addition, methods such as the Popularity strategy \cite{golbandi:2010bootstrapping,golbandi:2011adaptive} and the Coverage strategy \cite{golbandi:2010bootstrapping} always select the same set of users, which negatively influence the user experience.
\cite{anava:2015budget,aharon:2015excuseme} are works which also address the item cold-start problem in an active learning scheme. However, they both focus on the pure collaborative filtering model and do not consider the content information either.
\subsection{The Exploitation-exploration Trade-off}
Some works also consider the exploitation-exploration trade-off \cite{rubens:2007influence, rokach:2008pessimistic}. Many of the promising solutions come from the study of the multi-armed bandit problem \cite{feldman:2015recommendations}. The key idea of these solutions is to simultaneously optimize one's decisions based on existing knowledge (i.e. \emph{exploitation}) and new knowledge which would be acquired through these decisions (i.e. \emph{exploration}), in order to maximize the sum of rewards earned through a sequence of actions. The $\epsilon$-Greedy algorithm \cite{watkins:1989learning} selects the arm which has the best estimated mean reward with probability $1 - \epsilon$, and otherwise randomly selects an other arm. UCB-like (UCB refers to Upper Confidence Bound) algorithms \cite{auer:2002using, dani:2008stochastic, abbasi:2011improved} firstly calculate the confidence bound of all arms and then select the arm with the largest upper confidence bound. The insight is that arms with a large mean reward (exploitation) and high uncertainty (exploration) would have large upper confidence bound. Thompson Sampling algorithms \cite{thompson:1933likelihood, agrawal:2012analysis, russo:2014learning} firstly calculate the probability distribution of the mean reward for each arm, then draw a value from each distribution and finally select the arm with the largest drawn value.
Our task and the multi-armed bandit problem share some common features, e.g. both considering the exploitation-exploration trade-off. However, they have some key differences. In the multi-armed bandit problem, the arms are selected one by one and a reward is generated immediately after each arm is selected. Hence, many solutions (e.g. UCB-like algorithms, Thompson Sampling algorithms) design their selecting strategies based on previous rewards. However, in our setting, a batch of users are selected at the same time, without knowing other users' feedback (reward), thus many solutions for the multi-armed bandit problem cannot be applied to our task.
\section{preliminaries and model}
\subsection{Task Definition and Solution Overview}
We use $U$, $I$ and $A$ to denote the users, items and attributes set, respectively. Our task is: given a user-item rating matrix $\mathbf{R} \in R^{|U| \times |I|}$, an item-attribute matrix $\mathbf{T} \in R^{|I| \times |A|}$ and a new item $i_{new}$, whose attributes are denoted as a vector $\mathbf{i} \in R^{1 \times |A|}$, predict users' ratings on the new item $predict\_rating(u,i_{new}), u\in U$. $\mathbf{R}$, $\mathbf{T}$ and $\mathbf{i}$ are shown in Figure \ref{fig:matrix representation}. In this paper, the task is solved via two phases. The first one is the active learning phase, which is to solve: which users should be selected to rate $i_{new}$, so as to learn about $i_{new}$ as much as possible. The second one is the prediction phase, which is to solve: once given selected users' feedback, how to accurately predict the other users' ratings on $i_{new}$. For the active learning phase, we carefully select users based on four useful criteria, which involve both classification tasks and regression tasks. For the prediction phase, we model it as a pure regression task. We use Factorization Machines \cite{rendle:2012factorization} to model all classification tasks and regression tasks in these two phases.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.5]{ProblemDef_3.pdf}
\end{center}
\vspace*{-10pt}
\caption{Representations of user-item matrix $\mathbf{R} \in R^{|U| \times |I|}$, item-attributes matrix $\mathbf{T} \in R^{|I| \times |A|}$ and the vector $\mathbf{i} \in R^{1 \times |A|}$ representing a new item with attributes.
}
\label{fig:matrix representation}
\vspace*{-10pt}
\end{figure}
\subsection{Factorization Machines}
Factorization Machines(FM) is a state-of-art framework for latent factor models which can incorporate rich features. Here we briefly introduce it. Please refer to \cite{rendle:2012factorization} for a more detailed description. The prediction problem is described by a matrix $\mathbf{X} \in R^{N\times D}$ and a vector $\mathbf{y}\in R^{N\times 1}$, where each row $\mathbf{x}\in R^{1\times D}$ of $\mathbf{X}$ is one instance with $D$ real-valued features and each entry $y$ in $\mathbf{y}$ is the label of one instance. FM can model nested feature interactions up to an arbitrary order between the $D$ features of $\mathbf{x}$. For feature interactions up to $2$-order, they are modeled as:
\begin{equation}\label{eq:FM}
\begin{aligned}
y^\prime(\mathbf{x}) = &w(0) + \sum_{i=1}^D w(i)x(i)\\
&+\sum_{i=1}^D\sum_{j=i+1}^D x(i)x(j)\sum_{f=1}^k V(i,f)V(j,f),
\end{aligned}
\end{equation}
where $k$ is the dimensionality of the factorization and the model parameters $\mathbf{\Theta} = \{w(0),\mathbf{w},\mathbf{V}\}$ are: $w(0)\in R, \mathbf{w}\in R^{D\times 1}, \mathbf{V}\in R^{D\times k}$. $w(i)$, $x(i)$ and $V(i,f)$ are entries of $\mathbf{w}$, $\mathbf{x}$ and $\mathbf{V}$, respectively. The form of FM (Equation (\ref{eq:FM})) is very general and can be applied to many applications.
In our task, we need to predict 1) what rating a user will give to an item and 2) whether a user will rate an item.
For the regression task to predict what rating a user will give to an item, features can contain users, items and attributes of items, and labels are ratings. The first term of the right-hand side in Equation (\ref{eq:FM}) is a bias of the system. If $w(0)$ is large, then there is a bias towards high ratings, which may be due to the good user experience of the system design. The second term is a bias of unary features, that is, some optimistic users tend to give high ratings to every item, and some popular items or items with popular attributes always gain high ratings. The last term is a bias of feature interactions. Many users only give high ratings to certain items (or items with certain attributes) which they are really interested in.
For the classification task to predict whether a user will rate an item, the analysis of each term in Equation (\ref{eq:FM}) is similar, except that labels now represent whether users will rate items or not.
\section{Our method}
\subsection{Select Users to Rate the New Item}
As described in section 3.1, we first need to carefully select users to rate the new item $i_{new}$, so as to learn about $i_{new}$ as much as possible. Users are selected based on the following four criteria.
(1) Selected users are with high possibility to rate $i_{new}$. This can be modeled as a classification task. To achieve this, we first transform the user-item rating matrix $\mathbf{R}$ to a 0-1 matrix $\mathbf{R}^{01}$ \cite{koren:2008factorization}, where all entries with ratings are assigned to 1, and all entries with no ratings are assigned to 0 (see Figure \ref{fig:chooseToRate} (a)). Then we use FM to model them \cite{deldjoo:2016using}, where all entries equal to 1 are regarded as positive instances, and a same number of negative instances are selected from the entries equal to 0. Features contain users and attributes (without items). Labels are 1 for positive instances and 0 for negative instances. The classification model is trained based on $\mathbf{R}^{01}$ and $\mathbf{T}$. The general process is shown in Figure \ref{fig:chooseToRate} (b).
Finally, a vector $\mathbf{p}$ is defined as follows:
\begin{equation}\label{eq:high pro to rate}
\begin{aligned}
p(m) = willing\_score(u_m,i_{new}), u_m\in U,
\end{aligned}
\end{equation}
where $willing\_score(u_m,i_{new})$ is the possibility that user $u_m$ will rate item $i_{new}$, which is predicted by our learned classification model. We tend to select $u_m$ if $p(m)$ is large.
\begin{figure*}[htb!]
\begin{center}
\subfigure[Representations of $\mathbf{R}$, $\mathbf{R}^{01}$ and $\mathbf{T}$]{
\includegraphics[scale=0.45]{chooseToRatea_1.pdf}
}
\subfigure[Factorization Machines to model whether users will rate items]{
\begin{minipage}[b]{0.43\textwidth}
\centering
\includegraphics[scale=0.3]{chooseToRateb_1.pdf}
\end{minipage}
}
\subfigure[Factorization Machines to model what ratings users will give to items ]{
\begin{minipage}[b]{0.43\textwidth}
\centering
\includegraphics[scale=0.3]{chooseToRatec_1.pdf}
\end{minipage}
}
\end{center}
\vspace*{-10pt}
\caption{(a) shows how $\mathbf{R}$ is transformed to $\mathbf{R}^{01}$ \cite{koren:2008factorization}. (b) is the classification model. Each row is an instance, which corresponds to an entry of $\mathbf{R}^{01}$. For example, the first row corresponds to an entry of $\mathbf{R}^{01}$ with row 1 and column 1. Row 1 indicates the first user, thus the feature set labeled as `U' is (1,0,0,0) (one-hot representation). Column 1 indicates the first item, thus the feature set labeled as `A' is equal to the first row of $\mathbf{T}$, i.e. attributes of the first item. The first three rows correspond to $\mathbf{R}^{01}$'s three blue entries in (a). For testing, each user is predicted to show whether he will rate the new item. (c) is the regression model. Each row is an instance, which corresponds to an entry of $\mathbf{R}$. The first two rows correspond to $\mathbf{R}$'s two blue entries in (a). For testing, each user is predicted to show what rating he will give to the new item.
}
\label{fig:chooseToRate}
\vspace*{-10pt}
\end{figure*}
(2) Selected users' \emph{potential ratings} are diverse. Potential ratings are users' ratings on $i_{new}$ purely estimated according to $i_{new}$'s attributes (without feedback since there is no feedback yet). We expect selected users' potential ratings are diverse, so that: 1) selected users tend to have different interest. Ratings of these users would provide more information compared to ratings of similar users, and 2) the final prediction model trained on these users' feedback would not be biased to a fixed region of ratings. To choose users with diverse potential ratings, we firstly train a regression model based on $\mathbf{R}$ and $\mathbf{T}$, as shown in Figure \ref{fig:chooseToRate} (c). In this regression model, features contain users and attributes (without items). Labels are users' ratings. Once the regression model is learned, all users' potential ratings on the new item $P_r(u_m, i_{new}), u_m\in U$ can be estimated. Secondly, pair-wise diverse values among all these ratings are calculated to form the diverse matrix $\mathbf{D}$. The diverse value between potential ratings of $u_m$ and $u_n$ is defined as:
\begin{equation}\label{eq:diverse function}
\begin{aligned}
D(m,n) &= |P_r(u_m, i_{new}) - P_r(u_n, i_{new})|^{\frac{1}{2}}.
\end{aligned}
\end{equation}
Calculating $\mathbf{D}$ is computationally expensive. However, the calculations of diverse values are independent with each other. Therefore, when applied to real world recommender systems, they can be performed parallelly with acceleration techniques such as GPU acceleration \cite{tarditi:2006accelerator}, distributed computing \cite{bokhari:2012assignment}, etc. We tend to select $u_m$ and $u_n$ together if $D(m,n)$ is large.
(3) Selected users' generated ratings are \emph{objective}. A rating on an item is objective means that this rating approximates the average of all ratings on this item, which is a good estimation of this item's quality \cite{duan:2008online, chevalier:2006effect, dellarocas:2004exploring}. We favor selecting users who always generate objective ratings in the past. Then they are expected to also generate objective ratings for $i_{new}$. We form a vector $\mathbf{o}$, which consists of all users' objective values. The objective value of user $u_m$ is defined as:
\begin{equation}\label{eq:high objective}
\begin{aligned}
o(m) &= \frac{1}{\log |I(u_m)| + 1}\cdot \frac{1}{|I(u_m)|}\cdot \sum_{i_n\in I(u_m)} (R(m,n)-\overline{R(n)})^2,
\end{aligned}
\end{equation}
where $I(u_m)$ is the item set that $u_m$ has rated. $R(m,n)$ is $u_m$'s rating on $i_n$. $\overline{R(n)}$ is the mean rating on $i_n$. $\frac{1}{\log |I(u_m)|}$ is a penalty for users who have rated few items, since a user may generate a rating that approximates $\overline{R(n)}$ by coincidence. Note that a smaller $o(m)$ indicates that $u_m$ is more objective, so we tend to select $u_m$ if $o(m)$ is small.
Users selected with this criterion would give ratings that can better reflect the quality of items, i.e. higher for items with better quality and verse vice. Therefore, with this criterion, the re-trained prediction model would generate overall higher/lower prediction ratings for new items with better/worse quality, which is more reasonable. This criterion is a complement to Criterion (2). Criterion (2) encourages the feedback ratings of selected users to have a large variance, thus it could increase the model's differentiation power in terms of different users. With Criterion (3), we want the average of feedback ratings to be higher/lower for items with better/worse quality, which could increase the model's differentiation power in terms of different new items.
(4) Selected users are \emph{representative}. A selected user is representative means that this user is similar to unselected users. Selected users should be representative so that from their feedback, we can learn more about the preference of unselected users. To achieve this, we firstly construct a similarity matrix $\mathbf{S}$ from users' rating history. That is, each user is represented as a row vector of the user-item matrix $\mathbf{R}$, then the similarity between two users can be measured based on their vectors. $\mathbf{S}$ is defined as:
\begin{equation}\label{eq:representative}
\begin{aligned}
S(m,n)=
\begin{cases}
Sim(R(m,:),R(n,:))& \mbox{if $m\neq n$}\\
0 &\mbox{if $m = n$}
\end{cases},
\end{aligned}
\end{equation}
where $R(m,:)$ and $R(n,:)$ are vectors of users $u_m$ and $u_n$, and $Sim(R(m,:),R(n,:))$ is their similarity. In this paper, cosine similarity is used to measure it. Acceleration techniques described in Criterion (2) can also be applied to calculating $\mathbf{S}$. We tend to select one of $u_m$ and $u_n$ if $S(m,n)$ is large. Criterion (2) and Criterion (4) are highly related to the \emph{avoiding redundancy} and \emph{ensuring representativeness} described in \cite{chattopadhyay:2012batch}, which can be interpreted from the perspective of minimizing the distribution difference between the labeled and unlabeled data.
We now formulate the user selection task as an explicit mathematical optimization problem, where the objective is to select a batch of users based on above criteria. Specifically, we define a binary vector $\mathbf{q}$ with $|U|$ entries ($\mathbf{q} \in \{0, 1\}^{|U|\times 1}$), where each entry $q(m)$ denotes whether $u_m$ will be included in the batch ($q(m) = 1$) or not ($q(m) = 0$). Thus our user selection strategy (with given batch size $k$) can be expressed as the following integer quadratic programming (IQP) problem:
\begin{equation}\label{eq:initial objective function}
\begin{aligned}
\max_\mathbf{q} \mbox{~} &{\alpha\sum_{m=1}^{|U|} q(m)p(m) + \beta \sum_{m=1}^{|U|} \sum_{n=1}^{|U|} q(m)q(n)D(m,n)}\\
& - \gamma \sum_{m=1}^{|U|} q(m)o(m)+ \sigma \sum_{m=1}^{|U|} \sum_{n=1}^{|U|} q(m)(1-q(n))S(m,n)\\
&\mbox{s.t.~~~} q(m)\in \{0,1\}, \forall m\mbox{~~~and~~~} \sum_{m=1}^{|U|}q(m) = k.
\end{aligned}
\end{equation}
The first term is to satisfy Criterion (1). Supposing $u_m$ is with high possibility to rate $i_{new}$ ($p(m)$ is large), then in order to optimize the objective function, $u_m$ is encouraged to be selected ($q(m)$ is encouraged to be 1). The second term is to satisfy Criterion (2). Supposing potential ratings of $u_m$ and $u_n$ are very diverse ($D(m,n)$ is large), then $u_m$ and $u_n$ are encouraged to be selected together ($q(m)$ and $q(n)$ are encouraged to be 1 together) in order to optimize the objective function. The third term is to satisfy Criterion (3), whose analysis is similar to the analysis for the first term, except that we minus this term, since we want to select $u_m$ when $o(m)$ is small. The last term enforces selected users to be similar to unselected users, ensuring representativeness, which satisfies Criterion (4). This can be similarly analyzed as for the second term. $\alpha$, $\beta$, $\gamma$ and $\sigma$ are trade-off parameters.
Equation (\ref{eq:initial objective function}) can be reformulated as:
\begin{equation}\label{eq:initial objective function 2}
\begin{aligned}
&\alpha\mathbf{q}^T \mathbf{p} + \beta \mathbf{q}^T \mathbf{D} \mathbf{q} - \gamma\mathbf{q}^T \mathbf{o} + \sigma \mathbf{q}^T \mathbf{S} (\mathbf{1}-\mathbf{q})\\
=&\alpha\mathbf{q}^T \mathbf{p} + \beta \mathbf{q}^T \mathbf{D} \mathbf{q} - \gamma\mathbf{q}^T \mathbf{o} + \sigma \mathbf{q}^T \mathbf{S}\mathbf{1}-\sigma \mathbf{q}^T \mathbf{S}\mathbf{q}\\
=&\mathbf{q}^T(\alpha\mathbf{p} - \gamma\mathbf{o}+\sigma\mathbf{S}\mathbf{1}) + \mathbf{q}^T (\beta \mathbf{D} - \sigma\mathbf{S})\mathbf{q}\\
=&\mathbf{q}^T diag(\alpha\mathbf{p} - \gamma\mathbf{o}+\sigma\mathbf{S}\mathbf{1})\mathbf{q} + \mathbf{q}^T (\beta \mathbf{D} - \sigma\mathbf{S})\mathbf{q}\\
=&\mathbf{q}^T \mathbf{M} \mathbf{q}\\
&\mbox{s.t.~~~} q(m)\in \{0,1\}, \forall m\mbox{~~~and~~~} \sum_{m=1}^{|U|}q(m) = k,
\end{aligned}
\end{equation}
where $\mathbf{1}$ is a vector with all entries equal to $1$ and $diag(\alpha\mathbf{p} - \gamma\mathbf{o}+\sigma\mathbf{S}\mathbf{1})$ is a diagonal matrix, whose $(i,i)$-th entry is equal to the $i$-th entry of ($\alpha\mathbf{p} - \gamma\mathbf{o}+\sigma\mathbf{S}\mathbf{1}$). Since we have the constraint $q(m)\in \{0,1\}, \forall m$, thus we derive $\mathbf{q}^T(\alpha\mathbf{p} - \gamma\mathbf{o}+\sigma\mathbf{S}\mathbf{1}) = \mathbf{q}^T diag(\alpha\mathbf{p} - \gamma\mathbf{o}+\sigma\mathbf{S}\mathbf{1})\mathbf{q}$. $\mathbf{M}$ is defined as:
\begin{equation}\label{eq:final matrix}
\begin{aligned}
M(m,n)=
\begin{cases}
\beta D(m,n)-\sigma S(m,n)& \mbox{if $m\neq n$}\\
\alpha p(m) - \gamma o(m) + \sigma \mathbf{S} \mathbf{1}(m) &\mbox{if $m = n$}
\end{cases}.
\end{aligned}
\end{equation}
Finally, the objective function is transformed to:
\begin{equation}\label{eq:final objective function}
\begin{aligned}
&\max_\mathbf{q}{\mathbf{q}^T \mathbf{M} \mathbf{q}},\\
&\mbox{s.t.~~~} q(m)\in \{0,1\}, \forall m\mbox{~~~and~~~} \sum_{m=1}^{|U|}q(m) = k.
\end{aligned}
\end{equation}
Directly solving this integer quadratic programming (IQP) problem is NP-hard. However, it can be relaxed to 1) a convex quadratic programming (QP) problem by relaxing the constraint $q(m)\in \{0,1\}$ to $q(m)\in [0,1]$ \cite{chattopadhyay:2012batch}, or 2) a convex linear programming (LP) problem of two types, one from \cite{chattopadhyay:2012batch} and the other from \cite{chakraborty:2015batchrank}. Here we briefly describe the convex LP solution from \cite{chakraborty:2015batchrank}. It is by two steps. In step 1, compute a vector $\mathbf{v}\in R^{|U|\times 1}$ containing column sums of $\mathbf{M}$ and identify the k largest entries in $\mathbf{v}$ to derive the initial solution $\mathbf{q}_0$ (replace the k largest entries of $\mathbf{v}$ with value 1 and replace the other entries with value 0, then assign it to $\mathbf{q}_0$). In step 2, it is a iterative process as shown in \textbf{Algorithm \ref{algo:iterative process}}. Starting with initial solution $\mathbf{q}_0$, we generate a sequence of solutions $\mathbf{q}_1, \mathbf{q}_2, \cdots$ until convergence. Finally, we get the solution $\mathbf{q}$, whose 1-value entries indicate the selected user set to rate $i_{new}$.
If $\mathbf{M}$ is positive semi-definite, \textbf{Algorithm \ref{algo:iterative process}} has a guaranteed monotonic convergence. If $\mathbf{M}$ is not positive semi-definite, with a positive scalar added to the diagonal elements, this algorithm can still be run on the shifted quadratic function to guarantee a monotonic convergence \cite{yuan:2013truncated}. Due to the monotonic convergence, the quality of the solution can only improve over iterations. The iterative process converges fast, thus there is only a marginal increase of the running time. Therefore, the complexity of our algorithm is O($|U|^2$), where $|U|$ is the number of users. Refer to \cite{chakraborty:2015batchrank} for a more detailed description of the complexity analysis. In real recommender systems, $\mathbf{M}$ may be too large for the memory to load. Our algorithm can still work well in this situation. For step 1, the memory only needs to load one column of $\mathbf{M}$ at a time to calculate column sums of $\mathbf{M}$. For each iteration of step 2, the memory only needs to load one row (equal to corresponding column, $\mathbf{M}$ is symmetric) of $\mathbf{M}$ and $\mathbf{q}_{t-1}$ at a time to calculate $\mathbf{M} \cdot \mathbf{q}_{t-1}$.
\begin{algorithm}[htb!]
\caption{Iterative Process for LP Solution}\label{algo:iterative process}
\begin{algorithmic}[1]
\STATE t = 1
\STATE \textbf{repeat}
\STATE \ \ \ \ Compute $\mathbf{q}_t^\prime = \mathbf{M} \cdot \mathbf{q}_{t-1}$
\STATE \ \ \ \ Replace the k largest entries of $\mathbf{q}_t^\prime$ with value 1 and replace the other entries with value 0
\STATE \ \ \ \ $\mathbf{q}_t=\mathbf{q}_t^\prime$
\STATE \ \ \ \ t = t + 1
\STATE \textbf{until} Convergence ($\mathbf{q}_{t-1}$ is equal to $\mathbf{q}_{t-2}$)
\end{algorithmic}
\end{algorithm}
\subsection{Active Learning for a Batch of Items}
As described in the introduction section, the budget of active learning is fixed for each new item in previous active learning works. In this paper, we propose a dynamic active learning budget so that the limited active learning resources can be properly distributed. We use $new\_item_1,new\_item_2,\cdots ,new\_item_l$ to denote $l$ new items. The total budget is denoted as $k_{total}$. Budget for all new items is denoted as $\mathbf{k}\in R^{{l}\times 1}$, where $k(1), k(2), \cdots ,k(l)$ are corresponding numbers of selected users for each new item. Thus we have $k_{total} = \sum_{i=1}^l k(i)$. We propose that more budget is distributed to new items with following two features.
Firstly, these items are \emph{popular}, which means many people would be willing to rate them. In the active learning phase, since popular items tend to be rated by more selected users, thus we will get more feedback ratings if we require ratings on popular items rather than requiring ratings on unpopular items. In the prediction phase, since popular items also tend to receive more ratings from unselected users, learning more about popular items, rather than unpopular ones, will influence and generate accurate predictions for more ratings. This is a problem of whether users will rate items
(described in Criterion (1)). We use the mean of all users' willing scores to measure it:
\begin{equation}\label{eq:popular new item}
\begin{aligned}
&popular(new\_item_{i})\\
&= \frac{1}{|U|}\sum_{u_m\in U} willing\_score(u_m,new\_item_{i}),\\
& i\in \{1,2,\cdots ,l\},
\end{aligned}
\end{equation}
where $willing\_score(u_m,new\_item_{i})$ is defined in section 4.1.
Secondly, these items are \emph{controversial}, which means we are uncertain about whether they will be liked or disliked by users. For items which will be obviously favored by almost all users, we already have a high confidence what ratings users tend to give to them. In contrary, it is the controversial items that we need to learn more about. This is a problem of what ratings users will give to items (described in Criterion (2)). We use the standard deviation of potential ratings to measure it:
\begin{equation}\label{eq:fcontroversial new item}
\begin{aligned}
&controversial(new\_item_{i}) \\
&= \frac{1}{|U|}\sqrt{\sum_{u_m\in U} (P_r(u_m,new\_item_{i})-\overline{P_r(new\_item_{i})})^2},\\
& i\in \{1,2,\cdots ,l\},
\end{aligned}
\end{equation}
where $P_r(u_m,new\_item_{i})$ is defined in section 4.1. $\overline{P_r(new\_item_{i})}$ is the average of potential ratings on $new\_item_{i}$. A budget score for each new item is defined as:
\begin{equation}\label{eq:budget score}
\begin{aligned}
&budget\_score(new\_item_i)= \\
&popular(new\_item_i)+\lambda\cdot controversial(new\_item_{i}),
\end{aligned}
\end{equation}
where $\lambda$ is a parameter to balance importance of two features. Finally, budget is distributed as:
\begin{equation}\label{eq:ki}
\begin{aligned}
&k(i)= \\
&\frac{budget\_score(new\_item_i)}{\sum_{j=1}^l budget\_score(new\_item_j)}\cdot k_{total}, i\in \{1,2,\cdots ,l\}.
\end{aligned}
\end{equation}
$k(i)$ are rounded to be integers. This equation ensures that more popular and controversial items will get more budget, and meanwhile each item has the opportunity to gain some budget.
\subsection{Rating Prediction Based on Feedback}
\begin{figure*}[htb!]
\begin{center}
\subfigure[A feedback rating is obtained for the new item ]{\includegraphics[width=0.73\columnwidth]{feedback_1.pdf}}
\subfigure[Factorization Machines for final prediction]{
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[width=0.9\columnwidth]{Prediction_1.pdf}
\end{minipage}
}
\end{center}
\vspace*{-10pt}
\caption{We firstly pre-train the regression model using previous ratings. Once feedback ratings for the new item are obtained, we re-train the model to strengthen it. (a) is to show that a feedback rating from the second user (rating `3') is obtained. (b) is to show how we re-train the model by adding users' feedback ratings. Finally, ratings of all unselected users are predicted.
}
\label{fig:final prediction}
\vspace*{-10pt}
\end{figure*}
Once selected users' feedback is obtained, we use another regression model to predict unselected users' ratings. Features in this model contain not only users and items' attributes, but also items. The instances contain both previous ratings and the newly obtained feedback ratings. Again, this is modeled by the Factorization Machines. To reduce the iteration number and accelerate the convergence speed, we firstly pre-train the regression model using previous ratings to get pre-trained parameters. Then when feedback ratings are obtained, we use these pre-trained parameters as initial parameters, all previous ratings and feedback ratings as training data, to re-train the model. Finally, ratings of all unselected users are predicted. Figure \ref{fig:final prediction} shows the detailed procedure.
The procedure of firstly pre-training and then re-training is similar to the idea of Bayesian Analysis \cite{berger:2013statistical}. That is, by pre-training on previous items, we learn users' preferences on attributes, and gain a ``prior'' understanding of users' preferences on $i_{new}$ estimated according to $i_{new}$'s attributes. Then users' feedback on $i_{new}$ enhance our understanding of $i_{new}$ and allow us to give ``posterior'' estimations for users' preferences on $i_{new}$.
\subsection{Exploitation-exploration Analysis}
As described in the introduction section, there are two goals in our task, i.e. \emph{exploitation} (exploiting ``existing knowledge'' to select users who are willing to rate new items, in order to obtain good user experience in the active learning phase) and \emph{exploration} (selecting users whose feedback can provide as much ``new knowledge'' about unselected users' preferences as possible and generate accurate rating predictions for unselected users, in order to obtain good user experience in the prediction phase). 1) The strategy of dynamic budget distributes more budget to popular items on which people are more willing to give ratings. Criterion (1) encourages users who are more willing to rate a certain new item to be selected. Thus they both contribute to improving the user experience in the active learning phase. 2) The strategy of dynamic budget and four criteria all help us to learn more about unselected users' preferences and generate more accurate rating predictions. Thus they all contribute to improving the user experience in the prediction phase. Therefore, our method considers both of these two goals (\emph{exploitation} and \emph{exploration}). In addition, we are able to adjust the parameter setting to further balance their trade-off. Specifically, once the best prediction accuracy is obtained with all parameters assigned to appropriate values, if we want to attach more importance to the user experience in the active learning phase, we just need to simply increase $\alpha$ (the weight of Criterion (1)). The reason is that, putting more weight on Criterion (1) would result in a higher rate of feedback ratings. However, increasing $\alpha$ will destroy the optimized parameter setting for rating prediction, thus the prediction accuracy would decrease.
\section{Experiments}
\subsection{Dataset}
Our proposed algorithm is evaluated on two datasets, Movielens-IMDB and Amazon. For the Movielens-IMDB dataset, ratings are collected from \emph{Movielens} \footnote{http://grouplens.org/datasets/movielens/} and attributes of movies are collected from \emph{imdbpy} \footnote{http://imdbpy.sourceforge.net/}. \emph{Ratings} \footnote{http://jmcauley.ucsd.edu/data/amazon/links.html} and \emph{attributes} \footnote{https://developer.amazon.com/} are also collected for the Amazon dataset. The statistics of these two datasets are shown in Table \ref{table:Movielens-IMDB and Amazon datasets}. In Movielens-IMDB, the number of attributes is the total number of directors, actors, genres, etc. In Amazon, the number of attributes is the total number of authors, publishers, etc. We collect ratings from the Movielens dataset rather than the \emph{official Netflix dataset} \footnote{https://www.kaggle.com/netflix-inc/netflix-prize-data}. The reason is that, items' detailed attributes are required in our setting. The attributes of movies in the Movielens dataset can be collected from \emph{http://imdbpy.sourceforge.net/} by accurately linking the movie ids in Movielens and IMDB. However, we did not figure out how to accurately obtain the attributes of movies in the Netflix dataset. For each dataset, 20\% items are randomly chosen as ``new items'' (i.e. testing items) in our conducted experiments. Ratings and attributes of the other 80\% items are used to train different models. Our goal is to generate accurate rating predictions on the testing items. Following \cite{aharon:2015excuseme, anava:2015budget}, the training-testing experiments are done once (also called \emph{holdout} \cite{tan:2006introduction}). Inspired by \cite{harpale:2008personalized}, we randomly select half of all users as the active-selection set and the remaining users form the prediction set. For all tesing items, users are selected from the active-selection set in the active learning phase. The rating prediction and the evaluation are performed for users in the prediction set.
\begin{table}[htb]\caption{Movielens-IMDB and Amazon Datasets}\label{table:Movielens-IMDB and Amazon datasets}
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|}
\hline
&Movielens-IMDB&Amazon\\ \hline
Number of users&5000&973\\ \hline
Number of items&9998&5000\\ \hline
Number of ratings&5154925&97967\\ \hline
Number of attributes&255942&3840\\ \hline
\end{tabular}
\vspace*{-10pt}
\end{table}
\subsection{Compared Algorithms}
Since in this paper, we want to handle new items with no rating, thus many previous active learning recommendation methods \cite{hofmann:2003collaborative,jin:2004bayesian,schohn:2000less}, which require at least a small amount of initial ratings, are not applicable in our task. \cite{anava:2015budget,aharon:2015excuseme} are the most related works to ours, which also address the item cold-start problem in an active learning scheme. However, the approach proposed by \cite{aharon:2015excuseme} is under an online setting in which users arrive and are decided to be given rating requests one by one and apparently it is not applicable for our task. \cite{anava:2015budget} assumes that selected users will always rate the new item (the rate of feedback ratings is 100\%), while our task is under a more realistic setting that only a subset of selected users will give feedback ratings. Thus it is unfair to compare the method in \cite{anava:2015budget} with our method and other baselines. We denote our method without dynamic budget as FMFC (Factorization Machines with Four Criteria) and our method with dynamic budget as FMFC-DB. We try our best to adapt following baselines from previous literature to compare with our proposed methods. HBRNN, LCE and FM are hybrid methods which combine both content and collaborative information. The remaining algorithms all exploit active learning to perform recommendations. The pre-train schedule in Figure \ref{fig:final prediction} is the same for all active learning methods, but the re-train schedule differs since different active learning methods have different feedback ratings.
\noindent \textbf{\emph{Hybrid-based Recommendation with Nearest Neighbor (HBRNN)}:} This method \cite{iaquinta:2007hybrid} is a combination of content-based recommendation and item-based collaborative filtering. The similarity between two items $i_m$,$i_n$ (including training items and testing items) is defined as follows:
\begin{equation}\label{eq:similarity matrix}
\begin{aligned}
sim(i_m,i_n) = cos(T(m,:),T(n,:)).
\end{aligned}
\end{equation}
where $T(m,:)$ and $T(n,:)$ are row vectors of the item-attribute matrix $\mathbf{T}$, representing $i_m$ and $i_n$ based on attributes. Once similarities between items are obtained, all users' ratings on the new item $i_{new}$ are predicted using an item-based collaborative filtering idea:
\begin{equation}\label{eq:HBRNN}
\begin{aligned}
Rating(u_m,i_{new}) = &\frac{\sum_{i_n\in I(u_m)}R(u_m,i_n) sim(i_n,i_{new})}{\sum_{i_n\in I(u_m)} sim(i_n,i_{new})}, \\
&u_m\in U,
\end{aligned}
\end{equation}
where $I(u_m)$ is the item set that $u_m$ rates. $R(u_m,i_n)$ is the rating that $u_m$ gives to $i_n$.
\noindent \textbf{\emph{Local Collective Embeddings (LCE)}:} This method \cite{saveski:2014item} also combines content-based recommendation and collaborative filtering. Different from HBRNN, which is a hybrid-based recommendation method from the nearest neighbor perspective. LCE is a hybrid-based recommendation method from the perspective of matrix factorization. In addition, it exploits the manifold structure of the data to improve the performance. We use the publicly available Matlab implementation \footnote{https://github.com/msaveski/LCE} of the LCE algorithm. Parameters are set and tuned as recommended in \cite{saveski:2014item}.
\noindent \textbf{\emph{Factorization Machines without Active Learning phase (FM)}:} This method uses Factorization Machines \cite{rendle:2012factorization} to model user behaviours. We directly use the pre-trained model in Figure \ref{fig:final prediction} to predict users' ratings on the new item $i_{new}$.
\noindent \textbf{\emph{Factorization Machines with Random Sampling in the Active Learning phase (FMRSAL)}:} In this baseline, for the new item $i_{new}$, $k$ users are randomly selected from the active-selection set for rating requests. Since these users are randomly selected and ratings are sparse in our dataset, thus the rate of feedback ratings is expected to be low. The performance improvement may be limited when compared to FM. However, rating requests are given to users without bias to any type of users, thus no one is always selected for rating requests in this user selection strategy.
\noindent \textbf{\emph{Factorization Machines with $\epsilon$-Greedy in the Active Learning phase (FM$\epsilon$GAL)}:} The $\epsilon$-Greedy algorithm is from the study of the multi-armed bandit problem \cite{feldman:2015recommendations}. We adapt it to our task as follows. For the new item $i_{new}$, we select $k$ users by $k$ sequential actions. For each action, we select the user who has the highest possibility to rate $i_{new}$ (i.e. $u_m$ with the largest $p(m)$) with probability $1-\epsilon$, and otherwise randomly select other users. When one user is selected in one action, he/she does not participate in following actions. In this user selection strategy, one more parameter, i.e. $\epsilon$, needs to be tuned. When we set $\epsilon = 0$, it is equal to our FMFC with only Criterion (1). When we set $\epsilon = 1$, it is transformed to FMRSAL. When we set $0<\epsilon < 1$, due to the randomness, rating requests are distributed to a wide range of users. Meanwhile, it can ensure a rate of feedback ratings higher than FMRSAL. In our experiments, we find no matter how $\epsilon$ varies, the performance in terms of all metrics is always between the performance of FMFC with only Criterion (1) and FMRSAL, thus we only show the experiment results with $\epsilon$ equal to 0.5 for simplicity.
\noindent \textbf{\emph{Factorization Machines with Poplar Sampling in the Active Learning phase (FMPSAL)}:} Inspired by \cite{golbandi:2011adaptive,rubens:2009output}, for the new item $i_{new}$, $k$ users who have given the most ratings to the training items are selected for rating requests. Since these users are ``frequently" rating users, they also tend to rate $i_{new}$, which can ensure a high rate of feedback ratings. Note that different from our Criterion (1), which is ``personalized" for different new items, users selected in this strategy are always the same.
\noindent \textbf{\emph{Factorization Machines with Coverage Sampling in the Active Learning phase (FMCSAL)}:} Inspired by \cite{golbandi:2010bootstrapping,rubens:2009output}, for the new item $i_{new}$, $k$ users who have highly co-rated items with other users are selected for rating requests. We define $Coverage(u_i) =\sum_j n_{ij}$, where $n_{ij}$ is the number of items that are rated by both users $u_i$ and $u_j$. Users with high Coverage values are then selected. The heuristic used by this strategy is that users co-rate the same items with many other users can better reflect other users' interest, and thus their rating behaviors are more helpful for predicting rating behaviors of other users.
\noindent \textbf{\emph{Factorization Machines with Exploration Sampling in the Active Learning phase (FMESAL)}:} As described in \cite{rubens:2015active}, exploration is important for recommendation, especially for new items as in our task. Inspired by studies about exploration in \cite{rubens:2007influence, chattopadhyay:2012batch}, for the new item $i_{new}$, $k$ users are selected for rating requests ensuring that selected users are representative of unselected users, and at the same time, selected users themselves are with high diversity. This can be achieved by optimizing the following objective function:
\begin{equation}\label{eq:FMES}
\begin{aligned}
&\max_\mathbf{q}{-\mathbf{q}^T \mathbf{S} \mathbf{q} + \gamma \mathbf{q}^T \mathbf{S} (\mathbf{1}-\mathbf{q})},\\
&\mbox{s.t.~~~} q(i)\in \{0,1\}, \forall i\mbox{~~~and~~~} \sum_{i=1}^{|U|}q(i) = k,
\end{aligned}
\end{equation}
where $\mathbf{q}$ and $\mathbf{S}$ are defined as in Equation (\ref{eq:initial objective function 2}). The first term is to ensure ``diversity" (selected users are dissimilar to each other) and the second term is for ``representative" (selected users are similar to unselected users). This integer quadratic programming (IQP) problem can be relaxed to a standard quadratic problem (QP) and be solved by applying many existing solvers.
\subsection{Evaluations}
In the active learning phase, we use following two metrics to measure the user experience of selected users.
\noindent \textbf{percentage of feedback ratings ($\bm{PFR}$)}: the ratio of users who give feedback ratings among all users who receive rating requests. It is formally defined as:
\begin{equation}\label{eq:PFR}
\begin{aligned}
PFR = \frac{\mbox{Total number of feedback ratings}}{\mbox{Total number of rating requests}}.
\end{aligned}
\end{equation}
Users giving feedback ratings are more likely to be willing to rate the item than those who do not give feedback ratings. A higher $PFR$ means that more selected users give feedback ratings, which indicates a better user experience.
\noindent \textbf{Average Selecting Times ($\bm{AST}$)}: average selecting times per user after a certain selection strategy is applied for all testing items. It is formally defined as:
\begin{equation}\label{eq:AST}
\begin{aligned}
AST = \frac{\mbox{Total number of rating requests}}{\mbox{Total number of distinct users for rating requests}}.
\end{aligned}
\end{equation}
A higher $AST$ means some users are always selected for rating requests, which will quickly annoy them and indicates a poorer user experience. For algorithms without active learning, i.e. HBRNN, LCE and FM, these two metrics are not measured.
In the prediction phase, we use \textbf{Root Mean Square Error ($\bm{RMSE}$)} and \textbf{Mean Absolute Error ($\bm{MAE}$)} to measure the user experience of unselected users. They are defined as follows:
\begin{equation}\label{eq:RMSE}
\begin{aligned}
&RMSE = \\
&\sqrt {\frac{1}{|R|}\sum_{(u_m,i_{new})\in R}(R(u_m,i_{new})-\tilde{R}(u_m,i_{new})})^2,
\end{aligned}
\end{equation}
\begin{equation}\label{eq:MAE}
\begin{aligned}
&MAE = \\
&\frac{1}{|R|}\sum_{(u_m,i_{new})\in R} |R(u_m,i_{new})-\tilde{R}(u_m,i_{new})|,
\end{aligned}
\end{equation}
$R$ are sets of (user, item) pairs that users give ratings to new items. $R(u_m,i_{new})$ is the rating that $u_m$ actually gives to $i_{new}$ and $\tilde{R}(u_m,i_{new})$ is the predicted rating.
For methods with no active learning, i.e. HBRNN, LCE and FM, models are directly trained on training items. For the remaining methods, given a new testing item, we select some users from the active-selection set in the active learning phase to see whether they have actual ratings on the testing item. If yes, we regard these actual ratings as feedback ratings. In the prediction phase, we exploit all feedback ratings to re-train the model. For all methods, $RMSE$ and $MAE$ are evaluated in the prediction set.
Apart from rating prediction, we can mimic a setting of top-$N$ recommendations as follows. Firstly, for all testing items, we select users from the active-selection set to get feedback and predict ratings for users in the prediction set (for HBRNN, LCE and FM, we directly predict users' ratings). Secondly, for each user, we select $N$ (we set $N = 10$) testing items with the largest predicted ratings (i.e. top-$N$ items) as the recommendation list. Finally, we regard new items with actual ratings larger than 3 as users' preferred items \cite{guan:2016weakly}. Performance is evaluated based on how many preferred items existing in the recommendation list, their actual ratings and their ranking positions. Following ranking metrics are used to evaluate the performance of top-$N$ recommendations.
\noindent \textbf{Precision, Recall}: $Precision$ is defined as the number of correctly recommended items (i.e. the number of preferred items existing in the recommendation list) divided by the number of all recommended items. $Recall$ is defined as the number of correctly recommended items divided by the total number of items which should be recommended (i.e. the number of preferred items). $Precision@k$ and $Recall@k$ are corresponding values at ranking position $k$. In our setting, there are $N = 10$ items in the recommendation list, while the number of preferred items is relatively large, thus the original $Recall$ is too small. We multiply it by a appropriate factor to get the modified $Recall$ \cite{zhu:2016heterogeneous}.
\noindent \textbf{Normalized Discount Cumulative Gain ($\bm{NDCG}$)}:
$NDCG$ at position $k$ is defined as:
\begin{equation}\label{eq:NDCG}
NDCG@k = \frac{1}{IDCG} \times \sum_{i=1}^k \frac{2^{r_i}-1}{\log_2^{(i+1)}}
\end{equation}
where $r_i$ is the relevance rating of the item at position $i$. $IDCG$ is set so that the perfect ranking has a $NDCG$ value of 1. In our case, $r_i$ is set to be the actual rating for preferred items and 0 for the other items.
\subsection{Parameter Setting}
Before setting the parameters, we calibrate the four criteria first. The calibration contains following two steps.
Step 1: we normalize $\mathbf{p}$, $\mathbf{D}$, $\mathbf{o}$, $\mathbf{S}$ to be $\mathbf{p}^\prime$, $\mathbf{D}^\prime$, $\mathbf{o}^\prime$, $\mathbf{S}^\prime$ by standardization. Specifically, the normalization formula is: $p_i^\prime = \frac{p_i - \bar{\mathbf{p}}}{\sigma_{\mathbf{p}}}$, $D_{ij}^\prime = \frac{D_{ij} - \bar{\mathbf{D}}}{\sigma_{\mathbf{D}}}$, $o_i^\prime = \frac{o_i - \bar{\mathbf{o}}}{\sigma_{\mathbf{o}}}$, $S_{ij}^\prime = \frac{S_{ij} - \bar{\mathbf{S}}}{\sigma_{\mathbf{S}}}$, where $p_i^\prime$ and $p_i$ are the $i$-th entries of $\mathbf{p}^\prime$ and $\mathbf{p}$, respectively. $\bar{\mathbf{p}}$ and $\sigma_{\mathbf{p}}$ are the mean and standard deviation of all entries in $\mathbf{p}$. $D_{ij}^\prime$ and $D_{ij}$ are entries with the $i$-th row and $j$-th column in $\mathbf{D}^\prime$ and $\mathbf{D}$, respectively. $\bar{\mathbf{D}}$ and $\sigma_{\mathbf{D}}$ are the mean and standard deviation of all entries in $\mathbf{D}$. The denotations for $\mathbf{o}$ and $\mathbf{S}$ are similar to those in $\mathbf{p}$ and $\mathbf{D}$, respectively.
Step 2: we divide $\mathbf{D}^\prime$ and $\mathbf{S}^\prime$ by $|U|$ to be $\mathbf{D}_{new}$ and $\mathbf{S}_{new}$. There are $|U|$, $|U|*|U|$, $|U|$ and $|U|*|U|$ entries in $\mathbf{p}^\prime \in R^{|U|}$, $\mathbf{D}^\prime \in R^{|U|\times|U|}$, $\mathbf{o}^\prime \in R^{|U|}$ and $\mathbf{S}^\prime \in R^{|U|\times|U|}$, respectively. If we do not calibrate $\mathbf{p}^\prime$, $\mathbf{D}^\prime$, $\mathbf{o}^\prime$ and $\mathbf{S}^\prime$, then Eq. (\ref{eq:initial objective function}) will be vastly influenced by the second and fourth terms (due to much more entries in them). To prevent $\mathbf{D}^\prime$ and $\mathbf{S}^\prime$ from dominating $\mathbf{p}^\prime$ and $\mathbf{o}^\prime$ when optimizing Eq. (\ref{eq:initial objective function}), we divide $\mathbf{D}^\prime$ and $\mathbf{S}^\prime$ by $|U|$. $\alpha, \beta, \gamma$ and $\sigma$ in Eq. (\ref{eq:initial objective function}) are tuned based on $\mathbf{p}^\prime$, $\mathbf{D}_{new}$, $\mathbf{o}^\prime$, $\mathbf{S}_{new}$. Due to these two steps, the tuned $\alpha, \beta, \gamma$ and $\sigma$ could then have similar orders of magnitude.
\begin{table}[b]\caption{Performance Comparison of Active Learning and Rating Prediction on Movielens-IMDB}\label{table: performance comparision Movielens-IMDB}
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&$PFR(\%)$&$AST$&$RMSE$&$MAE$\\ \hline
HBRNN&x&x&0.8792&0.6738\\ \hline
LCE&x&x&0.8754&0.6712\\ \hline
FM&x&x&1.0364&0.7828\\ \hline
FMRSAL&5.17&\textbf{19.98}&0.9244&0.7320\\ \hline
FM$\epsilon$GAL($\epsilon = 0.5$)&14.24&23.62&0.8728&0.6701\\ \hline
FMPSAL&20.32&1998&0.8520&0.6562\\ \hline
FMCSAL&21.66&1998&0.8511&0.6549\\ \hline
FMESAL&6.35&1998&0.9149&0.7043\\ \hline
FMFC&23.04&163.76&0.8305&0.6388\\ \hline
FMFC-DB&\textbf{23.78}&140.03&\textbf{0.8231}&\textbf{0.6308}\\ \hline
\end{tabular}
\vspace*{-0pt}
\end{table}
\begin{table}[b]\caption{Performance Comparison of Active Learning and Rating Prediction on Amazon}\label{table: performance comparision Amazon}
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&$PFR(\%)$&$AST$&$RMSE$&$MAE$\\ \hline
HBRNN&x&x&0.8496&0.6565\\ \hline
LCE&x&x&0.8489&0.6564\\ \hline
FM&x&x&0.8781&0.6709\\ \hline
FMRSAL&2.09&\textbf{51.33}&0.8760&0.6619\\ \hline
FM$\epsilon$GAL($\epsilon = 0.5$)&8.04&55.87&0.8489&0.6561\\ \hline
FMPSAL&8.56&1000&0.8449&0.6533\\ \hline
FMCSAL&8.61&1000&0.8418&0.6511\\ \hline
FMESAL&7.63&1000&0.8540&0.6582\\ \hline
FMFC&13.87&82.24&0.8206&0.6360\\ \hline
FMFC-DB&\textbf{14.91}&71.02&\textbf{0.8055}&\textbf{0.6291}\\ \hline
\end{tabular}
\vspace*{-10pt}
\end{table}
\begin{table*}[t]\caption{Performance Comparison of top-$N$ recommendations on Movielens-IMDB}\label{table: performance comparision of top-$N$ recommendations Movielens-IMDB}
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
&$Precision@5$&$Precision@10$&$Recall@5$&$Recall@10$&$NDCG@5$&$NDCG@10$\\ \hline
HBRNN&0.3554&0.2775&0.1042&0.1791&0.2806&0.3617\\ \hline
LCE&0.3593&0.2804&0.1054&0.1810&0.2799&0.3653\\ \hline
FM&0.2835&0.2151&0.0831&0.1388&0.2062&0.2884\\ \hline
FMRSAL&0.3017&0.2324&0.0885&0.1499&0.2252&0.3087\\ \hline
FM$\epsilon$GAL($\epsilon = 0.5$)&0.3601&0.2821&0.1055&0.1821&0.2801&0.3679\\ \hline
FMPSAL&0.3787&0.3025&0.1111&0.1952&0.3302&0.4114\\ \hline
FMCSAL&0.3869&0.3086&0.1134&0.1991&0.3298&0.4182\\ \hline
FMESAL&0.3208&0.2519&0.0941&0.1625&0.2497&0.3312\\ \hline
FMFC&0.4486&0.3591&0.1316&0.2317&0.3916&0.4738\\ \hline
FMFC-DB&\textbf{0.4791}&\textbf{0.3898}&\textbf{0.1405}&\textbf{0.2515}&\textbf{0.4436}&\textbf{0.5241}\\ \hline
\end{tabular}
\vspace*{-0pt}
\end{table*}
\begin{table*}[htb!]\caption{Performance Comparison of top-$N$ recommendations on Amazon}\label{table: performance of top-$N$ recommendations Amazon}
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&$Precision@5$&$Precision@10$&$Recall@5$&$Recall@10$&$NDCG@5$&$NDCG@10$\\ \hline
HBRNN&0.2638&0.2061&0.2162&0.3747&0.2871&0.3732\\ \hline
LCE&0.2671&0.2080&0.2189&0.3782&0.2903&0.3764\\ \hline
FM&0.2004&0.1437&0.1643&0.2613&0.2121&0.2947\\ \hline
FMRSAL&0.2195&0.1603&0.1799&0.2915&0.2324&0.3176\\ \hline
FM$\epsilon$GAL($\epsilon = 0.5$)&0.2673&0.2101&0.2191&0.3820&0.2953&0.3845\\ \hline
FMPSAL&0.2879&0.2310&0.2360&0.4200&0.3345&0.4201\\ \hline
FMCSAL&0.2933&0.2385&0.2404&0.4336&0.3404&0.4199\\ \hline
FMESAL&0.2388&0.1825&0.1957&0.3318&0.2543&0.3393\\ \hline
FMFC&0.3432&0.2859&0.2813&0.5198&0.3994&0.4805\\ \hline
FMFC-DB&\textbf{0.3818}&\textbf{0.3205}&\textbf{0.3130}&\textbf{0.5827}&\textbf{0.4517}&\textbf{0.5335}\\ \hline
\end{tabular}
\vspace*{-10pt}
\end{table*}
$k$, $\alpha$, $\beta$, $\gamma$ and $\sigma$ are the main parameters in our paper. $k$ is the number of selected users for active learning. We empirically set $k = 25$ for each testing item to tune the other parameters. There are four parameters $\alpha$, $\beta$, $\gamma$ and $\sigma$ to trade off the importance of different criteria. In fact, they can be multiplied by an arbitrary scaling factor, so there are exactly three free parameters. We fix $\alpha = 1$ and tune the other three free parameters by grid search. We use $RMSE$ as the tuning metric, where $RMSE$ is measured by cross-validation on the training data (users are also split to the active-selection set and the prediction set). For the Movielens-IMDB dataset, the final tuned parameters are $\alpha = 1, \beta = 0.3, \gamma = 0.1, \sigma = 0.1$. We regard the performance measured on all testing items using this parameter setting as the performance of FMFC. Furthermore, we fix $\alpha = 1, \beta = 0.3, \gamma = 0.1, \sigma = 0.1$, and implement FMFC-DB, i.e. our method with dynamic active learning budget. The total budget $k_{total}$ (see section 4.2) is set to be $25 \times l$, where $l$ is the number of testing items. We regard the performance measured in this setting as the performance of FMFC-DB. For the other active learning baselines, the performance is measured with the number of selected users equal to $25$ for each testing item. For the Amazon dataset, the final tuned parameters are $\alpha = 1, \beta = 0.3, \gamma = 0.03, \sigma = 0.1$. The other settings are the same as for the Movielens-IMDB dataset.
\subsection{Results and Analysis}
\subsubsection{Algorithm Comparison}
We now compare our methods with all baselines. All the performance is measured on testing items. As shown in Table \ref{table: performance comparision Movielens-IMDB} and Table \ref{table: performance comparision Amazon}, our methods (FMFC and FMFC-DB) outperform the other baselines in terms of $RMSE$ and $MAE$, which indicates our methods have the highest prediction accuracy in the prediction phase. Our methods also perform the best in the task of top-$N$ recommendations according to Table \ref{table: performance comparision of top-$N$ recommendations Movielens-IMDB} and Table \ref{table: performance of top-$N$ recommendations Amazon}. Factorization Machines with different active learning strategies perform better than Factorization Machines without active learning (FM). This is easy to understand since feedback ratings give us more understanding of testing items, which can be exploited to enhance the prediction model. As methods without active learning, HBRNN and LCE perform better than FM and even better than active learning methods FMRSAL and FMESAL. The reason may be that HBRNN and LCE can make better use of both content and collaborative information than FM. LCE has a slightly better performance than HBRNN. The reason may be that it exploits the manifold structure of the data to improve the performance of hybrid recommendations. FMPSAL, FMCSAL, FMFC and FMFC-DB perform better than the other three active learning methods. The main reason is that these four methods can ensure a high rate of feedback ratings (high $PFR$), which is the domain factor that influences the prediction accuracy (we will analyze this in the next section). Our methods outperform FMPSAL and FMCSAL because 1) our methods achieve higher rates of feedback ratings than FMPSAL and FMCSAL, and 2) our methods consider not only the rate of feedback ratings (Criterion (1)), but also other three factors (Criteria (2), (3), (4)) to improve the prediction accuracy. $PFR$ and $AST$ are both metrics that measure the user experience in the active learning phase. There is no active learning phase for HBRNN, LCE and FM, thus we compare the other methods in terms of $PFR$ and $AST$. For $PFR$, FMRSAL and FMESAL have rather few feedback ratings, which indicates they always give rating requests to users who do not really want to rate them. Thus these two methods negatively influence the user experience. FM$\epsilon$GAL has a relatively higher $PFR$. The other active learning methods all have a considerable number of feedback ratings. For $AST$, due to the natural randomness, FMRSAL undoubtedly performs the best with the lowest $AST$ and FM$\epsilon$GAL performs the second best. FMPSAL, FMCSAL and FMESAL select the same user set to rate all testing items and it will certainly annoy them. Overall, our methods are the best when considering all these metrics. When comparing FMFC with FMFC-DB, it can be seen that our proposed dynamic active learning budget can further improve the performance in terms of all metrics.
The significant test is performed to show whether the differences of our experiment results are statistically significant. Paired t-tests are conducted to compare the differences between the experiment performance of (1) our two proposed methods (i.e. FMFC and FMFC-DB), (2) FMFC and all baselines, and (3) FMFC-DB and all baselines. Specifically, for each dataset, we repeat the training-testing experiments for $100$ times (the testing item set is independently chosen in different times). Then given a certain metric, each method would generate $100$ metric values. The inputs of paired t-test are two sets of metric values, with each set corresponding to one compared method. Since we want to validate that one method is better than the other, we use one-tailed hypothesis. Statistical significance is set at $p<0.05$. The results show that, in terms of all metrics except for $AST$, (1) FMFC-DB has significantly better performance than FMFC, and (2) compared to all baselines, both FMFC and FMFC-DB have significantly better performance.
\subsubsection{Criteria Analysis}
\begin{table}[tb]\caption{Percentage of Feedback Ratings on Movielens-IMDB and Amazon}\label{table: Percentage of Feedback Ratings on Movielens-IMDB and Amazon}
\scriptsize
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&\multicolumn{2}{|c|}{$PFR(\%)$ on Movielens-IMDB}&\multicolumn{2}{|c|}{$PFR(\%)$ on Amazon}\\ \hline
&FMFC&FMFC-DB&FMFC&FMFC-DB\\ \hline
No Criterion (1)&9.91&10.16&2.25&2.57\\ \hline
Original&\textbf{23.04}&\textbf{23.78}&\textbf{13.87}&\textbf{14.91}\\ \hline
\end{tabular}
\vspace*{-0pt}
\end{table}
\begin{table}[htb!]\caption{The Average Diverse Value of Selected Users' Ratings on Movielens-IMDB and Amazon}\label{table: The Average Diverse Value of Selected Users' Ratings on Movielens-IMDB and Amazon}
\scriptsize
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&\multicolumn{2}{|c|}{\multirow{2}{1.1in}{The Average Diverse Value on Movielens-IMDB}}&\multicolumn{2}{|c|}{\multirow{2}{1.1in}{The Average Diverse Value on Amazon}}\\
&\multicolumn{2}{|c|}{}&\multicolumn{2}{|c|}{} \\
\hline
&FMFC&FMFC-DB&FMFC&FMFC-DB\\ \hline
No Criterion (2)&1.21&1.23&1.13&1.16\\ \hline
Original&\textbf{1.37}&\textbf{1.39}&\textbf{1.19}&\textbf{1.22}\\ \hline
\end{tabular}
\vspace*{-0pt}
\end{table}
\begin{table}[htb!]\caption{The Difference Between the Average Rating of Selected Users and the Average Rating of All Users on Movielens-IMDB and Amazon}\label{table: The Objective Value of Selected Users' Ratings on Movielens-IMDB and Amazon}
\scriptsize
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&\multicolumn{2}{|c|}{\multirow{2}{0.9in}{The Difference on Movielens-IMDB}}&\multicolumn{2}{|c|}{\multirow{2}{1.1in}{The Difference on Amazon}}\\
&\multicolumn{2}{|c|}{}&\multicolumn{2}{|c|}{} \\
\hline
&FMFC&FMFC-DB&FMFC&FMFC-DB\\ \hline
\multirow{2}{0.8in}{No Criterion (3) $(|\bar{r}_{no\_c3} - \bar{r}_{all}|)$}&\multirow{2}{0.2in}{1.22}&\multirow{2}{0.2in}{1.19}&\multirow{2}{0.2in}{1.43}&\multirow{2}{0.2in}{1.39}\\
{}&{}&{}&{}&{} \\\hline
\multirow{2}{0.8in}{Original $(|\bar{r}_{ours} - \bar{r}_{all}|)$}&\multirow{2}{0.2in}{\textbf{0.97}}&\multirow{2}{0.2in}{\textbf{0.89}}&\multirow{2}{0.2in}{\textbf{1.22}}&\multirow{2}{0.2in}{\textbf{1.19}}\\
{}&{}&{}&{}&{} \\\hline
\end{tabular}
\vspace*{-0pt}
\end{table}
\begin{table}[htb!]\caption{The Average Similarity Value Between Selected and Unselected Users on Movielens-IMDB and Amazon}\label{table: The Average Similarity Value Between Selected and Unselected Users on Movielens-IMDB and Amazon}
\scriptsize
\vspace*{-10pt}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&\multicolumn{2}{|c|}{\multirow{2}{1.1in}{The Average Similarity Value on Movielens-IMDB}}&\multicolumn{2}{|c|}{\multirow{2}{1.1in}{The Average Similarity Value on Amazon}}\\
&\multicolumn{2}{|c|}{}&\multicolumn{2}{|c|}{} \\
\hline
&FMFC&FMFC-DB&FMFC&FMFC-DB\\ \hline
No Criterion (4)&0.54&0.56&0.46&0.49\\ \hline
Original&\textbf{0.61}&\textbf{0.67}&\textbf{0.52}&\textbf{0.58}\\ \hline
\end{tabular}
\vspace*{-0pt}
\end{table}
As mentioned in section 4.1, to generate more accurate rating predictions of the new item, our methods select users based on four criteria. In this section, we firstly validate whether each criterion works as they claim. Then we validate their contributions to the final prediction performance.\\\\
\textbf{Criterion (1): Selected users are with high possibility to rate $i_{new}$}
We remove Criterion (1) in FMFC and FMFC-DB to see how $PFR$ varies. The results are shown in Table \ref{table: Percentage of Feedback Ratings on Movielens-IMDB and Amazon}. Without Criterion (1), the $PFR$ decreases dramatically for both FMFC and FMFC-DB. Since a higher $PFR$ means that more selected users rate $i_{new}$, the result validates the effectiveness of Criterion (1).\\
\noindent\textbf{Criterion (2): Selected users' potential ratings are diverse}
The purpose of selecting users with diverse potential ratings is to ensure these users' actual ratings also tend to be diverse, so that the final prediction model is not biased to a fixed region of ratings. We remove Criterion (2) in FMFC and FMFC-DB to see how the average diverse value (defined in Eq. (\ref{eq:diverse function})) of selected users' actual ratings varies. As shown in Table \ref{table: The Average Diverse Value of Selected Users' Ratings on Movielens-IMDB and Amazon}, without Criterion (2), the average diverse value decreases for both FMFC and FMFC-DB. The result validates the effectiveness of Criterion (2).\\
\noindent\textbf{Criterion (3): Selected users' generated ratings are objective}
The insight of Criterion (3) is to let the average rating of users selected by our methods (denoted as $\bar{r}_{ours}$) approximate the average rating of all users (denoted as $\bar{r}_{all}$). We measure the difference between $\bar{r}_{ours}$ and $\bar{r}_{all}$, i.e. $|\bar{r}_{ours} - \bar{r}_{all}|$. Meanwhile, we calculate the average rating of users selected without Criterion (3) (denoted as $\bar{r}_{no\_c3}$) and measure $|\bar{r}_{no\_c3} - \bar{r}_{all}|$. The result is shown in Table \ref{table: The Objective Value of Selected Users' Ratings on Movielens-IMDB and Amazon}. We can see that $|\bar{r}_{no\_c3} - \bar{r}_{all}| > |\bar{r}_{ours} - \bar{r}_{all}|$ for both FMFC and FMFC-DB, which indicates that Criterion (3) can actually make the average rating of selected users closer to $\bar{r}_{all}$.\\
\noindent\textbf{Criterion (4): Selected users are representative}
The insight of Criterion (4) is to let the selected users similar to unselected users. We measure the average similarity value between selected and unselected users with/without Criterion (4) to validate this criterion. The result is shown in Table \ref{table: The Average Similarity Value Between Selected and Unselected Users on Movielens-IMDB and Amazon}. We can see that without Criterion (4), the average similarity value declines, which indicates that Criterion (4) can actually make the selected users more similar to unselected users.
We further validate the contribution of each criterion to the final prediction performance. The results are shown in Figure \ref{fig:featurecut}. $RMSE$ increases when we remove each criterion, which indicates each criterion contributes to the prediction improvement. $RMSE$ increases the most when we remove Criterion (1), which indicates this criterion is a domain factor that influences the prediction improvement. $k$ is the number of selected users. $RMSE$ decreases when $k$ increases. This is easy to understand, since larger $k$ leads to more feedback ratings, which will give us more understanding of the new item and thus generate more accurate predictions.
\begin{figure}[t!]
\begin{center}
\subfigure[Movielens-IMDB]{\includegraphics[scale=0.33]{featurecut.pdf}}
\subfigure[Amazon]{\includegraphics[scale=0.33]{featurecut_amazon.pdf}}
\end{center}
\vspace*{-10pt}
\caption{Performance of FMFC measured by $RMSE$ when we remove four criteria one at a time and vary $k$. \emph{All Criteria} refers to the performance measured when considering all four criteria. \emph{No Criterion (1)} refers to the performance measured when we remove Criterion (1) (i.e. $\alpha = 0$). Similarly, \emph{No Criterion (2)}, \emph{No Criterion (3)} and \emph{No Criterion (4)} refer to performance measured under corresponding settings.
}
\label{fig:featurecut}
\vspace*{-10pt}
\end{figure}
\subsubsection{Dynamic Budget Analysis}
As mentioned in section 4.2, we use the strategy of dynamic budget to properly distribute limited active learning resources. As shown in Figure \ref{fig:dynamic_budget}, for different values of the total budget, the dynamic budget all contributes to the performance improvement in terms of both $RMSE$ and $PFR$. The improvement is narrowed when the total budget increases. This is because the dynamic budget is proposed to address the problem of limited active learning resources. When the total budget is sufficient, this strategy provides less help.
\begin{figure}[t!]
\begin{center}
\subfigure[$RMSE$ on Movielens-IMDB]{\includegraphics[scale=0.3]{rmse_budget.pdf}}
\subfigure[$PFR$ on Movielens-IMDB]{\includegraphics[scale=0.3]{pfr_budget.pdf}}
\subfigure[$RMSE$ on Amazon]{\includegraphics[scale=0.3]{rmse_budget_amazon.pdf}}
\subfigure[$PFR$ on Amazon]{\includegraphics[scale=0.3]{pfr_budget_amazon.pdf}}
\end{center}
\vspace*{-10pt}
\caption{Performance measured by $RMSE$ (for the prediction phase) and $PFR$ (for the active learning phase) when we vary the total active learning budget for both FMFC and FMFC-DB.
}
\label{fig:dynamic_budget}
\end{figure}
\subsubsection{Exploitation-exploration Analysis}
Results in Table \ref{table: performance comparision Movielens-IMDB} and Table \ref{table: performance comparision Amazon} have shown that our method can achieve high performance for both \emph{exploitation} (high $PFR$ in the active learning phase) and \emph{exploration} (low $RMSE$ and $MAE$ in the prediction phase). Now we analyze how $\alpha$, i.e. the weight of Criterion (1), can further balance their trade-off. We vary $\alpha$ but fix other tuned parameters to see how the performance changes. As shown in Figure \ref{fig:performance vs parameters}, $RMSE$ of FMFC first decreases then increases when $\alpha$ varies, and obtains the best result when $\alpha$ is around $1$. $PFR$ keeps increasing when $\alpha$ varies. We certainly will not set $\alpha < 1$, which will achieve poor performance in terms of both $RMSE$ and $PFR$. For $\alpha \geq 1$, if we attach more importance to the active learning phase, we need to assign a larger value to $\alpha$. Similarly, if we pay more attention to the prediction phase, then a smaller value is assigned. These experiment results are consistent with the analysis of section 4.4.
\begin{figure}[t!]
\begin{center}
\subfigure[$RMSE$ on Movielens-IMDB]{\includegraphics[scale=0.3]{rmsevsalpha.pdf}}
\subfigure[$PFR$ on Movielens-IMDB]{\includegraphics[scale=0.3]{pfrvsalpha.pdf}}
\subfigure[$RMSE$ on Amazon]{\includegraphics[scale=0.3]{rmsevsalpha_amazon.pdf}}
\subfigure[$PFR$ on Amazon]{\includegraphics[scale=0.3]{pfrvsalpha_amazon.pdf}}
\end{center}
\vspace*{-10pt}
\caption{Performance of FMFC and FMCSAL (the best compared method) measured by $RMSE$ (for the prediction phase) and $PFR$ (for the active learning phase) when we vary $\alpha$.
}
\label{fig:performance vs parameters}
\vspace*{-10pt}
\end{figure}
\section{Conclusion}
In this paper, we propose a novel recommendation scheme for the item cold-start problem by leveraging both active learning and items' attribute information. We firstly pre-train the rating prediction model with users' historical ratings and items' attributes. Secondly, given a new item, a small portion of users are selected to rate this item based on four useful criteria. Thirdly, the prediction model is re-trained by adding feedback ratings. Finally, unselected users' ratings are predicted by the re-trained model. We further propose a dynamic active learning budget to properly distribute active learning resources, which contributes to better recommendation performance. The idea of dynamic active learning budget can also be applied to other active learning related tasks. Our methods are able to ensure a relatively good user experience for both of selected users in the active learning phase and unselected users in the prediction phase. For future work, we will explore more other criteria to improve our user selection strategy. In addition, libFM used in this paper is a regression model, which is suitable for the task of rating prediction. We will try to expend our method with a ranking model to better address the top-$N$ recommendation task.
\section{Acknowledgement}
This work was supported in part by the National Basic Research Program of China (973 Program) under Grant 2013CB336500, National Nature Science Foundation of China (Grant Nos: 61522206, 61379071, 61373118), and National Youth Top-notch Talent Support Program.
\ifCLASSOPTIONcaptionsoff
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\bibliographystyle{IEEEtran}
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Here's to You (Nicola and Bart) () е фолклорна песен, изпълнявана и в съпровод на симфоничен оркестър от американската певица и политическа активистка Джоан Байз. Издаден е през 1971 г. като последната част от саундтрака към филма Sacco e Vanzetti (Сако и Ванцети), филмова адаптация на процеса на Министерството на правосъдието на САЩ срещу Сако и Ванцети, който привлича международното внимание през 20-те години на миналия век. Текстът е на Байз, музиката е композирана от Енио Мориконе. Песента се превръща в химн на движението за правата на човека през 70-те години.
След премиерата на филма различни други видни изпълнители адаптират тази песен със свои собствени аранжименти и кавър версии, някои от които са многоезични. Със своята запомняща се мелодия, Here's to You се превръща в международно признат химн за жертвите на политическото правосъдие, който има въздействие и днес и днес се смята за евъргрийн.
Песента е в памет на родените в Италия анархисти Никола Сако и Бартоломео Ванцети, които са осъдени на смърт в сензационен процес от американски съдилища през 20-те години на миналия век и екзекутирани на 23 август 1927 г. В хода на процеса обаче в публичното пространство са изразени силни съмнения относно правилността на воденето на производството и последвалата осъдителна присъда, които са мотивирани предимно от отвращение към политическите нагласи и произхода на обвиняемите и по-малко от ясни доказателства за грабежа и убийството, в които са обвинени.
Песента на Баез е третата и последна част от Баладата за Сако и Ванцети, композирана за филма, режисиран от Джулиано Монталдо. Състои се от четири реда текст, повтарящи се осем пъти подред. Последните два реда заемат изречение от едно от писмата на Бартоломео Ванцети, в което той пише:
"Последният момент принадлежи на нас, тази агония е нашият триумф!"
"Последният момент е наш, тази агония е нашият триумф!"
Парчето е използвано, наред с други неща, в документалния епизод на филма Deutschland im Herbst (означава Германската есен на 1977 г.), придружаващ погребални сцени за терористите от RAF Андреас Баадер, Гудрун Енслин и Ян-Карл Пачне, които се самоубиват в Щутгарт (Станмхайм) затвор. Използвайки песента, създателите на филма правят директна препратка към случая с италианските анархисти, предоставяйки интерпретация на това как държавните власти по света ще се справят с антиавторитарните политически идеалисти.
Друга интерпретация вижда съзнателно сравнение и връзка на ситуацията в Германия с по-ранни леви политически движения и техните мъртви. В същото време, като преход към рамковото изречение на филма ("Когато стигнете до определен момент в жестокостта, няма значение кой го е извършил: трябва просто да спре.") Той изразява надеждата, че бъдещите дебати ще бъдат мирни начини за успех.
Песента е използвана и във филма The Deep Sea Divers и компютърните игри Metal Gear Solid 4: Guns of the Patriots и Metal Gear Solid V: Ground Zeroes.
Източници
Външни препратки
Here's to You. Диригент Енио Мориконе, на живо 10 ноември 2007
Песни от филми
Сако и Ванцети
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Flooding the airwaves and internet with discussions from experts and journalist alike, the historic bull run of the U.S. stock market is the longest period of growth in American history.
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\section*{Introduction}
One of the most useful process in Abelian category theory is the so-called localization of an abelian category $\cD$
to a quotient category $\cD/\cS$ by means of a Serre class $\cS$ in $\cD$.
When $\cS$ is a localizing subcategory in the sense of \cite{0201.35602}, the canonical exact functor $\cD \to \cD/\cS$ has a fully faithful right adjoint functor
$S: \cD/\cS \to \cD$ which allows to deal with $\cD/\cS$ as a full subcategory of $\cD$, which is called a Giraud subcategory of $\cD$.
Dualizing the context, one get the notion of a co-Giraud subcategory.
Giraud and co-Giraud subcategories very often appear in the literature in very different settings (see \ref{rG}).
On the other side, in 1981 Beilinson, Bernstein and Deligne introduced the notion of $t$-structure on a triangulated category
related to the study of the derived category of constructible sheaves on a stratified space.
Actually the notion of $t$-structure is a generalization of
the notion of torsion pair on an abelian category (see for example \cite{MR2327478}).
In their work \cite{MR1327209} Happel, Reiten and Smalo related the study of
torsion pairs to Tilting theory and $t$-structures.
In particular given an abelian category ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$
one can construct many non-trivial $t$-structures on its derived category
$D^b({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ by the procedure of tilting at a torsion pair (see \ref{Tstr}).
Inspired by the fundamental role of localizing subcategories in the study of
problems of gluing abelian categories or even triangulated categories we propose in this work a
bridge between the two previous abstract contexts.
The main progress in the present paper is to show how the process of (co-) localizing moves from a basic abelian category to the level of its tilt, with respect to a torsion pair, and viceversa.
On the one side we deal with a (co-) Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ of $\cD$, looking the way torsion pairs on $\cD$ reflect
on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ and, conversely, torsion pairs on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ extend to $\cD$: in particular we find a one to one correspondence
between arbitrary torsion pairs $({\mathcal T}, {\mathcal F})$ on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ and the torsion pairs $({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ on $\cD$
which are ``compatible'' with the (co-) localizing functor (Theorems~\ref{tt1-1} and \ref{ctt1-1}).
On the other side, we compare this action of ``moving'' torsion pairs from $\cD$ to ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ (and viceversa) with a ``tilting context'':
more precisely, we look at the associated hearts $\cH_{\cD}$ and $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ with respect to the torsion pairs $({\mathcal T}, {\mathcal F})$ on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ and $({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ on $\cD$, respectively, proving that $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ is still a (co-) Giraud subcategory of $\cH_{\cD}$, and that the ``tilted'' torsion pairs in the two hearts are still related (Theorems~\ref{adjhearts} and \ref{cadjhearts}). Here the ambient Abelian category $\cD$ is arbitrary, with the unique request that the inclusion functor of ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ into $\cD$ admits a right derived functor.
Finally given any Abelian category $\cD$ endowed with a torsion pair $({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$, and considering any Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}'$ of the associated heart
$\cH_{\cD}$ which is ``compatible'' with the ``tilted'' torsion pair on $\cH_{\cD}$, we prove in Theorem~\ref{reconstruction} how to recover
a Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ of $\cD$ such that
${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}'$ is equivalent to the heart $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ (with respect to the induced torsion pair).
\section{Serre, Giraud and co-Giraud subcategories}
We begin by fixing some notations on Serre, Giraud and co-Giraud subcategories. A complete account on quotient categories and Serre classes can be found in \cite[Chapter 3]{0201.35602} and \cite[Section 1.11]{MR0102537}.
\begin{definition}
Let $\cD$ be an abelian category. A {\it Serre} class $\cS$ in $\cD$ is a full subcategory $\cS$
of $\cD$ such that for any short exact sequence $0{\rightarrow} X_1{\rightarrow} X_2{\rightarrow} X_3{\rightarrow} 0$ in $\cD$ the middle term
$X_2$ belongs to $\cS$ if and only if $X_1, X_3$ belong to $\cS$.
\end{definition}
\smallskip
The data of an abelian category $\cD$ and a Serre class $\cS$ of $\cD$ allow to construct a new abelian category, denoted by $\cD/\cS$, called the {\it quotient category of $\cD$ by $\cS$} (see \cite{MR0102537}). It turns out that $\cD/\cS$ is abelian and the canonical functor $T \colon\cD \to \cD/\cS$ is exact. A Serre class $\cS$ in $\cD$ is called a {\it localizing subcategory} (resp. {\it co-localizing subcategory}) if the functor $T$ admits a right adjoint (resp.~left adjoint) {\it section functor} $S$. In this case, the left exact (resp.~right exact) functor $S\circ T$ is called the {\it localization functor}. This localization functor is exact if and only if $S$ is exact (see \cite[Chapter 3]{0201.35602}).
\begin{definition}
An abelian category with a {\it distinguished Giraud subcategory} is the data $(\cD,{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S},l,i)$ of two abelian categories
$\cD$ and ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ and two adjoint functors $\xymatrix{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \ar@<-0.5ex>[r]_{i}&\cD\ar@<-0.5ex>[l]_{l}}$
(with $l$ left adjoint of $i$) such that $l$ is exact and $i$ fully faithful.
Dually an abelian category with a {\it distinguished co-Giraud subcategory} is the data $(\cD,{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S},j,r)$ of two abelian categories
$\cD$ and ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ and two adjoint functors $\xymatrix{\cD \ar@<-0.5ex>[r]_{r}&{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \ar@<-0.5ex>[l]_{j}}$
(with $j$ left adjoint of $r$) such that $r$ is exact and $j$ fully faithful.
\end{definition}
\smallskip
Therefore a localizing subcategory $\cS$ of $\cD$ defines a distinguished Giraud subcategory
$(\cD, \cD/\cS, T, S)$. Conversely, given a distinguished Giraud subcategory $(\cD, {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}, l, i)$, the kernel of the functor $l$, i.e., the full subcategory $\cS$ of $\cD$ whose objects $S$ in $\cS$ satisfy $l(S)\cong0$, defines a localizing subcategory $\cS = \mathrm{Ker}(l)$ of $\cD$ whose associated quotient category is (equivalent to) ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Let us denote by $\eta \colon {\rm id}_D \to i\circ l$ the unit of the adjunction $(l,i)$, and by $\cS^\bot$ the full subcategory of
$\cD$ whose objects are defined by:
\[
\cS^\bot:=\{D\in\cD \,|\, \cD(S,D)=0, \forall S\in\cS\}.
\]
It turns out that
\[
\cS^\bot=\{D\in\cD \,|\, \eta_D: D{\rightarrow} il(D) \text{ is a monomorphism}\}.
\]
Moreover, let us notice that since $i$ is fully faithful the counit of the adjunction
$\varepsilon :l\circ i{\rightarrow} {\rm id}_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ is an isomorphism of functors.
Dually, starting from a distinguished co-Giraud subcategory $(\cD, {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}, j, r)$, and denoting by
$\varepsilon \colon j\circ r \to {\rm id}_D$ the counit of the adjunction $(j, r)$, the kernel of the functor $r$ defines a co-localizing subcategory $\cS = \mathrm{Ker}(r)$ of $D$ such that
\[\begin{matrix}
\;^\bot\cS:=&\{D\in\cD \,|\, \cD(D,S)=0, \forall S\in\cS\} \hfill \\
\hfill=&\{D\in\cD \,|\, \varepsilon_D: jr(D)\rightarrow D \text{ is an epimorphism}\}. \\
\end{matrix}
\]
Moreover, since $j$ is fully faithful, the unit of the adjunction
$\eta :{\rm id}_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}} {\rightarrow} r\circ j$ is an isomorphism of functors.
\smallskip
\begin{remark}\label{rG}
Giraud and co-Giraud subcategories very often appear in the literature in very different settings.
For example a well known result due to Popescu and Gabriel (see, for instance, \cite[Chapter 10]{MR0389953})
tells that any Grothendieck category is in a natural way a Giraud subcategory of the category $R$-Mod of all the left $R$-modules, for a suitable ring R.
On the other hand, the Yoneda tensor-embedding $M\mapsto {}-\otimes_R M$ naturally makes $R$-Mod to be a co-Giraud subcategory of the Grothendieck category $(\rm{FP}_R, \rm{Ab})$ whose objects are the covariant functors from the finitely presented right R-modules to the abelian groups, and the morphisms are the natural transformations between them.
This allows, for instance, to deal with the extensively studied notion of pure-injective module by means of injective objects in $(\rm{FP}_R, \rm{Ab})$, thanks to a result of Gruson and Jensen \cite{MR633523}. Dually, the Yoneda Hom-embedding $M\mapsto \mathrm{Hom}_R(-,M)$ naturally makes $R$-Mod to be a Giraud subcategory of the Grothendieck category of contravariant functors $({}_R\rm{FP}^{op}, \rm{Ab})$.
Auslander proposed to study the representation theory of $R$ in terms of the ambient category $({}_R\rm{FP}^{op}, \rm{Ab})$, and in \cite{MR0335575} and \cite{MR0379599} he and Reiten studied deeper the subcategory of the finitely presented objects of $({}_R\rm{FP}^{op}, \rm{Ab})$, developing the powerful theory of almost split sequences for Artin algebras.
\end{remark}
\smallskip
\section{Torsion and Torsion-free Classes}
\begin{definition}
Given an abelian category ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ a {\it torsion class} ${\mathcal T}$ is a full subcategory of ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ which is closed under taking inductive limits and extensions. Dually a {\it torsion free class} ${\mathcal F}$ is a full subcategory of ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ which is closed under taking projective limits and extensions.
\par
A {\it torsion pair} $({\mathcal T},{\mathcal F})$ in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is a the data of a torsion class ${\mathcal T}$ and a torsion free class ${\mathcal F}$ such that ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}({\mathcal T},{\mathcal F})=0$ and any object $C\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is the middle term of a short exact sequence $0 \to T \to C \to F \to 0$ with $T\in{\mathcal T}$ and $F\in{\mathcal F}$.
\end{definition}
\smallskip
A torsion class ${\mathcal T}$ {\it cogenerates} ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ when any object in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is a subobject of a suitable object in ${\mathcal T}$, and, dually, a torsion free class ${\mathcal F}$ {\it generates} ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ when any object in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is a factor of a suitable object in ${\mathcal F}$.
Typically, cogenerating torsion classes arise from Tilting theory and generating torsion free classes arise from Cotilting theory (see, for instance, \cite[Chapter I.3]{MR1327209} and \cite[Section 2]{MR2255195}).
\smallskip
\begin{remark}
If ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is a subcomplete abelian category in the sense of \cite{MR0191935} (that is, ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is an abelian category such that for any family $\{A_u \,|\, u\in U\}$ of subobjects of a fixed object $A$, the infinite sum
$\sum_{u\in U}A_u$ and the infinite product $\prod_{u\in U}(A/A_u)$ exist in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$), then any torsion class ${\mathcal T}$ (torsion-free class ${\mathcal F}$) on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ induces a torsion pair $({\mathcal T},{\mathcal F})$ on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
\end{remark}
\smallskip
The reader is referred to \cite[Chapter 1]{MR2327478} for more details.
\smallskip
In what follows, our aim is to move torsion class trough
exact functors
and subsequently trough a distinguished Giraud
(resp. co-Giraud)
subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ of $\cD$.
Since torsion classes (resp.\ torsion free classes)
are closed under inductive limits and extensions
(resp.\ projective limits and extensions),
it seems to us natural to use the left (resp.\ right)
adjoint functor $l$ (resp.\ $i$), which respects inductive limits
(resp.\ projective limits),
in order to move torsion classes
(resp.\ torsion free classes)
from ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ to $\cD$ (resp.\ from $\cD$ to ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$).
\bigskip
\begin{lemma}\label{MTC}(Dual to \ref{MTFC}).
Let ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be an abelian category and ${\mathcal T}$ a torsion class on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Let $l:\cD {\rightarrow} {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be a functor between abelian categories which respects
inductive limits. Then the class
$$l^{\scriptscriptstyle\leftarrow}({\mathcal T})=\{D\in\cD\; |\; l(D)\in {\mathcal T}\}$$
is a torsion class in $\cD$.
\end{lemma}
\begin{proof}
Clearly, the class $l^{\scriptscriptstyle\leftarrow}({\mathcal T})$ is closed under taking inductive limits, because so is ${\mathcal T}$ and $l$ respects inductive limits by assumption. Let us show that $l^{\scriptscriptstyle\leftarrow}({\mathcal T})$ is closed under extensions. Consider a short exact sequence in $\cD$
\begin{equation*}
\xymatrix{
0 \ar[r] & X_1\ar[r] & D\ar[r] & X_2\ar[r] & 0}
\end{equation*}
with $X_1, X_2\in l^{\scriptscriptstyle\leftarrow}({\mathcal T})$. By applying the functor $l$ (which is right exact) to this sequence we get an exact sequence in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$
\[\xymatrix{
l(X_1)\ar[r] & l(D)\ar[r] & l(X_2)\ar[r] &0}
\]
with $l(X_1), l(X_2)\in{\mathcal T}$. Taking the kernel $K$ of the morphism $l(D) {\rightarrow} l(X_2)$,
we see that $K$ is an epimorphic image of $l(X_1)$ and so $K\in{\mathcal T}$, therefore $l(D)\in{\mathcal T}$ as extension of objects in a torsion class. We conclude that $D\in l^{\scriptscriptstyle\leftarrow}({\mathcal T})$.
\end{proof}
\smallskip
\begin{lemma}\label{MTFC}(Dual to \ref{MTC}).
Let ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be an abelian category and ${\mathcal F}$ a torsion-free class on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Let $r:\cD {\rightarrow} {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be a functor between abelian categories which respects
projective limits. Then the class
$$r^{\scriptscriptstyle\leftarrow}({\mathcal F})=\{D\in\cD\; |\; r(D)\in {\mathcal F}\}$$
is a torsion-free class in $\cD$.
\end{lemma}
\section{Moving Torsion Pairs trough Giraud subcategories}
Given an abelian category $\cD$ with a distinguished Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$, by Lemma~\ref{MTC},
the class $l^{\scriptscriptstyle\leftarrow}({\mathcal T}):=\{D\in\cD \,|\, l(D)\in {\mathcal T}\}$ is a torsion class on $\cD$.
\smallskip
\begin{proposition}\label{ttD}
Let $\cD$ be an abelian category with a distinguished Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$. Suppose that ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is endowed with a torsion pair $({\mathcal T},{\mathcal F})$.
Then the classes $(\hat{{\mathcal T}},\hat{{\mathcal F}})$:
$$\begin{matrix}
\hat{{\mathcal T}}:= l^{\scriptscriptstyle\leftarrow}({\mathcal T})=\{X\in\cD \,|\, l(X)\in{\mathcal T}\} \hfill \\
\hat{{\mathcal F} }:= l^{\scriptscriptstyle\leftarrow}({\mathcal F})\cap\cS^\bot=\{Y\in {\cD} \,|\, Y\in{\cS^\bot} \text{ {\rm and} } l(Y)\in{\mathcal F} \} \\
\end{matrix}
$$
define a torsion pair on $\cD$ such that $i({\mathcal T})\subseteq \hat{{\mathcal T}}$, $i({\mathcal F})\subseteq \hat{{\mathcal F}}$,
$l(\hat{{\mathcal T}})={\mathcal T}$, $l(\hat{{\mathcal F}})={\mathcal F}$.
\end{proposition}
\begin{proof}
For any $T\in{\mathcal T}$ we have $li(T)\cong T$, which proves that $i({\mathcal T})\subseteq\hat{{\mathcal T}}$.
Moreover given $F\in{\mathcal F}$ it is clear that $i(F)\in\cS^\bot$ and $li(F)\cong F \in{\mathcal F}$,
hence $i({\mathcal F})\subseteq \hat{{\mathcal F}}$. We deduce that ${\mathcal T}=li({\mathcal T})\subseteq l(\hat{{\mathcal T}})\subseteq {\mathcal T}$ and
${\mathcal F}=li({\mathcal F})\subseteq l(\hat{{\mathcal F}})\subseteq {\mathcal F}$, which prove that
$l(\hat{{\mathcal T}})={\mathcal T}$ and $ l(\hat{{\mathcal F}})={\mathcal F}$.
Let us show that $(\hat{{\mathcal T}},\hat{{\mathcal F}})$ is a torsion pair on $\cD$.
Given $X\in\hat{{\mathcal T}}$ and $Y\in\hat{{\mathcal F}}$,
$$
{\cD}(X,Y)\hookrightarrow {\cD}(X,il(Y))\cong{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}(l(X),l(Y))=0.
$$
It remains to prove that for any $D$ in $\cD$ there exists a short exact sequence
\begin{equation*}\label{Dseq}
\xymatrix{
0 \ar[r] & X\ar[r] & D\ar[r] & Y\ar[r] & 0}
\end{equation*}
with $X\in\hat{{\mathcal T}}$ and $Y\in\hat{{\mathcal F}}$.
Given $D$ in $\cD$ there exist $T\in{\mathcal T}$ and $F\in{\mathcal F}$ such that
the sequence
\begin{equation}\label{l(D)seq}
\xymatrix{
0 \ar[r]& T\ar[r] & l(D) \ar[r] & F \ar[r] & 0
}
\end{equation}
is exact.
Let define $X:=i(T)\times_{il(D)}D$; then we obtain the diagram
\begin{equation}\label{1}
\xymatrix{
0 \ar[r]& i(T)\ar[r] & il(D) \ar[r] & i(F) & \\
0 \ar[r] & X \ar[r]\ar[u] & D\ar[r]\ar[u]^{\eta_D} & {D/X}\ar@{^{(}->}[u] \ar[r]& 0 \\
}
\end{equation}
whose rows are exact (the first because the functor $i$ is left exact since it is a right adjoint,
while the second by definition) and the map
$D/X\hookrightarrow i(F)$ is injective since the first square is cartesian.
Let us apply the functor $l$ to (\ref{1}) remembering that $l$ is exact (so in particular
it preserves pullbacks and exact sequences) and that
$l\circ i\cong {\rm id}_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$:
$$
\xymatrix{
0 \ar[r]& T\ar[r] & l(D) \ar[r] & F\ar[r] & 0 \\
0 \ar[r] & l(X) \ar[r]\ar[u]^{\cong} & l(D)\ar[r]\ar[u]_{{\rm id}_{l(D)}} & {l(D/X)}\ar[u]_{\cong} \ar[r]& 0. \\
}
$$
The first row coincides with (\ref{l(D)seq}) which is exact,
$l(X)\cong T\times_{l(D)}l(D)\cong T\in{\mathcal T}$, which proves that $X\in\hat{{\mathcal T}}$ and so
$l(D/X)\cong F\in{\mathcal F}$, and the third vertical arrow of (\ref{1}) proves that $D/X\in\cS^\bot$, thus $D/X\in \hat{{\mathcal F}}$.
\end{proof}
\smallskip
The following is a corollary of \ref{MTFC}:
\begin{corollary}\label{DCorGS}
Let $\cD$ be an abelian category with a distinguished Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$. Suppose that $\cD$ is endowed with a torsion pair $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$.
Then the class
$i^{\scriptscriptstyle\leftarrow}({\mathcal Y}}\def\cN{{\mathcal N}):=\{C\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \,|\, i(C)\in {\mathcal Y}}\def\cN{{\mathcal N}\}$
is a torsion free class on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
\end{corollary}
\smallskip
\begin{proposition}\label{GL}
Let $\cD$ be an abelian category with a distinguished Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Suppose that $\cD$ is endowed with a torsion pair $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$,and let
\[
\begin{matrix}
l({\mathcal X}):=\{T\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \,|\, T\cong l(X), \exists \,X\in{\mathcal X}\} \hfill \\
l({\mathcal Y}}\def\cN{{\mathcal N}):=\{F\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \,|\, F\cong l(Y), \exists \,Y\in{\mathcal Y}}\def\cN{{\mathcal N}\}\\
\end{matrix}
\]
Then $(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ defines a torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ if and only if $il({\mathcal Y}}\def\cN{{\mathcal N}) \subseteq {\mathcal Y}}\def\cN{{\mathcal N}$. In this case,
$i^{\scriptscriptstyle\leftarrow}({\mathcal Y}}\def\cN{{\mathcal N})=l({\mathcal Y}}\def\cN{{\mathcal N})$.
\end{proposition}
\begin{proof}
First let us suppose that $il({\mathcal Y}}\def\cN{{\mathcal N}) \subseteq {\mathcal Y}}\def\cN{{\mathcal N}$. Then since $l\circ i\cong {\rm id}_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ one has $i^{\scriptscriptstyle\leftarrow}({\mathcal Y}}\def\cN{{\mathcal N})=l({\mathcal Y}}\def\cN{{\mathcal N})$ and
by Corollary~\ref{MTFC} this is a torsion free class on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Given $T \in l({\mathcal X})$ (i.e., $T\cong l(X)$, with $X \in {\mathcal X}$) and $F \in i^{\scriptscriptstyle\leftarrow}({\mathcal Y}}\def\cN{{\mathcal N})$, one has
${{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}(X,F)={{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}(l(X), F)\cong{\cD}(X,i(F))=0$,
since $i(F)\in{\mathcal Y}}\def\cN{{\mathcal N}$ by the definition of $i^{\scriptscriptstyle\leftarrow}({\mathcal Y}}\def\cN{{\mathcal N})$.
Now let $C\in {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
There exist $X\in{\mathcal X}$, $Y\in{\mathcal Y}}\def\cN{{\mathcal N}$ and a short exact sequence in $\cD$
\[\xymatrix{
0 \ar[r] & X\ar[r] & i(C)\ar[r] & Y\ar[r] & 0.}
\]
Applying the functor $l$ to the previous sequence we get a short exact sequence in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$
\[\xymatrix{
0 \ar[r] & l(X)\ar[r] & C\ar[r] & l(Y)\ar[r] & 0}
\]
where $l(X)\in l({\mathcal X})$ and $l(Y)\in l({\mathcal Y}}\def\cN{{\mathcal N})$, which proves that $(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ is a torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Conversely, if $(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ is a torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ then for every $X \in {\mathcal X}$ and every $Y\in {\mathcal Y}}\def\cN{{\mathcal N}$ one has
$0={\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}(l(X),l(Y))\cong\cD(X,il(Y))$, therefore $il(Y)\in {\mathcal Y}}\def\cN{{\mathcal N}$.
\end{proof}
\smallskip
>From \ref{ttD} and \ref{GL} we derive the following correspondence:
\begin{theorem}\label{tt1-1}
Let $\cD$ be an abelian category with a distinguished Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
There exists a one to one correspondence between torsion pairs
$({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ on $\cD$ satisfying
$il({\mathcal Y}}\def\cN{{\mathcal N})\subseteq{\mathcal Y}}\def\cN{{\mathcal N}\subseteq \cS^\bot$
and torsion pairs $({\mathcal T},{\mathcal F})$ on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
\end{theorem}
\begin{proof}
From one side, taking a torsion pair $({\mathcal T}, {\mathcal F})$ in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$, the torsion pair $(\hat{{\mathcal T}}, \hat{{\mathcal F}})$
satisfies $il(\hat{{\mathcal F}})\subseteq \hat{{\mathcal F}}$ and one can easily verify that
$(l(\hat{{\mathcal T}}),l(\hat{{\mathcal F}}))=({\mathcal T},{\mathcal F})$.
On the other side given $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ a torsion pair on $\cD$
satisfying $il({\mathcal Y}}\def\cN{{\mathcal N})\subseteq{\mathcal Y}}\def\cN{{\mathcal N}\subseteq \cS^\bot$, its corresponding torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is
$(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ (by \ref{GL}) for whom
it is clear that $\widehat{l({\mathcal Y}}\def\cN{{\mathcal N})}:=l^{\scriptscriptstyle\leftarrow}(l({\mathcal Y}}\def\cN{{\mathcal N}))\cap \cS^\bot={\mathcal Y}}\def\cN{{\mathcal N}$
(since ${\mathcal Y}}\def\cN{{\mathcal N}\subseteq \cS^\bot$)
and so $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})=(\widehat{l({\mathcal X})},\widehat{l({\mathcal Y}}\def\cN{{\mathcal N})})$.
\end{proof}
\smallskip
We briefly list
the statements dual in the case of co-Giraud subcategories whose
proofs are simply the transcriptions of the previous ones in the opposite category.
\smallskip
First of all let us consider the following corollary of \ref{MTFC}:
\begin{corollary}\label{cCorcGS}
Let $\cD$ be an abelian category with a distinguished co-Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Suppose that ${\mathcal F}$ is a torsion-free class in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Then the class
$r^{\scriptscriptstyle\leftarrow}({\mathcal F}):=\{D\in\cD \,|\, r(D)\in {\mathcal F}\}$
is a torsion-free class on $\cD$.
\end{corollary}
\smallskip
\begin{proposition}\label{cttD}
Let $\cD$ be an abelian category with a distinguished co-Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$. Suppose that ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is endowed with a torsion pair $({\mathcal T},{\mathcal F})$.
Then the classes $(\hat{{\mathcal T}},\hat{{\mathcal F}})$:
$$\begin{matrix}
\hat{{\mathcal T}}:= r^{\scriptscriptstyle\leftarrow}({\mathcal T})\cap{}^\bot\cS=\{X\in\cD \,|\, X\in{}^\bot\cS \text{ {\rm and} } r(X)\in{\mathcal T}\} \hfill \\
\hat{{\mathcal F} }:= r^{\scriptscriptstyle\leftarrow}({\mathcal F})=\{Y\in {\cD} \,|\, r(Y)\in{\mathcal F} \} \\
\end{matrix}
$$
define a torsion pair on $\cD$ such that $j({\mathcal T})\subseteq \hat{{\mathcal T}}$, $j({\mathcal F})\subseteq \hat{{\mathcal F}}$,
$r(\hat{{\mathcal T}})={\mathcal T}$, $r(\hat{{\mathcal F}})={\mathcal F}$.
\end{proposition}
\smallskip
The following is a corollary of \ref{MTC}:
\begin{corollary}\label{cDCorGS}
Let $\cD$ be an abelian category with a distinguished co-Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Suppose that $\cD$ is endowed with a torsion pair $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$.
Then the class
$j^{\scriptscriptstyle\leftarrow}({\mathcal X}):=\{C\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \,|\, j(C)\in {\mathcal X}\}$
is a torsion class on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
\end{corollary}
\smallskip
\begin{proposition}\label{cGL}
Let $\cD$ be an abelian category with a distinguished co-Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Suppose that $\cD$ is endowed with a torsion pair $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$, and let
\[
\begin{matrix}
r({\mathcal X}):=\{T\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \,|\, T\cong r(X), \exists \,X\in{\mathcal X}\} \hfill \\
r({\mathcal Y}}\def\cN{{\mathcal N}):=\{F\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \,|\, F\cong r(Y), \exists \,Y\in{\mathcal Y}}\def\cN{{\mathcal N}\}\\
\end{matrix}
\]
Then $(r({\mathcal X}),r({\mathcal Y}}\def\cN{{\mathcal N}))$ defines a torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ if and only if $jr({\mathcal X}) \subseteq {\mathcal X}$. In this case,
$j^{\scriptscriptstyle\leftarrow}({\mathcal X})=r({\mathcal X})$.
\end{proposition}
\smallskip
>From \ref{cttD} and \ref{cGL} we derive the following correspondence:
\begin{theorem}\label{ctt1-1}
Let $\cD$ be an abelian category with a distinguished co-Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
There exists a one to one correspondence between torsion pairs
$({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ on $\cD$ satisfying
$jr({\mathcal X})\subseteq{\mathcal X}\subseteq \;^\bot\cS$
and torsion pairs $({\mathcal T},{\mathcal F})$ on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
\end{theorem}
\smallskip
\section{$t$-structures induced by torsion pairs}
\begin{definition}\label{deft-s
A {\it $t$-structure} on a triangulated category $\cD$ is a pair $t=(\cDl0,\cDg0)$ of strictly full subcategories of $\cD$ such that,
setting $\cDl n:=\cDl0[-n]$ and $\cDg n:=\cDg0[-n]$, one has
\begin{enumerate}
\item[\rm (0)] $\cDl0\subseteq\cDl1$ and $\cDg0\supseteq\cDg1$.
\item[\rm (i)] $\cD(X,Y)=0$ for every $X$ in $\cDl0$ and every $Y$ in $\cDg1$.
\item[\rm (ii)] For any object $X\in\cD$ there exists a distinguished triangle:
\[A\to X\to B \to A[1]
\]
in $\cD$ such that
$A\in\cDl0$
and $B\in\cDg1$.
\end{enumerate}
\end{definition}
\smallskip
\begin{proposition}{\rm\cite[Proposition 1.3.3]{MR751966}}\label{t-adj}
Let ${\rm t}=(\cDl0,\cDg0)$ be a $t$-structure on a triangulated category $\cD$.
\begin{enumerate}
\item[\rm (i)]The inclusion of $\cDl n$ in $\cD$ admits a right adjoint $\tau^{\leq n}$, and the inclusion of $\cDg n$ in $\cD$ a left adjoint $\tau^{\geq n}$,
called the truncation functors.
\item[\rm (ii)]For every $X$ in $\cD$ there exists a unique morphism
$d\colon \tau^{\geq 1}(X)\to \tau^{\leq 0}(X)[1]$ such that the triangle
\[
\tau^{\leq 0}(X){\rightarrow} X{\rightarrow} \tau^{\geq 1}(X)\overset{d}{\rightarrow}
\]
is distinguished. This triangle is (up to a unique isomorphism) the unique distinguished triangle
$(A,X,B)$ with $A$ in $\cDl0$ and $B$ in $\cDg1$.
\item[\rm (iii)]The category $\cH_t:=\cDl0\cap\cDg0$ is abelian, and the truncation functors induce a functor
$\mathrm{H}_t\colon \cD \to \cH_t$, called the $t$-cohomological functor
($H_t^0(X)=\tau^{\geq0}\tau^{\leq0}(X)\cong\tau^{\leq0}\tau^{\geq0}(X)$ and for every $i\in \Bbb Z$,
$\mathrm{H}_t^i(X)=\mathrm{H}_t^0(X[i])$, see {\rm\cite[Theorem~1.3.6]{MR751966}}).
\end{enumerate}
\end{proposition}
\smallskip
In particular, given an abelian category ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ its (unbounded) derived category $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ is a triangulated category
which admits a canonical $t$-structure, called the {\it natural $t$-structure}, whose class $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\leq 0}$
(resp. $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\geq 0}$) is that of complexes without cohomology in positive (resp. negative) degrees.
We will denote by $\mathrm{H}\colon D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}) \to {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ its cohomological functor and by
$t^{\leq n}$ resp. $t^{\geq n}$ its truncation functors.
As explained by A.~Beligiannis and I.~Reiten in their work \cite{MR2327478},
one can regard a $t$-structure on a triangulated category $\cD$ as a generalization
of a torsion pair, where the role of the torsion class is provided by $\cDl0$,
while that of the torsion free class is played by $\cDg1$.
Moreover, given a torsion pair on an abelian category ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ one can construct a
$t$-structure on its derived category $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$, as explained in \cite{MR1327209}.
\endgraf
Let us briefly recover this construction:
\begin{proposition}
Let $({\mathcal T},{\mathcal F})$ be a torsion pair on an abelian category ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
The classes
$$\begin{matrix}
t({\mathcal T})=\cD_{\rm t}^{\leq 0}= & \{ C^{\scriptscriptstyle\bullet}\in D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})\; | \; H^0(C^{\scriptscriptstyle\bullet})\in{\mathcal T},\; H^i(C^{\scriptscriptstyle\bullet})=0 \; \forall i>0 \} \hfill\\
t({\mathcal F})=\cD_{\rm t}^{\geq 0}= & \{C^{\scriptscriptstyle\bullet}\in D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})\; | \; H^{-1}(C^{\scriptscriptstyle\bullet})\in{\mathcal F},\; H^i(C^{\scriptscriptstyle\bullet})=0 \; \forall i<-1 \} \\
\end{matrix}
$$
define a $t$-structure on $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ which is called the $t$-structure induced by the torsion pair $t$.
\end{proposition}
\begin{proof}
It is straightforward to verify condition ($0$) of definition \ref{deft-s}.
Let us show that condition (i) holds. Indeed, given
$X^{\scriptscriptstyle\bullet} \in \cD_{\rm t}^{\leq 0}$ and $Y^{\scriptscriptstyle\bullet}\in \cD_{\rm t}^{\geq 1}$ (i.e.,$H^0(Y^{\scriptscriptstyle\bullet})\in{\mathcal F}$, and
$H^i(Y^{\scriptscriptstyle\bullet})=0 \; \forall i<0$) one has
\begin{eqnarray*}
D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})(X^{\scriptscriptstyle\bullet},Y^{\scriptscriptstyle\bullet})\cong & D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})\left(\tau^{\geq 0}(X^{\scriptscriptstyle\bullet}),Y^{\scriptscriptstyle\bullet}\right)\cong
D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})\left(\tau^{\geq 0}(X^{\scriptscriptstyle\bullet}),\tau^{\leq 0}(Y^{\scriptscriptstyle\bullet})\right)\cong \hfill \cr
\hfill \cong & D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})\left(H^0(X^{\scriptscriptstyle\bullet})[0],H^0(Y^{\scriptscriptstyle\bullet})[0]\right)\cong {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}\left(H^0(X^{\scriptscriptstyle\bullet}),H^0(Y^{\scriptscriptstyle\bullet})\right)=0 \hfill \cr
\end{eqnarray*}
Finally, let us prove condition (ii). Given any $C^{\scriptscriptstyle\bullet}\in D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$, let us consider the object $T=t(H^0(C^{\scriptscriptstyle\bullet}))$ which is the torsion part of the zero cohomology
$H^0(C^{\scriptscriptstyle\bullet})={{\mathrm{Ker}}(d^0_C)\over{{\mathrm{Im}}(d^{-1}_C)} }$,
and let us define $X$ to be the fiber product
\[
X=T\times_{H^0(C^{\scriptscriptstyle\bullet})}{\mathrm{Ker}}(d^0_C).
\]
Then we obtain the following functorial construction:
\\
\[
\xymatrix@C=1.7em{
\mathrm{Im}(d_C^{-1}) \ar@{^{(}->}@/^2pc/[rr]\ar[rd]_{0} \ar@{^{(}.>}[r]^(0.6){\exists !}
& X \ar@{^{(}->}[r]\ar@{->>}[d] & {\mathrm{Ker}}(d^0_C) \ar@{->>}[d] \\
& T\ar@{^{(}->}[r] & H^0(C^{\scriptscriptstyle\bullet})\\
}
\]
where ${X\over{\mathrm{Im}(d^{-1}_C)}}\cong T$.
This permits to define the short exact sequence of complexes:
\[
\xymatrix@C=1.7em{
0= \ar[d] &
[\cdots \ar[r] & 0 \ar[r]\ar[d] &
0 \ar[r]\ar[d] & 0 \ar[r]\ar[d] & 0 \ar[r]\ar[d] & 0\ar[r]\ar[d] &
\cdots ] \hfill \\
\tau_t^{\leq 0}(C^{\scriptscriptstyle\bullet})= \ar[d] &
[\cdots \ar[d]\ar[r] & C^{-2}\ar[d]^{\cong} \ar[r]^{d_C^{-2}} &
C^{-1} \ar[d]^{\cong} \ar[r]^{d_C^{-1}} & X \ar[d] \ar[r] & 0 \ar[d]\ar[r] & 0\ar[d]\ar[r] &
\cdots ] \hfill \\
{\rm id}(C^{\scriptscriptstyle\bullet})= \ar[d]& [\cdots \ar[r]\ar[d] & C^{-2}\ar[d] \ar[r]^{d_C^{-2}} &
C^{-1} \ar[d] \ar[r]^{d_C^{-1}} & C^0\ar[d]\ar[r]^{d^0_C} & C^1\ar[d]^{\cong}\ar[r]^{d^1_C} & C^2 \ar[d]^{\cong}\ar[r] &
\cdots ] \hfill \\
\tau_t^{\geq 1}(C^{\scriptscriptstyle\bullet})= \ar[d]&
[\cdots \ar[r] & 0 \ar[r]\ar[d] & 0\ar[r]\ar[d] & {C^0\over{X}} \ar[r]^{d^0_C}\ar[d] & C^1\ar[r]^{d^1_C}\ar[d] & C^2 \ar[r] \ar[d]&
\cdots ] \hfill \\
0= &
[\cdots \ar[r] & 0 \ar[r] &
0 \ar[r] & 0 \ar[r] & 0 \ar[r]& 0\ar[r] &
\cdots ] \hfill \\
}
\]
such that
$\tau_t^{\leq 0}(C^{\scriptscriptstyle\bullet})\in \cD_{\rm t}^{\leq 0}$
(since
$H^0(\tau_t^{\leq 0}(C^{\scriptscriptstyle\bullet}))={X\over{\mathrm{Im}(d^{-1}_C)}}\cong T\in {\mathcal T}$ and $H^i(\tau_t^{\leq 0}(C^{\scriptscriptstyle\bullet}))=0$
for any $i>0$)
and
$\tau_t^{\geq 1}(C^{\scriptscriptstyle\bullet})\in \cD_{\rm t}^{\geq 1}$
(since
$H^0(\tau_t^{\geq 1}(C^{\scriptscriptstyle\bullet}))={{\mathrm{Ker}}(d^0_C)\over{X}}\cong {H^0(C)\over T}\in {\mathcal F}$ and $H^i(\tau_t^{\leq 0}(C^{\scriptscriptstyle\bullet}))=0$
for any $i<0$).
\endgraf
Let us recall that any short exact sequence of complexes in an abelian category ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ induces a distinguished triangle
in its derived category $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ (see \cite[Section 2.4.2]{MR2286904}).
In our case the previous exact sequence provides a distinguished triangle
$\tau_t^{\leq 0}(C^{\scriptscriptstyle\bullet})\to C^{\scriptscriptstyle\bullet}\to \tau_t^{\geq 1}(C^{\scriptscriptstyle\bullet}) \to \tau_t^{\leq 0}(C^{\scriptscriptstyle\bullet})[1]$,
and this concludes the proof.
\end{proof}
\smallskip
\begin{remark}
Let ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be an abelian category endowed with the trivial torsion pair
$({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}, 0)$. The the $t$-structure associated to this trivial torsion pair is the trivial $t$-strucuture
on $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ (see \ref{t-adj}).
\end{remark}
\smallskip
\begin{remark}\label{Tstr}\label{RTS}
Let ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be an abelian category with a torsion pair $({\mathcal T},{\mathcal F})$.
The {\it heart} associated to the $t$-structure
$(\cD_{\rm t}^{\leq 0},\cD_{\rm t}^{\geq 0})$ on $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$
is the full subcategory $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}} := t({\mathcal T})\cap t({\mathcal F})$ of $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ called the tilt of ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ by the torsion pair
$({\mathcal T},{\mathcal F})$.
It is shown in
\cite{MR751966} that $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ is an abelian category where short exact sequences are deduced by distinguished triangles in $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$.
The objects of $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ are represented, up to isomorphism, by complexes of the form
\begin{equation*}
X:\ X^{-1}\overset{x}{\longrightarrow }X^{0},\;\text{ with }\mathrm{Ker}(x)\in {\mathcal F}\text{ and }
\mathrm{Coker}(x)\in {\mathcal T},
\end{equation*}
while a morphism $\phi : X {\rightarrow} Y$ in $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ is a formal fraction $\phi = (s)^{-1}\circ f$, where:
\begin{enumerate}
\item $X\overset{f}{\longrightarrow}Z$ is a representative of a homotopy class of maps of complexes
$$
\xymatrix{X^{-1}\ar[d]_{f^{-1}}\ar[r]^x & X^0\ar[d]^{f^0} \\
Z^{-1}\ar[r]^z & Z^0
}
$$
where we recall that $X\overset{f}{\longrightarrow }Z$ is null-homotopic if there is
a map $r^{0}:X^{0}\rightarrow Z^{-1}$ such that%
\begin{equation*}
f^{0}=zr^{0}\text{ \ and \ }f^{-1}=r^{0}x
\end{equation*}
\item $Y\overset{s}{\longrightarrow }Z$ is a quasi-isomorphism, i.e., it is a map of complexes
which induces isomorphism in cohomology:
$$
\xymatrix{0\ar[r] & \mathrm{Ker}(y) \ar[d]_{\cong} \ar[r] & Y^{-1}\ar[d]_{s^{-1}} \ar[r]^y & Y^0 \ar[d]^{s^0} \ar[r] & \mathrm{Coker}(y)\ar[d]^{\cong} \ar[r] & 0 \\
0\ar[r] & \mathrm{Ker}(z) \ar[r] & Z^{-1} \ar[r]^z & Z^0 \ar[r] & \mathrm{Coker}(z) \ar[r] & 0
}
$$
\end{enumerate}
Every distinguished triangle $X_1^{\scriptscriptstyle\bullet}{\rightarrow} X_2^{\scriptscriptstyle\bullet} {\rightarrow} X_3^{\scriptscriptstyle\bullet} \buildrel{[+1]}\over{\rightarrow} X_1^{\scriptscriptstyle\bullet}[1]$
in $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ provides a long exact sequence of $t$-cohomology
in the heart $\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$:
\begin{equation*}
\cdots H_t^{-1}(X_3){\rightarrow} H_t^0(X_1){\rightarrow} H_t^0(X_2){\rightarrow} H_t^0(X_3){\rightarrow}
H_t^1(X_1)\cdots
\end{equation*}
Moreover given an object $C$ in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$, its $t$-cohomology objects in $\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ are
$H^i_t(C)=0$ for any $t<0$, $t>1$; $H^0_t(C)=t(C)[0]$ is the torsion part of $C$ (with respect
to the torsion pair $({\mathcal T},{\mathcal F})$) placed in degree zero, while $H^1_t(C)=\frac{C}{t(C)}[1]$.
The tilted pair $({\mathcal F}[1],{\mathcal T}[0])$ is a torsion pair in $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ with category equivalences
${\mathcal F}[1]\cong{\mathcal F}$ and ${\mathcal T}[0]\cong{\mathcal T}$ (see \cite[Corollary~2.2]{MR1327209}).
\end{remark}
\smallskip
\begin{remark}
In \cite{MR2255195} the authors introduced the notion of a tilting object for an arbitrary Abelian category, proving that for any ring
$R$ and for any faithful torsion pair $({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ in $R$-Mod the heart $\cH({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ of the t-structure in
$D(R)$ associated to $({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ is (an abelian category) with a tilting object $T=R[1]$.
Then, again the first author with Gregorio and Mantese in \cite{MR2275375} showed that the heart is a prototype for these categories, in the sense that an Abelian category
$\cD$ admits a tilting object $T$ if and only if $\cD$ is equivalent to the category $\cH({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ for a suitable torsion pair $({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ in $\mathrm{End}(T)$-Mod which is ``tilted'' by $T$, and with Gregorio in \cite{PREPRINT} proved that $\cH({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ is a Grothendieck category if and only if the torsion pair is cogenerated by a cotilting module in the sense of \cite{MR1448804}.
This allows us to deal with a more general notion of a``tilting context'': given an Abelian category $\cD$ endowed with a faithful torsion pair $({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ (i.e., such that ${\mathcal Y}}\def\cN{{\mathcal N}$ generates $\cD$), we get a new Abelian category $\cH({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ endowed with a torsion pair $({\mathcal Y}}\def\cN{{\mathcal N}[1], {\mathcal X}[0])$ which is ``tilting'', in the sense that the torsion class ${\mathcal Y}}\def\cN{{\mathcal N}[1]$ cogenerates the category $\cH({\mathcal X}, {\mathcal Y}}\def\cN{{\mathcal N})$ and there are category equivalences ${\mathcal Y}}\def\cN{{\mathcal N}[1]\cong{\mathcal Y}}\def\cN{{\mathcal N}$ and ${\mathcal X}[0]\cong{\mathcal X}$ induced by exact functors.
\end{remark}
\smallskip
Let ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be an abelian category endowed with a torsion pair $({\mathcal T},{\mathcal F})$.
Since we need to use different torsion pairs we would use the notation
$(t({\mathcal T}),t({\mathcal F}))$ instead of $(\cD_{\rm t}^{\leq 0},\cD_{\rm t}^{\geq 0})$ to denote the $t$-structure
associated to the torsion pair $({\mathcal T},{\mathcal F})$.
For the same reason when we need to
clarify the torsion pair we would denote by $\tau_{t({\mathcal T})}, \tau_{t({\mathcal F})}$ the truncation functors
instead of $\tau_t^{\leq 0},\tau_t^{\geq 1}$.
\smallskip
As showed in \cite{MR2327478}, there exists an injective function between the poset of torsion pairs in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ and that of $t$-structures in $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$.
Moreover one can recover those $t$-structures on $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ which are induced by torsion pairs by means of the following fact proved by Keller and Vossieck in \cite{0663.18005}, and Polishchuk in \cite[Lemma 1.2.2]{MR2324559}.
\smallskip
\begin{theorem}
Given ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ an abelian category. There exists a bijection between
\begin{enumerate}
\item torsion pairs on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$
\item $t$-structures $({\mathcal T}^{\leq 0},{\mathcal T}^{\geq 0})$ on $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ such that
$D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\leq -1}\subset {\mathcal T}^{\leq 0}\subset D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\leq 0}$.
\end{enumerate}
\end{theorem}
\begin{proof}
We have just seen that any torsion pair $({\mathcal T},{\mathcal F})$ induces
the $t$-structure $(t({\mathcal T}),t({\mathcal F})[1])$ on $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ which, by definition, satisfies
$D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\leq -1}\subset t({\mathcal T})\subset D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\leq 0}$.
\\
On the other side given $({\mathcal T}^{\leq 0},{\mathcal T}^{\geq 0})$ a
$t$-structure on $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ such that
$D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\leq -1}\subset {\mathcal T}^{\leq 0}\subset D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\leq 0}$ we obtain (by orthogonality) that
$D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\geq 1}\subset {\mathcal T}^{\geq 1}\subset D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\geq 0}$.
Then the classes ${\mathcal T}={\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}[0]\cap {\mathcal T}^{\leq 0}$ and
${\mathcal F}={\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}[0]\cap {\mathcal T}^{\geq 1}$ define a torsion pair in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$
whose approximation short exact sequence for an object $C\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$
is given by the long exact sequence of $\mathrm{H}$ cohomology for the distinguished triangle $\tau_{t({\mathcal T})}(C){\rightarrow} C{\rightarrow} \tau_{t({\mathcal F})}(C)$
since $\tau_{t({\mathcal T})}(C)\in \subset D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\leq 0}$ (resp. $\tau_{t({\mathcal F})}(C)\in \subset D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})^{\geq 0}$).
\end{proof}
\smallskip
\section{Tilted Giraud subcategories}
\begin{lemma}\label{DlComm}
Let $\cD$ and ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be abelian categories and $l \colon \cD \to {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be an exact functor. Suppose that $\cD$ is endowed with a torsion pair $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ and that $({\mathcal T},{\mathcal F})=(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ defines a torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$. Then
$Dl\circ\tau_{t({\mathcal Y}}\def\cN{{\mathcal N})}=\tau_{t({\mathcal F})}\circ Dl$ and $Dl\circ\tau_{t({\mathcal X})}=\tau_{t({\mathcal T})}\circ Dl$. In particular, $Dl$ commutes with the functors $H^0_t$.
\end{lemma}
\begin{proof}
Since $l$ is exact it admits a total derived functor $Dl \colon D(\cD) \to D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$. Moreover, from $l({\mathcal X})={\mathcal T}$ and $l({\mathcal Y}}\def\cN{{\mathcal N})={\mathcal F}$ we derive that $Dl(t({\mathcal X}))\subseteq t({\mathcal T})$ and $Dl(t({\mathcal Y}}\def\cN{{\mathcal N}))\subseteq t({\mathcal F})$, i.e.,$Dl$ is an exact functor for the $t$-structure $(t({\mathcal X}),t({\mathcal Y}}\def\cN{{\mathcal N}))$ on $D(\cD)$ (see \cite[1.3.16]{MR751966}).
Let $D^{\scriptscriptstyle\bullet}\in D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$ and
\begin{equation}\label{triaD}
\xymatrix{ \tau_{t({\mathcal X})}(D^{\scriptscriptstyle\bullet}) \ar[r]& D^{\scriptscriptstyle\bullet} \ar[r]& \tau_{t({\mathcal Y}}\def\cN{{\mathcal N})}(D^{\scriptscriptstyle\bullet}) \ar[r]^{\qquad+1}&
}
\end{equation}
its distinguished triangle, with $\tau_{t({\mathcal X})}(D^{\scriptscriptstyle\bullet}) \in t({\mathcal X})$ and $\tau_{t({\mathcal Y}}\def\cN{{\mathcal N})}(D^{\scriptscriptstyle\bullet}) \in t({\mathcal Y}}\def\cN{{\mathcal N})$. By applying the functor $Dl$ to (\ref{triaD}) we get the triangle in $D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$
\begin{equation}\label{triaC}
\xymatrix{ Dl(\tau_{t({\mathcal X})}(D^{\scriptscriptstyle\bullet})) \ar[r]& Dl(D^{\scriptscriptstyle\bullet}) \ar[r]& Dl(\tau_{t({\mathcal Y}}\def\cN{{\mathcal N})}(D^{\scriptscriptstyle\bullet})) \ar[r]^{\qquad+1}&
}
\end{equation}
with $Dl(\tau_{t({\mathcal X})}(D^{\scriptscriptstyle\bullet})) \in t({\mathcal T})$ and $Dl(\tau_{t({\mathcal Y}}\def\cN{{\mathcal N})}(D^{\scriptscriptstyle\bullet})) \in t({\mathcal F})$, so (\ref{triaC}) is the distinguished triangle associated to $Dl(D^{\scriptscriptstyle\bullet})$, which proves that $Dl\circ\tau_{t({\mathcal Y}}\def\cN{{\mathcal N})}=\tau_{t({\mathcal F})}\circ Dl$ and $Dl\circ\tau_{t({\mathcal X})}=\tau_{t({\mathcal T})}\circ Dl$.
\end{proof}
\smallskip
\begin{proposition}\label{SonH}
Let $\cS$ be a Serre subclass in an abelian category $\cD$, and suppose that $\cD$ is endowed with a torsion pair
$({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ such that $(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ is a torsion pair on the quotient category
${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} :=\cD/\cS$.
Then $l$ induces a functor $l_\cH \colon \cH_{\cD} \to \cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ on the associated hearts which
is exact and essentially surjective and so, denoted by $\cS_\cH$ the kernel of $l_\cH$, one has that $\cS_\cH$
is a Serre subclass of $\cH_\cD$ and $\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}\cong \cH_\cD/\cS_\cH$.
\end{proposition}
\begin{proof}
As $l$ is exact and it respects the torsion pairs
its total derived functor $Dl \colon D(\cD) \to D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$
is exact with respect to the $t$-structures associated to the torsion pairs so the
the restriction of $Dl$ to $\cH_{\cD}$ defines a functor
$l_\cH \colon \cH_{\cD} \to \cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ on the hearts which is exact.
In particular the kernel $\cS_\cH$ of $l_\cH$ is a Serre subclass of $\cH_\cD$.
The functor $l_\cH$ is essentially surjective since given an object $X^{\scriptscriptstyle\bullet} \in \cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$,
there exists $D^{\scriptscriptstyle\bullet} \in \cH_{\cD}$ such that $X^{\scriptscriptstyle\bullet} \cong Dl(D^{\scriptscriptstyle\bullet})$.
Therefore using Lemma~\ref{DlComm} we find that
\[
\begin{matrix}
X^{\scriptscriptstyle\bullet} =& H^0_t(X^{\scriptscriptstyle\bullet})
\hfill\cong& H^0_t\circ Dl(D^{\scriptscriptstyle\bullet}) \hfill \\
\hfill\cong& Dl\circ H^0_t(D^{\scriptscriptstyle\bullet}) \cong& l_{\cH}\circ H^0_t(D^{\scriptscriptstyle\bullet}), \hfill \\
\end{matrix}
\]
which proves that $l_{\cH}$ is essentially surjective, so that
by \cite[Chapter 3, Section 1, Corollary~2]{0201.35602} we get
$\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}\cong \cH_\cD/\cS_\cH$.
\end{proof}
\smallskip
\begin{theorem}\label{adjhearts}
Let $\cD$ be an abelian category with a distinguished Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ such that
$i$ admits a right derived functor ${Ri}$.
Let $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ be a torsion pair on $\cD$ such that $il({\mathcal Y}}\def\cN{{\mathcal N})\subseteq{\mathcal Y}}\def\cN{{\mathcal N}$, and let $({\mathcal T},{\mathcal F})=(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ be the induced torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
Let us denote by $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ and $\cH_{\cD}$ the associated hearts.
Then there exists a distinguished Giraud subcategory
$(\cH_\cD,\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S},l_\cH,i_\cH)$ such that
$i_\cH(l_\cH({\mathcal X}[0]))\subseteq {\mathcal X}[0]$.
\end{theorem}
\begin{proof}First we remark that since $l$ and $i$ are additive, they extend to an adjunction $\xymatrix{K({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}) \ar@<-0.5ex>[r]_{i}& K(\cD)\ar@<-0.5ex>[l]_{l}}$ between the homotopy categories. Moreover, since $l$ is exact it admits a total derived functor $Dl\colon D(\cD)\to D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})$. Therefore $\xymatrix{D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}) \ar@<-0.5ex>[r]_{Ri}&D(\cD)\ar@<-0.5ex>[l]_{Dl}}$ are two adjoint functors (with $Dl$ left adjoint of $Ri$) by \cite[Section 3.1]{MR2384608}, and $Dl\circ Ri\cong R(l\circ i)\cong id_{D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S})}$.
By Proposition~\ref{SonH}, $l$ induces a functor $l_\cH \colon \cH_{\cD} \to \cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ on the associated hearts which
is exact and essentially surjective and so $\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}\cong \cH_\cD/\cS_\cH$.
On the other hand, the fact that $i$ is left exact ensures that $Ri$ takes $t({\mathcal F})$ inside $t({\mathcal Y}}\def\cN{{\mathcal N})$.
Let $\tau_{t({\mathcal X})} \colon D(\cD)\to t({\mathcal X})$ be the right adjoint of the inclusion
$t({\mathcal X})\to D(\cD)$ (see \cite[Proposition~1.3.3.(i)]{MR751966}). Then the restriction of the composition $\tau_{t({\mathcal X})}\circ Ri$ to $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ gives a functor $i_\cH \colon \cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}\to \cH_{\cD}$ and it is easy to see that $l_\cH$ is left adjoint of $i_\cH$ by composing the previous adjunctions.
Next, using Lemma~\ref{DlComm} we have that
$$\begin{matrix}
l_\cH \circ i_\cH =& Dl \circ \tau_{t({\mathcal X})}\circ Ri_{|\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}} \cong&
\tau_{t({\mathcal T})}\circ Dl \circ Ri_{|\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}}\hfill \\
\hfill\cong& \tau_{t({\mathcal T})}\circ D(l \circ i)_{|\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}}\cong& \tau_{t({\mathcal T})}\circ id_{\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}} \hfill \\
\hfill\cong& id_{\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}\hfill \\
\end{matrix}
$$
and from this we conclude that $i_\cH$ is fully faithful.
Finally,
\[
i_\cH \circ l_\cH ({\mathcal X}[0]) \subseteq \tau_{t({\mathcal X})}\circ (Ri \circ Dl)(D^{\geq0}(\cD)) \subseteq
\tau_{t({\mathcal X})}(D^{\geq0}(\cD)) \subseteq {\mathcal X}[0].
\]
\end{proof}
\smallskip
\begin{remark}
Let us explain two examples in which one can apply the previous result.
As a first example let ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ be an abelian category satisfying $AB4^*$ (that is, small products exist in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ and such products are exact in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$) and with enough injectives, and
$i \colon {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S} \to \cD$ is an additive functor. Then the right derived functor $Ri \colon D({\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}) \to D(\cD)$ exists by \cite[APPLICATION 2.4]{MR1214458}.
Another interesting case is the one in which the category ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ admits enough $i$-acyclic objects.
In this case one can use the same argument as in Proposition~\ref{adjhearts} restricted to the bounded below derived categories in order to obtain the same result.
\end{remark}
\smallskip
Dually, we have:
\begin{theorem}\label{cadjhearts}
Let $\cD$ be an abelian category with a distinguished co-Giraud subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ such that
$j$ admits a left derived functor ${Lj}$.
Let $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ a torsion pair on $\cD$ such that $jr({\mathcal X})\subseteq{\mathcal X}$, and let $({\mathcal T},{\mathcal F})=(r({\mathcal X}),r({\mathcal Y}}\def\cN{{\mathcal N}))$ be the induced torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$. Let us denote by $\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}$ and $\cH_{\cD}$ the associated hearts.
Then there exists a distinguished co-Giraud subcategory
$(\cH_\cD,\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S},r_\cH,j_\cH)$ such that
$j_\cH(r_\cH({\mathcal Y}}\def\cN{{\mathcal N}[1]))\subseteq {\mathcal Y}}\def\cN{{\mathcal N}[1]$.
\end{theorem}
\smallskip
The next result shows that the contexts described in Theorems~\ref{adjhearts} and \ref{cadjhearts} are as general as possible.
\begin{theorem}\label{reconstruction}
Let $\cD$ be an abelian category endowed with a torsion pair
$({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ and let $\cH_{\cD}$ be the corresponding heart with respect to the
$t$-structure on $D(\cD)$ induced by $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$.
Let $\cS'$ be a Serre subcategory of $\cH_\cD$ and
$l':\cH_\cD\to {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}':=\cH_\cD/\cS'$ be its corresponding quotient functor,
and let us suppose that $i'l'({\mathcal X}[0])\subseteq X[0]$ ( i.e., $(l'({\mathcal Y}}\def\cN{{\mathcal N}[1]),l'({\mathcal X}[0]))$ is a torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}'$).
Then:
\begin{enumerate}
\item The class
$\cS=\{ D\in \cD\; | \; l'(H_t^i(D))=0 \; \forall i\in \Bbb Z\}$
is a Serre subcategory
of $\cD$.
\item Denoted by ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}:=\cD/\cS$ the quotient category and by
$l:\cD{\rightarrow} {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ the quotient functor, then $l$ is exact
and the classes $(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ define a torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
\item There is an equivalence of categories
${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}' \buildrel{\cong}\over\rightarrow\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$
for whom $l_\cH$ (defined in \ref{adjhearts}) is identified with $l'$.
\item
Moreover in the case in which the torsion-free class ${\mathcal Y}}\def\cN{{\mathcal N}$ generates $\cD$
(or dually if the torsion class ${\mathcal X}$ cogenerates $\cD$) and
if $(\cH_\cD,{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}',l',i')$ is a distinguished Giraud (resp. $(\cH_\cD,{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}',l',j')$ co-Giraud)
subcategory such that $i'$ (resp. $j'$) admits a derived functor,
then
the functor $l$ admits a right adjoint $i$ (resp. left adjoint $j$)
such that the $(\cD,{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S},l,i)$ is a
distinguished Giraud subcategory of $\cD$ which induces the distinguished Giraud
(resp. co-Giraud)
subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}'$
of $\cH_\cD$.
\end{enumerate}
\end{theorem}
\begin{proof}
{$1.$}
We have to prove that given a short exact sequence $0{\rightarrow} S_1{\rightarrow} S{\rightarrow} S_2{\rightarrow} 0$ in $\cD$
the middle term $S$ belongs to $\cS$ if and only if $S_1, S_2\in \cS$ where $\cS$
is defined as $\cS=\{ D\in \cD\; | \; l'(H_t^i(D))=0 \; \forall i\in \Bbb Z\}$.
Now, any short exact sequence on $\cD$ defines a distinguished triangle in $D(\cD)$ and so
one obtain the long exact sequence in $\cH_\cD$
\begin{equation}\label{LES}
\scriptstyle{\cdots H_t^{-1}(S_2){\rightarrow} H_t^0(S_1){\rightarrow} H_t^0(S){\rightarrow} H_t^0(S_2){\rightarrow}
H_t^1(S_1){\rightarrow} H_t^1(S){\rightarrow} H_t^1(S_2){\rightarrow} H_t^2(S_1)\cdots}
\end{equation}
By \ref{RTS},
$H_t^{-1}(S_2)=0=H_t^2(S_1)$ and for any $D\in\cD$ one has
$H^0(D)=t(D)[0]$ as a complex concentrated in degree $0$ while
$H^1(D)={D\over{t(D)}}[1]$.
So the sequence (\ref{LES}) reduces to the sequence in $\cH_\cD$
\begin{equation}\label{LES1}
0{\rightarrow} t(S_1)[0]{\rightarrow} t(S)[0]{\rightarrow} t(S_2)[0]{\rightarrow}
{S_1\over{t(S_1)}}[1]{\rightarrow} {S\over{t(S)}}[1]{\rightarrow} {S_2\over{t(S_2)}}[1]{\rightarrow} 0.
\end{equation}
Let us recall that the class
\begin{equation}\label{SerreSC}
\cS'=\{E\in \cH_\cD\; | \; l'(E)=0\}
\end{equation}
is a Serre subcategory of $\cH_\cD$.
So from one side it is clear that if $S_1, S_2\in\cS$ then
$t(S_i)[0], {S_i\over{t(S_i)}}[1]\in \cS'$ for any $i\in \{1,2\}$,
which implies that $t(S)[0]$ and $\frac{S}{t(S)}[1]$ belong to $\cS'$,
and so $S\in \cS$.
On the other side if $S\in\cS$ then $t(S)[0], {S\over{t(S)}}[1]\in \cS'$,
and by applying the functor $l'$ (which is exact by hypothesis) to (\ref{LES})
we obtain the exact sequence in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}'$
$$0{\rightarrow} l'(t(S_1)[0]){\rightarrow} 0 {\rightarrow} l'(t(S_2)[0]) {\rightarrow} l'\left({S_1\over{t(S_1)}}[1]\right){\rightarrow} 0{\rightarrow}
l'\left({S_2\over{t(S_2)}}[1]\right){\rightarrow} 0.
$$
This proves that $t(S_1)[0],{S_2\over{t(S_2)}}[1]\in \cS'$ and
$ l'(t(S_2)[0])\cong l'\left({S_1\over{t(S_1)}}[1]\right)\in
l'({\mathcal X}[0])\cap l'({\mathcal Y}}\def\cN{{\mathcal N}[1])=0
$
which proves that
$t(S_2)[0], {S_1\over{t(S_1)}}[1]\in \cS'$
and so $S_2\in\cS$ and $S_1\in \cS$.
\smallskip
{$2.$}
Let us show that the classes $(l({\mathcal X}),l({\mathcal Y}}\def\cN{{\mathcal N}))$ define a torsion pair on ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$. First of all, since any object of ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ may be regarded as an object of $\cD$ and the functor $l$ is exact, it is clear that any object $C\in{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ is the middle term of a short exact sequence $0 \to X \to C \to Y \to 0$ with $X\in l({\mathcal X})$ and $Y\in l({\mathcal Y}}\def\cN{{\mathcal N})$. It remains to show that ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}(X,Y)=0$,for every $X\in {\mathcal X}$ and every $Y\in {\mathcal Y}}\def\cN{{\mathcal N}$. So let $X\in l({\mathcal X})$ and $Y\in l({\mathcal Y}}\def\cN{{\mathcal N})$. A morphism $\varphi\colon X \to Y$ in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$ may be viewed as the class of a morphism $X' \to Y/Y'$ in $\cD$, where $X/X'$ and $Y'$ are in $\cS$. Let $t(X')$ be the torsion part of $X'$ (viewed as an object of $\cD$) with respect to the torsion pair $({\mathcal X},{\mathcal Y}}\def\cN{{\mathcal N})$ in $\cD$ and $Y/Y''$ be the torsion-free quotient of $Y/Y'$. We show that the composite morphism $t(X')\to X' \to Y/Y' \to Y/Y''$ also represents the morphism $\varphi$
in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$, i.e., $X/t(X')\in \cS$ and $Y''\in\cS$. Hence $\varphi=0$, since it is a morphism from a torsion to a torsion-free object. Now, the short exact sequence in $\cD$
\[
0 \to {X'\over t(X')} \to {X\over t(X')} \to {X\over X'} \to 0
\]
defines a distinguished triangle in $D(\cD)$ and so one obtains the long exact sequence of cohomology in $\cH_\cD$
\begin{equation*}
\textstyle
{\cdots\ H_t^0\left({X'\over t(X')}\right){\rightarrow} H_t^0\left({X\over t(X')}\right){\rightarrow} H_t^0\left({X\over X'}\right){\rightarrow}
H_t^1\left({X'\over t(X')}\right){\rightarrow} H_t^1\left({X\over t(X')}\right)\ \cdots}
\end{equation*}
which reduces to
\begin{equation*}
0{\rightarrow} {X\over t(X')}[0]{\rightarrow} {X\over X'}[0]{\rightarrow} {X'\over t(X')}[1]{\rightarrow} 0
\end{equation*}
since $H_t^0({X'\over t(X')})=t\left(X'\over t(X')\right)[0]=0$ and
$H_t^1({X\over t(X')}) = {{X/t(X')}\over{t(X/t(X'))}}[1]=0$ (since $X\in{\mathcal X}$).
By applying the exact functor $l'$ we obtain the exact sequence in ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}'$
\begin{equation*}
0{\rightarrow} l'\left({X\over t(X')}[0]\right){\rightarrow} l'\left({X\over X'}[0]\right){\rightarrow} l'\left({X'\over t(X')}[1]\right) {\rightarrow} 0
\end{equation*}
where $l'\left({X\over X'}[0]\right)=0$ because $X/X' \in \cS$.
Hence $l'\left({X\over t(X')}[0]\right)=0=l'\left({X'\over t(X')}[1]\right)$.
In particular, $X/t(X') \in \cS$. A dual argument shows that $Y'' \in \cS$.
\smallskip
{$3.$} Given a distinguished Giraud subcategory
$(\cH_{\cD},{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}',l',i')$, one can identify ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}'$ with the quotient category of
$\cH_\cD$ with respect to its Serre subcategory $\cS'$ defined in \ref{SerreSC}.
Applying Proposition~\ref{SonH} we see that the functor $l$ previously defined induces an exact essentially surjective functor
$l_{\cH}:\cH_\cD\rightarrow \cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$, and this proves that $\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}\cong \cH_\cD/\cS_\cH$
where $\cS_{\cH}$ is the kernel of the functor $l_\cH$.
In order to conclude the proof of this third statement it is enough to prove that
$\cS_{\cH}$ coincides with $\cS'$.
An object $X^{-1}\overset{x}{\longrightarrow }X^{0}$ in $\cH_\cD$ is in the kernel of $l_\cH$ if and only if the complex $l(X^{-1})\overset{l(x)}{\longrightarrow }l(X^{0})$ is zero in $\cH_{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$, that is:
$\mathrm{Ker}(l(x))=l(\mathrm{Ker}(x))=0$ and $\mathrm{Coker}(l(x))=l(\mathrm{Coker}(x))=0$.
This proves that $\mathrm{Ker}(x) \in \cS\cap{\mathcal Y}}\def\cN{{\mathcal N}$ which is equivalent to $\mathrm{Ker}(x)[1]=H^1_t(\mathrm{Ker}(x))\in \cS'$,
and $\mathrm{Coker}(x) \in \cS\cap{\mathcal X}$ which is equivalent to $\mathrm{Coker}(x)[0]=H^0_t(\mathrm{Coker}(x))\in \cS'$.
So $X^{-1}\overset{x}{\longrightarrow }X^{0}$ belongs to $\cS'$.
\smallskip
{$4.$}
Let us suppose that the torsion-free class ${\mathcal Y}}\def\cN{{\mathcal N}$ generates $\cD$.
Then it is clear that $l({\mathcal Y}}\def\cN{{\mathcal N})$ generates the quotient category $\cD/\cS$
and so by \cite[Theorem 8.2]{MR2486794}
the double heart $\cH_{\cH_{\cD}}$ is equivalent to $\cD$ and
$\cH_{\cH_{{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}}}\cong {\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}$.
If, moreover, $(\cH_\cD,{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}',l',i')$ is a distinguished Giraud subcategory
such that $i'$ admits a derived functor,
then we can apply
Theorem~\ref{adjhearts} in order to obtain a distinguished Giraud subcategory on the associated hearts.
This proves that
the functor $l\cong l_{l_{\cH}}$ admits a right adjoint $i$
such that $(\cD,{\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S},l,i)$ is a
distinguished Giraud subcategory of $\cD$ which induces the distinguished Giraud
subcategory ${\mathcal C}}\def\cD{{\mathcal D}}\def\cH{{\mathcal H}}\def\cS{{\mathcal S}'$
of $\cH_\cD$.
\end{proof}
\smallskip
\bibliographystyle{amsplain}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Adding timeout to avformat_open_input I am trying to add timeout to the function avformat_open_input using dictionary options.
here is my code.
AVDictionary *dict = NULL; // "create" an empty dictionary
av_dict_set(&dict, "timeout", "6000", 0); // add an entry
if (avformat_open_input(&av_format_ctx, "rtmp://192.168.1.2:1935/live/sum", NULL, &dict) < 0) {
return false;
}
but the function does not wait for 6 seconds it exits immediately after being called.
A: I know it is an old question, but anyhow...
The timeout option is used in listen mode.
According to the ffmpeg documentation:
timeout Set maximum timeout (in seconds) to wait for incoming connections.
A value of -1 means infinite (default). This option implies the rtsp_flags set to listen.
Check the console after the error when trying to set the timeout option. I bet it is saying that it could not open the rtsp for listening.
Use stimeout:
stimeout Set socket TCP I/O timeout in microseconds.
Pay double attention on the number of zeros (microseconds)
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,529
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"Heartless" bride-to-be slammed online after banning anyone elderly guests
Elderly resident ate soap and had allergic reaction after carer made…
Royal Mail customers fuming after being sent stamps as compensation
Tiny London "jail cell" office listed for £250-per-month
Carer wanted elderly resident to "f***ing die"
Over half a million people sign petition against cervical cancer screening…
Visa changes in popular destinations could see digital nomads thrive in…
High society photoshoot celebrates top female entrepreneurs
Home Sport Hibs Hibs striker Christian Doidge desperate to go from wearing Ryan Giggs' shirt...
Hibs striker Christian Doidge desperate to go from wearing Ryan Giggs' shirt to the Man United legend's Wales squad
Iain Collin
CHRISTIAN DOIDGE grew up idolising Ryan Giggs, wearing the Manchester United legend's name across the back of his shirt. Now, the Hibernian striker is eager to grab the attention of the Wales manager and earn a dream international call-up.
With 15 goals from 31 appearances for Hibs so far, Doidge has netted more times than the other forwards in the Wales squad – including Gareth Bale – combined. The 27-year-old saw that statistic doing the rounds on social media recently and is hopeful Giggs has too.
He has confessed that one motivation for his move north in the summer was to try to further his international ambitions, with games against the likes of Celtic and Rangers more high profile than playing for previous club Forest Green Rovers in League Two.
And, although it may be against Lowland League opposition, Sunday's televised Scottish Cup encounter against BSC Glasgow affords Doidge another opportunity to catch the eye, with a Euro 2020 call-up firmly in his sights.
"I always had that hope that maybe they'd see I'd scored a lot of goals," he says of his time down south. "But I think being in Scotland, doing well against good sides, is hopefully enough to get that opportunity. We'll have to see what happens.
"I've timed it quite well [with Euro 2020], haven't I? Playing for Wales is all I have ever wanted to do, to be honest. I would be very proud and my family would be, it would be unbelievable. But I have to do it for Hibs.
"I was shown that stat, it's obviously nice to see. There's some great players on that list, so to be on top is a good feeling. But I want to score more goals, keep pushing and help Hibs win. I just want to do well for Hibs and enjoy it.
"In the attacking areas Wales have always been good. There's been a few, there's Giggs, the manager, then John Hartson, Dean Saunders, Ian Rush, Craig Bellamy, some fantastic players who had great careers, so obviously you look up to them as kids and it's something I'd like to do myself.
"I was a Man U fan as a kid. With the whole Welsh thing, Giggsy was a legend and being a Welsh lad you wanted to get the Welshman on the back of your shirt. He was an idol growing up."
Doidge suffered a frustrating start to life with Hibs but has six goals from his last six games and will be hopeful of getting back on the scoresheet this weekend. However, having originally come through from the part-time ranks with the likes of Barry Town and Carmarthen Town, the Newport-born marksman will be wary of the Easter Road side being embarrassed against BSC.
"Of course, on paper, the two teams are a lot different," he added. "But it's going to be a tough game, they're going to really up for it. It's a great opportunity for their players and with the magic of the cup anything can happen. We need to be fully focused and go there to win.
"I understand what they're going through. They've probably worked all week and they've got the game on the weekend. I'm sure they'll have been looking forward to it all week and all their friends will have been talking about it.
"It's a great opportunity for them. I've been in their shoes so I know how they're feeling. They're going to be bringing the best of what they can bring and we've got to make sure we do as well."
BSC Glasgow
Previous articleCeltic legend Paul McStay's son looking to upset the Hoops on Sunday
Next articleAfter St Mirren experience at Saughton, Jack Ross happy to be back on familiar territory with Hibs for Scottish Cup clash against BSC Glasgow
https://www.deadlinenews.co.uk/2020/02/07/hibs-striker-christian-doidge-desperate-to-go-from-wearing-ryan-giggs-shirt-to-the-man-united-legends-wales-squad/
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,777
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Q: How to use Google Apps Script as a submission form I'm relatively new to Google Apps Script. I'm trying to build a script, as well as Html, to create a site where I have people submit their names, age and so on (basic info for a test site). After they do so I want it the information to be automatically recorded into a google sheets file. Issue I'm having is that I don't know how to record the submissions using Google Apps Script to the google sheets file. I also want an email sent to me automatically after someone submits something. I tried looked at the API Ref but don't really have a solid starting point. I'm learning Javascript now but would like an idea of how Google Apps Script would work if I tried making a program like this. Thank you
A: Google Forms (https://www.google.com/forms/about/) will do most of this for you without any code. Once you have a basic form working, you can enhance that form with Google Apps Script if needed.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//JP">
<!--Converted with LaTeX2HTML 2008 (1.71)
original version by: Nikos Drakos, CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Text View Panel</TITLE>
<META NAME="description" CONTENT="Text View Panel">
<META NAME="keywords" CONTENT="manual">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
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<LINK REL="up" HREF="manual-node154.html">
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</HEAD>
<BODY >
<DIV CLASS="navigation"><!--Navigation Panel-->
<A NAME="tex2html2783"
HREF="manual-node158.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
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<!--End of Navigation Panel-->
<H3><A NAME="SECTION04092300000000000000">
Text View Panel</A>
</H3>
<P>
TextViewPanel is an application window class to display text files
(Fig. <A HREF="#textviewpanel">19</A>).
The program text is shown to demonstrate how
one of the simplest application windows is described.
In the <TT>:create</TT> method, the quit button and find button,
and a text-item to feed the string to be searched for in the file
are created.
The view-window is a ScrollTextWindow that displays the file
with the vertical and horizontal scroll-bars.
The TextViewPanel captures <TT>:ConfigureNotify</TT> event
to resize the view-window when the outermost title window is resized
by the window manager.
<P>
<DIV ALIGN="CENTER"><A NAME="textviewpanel"></A><A NAME="46971"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 19:</STRONG>
TextViewPanel window</CAPTION>
<TR><TD><IMG
WIDTH="317" HEIGHT="317" BORDER="0"
SRC="manual-img110.png"
ALT="\begin{figure}\begin{center}
\mbox{\epsfysize =7cm
\epsfbox{fig/textviewpanel.ps}
}
\end{center}
\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
<PRE>
(defclass TextViewPanel :super panel
:slots (quit-button find-button find-text view-window))
(defmethod TextViewPanel
(:create (file &rest args &key (width 400) &allow-other-keys)
(send-super* :create :width width args)
(setq quit-button
(send self :create-item panel-button "quit" self :quit))
(setq find-button
(send self :create-item panel-button "find" self :find))
(setq find-text
(send self :create-item text-item "find: " self :find))
(setq view-window
(send self :locate-item
(instance ScrollTextWindow :create
:width (setq width (- (send self :width) 10))
:height (- (setq height (send self :height)) 38)
:scroll-bar t :horizontal-scroll-bar t
:map nil :parent self)))
(send view-window :read-file file))
(:quit (event) (send self :destroy))
(:find (event)
(let ((findstr (send find-text :value)) (found)
(nlines (send view-window :nlines)))
(do ((i 0 (1+ i)))
((or (>= i nlines) found))
(if (substringp findstr (send view-window :line i)) (setq found i)))
(when found
(send view-window :display-selection found)
(send view-window :locate found))))
(:resize (w h)
(setq width w height h)
(send view-window :resize (- w 10) (- h 38)))
(:configureNotify (event)
(let ((newwidth (send self :width))
(newheight (send self :height)))
(when (or (/= newwidth width) (/= newheight height))
(send self :resize newwidth newheight))) ) )
</PRE>
<P>
<BR><HR>
<ADDRESS>
2015-07-31
</ADDRESS>
</BODY>
</HTML>
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{
"redpajama_set_name": "RedPajamaGithub"
}
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\section{Introduction}
Let $X$ be a smooth projective variety and we denote by $\mathrm{D}(X)$ the bounded derived category of coherent sheaves on $X$.
It is noteworthy that $\mathrm{D}(X)$ is one of the most important invariant of $X$.
For example, $X$ is uniquely determined by $\mathrm{D}(X)$ if the canonical bundle of $X$ is ample or anti-ample (see \cite{BO01}).
To investigate varieties via their derived categories,
Bondal et al. \cite{BK90, BO95} introduce
the notion of {\it semiorthogonal decomposition} which has become an important tool in algebraic geometry.
In particular, the notion of semiorthogonal decomposition includes {\it full exceptional collection} as a specially important example.
More general,
any exceptional collection $\{E_1, E_2, \ldots, E_l\}$ on $\mathrm{D}(X)$ gives a semiorthogonal decomposition for $\mathrm{D}(X)$ of the form
\begin{equation}\label{exceptional-sod}
\mathrm{D}(X)=
\langle
\mathcal{A}_{X}, E_1, E_2, \ldots,E_l\rangle,
\end{equation}
where $\mathcal{A}_{X}$ is an admissible subcategory of $\mathrm{D}(X)$ (see \cite{Bon90}).
Then $\mathcal{A}_{X}$ is trivial if and only if the exceptional collection $\{E_1,E_2,\ldots,E_l\}$ is {\it full}.
Naturally, there is a crucial problem as follows (see \cite{BO95} or \cite[Question 1.9]{Kuz-ICM}):
\textit{
Find a good condition for an exceptional collection to be full,
or find a good condition for $\mathcal{A}_{X}$ to be trivial.
}
Given an admissible subcategory $\mathcal{A}_{X}\subset \mathrm{D}(X)$,
we say that $\mathcal{A}_{X}$ is \textit{quasi-phantom} if its Hochschild homology is trivial and its Grothendieck group is of finite rank;
moreover, it is called \textit{phantom} if the Grothendieck group is trivial.
In \cite{Kuz09}, Kuznetsov proposes a nonvanishing conjecture which asserts that
if $\mathcal{A}_{X}$ is a quasi-phantom or phantom category
then $\mathcal{A}_{X}$ is trivial.
However, for some surfaces $X$ of general type with $q=p_g=0$,
$\mathrm{D}(X)$ admits a semiorthogonal decomposition consisting of an exceptional collection of line bundles and its orthogonal complement which is a quasi-phantom category or phantom category (\cite{GO13, AO13, BGvBS13, GS13, Keum17, BGvBKS15, GKMS15, Lee15, LS14, KKL17,Lee16, AB17} etc).
Motivated by those examples, the existence problem of (quasi-)phantom category on smooth projective varieties has become a much more interesting topic.
Let us now suppose that $X$ admits a full exceptional collection of length $l$.
Since the Grothendieck group is preserved under semiorthogonal decompositions,
as a consequence, $\mathcal{A}_{X}$ in \eqref{exceptional-sod} is a phantom category.
Although Kuznetsov's nonvanishing conjecture is not true in general,
there is still an interesting fullness conjecture attributed to Kuznetsov:
{\it any exceptional collection of length $l$ on $X$ is full} (see \cite[Conjecture 1.10]{Kuz-ICM}).
In dimension $1$, Kuznetsov's fullness conjecture is trivially true;
in dimension $2$,
a result of Kuleshov-Orlov \cite{KO94}
asserts that any exceptional collection on a del Pezzo surface is contained in a full exceptional collection and hence Kuznetsov's fullness conjecture holds for del Pezzo surfaces.
To our best knowledge, there is no common method which can be used to address Kuznetsov's fullness conjecture.
More specifically, by considering a smooth projective variety which already admits a full exceptional collection of line bundles, there is a weak version of Kuznetsov's fullness conjecture.
\begin{conj}[Kuznetsov \cite{Kuz-ICM}]\label{Kuz-icm-conj-linebundles}
If a smooth projective variety $X$ admits a full exceptional collection of line bundles of length $l$, then any exceptional collection of line bundles of length $l$ is full.
\end{conj}
This conjecture is true for $\mathbb{P}^n$ (cf. \cite{Bei78}), del Pezzo surfaces (cf. \cite{KO94}), Hirzebruch surfaces (cf. \cite{H04}) and smooth projective toric surface $X$ of Picard rank $3$ or $4$ (cf. \cite{HI13}).
To the best of our knowledge, there are no more cases supporting this conjecture.
In this paper, we show that the Conjecture \ref{Kuz-icm-conj-linebundles} holds for $X$. To be more precisely, the main result of this paper is stated as follows.
\begin{thm}[Theorem \ref{main-thm-point}+\ref{main-thm-line}+\ref{main-thm-cubic}]\label{main-thm}
Let $X$ be the blow-up of $\mathbb{P}^{3}$ at a point, or a line, or a twisted cubic curve.
Then any exceptional collection of line bundles of length $6$ on $X$ is full. Moreover, we obtain an explicit classification \footnote{ See Theorem \ref{Classify-Blowup-p-P3}, \ref{Classify-Blowup-line-P3}, \ref{Classify-Blowup-twisted-cubic-P3}. }
of full exceptional collection of line bundles on such $X$.
\end{thm}
\subsection*{Idea of Proof}
It is known that $X$ is either a $\mathbb{P}^1$-bundle over $\mathbb{P}^2$ (for the blow-up at a point or a twisted cubic curve) or $\mathbb{P}^2$-bundle over $\mathbb{P}^1$ (for the blow-up at a line). By Orlov's projective bundle formula (Theorem \ref{Orlov-projbundle-formula}), we already know a family of full exceptional collection of line bundles. However, it is not clear that any exceptional collection of line bundles could be obtained by this method. Instead, we first classify cohomologically zero line bundles. We then classify the exceptional collection of line bundles of length $6$ on $X$ up to mutations and normalizations. It turns out that in the case of blow-up a point or a twisted cubic curve, some exceptional collections of line bundles are not from the projective bundle formula. To show the fullness in the blow-up of a point case, we generalize Hille-Perling's construction of augmentation \cite{HP11} from surface case to higher dimensional case (Theorem~\ref{blowup-point-fullness-lemma}). The fullness in the blow-up of a twisted cubic case is proved by using a technical lemma that if there are two exceptional collections of the same length with only one exceptional object different, the one is full if and only if the other is full (Lemma~\ref{lem:exlb}).
The outline of this paper is organized as follows.
We devote Section \ref{preliminaries-sod} to some basic definitions and important results on semiorthogonal decompositions and full exceptional collections.
In Section \ref{Exceptional-line-bundles},
we give a rapid review of exceptional collection of line bundles on smooth projective varieties and mainly give a high dimensional augmentation.
In Section \ref{blow-up-one-point}, \ref{blow-up-one-line} and \ref{blow-up-one-cubic}, we study the three cases of blow-ups in Theorem \ref{main-thm}. In the last section, we give some remarks on related and further problems which we are interested.
\subsection*{Acknowledgement}
The first author is supported by IBS-R003-D1. He thanks Kyoung-Seog Lee and Jihun Park for useful discussions. The second and third authors are partially supported by NSFC (Grant No. 11626250). They would like to thank Xiangdong Yang for giving some useful comments. The second author also thanks the IBS Center for Geometry and Physics in Pohang, South Korea, for the hospitality during his visit on August in 2017.
\subsection*{Notation and convention}
Throughout the paper,
we work over the complex number field $\mathbb{C}$.
A variety is an integral separated schemes of finite type over $\mathbb{C}$.
Let $X$ be a smooth projective variety,
and we will use the following notations:
\begin{center}
\begin{tabular}{ r l }
$\omega_X$ (resp. $K_{X}$) & the canonical bundle of $X$ (resp. the corresponding canonical divisor) \\
$\mathrm{D}(X)$ & the bounded derived category of coherent sheaves on $X$ \\
$\mathcal{O}_{X}(D)$ & the associated line bundle of a divisor $D$ on $X$, then $\omega_{X} =\mathcal{O}_{X}(K_X)$ \\
$H^{i}(L)$ & $H^{i}(X, L)$ the $i$-th sheaf cohomology of line bundle $L$ \\
$h^{i}(L)$ & $\mathrm{dim}\, H^{i}(L)$ the dimension of the $i$-th sheaf cohomology of line bundle $L$ \\
$\mathbb{P}(\mathcal{E})$ & $\mathrm{Proj}(\mathrm{Sym\, \mathcal{E}^{\vee}})$ the projective bundle of a vector bundle $\mathcal{E}$ \\
$f_{\ast}$ (resp. $f^{\ast}$) & the derived pushforward (resp. pullback) functor
of $f: X \to Y$ \\
& of smooth projective varieties \\
$\mathcal{F}\otimes \mathcal{G}$ & the derived tensor product of $\mathcal{F}, \mathcal{G}\in \mathrm{D}(X)$
\end{tabular}
\end{center}
\section{Preliminaries}\label{preliminaries-sod}
In this section, we review some of basic definitions on semiorthogonal decompositions and gather the results that will be of importance to us later on.
Most of the materials presented here are standard,
we refer to \cite{Bon90, BK90, BO95, Kuz-ICM, Orlov93} for more detailed discussions.
\subsection{Semiorthogonal decompositions}
Let $\mathcal{T}$ be a triangulated category.
For any morphism $f:E\to F$ between two objects $E$ and $F$ of $\mathcal{T}$,
there exists a distinguished triangle
$$
E \to F \to \mathrm{Cone}(f) \to E[1],
$$
where $\mathrm{Cone}(f)$ is called the \textit{cone} of the morphism $f:E\to F$.
\begin{defn}
A \textit{semiorthogonal decomposition} of $\mathcal{T}$
is an ordered full triangulated subcategories $\{\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{l}\}$ of $\mathcal{T}$ such that
\begin{enumerate}
\item[(1)] $\mathrm{Hom}(\mathcal{A}_{i},\mathcal{A}_{j})=0$ for $i>j$, and
\item[(2)] for any object $T\in \mathcal{T}$, there exist $T_{i}\in \mathcal{T}$ and a sequence
$$
0=T_{l}\to T_{l-1} \to\cdots \to T_{1}\to T_{0}=T
$$
such that the cone $\mathrm{Cone}(T_{i}\to T_{i-1})\in \mathcal{A}_{i}$, for all $1\leq i\leq l$.
\end{enumerate}
For convenience, we denote by
$$
\mathcal{T}
=\langle \mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{l}\rangle,
$$
the semiorthogonal decomposition of $\mathcal{T}$ with the components $\{\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{l}\}$.
\end{defn}
To construct semiorthogonal decompositions,
a notion of admissible category plays an important role.
Let us recall the definition.
\begin{defn}
Let $\mathcal{A}$ be a full triangulated subcategory $\mathcal{T}$.
We say that $\mathcal{A}$ is {\it right admissible} if the inclusion functor
$i: \mathcal{A} \hookrightarrow \mathcal{T}$ admits a right adjoint functor $i^{!}: \mathcal{T}\to \mathcal{A}$.
If the functor $i$ has a left adjoint functor $i^{\ast}: \mathcal{T}\to \mathcal{A}$,
then $\mathcal{A}$ is called a {\it left admissible category}.
We say that $\mathcal{A}$ is {\it admissible} if $\mathcal{A}$ is both left and right admissible.
\end{defn}
Now, suppose $\mathcal{T}=\langle \mathcal{A},\mathcal{B}\rangle$ is a semiorthogonal decomposition.
Then $\mathcal{A}$ is a left admissible category and $\mathcal{B}$ is a right admissible category (\cite[Lemma 3.1]{Bon90}).
Conversely, for any given left admissible category $\mathcal{A}\subset \mathcal{T}$ and right admissible category $\mathcal{B}\subset \mathcal{T}$, there exist two semiorthogonal decompositions
$$
\mathcal{T}
=\langle \mathcal{B}^{\bot},\mathcal{B}\rangle
=\langle \mathcal{A},{}^{\bot}\mathcal{A}\rangle,
$$
where
$
\mathcal{B}^{\bot}:=\{ T \in \mathcal{T} \mid \mathrm{Hom}(B, T)=0, \forall B\in \mathcal{B}\}
$
is the right orthogonal complement of $\mathcal{B}$,
and
$
{}^{\bot}\mathcal{A}:=\{ T \in \mathcal{T} \mid \mathrm{Hom}(T, A)=0, \forall A\in \mathcal{A}\}
$
is the left orthogonal complement of $\mathcal{A}$ (\cite{BK90}).
In particular, for a semiorthogonal decomposition $\mathcal{T}=\langle \mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{l}\rangle$, we have $\mathcal{A}_{j} \subset \mathcal{A}_{i}^{\bot}$ and $\mathcal{A}_{i} \subset {}^{\bot}\mathcal{A}_{j}$ for all $i>j$.
Much more specially, we have the following.
\begin{prop}[\cite{Bon90}, Theorem 3.2]
Let $X$ be a smooth projective variety.
If there exists a semiorthogonal decomposition,
$$
\mathrm{D}(X)=\langle \mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{l}\rangle,
$$
then each $\mathcal{A}_{i}$ is an admissible category.
\end{prop}
\subsection{Orlov's semiorthogonal decompositions}
The first and simplest example of semiorthogonal decomposition (full exceptional collection) is Beilinson's exceptional collection (\cite{Bei78}).
Accurately,
there is a semiorthogonal decomposition of $\mathrm{D}(\mathbb{P}^{n})$,
$$
\mathrm{D}(\mathbb{P}^{n})
=\langle
\mathcal{O}_{\mathbb{P}^{n}},\mathcal{O}_{\mathbb{P}^{n}}(H),\ldots,\mathcal{O}_{\mathbb{P}^{n}}(nH)
\rangle,
$$
where $H$ is the hyperplane of $\mathbb{P}^{n}$.
Orlov generalizes Beilinson's semiorthogonal decomposition to relative cases (\cite[Corollary 2.7]{Orlov93}).
\begin{thm}[Orlov's projective bundle formula \cite{Orlov93}]\label{Orlov-projbundle-formula}
Suppose $\mathcal{E}$ is a vector bundle of rank $r+1$ on a smooth projective variety $Y$,
and let $\rho:X:=\mathbb{P}(\mathcal{E})\to Y$ be the projective bundle of $\mathcal{E}$.
Then there is a semiorthogonal decomposition
$$
\mathrm{D}(X)=
\langle \rho^{\ast}\mathrm{D}(Y), \rho^{\ast}\mathrm{D}(Y)\otimes \mathcal{O}_{X}(1),
\ldots,
\rho^{\ast}\mathrm{D}(Y)\otimes \mathcal{O}_{X}(r)\rangle,
$$
where $\mathcal{O}_{X}(1)$ is the Grothendieck line bundle of $X$.
\end{thm}
Let $Y$ be a smooth projective variety.
Suppose $i:Z \hookrightarrow Y$ is a closed smooth subvariety of $Y$ of codimension $r+1$.
The normal bundle of $Z$ in $Y$, denoted by $\mathcal{N}_{Z/Y}$,
is a vector bundle of rank $r+1$ on $Z$.
The blow-up $ \mathrm{Bl}_{Z}Y$ of $Y$ at center $Z$ is a projective morphism $\pi: \mathrm{Bl}_{Z}Y\to Y$, and the exceptional divisor of the blow-up is $E:=\pi^{-1}(Y)\cong \mathbb{P}(\mathcal{N}_{Z/Y})$.
Then one has the following blow-up diagram
$$
\xymatrix{
E\cong \mathbb{P}(\mathcal{N}_{Z/Y}) \ar[d]_{\rho} \ar[r]^{j} & X:=\mathrm{Bl}_{Z}Y \ar[d]^{\pi}\\
Z \ar[r]^{i} & Y.
}
$$
It is important to notice that the canonical divisor of the blow-up $X$ is determined by the following formula
$$
K_{X}=\pi^{\ast}K_{Y}+rE,
$$
and the restriction of $\mathcal{O}_{X}(E)$ on each fiber of $\pi$ is isomorphic to $\mathcal{O}_{\mathbb{P}^{r}}(-1)$.
Furthermore, $\mathcal{O}_{E}(E)\cong \mathcal{O}_{E}(-1)$.
\begin{thm}[Orlov's blow-up formula \cite{Orlov93}]\label{Orlov-blowup-formula}
Using notations as above, there is a semiorthogonal decomposition
\begin{eqnarray*}
\mathrm{D}(X)
&=&
\langle
\pi^{\ast}\mathrm{D}(Y), j_{\ast}\rho^{\ast}\mathrm{D}(Z)\otimes \mathcal{O}_{E}(1), \ldots,
j_{\ast}\rho^{\ast}\mathrm{D}(Z)\otimes \mathcal{O}_{E}(r)
\rangle\\
&=&
\langle
j_{\ast}\rho^{\ast}\mathrm{D}(Z)\otimes \mathcal{O}_{E}(-r), \ldots,
j_{\ast}\rho^{\ast}\mathrm{D}(Z)\otimes \mathcal{O}_{E}(-1), \pi^{\ast}\mathrm{D}(Y)
\rangle
\end{eqnarray*}
where $\mathcal{O}_{E}(1)$ is the Grothendieck line bundle of $E$.
\end{thm}
\begin{proof}
See for example \cite[Theorem 4.3]{Orlov93} or \cite[Proposition 11.18]{Huy06}.
\end{proof}
\subsection{Full exceptional collections}
This subsection reviews some of basic facts about full exceptional collections.
The full exceptional collections are the most important examples of semiorthogonal decompositions.
\begin{defn}
(1) An object $E\in \mathcal{T}$ is called an {\it exceptional object} if
$$
\mathrm{Hom}(E,E[k])
=\left\{
\begin{array}{ll}
0, & k\neq 0, \\
\mathbb{C}, & k=0.
\end{array}
\right.
$$
(2) An ordered sequence of exceptional objects $\{E_1,E_2,\ldots,E_{l}\}$
is called an {\it exceptional collection} if
$$
\mathrm{Hom}(E_i,E_j[k])=0\;\; \textrm{for any}\; i>j\, \textrm{and for all}\; k\in \mathbb{Z}.
$$
In particular, a pair $(E, F)$ is said to be an {\it exceptional pair} if the sequence $\{E, F\}$ is an exceptional collection.
Moreover, we say that the exceptional collection $\{E_1,E_2,\ldots,E_{l}\}$ is {\it strong} if
$$
\mathrm{Hom}(E_i,E_j[k])=0,
$$
for $k\neq 0$ and for all $i, j$.
(3) An exceptional collection $\{E_1,E_2,\ldots,E_{l}\}$ is said to be {\it full}
if the smallest full triangulated subcategory, denoted by $\langle E_1,E_2,\ldots,E_{l} \rangle$, of $\mathcal{T}$
containing $E_1,E_2,\ldots,E_{l}$ is $\mathcal{T}$ itself, i.e., $\mathcal{T}=\langle E_1,E_2,\ldots,E_{l} \rangle$,
and then we say that $\mathcal{T}$ has a full exceptional collection of {\it length $l$}.
\end{defn}
\begin{defn}
Let $X$ be a smooth projective variety.
We say that $X$ admits a \textit{(full) exceptional collection of length $l$} if its derived category $\mathrm{D}(X)$ has a (full) exceptional collection of length $l$.
\end{defn}
It is not difficult to see that if $E\in \mathrm{D}(X)$ is an exceptional object
then $\langle E\rangle\cong \mathrm{D}(\mathrm{Spec}\,\mathbb{C})$.
We see that an exceptional collection gives a splitting off copies of $\mathrm{D}(\mathrm{Spec}\,\mathbb{C})$ in $\mathrm{D}(X)$ in the sense of semiorthogonal decompositions.
Precisely, we have
\begin{prop}[\cite{Bon90}, Theorem 3.2]
Let $X$ be a smooth projective variety.
If $\{E_1,\ldots, E_l\}$ is an exceptional collection of $X$,
then there exists a semiorthogonal decomposition
$$
\mathrm{D}(X)=\langle \mathcal{A}, E_1, E_2,\ldots, E_l\rangle,
$$
where $\mathcal{A}:=\langle E_1, E_2,\ldots, E_l\rangle^{\bot}\cong\{ F\in \mathrm{D}(X) \mid \mathrm{Hom}(E_i, F[k])=0, \forall k\in \mathbb{Z}, 1\leq i\leq l\}$.
\end{prop}
For our purpose, we need the following easy but useful result.
\begin{lem}\label{usefull-fullness-lem}
If there are two exceptional collections of the same length with only one exceptional object different, then one is full if and only if the other is full.
\end{lem}
\begin{proof}
Without loss of generality, we assume $\{E_{1}, E_{2}, \cdots ,E_{n}\}$ is a full exceptional collection and $\{E_{1}', E_{2}, \ldots ,E_{n}\}$ is another exceptional collection.
Suppose $A\in \langle E_{1}', E_{2}, \ldots ,E_{n}\rangle^{\bot}$ and $A\neq 0$.
Since there is a semiorthogonal decomposition
$$
\mathrm{D}(X)=\langle E_{1}, E_{2}, \cdots ,E_{n}\rangle,
$$
and hence $\langle E_1\rangle \cong \langle E_{2}, \cdots ,E_{n} \rangle^{\bot}$.
Hence $A\in \langle E_{2}, \ldots ,E_{n} \rangle^{\bot}=\langle E_1\rangle$
and $A$ can be written as $A\cong \bigoplus E_1[i]^{\oplus j_i}$.
By the assumption, $(E_{1}, E_{1}')$ and $(E_{1}', E_{1})$ are not exceptional pair.
If not, the full exceptional collection $\{E_{1}, E_{2}, \cdots ,E_{n}\}$ will be extended.
Then there some $i_0\in \mathbb{Z}$ such that $\mathrm{Hom}(E_{1}', E_1[i_0]) \neq 0$.
Therefore, we obtain
$$
\mathrm{Hom}(\bigoplus E_{1}'[i]^{\oplus j_i}, A[i_0])
\cong
\mathrm{Hom}(\bigoplus E_{1}'[i]^{\oplus j_i}, \bigoplus E_1[i]^{\oplus j_i}[i_0])
\neq 0.
$$
This is a contradiction and the lemma follows.
\end{proof}
\subsection{Mutations}
In this subsection, we will recall some basic facts upon the mutations which we refer to \cite{Bon90, BK90} for more details.
Let $\mathcal{T}$ be a triangulated category.
If $i: \mathcal{A} \hookrightarrow \mathcal{T}$ is an admissible category,
then there exist two functors, for any $F\in \mathcal{T}$,
$$
\mathrm{L}_{\mathcal{A}}(F)
:=\mathrm{Cone}(ii^{!}(F)\to F)
\;\;\textrm{and}\;\;
\mathrm{R}_{\mathcal{A}}(F)
:=\mathrm{Cone}(F\to ii^{\ast}(F))[-1],
$$
which are called the {\it left} and the {\it right mutation functors} respectively.
More specifically, if $\mathcal{A}$ is generated by an exceptional object $E$,
then
$$
\mathrm{L}_{\mathcal{A}}(F)=\mathrm{Cone}(\mathrm{RHom}(E, F)\otimes E\to F)
$$
and
$$
\mathrm{R}_{\mathcal{A}}(F)=\mathrm{Cone}(F\to \mathrm{RHom}(F,E)^{\ast}\otimes E)[-1].
$$
Suppose $\mathcal{T}=\langle \mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{l}\rangle$ is a semiorthogonal decomposition.
Then there exist two semiorthogonal decompositions (see \cite{Bon90} or \cite[Lemma 1.9]{BK90}):
for $1\leq j \leq l-1$,
$$
\mathcal{T}
=\langle \mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{j-1},\mathrm{L}_{\mathcal{A}_{j}}(\mathcal{A}_{j+1}),\mathcal{A}_{j},\mathcal{A}_{j+2},\ldots,\mathcal{A}_{l}\rangle;
$$
and for any $2\leq j \leq l$,
$$
\mathcal{T}
=\langle \mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{j-2}, \mathcal{A}_{j},\mathrm{R}_{\mathcal{A}_{j}}(\mathcal{A}_{j-1}),\mathcal{A}_{j+1},\ldots,\mathcal{A}_{l}\rangle,
$$
In particular,
one has
\begin{lem}[\cite{Bon90,BK90}]\label{mutation-fullness-lem}
Let $X$ be a smooth projective variety.
If $\mathrm{D}(X)=\langle \mathcal{A},\mathcal{B} \rangle$ is a semiorthogonal decomposition, then
$\mathrm{L}_{\mathcal{A}}(\mathcal{B})=\mathcal{B}\otimes \omega_{X}$
and $\mathrm{R}_{\mathcal{B}}(\mathcal{A})=\mathcal{A} \otimes \omega_{X}^{\vee}$.
In particular, the sequence $\{ E_1,\ldots,E_n\}$ is a full exceptional collection
if and only if
$\{E_2,\ldots,E_n,E_{n+1}\}$
is a full exceptional collection,
where $E_{n+1}:=E_1\otimes \omega_{X}^{\vee}$.
\end{lem}
\section{Exceptional collection of line bundles}\label{Exceptional-line-bundles}
In this section, we review some basic aspects of exceptional collection of line bundles on smooth projective varieties, which will be used in the sequel.
Let $X$ be a smooth projective variety of dimension $n$.
A line bundle on $X$ is an exceptional object if and only if the structure sheaf $\mathcal{O}_X$ is an exceptional object.
Recently, Sosna shows that if the Grothendieck group of $X$ is of finite rank then $\mathcal{O}_X$ is an exceptional object (\cite[Proposition 3.1]{Sosna}).
More interesting, if $X$ is a smooth Fano variety, i.e., the canonical bundle $\omega_X$ is anti-ample, then by Kodaira vanishing theorem any line bundle on $X$ is an exceptional object. Moreover, suppose $X$ is a smooth Fano variety of Picard rank one
and of index $r$ (i.e., $\omega_X=\mathcal{O}_{X}(-rH)$, where $H$ is the positive generator of $\mathrm{Pic}(X)$); for example, a smooth cubic fourfold is of Picard
rank one and index $3$.
Then $r\leq n+1$ and there is a semiorthogonal decomposition of D(X),
$$
\mathrm{D}(X)=
\langle
\mathcal{A}_{X}, \mathcal{O}_{X}, \mathcal{O}_{X}(1), \ldots, \mathcal{O}_{X}(r-1)\rangle,
$$
where $\mathcal{A}_{X} = \langle \mathcal{O}_{X}, \mathcal{O}_{X}(1), \ldots, \mathcal{O}_{X}(r-1)\rangle^{\bot}$ (\cite[Corollary 3.5]{Kuz09a}).
In particular, if $r=n+1$ then $X\cong\mathbb{P}^n$, and $\mathcal{A}_X=0$.
Basing on a result of Bondal-Polishchuk \cite[Theorem 3.4]{BP94},
Vial \cite{Vial} shows that for a smooth projective variety $X$ of dimension $n$ which admits a full exceptional collection of line bundles of length $n+1$ then $X\cong\mathbb{P}^n$ (see \cite[Proposition 1.3]{Vial}).
\subsection{Basics of cohomologically zero line bundles}
Now suppose $X$ is a smooth projective variety and its structure sheaf is an exceptional object.
In order to classify the exceptional collection of line bundles on $X$,
we will first classify the line bundles which are cohomologically zero.
Let us recall the definition of cohomologically zero line bundles.
\begin{defn}\label{def:cohzero}
Let $X$ be a smooth projective variety.
A line bundle $L\in \mathrm{Pic}(X)$ is called {\it cohomologically zero} if $H^{i}(X,L)=0$ for all $i\in \mathbb{Z}$.
\end{defn}
Note that a line bundle $L$ is cohomologically zero if and only if
the pair $(\mathcal{O}_{X}, L^{\vee})$ is an exceptional pair if and only if
its dual line bundle $L^{\vee}$ is {\it left-orthogonal} to $\mathcal{O}_X$ in the sense of Hille-Perling \cite[Definition 3.1 (ii)]{HP11}.
\begin{example}\label{cohom-zero-F_1}
The Picard group of $\mathbb{P}^1\times\mathbb{P}^1$ is
$$
\mathrm{Pic}(\mathbb{P}^1\times\mathbb{P}^1)\cong \mathbb{Z}[S] \oplus \mathbb{Z}[F],
$$
where $S$ is the pullback of hyperplane class
and $F$ is a fiber of $\mathbb{P}^1\times\mathbb{P}^1\to \mathbb{P}^1$.
Then a line bundle $\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(aS+bF)$
is cohomologically zero if and only if $a=-1$ or $b=-1$.
\end{example}
In terms of the notion of cohomologically zero line bundles,
one has the following well-known results; see for example \cite{HP11}.
\begin{lem}\label{lem:exlb}
Let $X$ be a smooth projective variety.
Then the sequence $\{L_1, L_2,\ldots, L_l\}$ is an exceptional collection of line bundles on $X$ if and only if $L_j\otimes L_i^{-1}$ are cohomologically zero for all $i>j$.
\end{lem}
\begin{lem}\label{normalization-lem}
Let $X$ be a smooth projective variety.
Then the sequence $\{L_1, L_2, \ldots, L_l\}$ is a (full) exceptional collection of line bundles if and only if the normalized sequence $\{\mathcal{O}_X, L_1^{-1}\otimes L_2, \ldots, L_1^{-1}\otimes L_l\}$ is a (full) exceptional collection.
\end{lem}
\begin{rem}
As a consequence of the above lemma,
classifying (full) exceptional collection of line bundles
is the same as classifying (full) exceptional collection of \emph{normalized} line bundles of type
$\{\mathcal{O}_X,L_2, L_3,\ldots, L_l\}$. We also call the operation of tensoring $L_1^{-1}$ as \emph{normalization}.
\end{rem}
\subsection{Augmentation}
In the following,
we will discuss the behavior of full exceptional collections of line bundles
under the blow-up of a point.
To begin with, we will review the case of smooth projective surfaces.
Let $\pi: X\to Y$ be the blow-up of a smooth projective surface $Y$ at a point $p\in Y$,
and denote by $E$ the exceptional divisor of $\pi$.
A collection of line bundles,
$$
\{\mathcal{O}_{Y}(D_1), \mathcal{O}_{Y}(D_2),\ldots, \mathcal{O}_{Y}(D_l)\}
$$
is a full exceptional collection if and only if,
for any $1\leq i \leq l$,
$$
\{\mathcal{O}_{X}(D_1+E),\ldots ,\mathcal{O}_{X}(D_{i-1}+E), \mathcal{O}_{X}(D_i), \mathcal{O}_{X}(D_i+E), \mathcal{O}_{X}(D_{i+1}), \ldots, \mathcal{O}_{X}(D_l)\}
$$
is a full exceptional collection (\cite[Proposition 2.4]{HP14}).
This process is called \textit{augmentation} (cf. \cite{HP11}).
In fact, the augmentations allow us to construct many examples of full exceptional collections of line bundles from a known one.
In \cite{HP11}, Hille-Perling give the first systematic study of full exceptional
collections of line bundles on smooth projective surfaces and introduce the notion of {\it standard augmentation} of an exceptional collection.
Recently, Elagin-Lunts \cite{EL15} show that any full exceptional collection of line bundles on a smooth del Pezzo surface is a standard augmentation.
Next, we will generalize this augmentation from surface caes to higher dimensional case.
Now we assume $Y$ is a smooth projective variety of dimension $n$.
Let $\pi: X\to Y$ be the blow-up of $Y$ at a point $p\in Y$ with the exceptional divisor $E$.
Let
\begin{equation}\label{blowup-point-FCE-before}
\{\mathcal{O}_{Y}(D_1), \mathcal{O}_{Y}(D_2),\ldots, \mathcal{O}_{Y}(D_l)\}
\end{equation}
be a collection of line bundles of length $l$, here $l$ must be no less than $n+1$.
Then, for any $n-1\leq i \leq l$,
we consider the following collection
\begin{eqnarray}\label{blowup-point-FCE}
\{\mathcal{O}_{X}(D_1+(n-1)E), \ldots,\mathcal{O}_{X}(D_{i-n+1}+(n-1)E), \nonumber\\
\mathcal{O}_{X}(D_{i-n+2}+(n-2)E), \mathcal{O}_{X}(D_{i-n+2}+(n-1)E),\nonumber\\
\ldots,\mathcal{O}_{X}(D_{i-1}+E), \mathcal{O}_{X}(D_{i-1}+2E), \nonumber \\
\mathcal{O}_{X}(D_{i}), \mathcal{O}_{X}(D_{i}+E), \nonumber \\
\mathcal{O}_{X}(D_{i+1}), \mathcal{O}_{X}(D_{i+2}), \ldots, \mathcal{O}_{X}(D_{l})\}.
\end{eqnarray}
\begin{thm}\label{blowup-point-fullness-lemma}
The collection \eqref{blowup-point-FCE-before} is a full exceptional collection if and only if the collection \eqref{blowup-point-FCE} is also.
In particular, if $\mathrm{D}(Y)$ has a full exceptional collection of line bundles, so does $\mathrm{D}(X)$.
\end{thm}
\begin{proof}
\textbf{Step 1: \eqref{blowup-point-FCE-before} is an exceptional collection if and only if \eqref{blowup-point-FCE} is an exceptional collection.}
This claim is a direct consequence from the following:
(1) From the short exact sequence,
\begin{equation}\label{effective-exact-sequence}
0\to \mathcal{O}_{X}(-E)\to \mathcal{O}_{X}\to \mathcal{O}_{E}\to 0,
\end{equation}
we have a long exact sequence of sheaf cohomology
$$
0\to H^{0}(\mathcal{O}_{X}(-E)) \to H^{0}(\mathcal{O}_{X}) \to H^{0}(\mathcal{O}_{E}) \to H^{1}(\mathcal{O}_{X}(-E)) \to H^{1}(\mathcal{O}_{X}) \to H^{1}(\mathcal{O}_{E})\to \cdots .
$$
From this, we obtain $H^{k}(\mathcal{O}_{X}(-E))=0$ for any $k\in \mathbb{Z}$ if $\mathcal{O}_X\in D(X)$ is an exceptional object.
(2) For any $1\leq p \leq n-1$, from \eqref{effective-exact-sequence},
we tensor with $\mathcal{O}_{X}(D_j-D_i+pE)$ to obtain an exact sequence,
$$
0\to \mathcal{O}_{X}(D_j-D_i+(p-1)E)\to \mathcal{O}_{X}(D_j-D_i+pE)\to \mathcal{O}_{E}(pE)\to 0.
$$
Since for any $1\leq p \leq n-1$,
$\mathcal{O}_{E}(pE)\cong \mathcal{O}_{\mathbb{P}^{n-1}}(-p)$
is cohomologically zero on $E$,
thus we have
$$
H^{k}(\mathcal{O}_{X}(D_j-D_i+pE))
\cong H^{k}(\mathcal{O}_{X}(D_j-D_i+(p-1)E))\cong\cdots\cong H^{k}(\mathcal{O}_{X}(D_j-D_i)),
$$
for any $i>j$ and all $k\in \mathbb{Z}$.
\textbf{Step 2: \eqref{blowup-point-FCE-before} is full if and only if \eqref{blowup-point-FCE} is a full.}
By Orlov's blow-up formula (Theorem \ref{Orlov-blowup-formula}),
there is a semiorthogonal decomposition of $\mathrm{D}(X)$,
\begin{equation}\label{Orlov-blowup-point-SOD}
\mathrm{D}(X)
=\langle \pi^{\ast}\mathrm{D}(Y), \mathcal{O}_{E}(E), \ldots, \mathcal{O}_{E}((n-1)E) \rangle.
\end{equation}
From \eqref{effective-exact-sequence},
we tensor $\mathcal{O}_{X}(D_{s}+tE)$ to have an exact sequence
\begin{equation}\label{effective-exact-sequence-twisted}
0\to \mathcal{O}_{X}(D_{s}+(t-1)E)\to \mathcal{O}_{X}(D_{s}+tE)\to \mathcal{O}_{E}(tE)\to 0.
\end{equation}
\begin{enumerate}
\item[(i)]
Suppose the collection \eqref{blowup-point-FCE-before} is a full exceptional collection.
In \eqref{effective-exact-sequence-twisted},
if $s=i, i-1,\ldots ,i-n+2$ and $t=i-s+1$,
then we have
$$
0\to \mathcal{O}_{X}(D_{s}+(i-s)E)\to \mathcal{O}_{X}(D_{s}+(i-s+1)E)
\to \mathcal{O}_{E}((i-s+1)E)\to 0,
$$
and then $\mathcal{O}_{E}(pE)\in \langle \eqref{blowup-point-FCE}\rangle$ for $1\leq p \leq n-1$.
Therefore, inductively,
we may use the exact sequence \eqref{effective-exact-sequence-twisted}
to show $\mathcal{O}_{X}(D_{s})\in \langle \eqref{blowup-point-FCE}\rangle$
for $s=1, 2, \ldots,i-1$.
For example, for $s=1$ and $1\leq p \leq n-1$,
from the exact sequence \eqref{effective-exact-sequence-twisted},
we have
$$
0\to \mathcal{O}_{X}(D_{1}+(p-1)E)\to \mathcal{O}_{X}(D_{1}+pE)
\to \mathcal{O}_{E}(pE)\to 0,
$$
Therefore, if $\mathcal{O}_{X}(D_{1}+pE)\in \langle \eqref{blowup-point-FCE}\rangle$,
then $\mathcal{O}_{X}(D_{1}+(p-1)E)\in \langle \eqref{blowup-point-FCE}\rangle$
and thus
$\mathcal{O}_{X}(D_{1})\in \langle \eqref{blowup-point-FCE}\rangle$.
\item[(ii)] Conversely, suppose the collection \eqref{blowup-point-FCE} is a full exceptional collection.
Assume $A\in \langle \eqref{blowup-point-FCE-before}\rangle^{\bot}$.
Next we want to show $\pi^{\ast}A \in \langle \eqref{blowup-point-FCE}\rangle^{\bot}$ and hence $A=0$.
To this end, from \eqref{effective-exact-sequence-twisted},
there is a long exact sequence
\begin{eqnarray}\label{effective-exact-sequence-twisted-long}
&\cdots &\to \mathrm{Hom}(\mathcal{O}_{E}(tE)[j+1], \pi^{\ast}A[k])
\to \mathrm{Hom}(\mathcal{O}_{X}(D_{s}+(t-1)E)[j], \pi^{\ast}A[k]) \nonumber \\
&& \to\mathrm{Hom}(\mathcal{O}_{X}(D_{s}+tE)[j], \pi^{\ast}A[k]) \to \cdots,
\end{eqnarray}
for any $j, k\in \mathbb{Z}$.
By \eqref{Orlov-blowup-point-SOD},
for any $1\leq t\leq n-1$,
$\mathcal{O}_{E}(tE)\in {}^{\bot}\langle \pi^{\ast}\mathrm{D}(Y)\rangle$,
we obtain
$$
\mathrm{Hom}(\mathcal{O}_{E}(tE)[j], \pi^{\ast}A[k])=0, \; \textrm{for any}\; j, k\in \mathbb{Z}.
$$
From this and \eqref{effective-exact-sequence-twisted-long},
we have
\begin{eqnarray*}
\mathrm{Hom}(\mathcal{O}_{X}(D_{s}), \pi^{\ast}A[k])\cong \cdots
&\cong& \mathrm{Hom}(\mathcal{O}_{X}(D_{s}+(t-1)E), \pi^{\ast}A[k]) \\
&\cong& \mathrm{Hom}(\mathcal{O}_{X}(D_{s}+tE), \pi^{\ast}A[k]),
\end{eqnarray*}
for any $k\in \mathbb{Z}$, $1\leq s\leq l$ and $1\leq t\leq n-1$.
Therefore, we obtain that $\pi^{\ast}A \in \langle \eqref{blowup-point-FCE}\rangle^{\bot}$.
\end{enumerate}
This completes the proof.
\end{proof}
\section{Blow-up a point in $\mathbb{P}^{3}$}
\label{blow-up-one-point}
In this section, we will classify cohomologically zero line bundles and hence the exceptional collection of line bundles of length $6$ on
the blow-up of $\mathbb{P}^3$ at a point and show these exceptional collections are full.
\subsection{Geometry of $X$}
Let $\pi: X\to \mathbb{P}^{3}$ be the blow-up of $\mathbb{P}^{3}$ at a point $p\in \mathbb{P}^{3}$, and let $E$ be the exceptional divisor.
Then $X$ is a toric smooth Fano threefold with the canonical divisor
$$
K_{X}=\pi^{\ast} K_{\mathbb{P}^{3}}+2E=-4H+2E,
$$
where $H$ is the pullback of hyperplane class in $\mathbb{P}^3$.
The Picard group of $X$ is
$$
\mathrm{Pic}(X)\cong \mathrm{Pic}(\mathbb{P}^{3})\oplus \mathbb{Z}[E]
= \mathbb{Z}[H]\oplus \mathbb{Z}[E]
$$
with the intersection numbers
$$
H^3=1, H^2E=0, HE^2=0, E^3=1.
$$
Let $a$ be an integer. Then $X$ is also the projective bundle
\begin{equation}\label{eq:projective bundle 1}
X\cong \mathbb{P}(\mathcal{E}) \stackrel{\rho}\to \mathbb{P}^2, \text{ with } \mathcal{E}=\mathcal{O}_{\mathbb{P}^2}(-a+1)\oplus \mathcal{O}_{\mathbb{P}^2}(-a).
\end{equation}
Moreover, since $\rho^{\ast}\mathcal{O}_{\mathbb{P}^2}(1)=[H-E]$ and
$
K_X=\rho^{\ast}(K_{\mathbb{P}^{2}} +\det(\mathcal{E}^{\vee}))
\otimes \mathcal{O}_{X}(-2),
$
we have
$$\mathcal{O}_X(1)=\mathcal{O}_X(aH-(a-1)E).$$
\subsection{Cohomologically zero line bundles}
\begin{lem}\label{Char-H0-H3-point}
$H^{0}(\mathcal{O}_{X}(aH+bE))=0$ if and only if $a<0$ or $a+b<0$.
Consequently, $H^{3}(\mathcal{O}_{X}(aH+bE))=0$ if and only if $a>-4$ or $a+b>-2$.
\end{lem}
\begin{proof}
First, we show that $H^{0}(\mathcal{O}_{X}(aH+bE))>0$ if and only if $a\geq 0$ and $a+b\geq0$. In fact, if $a\geq 0$ and $a+b\geq0$, then $aH+bE$ is an effective divisor.
Conversely, suppose $aH+bE$ is an effective divisor ($a, b\in \mathbb{Z}$).
Since $H$ is a nef divisor,
then the intersection number
$$
H^{2}(aH+bE)=a\geq 0.
$$
Since $H-E$ is base-point free and hence a nef divisor,
and then the intersection number
$$
(H-E)^{2}(aH+bE)=a+b\geq 0.
$$
For the second part, by Serre duality, we have
$$
H^{3}(\mathcal{O}_{X}(aH+bE))\cong H^{0}(\mathcal{O}_{X}((-4-a)H-(b-2)E)).
$$
Then the lemma follows.
\end{proof}
To classify the cohomologically zero line bundles,
we have the following observation.
\begin{lem}\label{H1H2-zero-lem-point}
For any $a, b\in \mathbb{Z}$,
$h^{1}(\mathcal{O}_{X}(aH+bE))h^{2}(\mathcal{O}_{X}(aH+bE))=0$.
\end{lem}
\begin{proof}
From the short exact sequence,
$$
0\to \mathcal{O}_{X}(-E) \to \mathcal{O}_{X} \to \mathcal{O}_{E} \to 0,
$$
we tensor with $\mathcal{O}_{X}(aH+bE)$ to obtain an exact sequence of sheaves,
$$
0\to \mathcal{O}_{X}(aH+(b-1)E)\to \mathcal{O}_{X}(aH+bE)\to \mathcal{O}_{E}(aH+bE)\to 0.
$$
Taking sheaf cohomology,
we obtain a long exact sequence,
$$
\cdots \to H^{1}(\mathcal{O}_{X}(aH+(b-1)E)) \to H^{1}(\mathcal{O}_{X}(aH+bE))
\to H^{1}( \mathcal{O}_{E}(aH+bE)) \to \cdots.
$$
Since $H^{1}(\mathcal{O}_{X}(aH))\cong H^{1}(\mathcal{O}_{\mathbb{P}^{3}}(a))=0$
and $H^{1}(\mathcal{O}_{E}(aH+bE))\cong H^{1}(\mathcal{O}_{\mathbb{P}^{2}}(-b))=0$,
then we have the following inequalities
$$
0=h^{1}(\mathcal{O}_{X}(aH))
\geq h^{1}(\mathcal{O}_{X}(aH+E))
\geq \cdots
\geq h^{1}(\mathcal{O}_{X}(aH+(b-1)E))
\geq h^{1}(\mathcal{O}_{X}(aH+bE))
$$
for $b\geq 0$ and $a\in \mathbb{Z}$,
and hence $H^{1}(\mathcal{O}_{X}(aH+bH))=0$ for $b\geq0$ and $a\in \mathbb{Z}$.
Therefore, if $b< 0$ and thus $2-b>0$,
by Serre duality,
then we have
$$
H^{2}(\mathcal{O}_{X}(aH+bE))\cong H^{1}(\mathcal{O}_{X}((-a-4)H-(b-2)E))=0.
$$
This completes the proof.
\end{proof}
Now we give the characterization of cohomologically zero line bundles on the blow-up of $\mathbb{P}^3$ at a point.
\begin{prop}\label{Coh-zero-line-bundles-point}
A line bundle $\mathcal{O}_{X}(aH+bE)$ is cohomologically zero if and only if one of the following holds:
\begin{enumerate}
\item[(1)] $a+b=-1$;
\item[(2)] $a=-1$, $b=1$;
\item[(3)] $a=-1$, $b=2$;
\item[(4)] $a=-2$, $b=0$;
\item[(5)] $a=-2$, $b=2$;
\item[(6)] $a=-3$, $b=0$;
\item[(7)] $a=-3$, $b=1$.
\end{enumerate}
\end{prop}
\begin{proof}
Suppose $H^{0}(\mathcal{O}_{X}(aH+bE))= H^{3}(\mathcal{O}_{X}(aH+bE))=0$.
Then $\mathcal{O}_{X}(aH+bE)$ is cohomologically zero if and only if $\chi(\mathcal{O}_{X}(aH+bE))=0$.
In fact, by Lemma \ref{H1H2-zero-lem-point},
$h^{1}(\mathcal{O}_{X}(aH+bE))h^{2}(\mathcal{O}_{X}(aH+bE))=0$,
then we have
$$
\chi(\mathcal{O}_{X}(aH+bE))^2=(h^{1}(\mathcal{O}_{X}(aH+bE))^2+(h^{2}(\mathcal{O}_{X}(aH+bE))^2.
$$
Then $\chi(\mathcal{O}_{X}(aH+bE))=0$ if and only if $h^{1}(\mathcal{O}_{X}(aH+bE))=h^{2}(\mathcal{O}_{X}(aH+bE)=0$.
By Riemann-Roch Theorem,
we have
\begin{eqnarray}\label{RR-formula}
\chi(\mathcal{O}_{X}(aH+bE))
&=&\int_{X} \mathrm{ch}(\mathcal{O}_{X}(aH+bE))\mathrm{Td}(X) \nonumber\\
&=& \frac{c_{1}(X)c_{2}(X)}{24}+\frac{(c_{1}^{2}(X)
+c_{2}(X))c_{1}(\mathcal{O}_{X}(aH+bE))}{12} \nonumber\\
&& +\frac{c_{1}(X)c_{1}^{2}(\mathcal{O}_{X}(aH+bE))}{4}
+\frac{c_{1}^{3}(\mathcal{O}_{X}(aH+bE))}{6}.
\end{eqnarray}
By the blow-up formula of Chern classes,
$$
c_{2}(X)=\pi^{\ast}c_{2}(\mathbb{P}^{3})=6H^2,
$$
and by Riemann-Roch formula \eqref{RR-formula},
we obtain
\begin{eqnarray*}
\chi(\mathcal{O}_{X}(aH+bE))
&=&\frac{1}{6}(a^3+6a^2+11a+b^3-3b^2+2b+6) \\
&=&\frac{1}{6}((a+1)(a+2)(a+3)+b(b-1)(b-2)).
\end{eqnarray*}
Therefore, by Lemma \ref{Char-H0-H3-point},
$\mathcal{O}_{X}(aH+bE)$ is cohomologically zero if and only if
the following condition hold:
\begin{enumerate}
\item[(i)] $a<0$, or $a+b<0$;
\item[(ii)] $a>-4$, or $a+b>-2$;
\item[(iii)] $(a+1)(a+2)(a+3)+b(b-1)(b-2)=0$.
\end{enumerate}
Next, we denote $x:=a+1$ and $y:=-b$,
then $(a+1)(a+2)(a+3)+b(b-1)(b-2)=0$ is equivalent to
$x(x+1)(x+2)=y(y+1)(y+2)$.
Since
\begin{eqnarray*}
2(x(x+1)(x+2)-y(y+1)(y+2))
&=& (x-y)(x^2+xy+y^2+3x+3y+2)\\
&=& (x-y)((x+y)^2+(x+3)^2+(y+3)^2-14),
\end{eqnarray*}
and hence if $x\neq y$ and $(x+y)^2+(x+3)^2+(y+3)^2=14(=1^2+2^2+3^2)$,
we have
$$
y=0, x=-1, -2; y= -1, x=0, -1; y=-2, x=0, -1,
$$
i.e., $b=0, a=-2, -3; b=1, a=-1, -2; b=2, a=-1, -2$.
This completes the proof.
\end{proof}
\subsection{Classification results}
\begin{thm}\label{Classify-Blowup-p-P3}
Let $X$ be the blow-up of $\mathbb{P}^{3}$ at a point.
Then the normalized sequence
\begin{equation}\label{basic-type-of-EFC-linebundle}
\{\mathcal{O}_{X},\mathcal{O}_{X}(D_1), \mathcal{O}_{X}(D_2),\mathcal{O}_{X}(D_3), \mathcal{O}_{X}(D_4), \mathcal{O}_{X}(D_5)\}
\end{equation}
is an exceptional collection of line bundles
if and only if the ordered set of divisors $\{D_1, D_2, D_3, D_4, D_5\}$ is one of the following types:
\begin{enumerate}
\item[$(1)_a$] $\{H-E, 2H-2E,
aH-(a-1)E, (a+1)H-aE,
(a+2)H-(a+1)E \}$;
\item[$(2)_a$] $\{H-E, aH-(a-1)E,
(a+1)H-aE, (a+2)H-(a+1)E,
3H-E\}$;
\item[$(3)_a$] $\{aH-(a-1)E, (a+1)H-aE,
(a+2)H-(a+1)E, 2H,3H-E\}$;
\item[$(4)$] $\{H-E, H,
2H-2E, 2H-E, 3H-2E\}$;
\item[$(5)$] $\{E, H-E,
H, 2H-E, 3H-E\}$;
\item[$(6)$] $\{H-2E, H-E,
2H-2E, 3H-2E,4H-3E\}$;
\item[$(7)$] $\{E, H,
2H, 3H-E,3H\}$;
\item[$(8)$] $\{H-E, 2H-E,
3H-2E, 3H-E, 4H-3E\}$;
\item[$(9)$] $\{H, 2H-E,
2H, 3H-2E, 3H-E\}$,
\end{enumerate}
where $a\in \mathbb{Z}$.
Moreover, by mutations and normalizations, they are related as:
\begin{eqnarray*}
& & (1)_a\Rightarrow (2)_a\Rightarrow (3)_a \Rightarrow (1)_{4-a} \Rightarrow (2)_{4-a} \Rightarrow (3)_{4-a} \Rightarrow (1)_{a}; \\
& & (4)\Rightarrow (5)\Rightarrow (6)\Rightarrow (7)\Rightarrow
(8) \Rightarrow (9)\Rightarrow (4).
\end{eqnarray*}
\end{thm}
\begin{proof}
Write $D_0=0$. By Lemma \ref{lem:exlb},
the sequence \eqref{basic-type-of-EFC-linebundle}
is an exceptional collection if and only if for any integers $0\leq j<i\leq 5$ the line bundles $\mathcal{O}_{X}(D_j-D_i)$ are cohomologically zero.
Now suppose the sequence \eqref{basic-type-of-EFC-linebundle} is an exceptional collection.
First of all, by Proposition \ref{Coh-zero-line-bundles-point}, we notice that $\mathcal{O}_{X}(D_{i})$ must be one of the following line bundles:
$B_0=\mathcal{O}_{X}(aH-(a-1)E)$ ($\forall a \in \mathbb{Z}$),
$B_1=\mathcal{O}_{X}(H-E)$,
$B_2=\mathcal{O}_{X}(H-2E)$,
$B_3=\mathcal{O}_{X}(2H)$,
$B_4=\mathcal{O}_{X}(2H-2E)$,
$B_5=\mathcal{O}_{X}(3H)$,
$B_6=\mathcal{O}_{X}(3H-E)$.
Furthermore,
to find out all the exceptional collections \eqref{basic-type-of-EFC-linebundle},
it suffices to pick up any five $\{B_{l_1},\cdots ,B_{l_5}\}$ such that each pair $(B_{l_j},B_{l_i})$ ($j<i$) is an exceptional pair.
In order to achieve this goal, we will build up a table of all exceptional pairs which consists of $B_i$ ($i=0,1,\cdots ,6$).
Since a pair $(B_s, B_t)$ is an exceptional pair if and only if the line bundle $B_s \otimes B_t^{\vee}$ is cohomologically zero, by Proposition \ref{Coh-zero-line-bundles-point} we have the following table:
\begin{table}[ht]
\caption{Exceptional pairs $(B_s, B_t)$ for blow-up of $\mathbb{P}^3$ at a point }\label{tab1:g}
\begin{center}
{\tiny
\begin{tabular}{|c| p{20mm}|c|c|c|c|c|c|}
\hline
& $B_0^{\prime}$ &$B_1$&$B_2$&$B_3$&$B_4$&$B_5$&$B_6$\\
\hline
$B_0$& $a^{\prime}=a+1$, $a+2$& $a=0$& & $\forall a$& $a=1$&$a=0$, $1$&$\forall a$\\
\hline
$B_1$& $\forall a^\prime$& & & & $\surd$& &$\surd$\\
\hline
$B_2$& $a^{\prime}=3$, $4$& $\surd$& & & $\surd$& & \\
\hline
$B_3$&$a^\prime=3$& & & & &$\surd$&$\surd$\\
\hline
$B_4$& $\forall a^{\prime}$& & & & & & \\
\hline
$B_5$& & & & & & & \\
\hline
$B_6$& $a^{\prime}=4$& & & & &$\surd$& \\
\hline
\end{tabular}
}
\end{center}
\;
\;
\small{
In the table, the first column stand for $B_s$ and the first row stand for $B_t$, and the blank means that $(B_s,B_t)$ is not an exceptional pair; otherwise is.
For example, $(B_1, B_3)$ is not an exceptional pair, $(B_0, B_0')$ is an exceptional pair if and only if $a'=a+1$ or $a'=a+2$, and $(B_2, B_4)$ is an exceptional pair.}
\end{table}
Before we continue the proof,
we need the following.
\begin{claim}\label{small-lem1}
$\{\mathcal{O}_{X}, \mathcal{O}_{X}(a_1H+(1-a_1)E), \cdots, \mathcal{O}_{X}(a_i H+(1-a_i )E)\}$ is an exceptional collection if and only if one of the following three conditions holds:
\begin{enumerate}
\item[(1)] $i=1$, $a_1\in \mathbb{Z}$;
\item[(2)] $i=2$, $a_1=a_2-1$ or $a_1=a_2-2$;
\item[(3)] $i=3$, $a_1+1=a_2=a_3-1$.
\end{enumerate}
\end{claim}
\begin{proof}[Proof of Claim \ref{small-lem1} ]
The pair $\{\mathcal{O}_{X}(aH-(a-1)E), \mathcal{O}_{X}(bH-(b-1)E)\}$ is an exception collection if and only if $\mathcal{O}_{X}((a-b)H-(a-b)E)$ is cohomologically zero.
Then, by Proposition \ref{Coh-zero-line-bundles-point} (2) and (5),
we get $a-b=-1$ or $a-b=-2$ and the lemma follows.
\end{proof}
From Claim \ref{small-lem1} and Table \ref{tab1:g},
we observe that the line bundle $\mathcal{O}_{X}(D_1)$ in the sequence \eqref{basic-type-of-EFC-linebundle} may be only one of $B_0$, $B_1$ and $B_2$.
To finish the proof,
we shall discuss $\mathcal{O}_{X}(D_1)$ case-by-case:
\begin{enumerate}
\item[(I)] Suppose $\mathcal{O}_{X}(D_1)=B_0$.
From Table \ref{tab1:g}, for $t>0$, the exceptional pairs of type $(B_0, B_t)$ are: $(B_0, B_1)$, $(B_0,B_3)$, $(B_0, B_4)$, $(B_0, B_5)$ and $(B_0, B_6)$; then for $t_1, t_2>0$, the exceptional collections of type $\{\mathcal{O}_{X},B_0, B_{t_1}, B_{t_2}\}$ are : \underline{$\{\mathcal{O}_{X}, B_0, B_1, B_6\}$}, \underline{$\{\mathcal{O}_{X}, B_0, B_3, B_5\}$}, \underline{$\{\mathcal{O}_{X}, B_0, B_3, B_6\}$}, $\{\mathcal{O}_{X}, B_0, B_6, B_5\}$; then for $t_1, t_2, t_3>0$, the only exceptional collection of type $\{\mathcal{O}_{X}, B_0$, $B_{t_1}, B_{t_2}, B_{t_3}\}$ is : \underline{$\{\mathcal{O}_{X}, B_0, B_3, B_6, B_5\}$}.
Then we add more divisors of type $B_0$ into the above five exceptional collections to obtain exceptional collections of length 6:
\begin{enumerate}
\item For $\{\mathcal{O}_{X},B_0, B_1,B_6\}$, we need to add two divisors of type $B_0$,
then we have \underline{$\{\mathcal{O}_{X},B_0, B_1$}, \underline{$B_0', B_0'', B_6\}$} (i.e., $(5)$ in Theorem \ref{Classify-Blowup-p-P3});
\item For $\{\mathcal{O}_{X}, B_0, B_3, B_5\}$, we need to add two divisors of type $B_0$, which is impossible by Table \ref{tab1:g};
\item For $\{\mathcal{O}_{X}, B_0, B_3, B_6\}$, we need to add two divisors of type $B_0$, then we have two cases: \underline{$\{\mathcal{O}_{X}, B_0, B_0', B_3, B_0'', B_6\}$} (i.e., $(9)$ in Theorem \ref{Classify-Blowup-p-P3}), and \underline{$\{\mathcal{O}_{X}, B_0, B_0', B_0'', B_3, B_6\}$} (i.e., $(3)$ in Theorem \ref{Classify-Blowup-p-P3});
\item For $\{\mathcal{O}_{X}, B_0, B_6, B_5\}$, we need to add two divisors of type $B_0$, which is impossible by Table \ref{tab1:g};
\item For $\{\mathcal{O}_{X}, B_0, B_3, B_6, B_5\}$, we need to add one divisor of type $B_0$, then we have \underline{$\{\mathcal{O}_{X}, B_0, B_0'$}, \underline{$B_3, B_6, B_5\}$} (i.e., $(7)$ in Theorem \ref{Classify-Blowup-p-P3}).
\end{enumerate}
\item[(II)] Suppose $\mathcal{O}_{X}(D_1)=B_1$. From Table \ref{tab1:g}, for $t> 0$, the exceptional pairs of type $(B_1, B_t)$ are: $(B_1,B_4)$, $(B_1, B_6)$; then for $t_1, t_2>0$, there are no exceptional collections of type $(B_1, B_{t_1}, B_{t_2})$.
Then we need to add divisors of type $B_0$ into the following two exceptional collections of length $3$:
\begin{enumerate}
\item For $(\mathcal{O}_{X}, B_1,B_4)$, we need to add three divisors of type $B_0$, then we have two cases: \underline{$\{\mathcal{O}_{X},B_1, B_0, B_4, B_0',B_0''\}$} (i.e., $(4)$ in Theorem \ref{Classify-Blowup-p-P3}),
and \underline{$\{\mathcal{O}_{X},B_1, B_4, B_0, B_0',B_0''\}$} (i.e., $(1)$ in Theorem \ref{Classify-Blowup-p-P3});
\item For $(\mathcal{O}_{X}, B_1,B_6)$, we need to add three divisors of type $B_0$,
then there are also two cases: \underline{$\{\mathcal{O}_{X},B_1, B_0, B_0',B_0'', B_6\}$} (i.e., $(2)$ in Theorem \ref{Classify-Blowup-p-P3}),
and \underline{$\{\mathcal{O}_{X},B_1, B_0, B_0', B_6, B_0''\}$} (i.e., $(8)$ in Theorem \ref{Classify-Blowup-p-P3}).
\end{enumerate}
\item[(III)] Suppose $\mathcal{O}_{X}(D_1)=B_2$.
From Table \ref{tab1:g}, there are at most two divisors of type $B_0$ after $D_1$ in the sequence \eqref{basic-type-of-EFC-linebundle}. On the other hand, for $t_1, t_2>0$, there is only one exceptional collection of type $(B_2, B_{t_1}, B_{t_2})$: $(B_2, B_1, B_4)$. Thus, we get only one exceptional collection of length 6: \underline{$\{\mathcal{O}_{X},B_2, B_1,B_4, B_0, B_0'\}$} (i.e., $(6)$ in Theorem \ref{Classify-Blowup-p-P3}).
\end{enumerate}
Recall that $\omega_X^{\vee}=\mathcal{O}_X(4H-2E)$. To check the two types of relations:
\begin{eqnarray*}
& & (1)_a\Rightarrow (2)_a\Rightarrow (3)_a \Rightarrow (1)_{4-a} \Rightarrow (2)_{4-a} \Rightarrow (3)_{4-a} \Rightarrow (1)_{a}; \\
& & (4)\Rightarrow (5)\Rightarrow (6)\Rightarrow (7)\Rightarrow
(8) \Rightarrow (9)\Rightarrow (4).
\end{eqnarray*}
we just use mutations (Lemma \ref{mutation-fullness-lem}) and normalizations (Lemma \ref{normalization-lem}) repeatedly.
This completes the proof of Theorem \ref{Classify-Blowup-p-P3}.
\end{proof}
Now we are ready to give the proof of the main result of this section.
\begin{thm}\label{main-thm-point}
Let $X$ be the blow-up of $\mathbb{P}^{3}$ at a point.
Then any exceptional collection of line bundles of length $6$ on $X$ is full.
\end{thm}
\begin{proof}
According to Lemma \ref{normalization-lem},
to prove Theorem \ref{main-thm-point},
it suffices to show that any exceptional collection of line bundles of length $6$
in Theorem \ref{Classify-Blowup-p-P3} is full.
By Lemma \ref{mutation-fullness-lem},
it suffices to show that the exceptional collections of type $(1)$ and $(4)$ in Theorem \ref{Classify-Blowup-p-P3} are full.
(i) To prove the fullness of type $(4)$ in Theorem \ref{Classify-Blowup-p-P3},
we will use the Beilinson's semiorthogonal decomposition of $\mathrm{D}(\mathbb{P}^3)$,
$$
\langle \mathcal{O}_{\mathbb{P}^3}, \mathcal{O}_{\mathbb{P}^3}(H),
\mathcal{O}_{\mathbb{P}^3}(2H), \mathcal{O}_{\mathbb{P}^3}(3H)\rangle.
$$
By Theorem \ref{blowup-point-fullness-lemma},
we obtain a semiorthogonal decomposition of $\mathrm{D}(X)$,
$$
\langle \mathcal{O}_{X}(2E), \mathcal{O}_{X}(H+E), \mathcal{O}_{X}(H+2E), \mathcal{O}_{X}(2H), \mathcal{O}_{X}(2H+E), \mathcal{O}_{X}(3H)\rangle.
$$
Then, by Lemma \ref{normalization-lem}, this turns into the case $(4)$ in Theorem \ref{Classify-Blowup-p-P3}, and hence
$$
\mathrm{D}(X)
=\langle
\mathcal{O}_{X}, \mathcal{O}_{X}(H-E), \mathcal{O}_{X}(H), \mathcal{O}_{X}(2H-2E),
\mathcal{O}_{X}(2H-E), \mathcal{O}_{X}(3H-2E)
\rangle.
$$
(ii) To show the fullness of type $(1)_a$ in Theorem \ref{Classify-Blowup-p-P3}, we shall use the projective bundle structure of $X$ as (\ref{eq:projective bundle 1}).
By Orlov's projective bundle formula (Theorem \ref{Orlov-projbundle-formula}),
we have the following semiorthogonal decompositions
\begin{eqnarray*}
\mathrm{D}(X)
&=&\langle \rho^{\ast}\mathrm{D}(\mathbb{P}^2),
\rho^{\ast}\mathrm{D}(\mathbb{P}^2)\otimes \mathcal{O}_X(1) \rangle\\
&=& \langle
\rho^{\ast}\mathcal{O}_{\mathbb{P}^2}, \rho^{\ast}\mathcal{O}_{\mathbb{P}^2}(1),
\rho^{\ast}\mathcal{O}_{\mathbb{P}^2}(2),\\
&&\;\;\;\; \rho^{\ast}\mathcal{O}_{\mathbb{P}^2}\otimes \mathcal{O}_X(1),
\rho^{\ast}\mathcal{O}_{\mathbb{P}^2}(1)\otimes \mathcal{O}_X(1),
\rho^{\ast}\mathcal{O}_{\mathbb{P}^2}(2)\otimes \mathcal{O}_X(1)\rangle \\
&=& \langle
\mathcal{O}_{X}, \mathcal{O}_{X}(H-E), \mathcal{O}_{X}(2H-2E),
\mathcal{O}_{X}\otimes \mathcal{O}_{X}(aH-(a-1)E), \\
&& \;\;\;\; \mathcal{O}_{X}(H-E)\otimes \mathcal{O}_{X}(aH-(a-1)E),
\mathcal{O}_{X}(2H-2E)\otimes \mathcal{O}_{X}(aH-(a-1)E)\rangle \\
&=& \langle
\mathcal{O}_{X}, \mathcal{O}_{X}(H-E), \mathcal{O}_{X}(2H-2E),
\mathcal{O}_{X}(aH-(a-1)E), \\
&&\;\;\;\; \mathcal{O}_{X}((a+1)H-aE),\mathcal{O}_{X}((a+2)H-(a+1)E) \rangle.
\end{eqnarray*}
This completes the proof of Theorem \ref{main-thm-point}.
\end{proof}
\section{Blow-up a line in $\mathbb{P}^3$}
\label{blow-up-one-line}
In this section, we will classify the cohomologically zero line bundles
and the exceptional collection of line bundles of length $6$ on the blow-up of $\mathbb{P}^3$ at a line and show they are full.
\subsection{Geometry of $X$}
Let $\pi: X\to \mathbb{P}^3$ be the blow-up of $\mathbb{P}^3$
at a line $\mathbb{P}^1 \cong \mathbb{L} \subset \mathbb{P}^3$.
The exceptional divisor of $\pi$ is
$E\cong\mathbb{P}(\mathcal{N}_{\mathbb{P}^1/ \mathbb{P}^3})\cong\mathbb{P}^{1}\times \mathbb{P}^{1}$.
Then $X$ is a toric smooth Fano threefold with the canonical divisor
$$
K_{X}=\pi^{\ast}K_{\mathbb{P}^3}+E=-4H+E,
$$
where $H$ is the pullback of hyperplane class in $\mathbb{P}^3$.
The Picard group of $X$ is
$$
\mathrm{Pic}(X)\cong \mathrm{Pic}(\mathbb{P}^3)\oplus \mathbb{Z}[E]=\mathbb{Z}[H]\oplus \mathbb{Z}[E]
$$
with intersection numbers
$$
H^3=1, H^2E=0, HE^2=-1, E^3=-2,
$$
and we may assume $\mathcal{O}_{E}(E)\cong \mathcal{O}_{E}(-S+F)$ (note that $(aS+bF)^2=-2$ implies $aS+bF=\pm (-S+F)$),
$\mathcal{O}_{E}(H)\cong \mathcal{O}_{E}(F)$, where $S$ and $F$ are given in Example \ref{cohom-zero-F_1}.
Let $a$ be an integer. Then $X$ is also the projective bundle
\begin{equation}\label{eq:projective bundle 2}
X\cong \mathbb{P}(\mathcal{V})
\stackrel{\rho}\to \mathbb{P}^{1}, \text{ with }
\mathcal{V}:=\mathcal{O}_{\mathbb{P}^{1}}(-a+1)^{\oplus 2} \oplus \mathcal{O}_{\mathbb{P}^1}(-a).
\end{equation}
Since $\rho^{\ast}\mathcal{O}_{\mathbb{P}^{1}}(1)=[H-E]$ and
$
K_X=\rho^{\ast}(K_{\mathbb{P}^{1}}+\det(\mathcal{V}^{\vee}))
\otimes \mathcal{O}_{X}(-3),
$
we have
$$\mathcal{O}_X(1)=\mathcal{O}_X(aH-(a-1)E).$$
\subsection{Cohomologically zero line bundles}
\begin{lem}\label{Char-H0-H3-line}
$H^{0}(\mathcal{O}_{X}(aH+bE))=0$ if and only if $a<0$ or $a+b<0$.
Consequently, $H^{3}(\mathcal{O}_{X}(aH+bE))=0$ if and only if $a>-4$ or $a+b>-3$.
\end{lem}
\begin{proof}
Similar to Lemma \ref{Char-H0-H3-point},
it suffices to show that if $aH+bE$ ($a, b\in \mathbb{Z}$) is an effective divisor,
then $a\geq 0$ and $a+b\geq0$.
Suppose $aH+bE$ is an effective divisor, $a, b\in \mathbb{Z}$.
Since $H$ is a nef divisor,
then the intersection number
$$
H^{2}(aH+bE)=a\geq 0.
$$
Since $H, H-E$ are base-point free and hence are nef divisors,
and then
$$
H(H-E)(aH+bE)=a+b\geq 0.
$$
By Serre duality,
we have
$$
H^{3}(\mathcal{O}_{X}(aH+bE))\cong H^{0}(\mathcal{O}_{X}((-4-a)H-(b-1)E)).
$$
It follows the proposition.
\end{proof}
Let $P\cong \mathbb{P}^2$ be the proper transform of a plane containing the line
$\mathbb{L}$.
Then $\mathcal{O}_{X}(P)\cong\mathcal{O}_{X}(H-E)$.
\begin{lem}\label{divisor-P-lem}
$H|_{P}=\mathcal{O}_{P}(H)\cong \mathcal{O}_{P}(1)$
and $E|_{P}=\mathcal{O}_{P}(H)\cong \mathcal{O}_{P}(1)$.
\end{lem}
\begin{proof}
Since the divisor $H$ is nef,
then the line bundle $\mathcal{O}_{P}(H)$ is also nef.
Thus $\mathcal{O}_{P}(H)\cong \mathcal{O}_{P}(k)$ for some $k\geq 0$.
We obtain intersection numbers
$$
k^2=(H|_{P})^{2}=H^2(H-E)=H^3-H^2E=1,
$$
and hence $k=1$.
Suppose $E|_{P}\cong \mathcal{O}_{P}(k')$.
Then the intersection numbers
$$
k'=(E|_{P})(H|_{P})=EH(H-E)=1,
$$
and we have $k'=1$.
\end{proof}
Analogous to Lemma \ref{H1H2-zero-lem-point},
we have the following result.
\begin{lem}\label{H1H2-zero-lem-line}
For any $a, b\in \mathbb{Z}$,
$h^{1}(\mathcal{O}_{X}(aH+bE))h^{2}(\mathcal{O}_{X}(aH+bE))=0$.
\end{lem}
\begin{proof}
We first show that $H^{1}(\mathcal{O}_{X}(sP+tH))=0$ if $s\geq 0$.
From the short exact sequence
$$
0 \to \mathcal{O}_{X}(-P)\to \mathcal{O}_{X}\to \mathcal{O}_{P} \to 0,
$$
we tensor $\mathcal{O}_{X}(sP+tH)$ to obtain a short exact sequence
$$
0 \to \mathcal{O}_{X}((s-1)P+tH)\to \mathcal{O}_{X}(sP+tH)\to \mathcal{O}_{P}(sP+tH) \to 0.
$$
Taking cohomology,
we have a long exact sequence
$$
\cdots \to H^1(\mathcal{O}_{X}(s-1)P+tH))\to H^1(\mathcal{O}_{X}(sP+tH))\to H^{1}(\mathcal{O}_{P}(sP+tH)) \to \cdots.
$$
Since $P\cong \mathbb{P}^{2}$, by Lemma \ref{divisor-P-lem},
we obtain $H^{1}(\mathcal{O}_{P}(sP+tH))\cong H^{1}(\mathcal{O}_{P}(t))=0$ for $t\in \mathbb{Z}$.
Then we have the following inequalities
$$
0=h^{1}(\mathcal{O}_{X}(tH))
\geq h^{1}(\mathcal{O}_{X}(P+tH))
\geq \cdots
\geq h^{1}(\mathcal{O}_{X}((s-1)P+tH))
\geq h^{1}(\mathcal{O}_{X}(sP+tH)),
$$
and hence $H^{1}(\mathcal{O}_{X}(sP+tH))=0$ for $s\geq0$ and $t\in \mathbb{Z}$.
Secondly, since $aH+bE=-b(H-E)+(a+b)H$ and if $-b\geq 0$,
then $H^{1}(\mathcal{O}_{X}(aH+bE))=0$ for $b\leq 0$ and $a\in \mathbb{Z}$.
If $-b<0$, i.e., $b\geq 1$,
by Serre duality, we have
\begin{eqnarray*}
H^{2}(\mathcal{O}_{X}(aH+bE))
&\cong& H^{1}(\mathcal{O}_{X}((-4-a)H-(b-1)E)) \\
&=& H^{1}(\mathcal{O}_{X}((b-1)(H-E)+(-3-a-b)H))=0.
\end{eqnarray*}
This completes the proof.
\end{proof}
Next, we give the characterization of cohomologically zero line bundles on the blow-up of $\mathbb{P}^3$ at a line.
\begin{prop}\label{Coh-zero-line-bundles-line}
A line bundle $\mathcal{O}_{X}(aH+bE)$ is cohomologically zero if and only if one of the following conditions hold:
\begin{enumerate}
\item[(1)] $a+b=-1$;
\item[(2)] $a+b=-2$;
\item[(3)] $a=-1$, $b=1$;
\item[(4)] $a=-3$, $b=0$.
\end{enumerate}
\end{prop}
\begin{proof}
By Blow-up formula of Chern classes,
we have
$$
c_2(X)
=\pi^{\ast}(c_2(\mathbb{P}^3)+\mu_{\mathbb{P}^1})-\pi^{\ast}c_{1}(\mathbb{P}^3)E
=7H^2-4HE,
$$
where $\mu_{\mathbb{P}^1}=H^2$.
By Riemann-Roch formula \eqref{RR-formula},
we have
\begin{eqnarray*}
\chi(\mathcal{O}_{X}(aH+bE)) &=& \frac{1}{6}(a^3+6a^2+11a+6-2b^3-3b^2+5b+3ab-3ab^2)\\
&=& \frac{1}{6}(a-2b+3)(a+b+1)(a+b+2).
\end{eqnarray*}
Therefore, by Lemma \ref{Char-H0-H3-line} and Lemma \ref{H1H2-zero-lem-line},
$\mathcal{O}_{X}(aH+bE)$ is cohomologically zero
if and only if $\chi(\mathcal{O}(aH+bE))=0$ and one of the following conditions hold:
\begin{enumerate}
\item[(1)] $-4<a<0$;
\item[(2)] $a<0$, $a+b>-3$;
\item[(3)] $a+b<0$, $a>-4$;
\item[(4)] $-3<a+b<0$.
\end{enumerate}
Then the proposition follows the case by case:
(I) If $-3<a+b<0$, i.e., $a+b=-1$ or $a+b=-2$, then $\chi(\mathcal{O}_{X}(aH+bE))=0$.
(II) We assume $(a+b+1)(a+b+2)\neq 0$ and $\chi(\mathcal{O}_{X}(aH+bE))=0$ (i.e., $a-2b+3=0$).
Then we obtain:
\begin{enumerate}
\item[(i)] if $-4<a<0$, then $(a=-1, b=1)$ and $(a=-3, b=0)$;
\item[(ii)] if $a<0$ and $a+b>-3$, then $a=-1, b=1$;
\item[(iii)] if $a+b<0$ and $a>-4$, then $a=-3, b=0$;
\end{enumerate}
This proves the proposition.
\end{proof}
\subsection{Classification results}
\begin{thm}\label{Classify-Blowup-line-P3}
Let $X$ be the blow-up of $\mathbb{P}^{3}$ at a line.
Then the normalized sequence
\begin{equation}\label{basic-type-of-EFC-linebundle2}
\{\mathcal{O}_{X},\mathcal{O}_{X}(D_1), \mathcal{O}_{X}(D_2),\mathcal{O}_{X}(D_3), \mathcal{O}_{X}(D_4), \mathcal{O}_{X}(D_5)\}
\end{equation}
is an exceptional collection of line bundles
if and only if the ordered set of divisors $\{D_1, D_2, D_3, D_4, D_5\}$ is one of the following two types:
\begin{enumerate}
\item[$(1)_{a,b}$] $\{aH-(a-1)E, (a+1)H-aE,
bH-(b-2)E, (b+1)H-(b-1)E,
3H\}$;
\item[$(2)_{a,b}$] $\{H-E,aH-(a-1)E,
(a+1)H-aE, bH-(b-2)E,
(b+1)H-(b-1)E\}$,
\end{enumerate}
where $a, b\in \mathbb{Z}$. Moreover, by mutations and normalizations, they are related as
$$
(1)_{a,b}\Rightarrow (2)_{b-a,3-a}\Rightarrow (1)_{b-a-1,2-a}.
$$
\end{thm}
\begin{proof}
The idea of the proof is analogous to that of Theorem \ref{Classify-Blowup-p-P3} and it is relatively easy. Write $D_0=0$. By Lemma \ref{lem:exlb},
the sequence \eqref{basic-type-of-EFC-linebundle2}
is an exceptional collection if and only if for any integers $0\leq j<i\leq 5$ the line bundles $\mathcal{O}_{X}(D_j-D_i)$ are cohomologically zero.
Suppose the sequence \eqref{basic-type-of-EFC-linebundle2} is an exceptional colection. According to Proposition \ref{Coh-zero-line-bundles-line},
for any $1 \leq i \leq 5$,
$\mathcal{O}_{X}(D_{i})$ must be one of the line bundles:
$B_0=\mathcal{O}_{X}(aH-(a-1)E)$,
$B_1=\mathcal{O}_{X}(bH-(b-2)E)$,
$B_2=\mathcal{O}_{X}(H-E)$,
and $B_3=\mathcal{O}_{X}(3H)$,
where $a, b\in \mathbb{Z}$.
To determine all the exceptional collections of line bundle of length $6$,
analogously, by Proposition \ref{Coh-zero-line-bundles-line}
we have the following table of exceptional pairs $(B_s, B_t)$ which consists of $B_0, B_1, B_2$ and $B_3$:
\begin{table}[ht]
\caption{Exceptional pairs $(B_s, B_t)$ for blow-up of $\mathbb{P}^3$ at a line }\label{tab2:g}
\begin{center}
{\tiny
\begin{tabular}{|c| p{12mm}|c|c|c|c|}
\hline
& $B_0^{\prime}$ &$B_1'$&$B_2$&$B_3$\\
\hline
$B_0$& $a'=a+1$ & $\forall a, b'$& & $\forall a$ \\
\hline
$B_1$& & $b'=b+1$ & & $\forall b$ \\
\hline
$B_2$& $\forall a$ & $\forall b'$ & & \\
\hline
$B_3$& & & & \\
\hline
\end{tabular}
}
\end{center}
\end{table}
Consequently, $\mathcal{O}_{X}(D_1)$ may be one of $B_0$ and $B_2$,
and we have the following two cases:
\begin{enumerate}
\item if $\mathcal{O}_{X}(D_1)=B_0$, then we get the exceptional collection:
$\{\mathcal{O}_{X}, B_0, B_0', B_1, B_1', B_3\} $ (i.e., $(1)$ in Theorem \ref{Classify-Blowup-line-P3});
\item if $\mathcal{O}_{X}(D_1)=B_2$, then we obtain the exceptional collection:
$\{\mathcal{O}_{X}, B_2, B_0, B_0', B_1, B_1' \}$ (i.e., $(2)$ in Theorem \ref{Classify-Blowup-line-P3}).
\end{enumerate}
Recall that $\omega_X^{\vee}=\mathcal{O}_X(4H-E)$. To check the relations
$$
(1)_{a,b}\Rightarrow (2)_{b-a,3-a}\Rightarrow (1)_{b-a-1,2-a},
$$
we just use mutations (Lemma \ref{mutation-fullness-lem}) and normalizations (Lemma \ref{normalization-lem}) repeatedly.
This completes the proof of Theorem \ref{Classify-Blowup-line-P3}.
\end{proof}
Now we are ready to prove the main result of this section.
\begin{thm}\label{main-thm-line}
Let $X$ be the blow-up of $\mathbb{P}^{3}$ at a line.
Then any exceptional collection of line bundles of length $6$
on $X$ is full.
\end{thm}
\begin{proof}
By Lemma \ref{normalization-lem} and Lemma \ref{mutation-fullness-lem},
it suffices to show the exceptional collection $(2)$ in Theorem \ref{Classify-Blowup-line-P3} is full.
By using the projective bundle structure of $X$ (\ref{eq:projective bundle 2}) and Orlov's projective bundle formula (Theorem \ref{Orlov-projbundle-formula}),
we have semiorthogonal decompositions of $\mathrm{D}(X)$,
\begin{eqnarray*}
\mathrm{D}(X)
&=&
\langle
\rho^{\ast}\mathrm{D}(\mathbb{P}^1),
\rho^{\ast}\mathrm{D}(\mathbb{P}^1)\otimes \mathcal{O}_X(1),
\rho^{\ast}\mathrm{D}(\mathbb{P}^1)\otimes \mathcal{O}_X(2)
\rangle \\
&=&
\langle
\rho^{\ast}\mathcal{O}_{\mathbb{P}^1}, \rho^{\ast}\mathcal{O}_{\mathbb{P}^1}(1),
\rho^{\ast}\mathcal{O}_{\mathbb{P}^1}\otimes\mathcal{O}_X(1), \rho^{\ast}\mathcal{O}_{\mathbb{P}^1}(1)\otimes \mathcal{O}_X(1), \\
&& \;\;\;\; \rho^{\ast}\mathcal{O}_{\mathbb{P}^1}\otimes \mathcal{O}_X(2),
\rho^{\ast}\mathcal{O}_{\mathbb{P}^1}(1)\otimes \mathcal{O}_X(2)\rangle \\
&=&
\langle
\mathcal{O}_{X}, \mathcal{O}_{X}(H-E),
\mathcal{O}_X(aH-(a-1)E), \mathcal{O}_{X}((a+1)H-aE) , \\
&& \;\;\;\; \mathcal{O}_X(2aH-(2a-2)E), \mathcal{O}_{X}((2a+1)H-(2a-1)E)
\rangle.
\end{eqnarray*}
Based on this semiorthogonal decomposition, we may inductively show that the exceptional collection $(2)$ in Theorem \ref{Classify-Blowup-line-P3} is full.
For any given $a\in \mathbb{Z}$, inductively,
we assume that for $b=k$ the exceptional collection
\begin{eqnarray}\label{main-thm-line-equ1}
&& \{\mathcal{O}_{X}, \mathcal{O}_{X}(H-E),
\mathcal{O}_{X}(aH-(a-1)E), \mathcal{O}_{X}((a+1)H-aE),\nonumber \\
&& \;\;\;\; \mathcal{O}_{X}(kH+(2-k)E), \mathcal{O}_{X}((k+1)H+(1-k)E)\}
\end{eqnarray}
is full.
Then, for $b=k-1$, we have the exceptional collection
\begin{eqnarray}\label{main-thm-line-equ2}
&& \{\mathcal{O}_{X}, \mathcal{O}_{X}(H-E),
\mathcal{O}_{X}(aH-(a-1)E), \mathcal{O}_{X}((a+1)H-aE),\nonumber \\
&& \;\;\;\; \mathcal{O}_{X}((k-1)H-(k-3)E), \mathcal{O}_{X}(kH-(k-2)E)\},
\end{eqnarray}
and for $b=k+1$ we have the exceptional collection
\begin{eqnarray}\label{main-thm-line-equ3}
&& \{\mathcal{O}_{X}, \mathcal{O}_{X}(H-E),
\mathcal{O}_{X}(aH-(a-1)E), \mathcal{O}_{X}((a+1)H-aE), \nonumber \\
&& \;\;\;\; \mathcal{O}_{X}((k+1)H-(k-1)E), \mathcal{O}_{X}((k+2)H-kE)\}.
\end{eqnarray}
Comparing the exceptional collections \eqref{main-thm-line-equ2} and \eqref{main-thm-line-equ3} with \eqref{main-thm-line-equ1},
by Lemma \ref{usefull-fullness-lem},
the exceptional collection \eqref{main-thm-line-equ2} and \eqref{main-thm-line-equ3} are full.
This finishes the proof.
\end{proof}
\section{Blow-up a twisted cubic curve in $\mathbb{P}^3$}
\label{blow-up-one-cubic}
In this section, we shall classify the exceptional collection of line bundles of length $6$ on
the blow-up of $\mathbb{P}^3$ at a twisted cubic curve and show they are all full.
\subsection{Geometry of $X$}
Let $C\subset \mathbb{P}^3$ be a smooth rational curve of degree $3$ (i.e., a twisted cubic curve).
The normal bundle of $C$ in $\mathbb{P}^3$ is $\mathcal{N}_{C/\mathbb{P}^3}\cong \mathcal{O}_{C}(5)\oplus \mathcal{O}_{C}(5)$ (see \cite[Proposition 6]{EV81}).
Let $\pi:X \to \mathbb{P}^3$ be the blow-up of $\mathbb{P}^3$ at $C$.
Then $X$ is a (non-toric) smooth Fano threefold.
Let $E$ be the exceptional divisor of $\pi$,
that is, $E:=\mathbb{P}(\mathcal{N}_{C/\mathbb{P}^3})\cong \mathbb{P}^1\times \mathbb{P}^1$.
We denote by $H$ the pull back of hyperplane class in $\mathbb{P}^3$,
and then the canonical divisor of $X$ is
$$
K_{X}=\pi^{\ast}K_{\mathbb{P}^3}+E=-4H+E.
$$
The Picard group of $X$ is
$$
\mathrm{Pic}(X)\cong \mathrm{Pic}(\mathbb{P}^3)\oplus \mathbb{Z}[E]
=\mathbb{Z}[H]\oplus \mathbb{Z}[E]
$$
with intersection numbers
$$
H^3=1, H^2E=0, HE^2=-3, E^3=-10,
$$
and $\mathcal{O}_{E}(E)\cong \mathcal{O}_{E}(-S+5F)$,
$\mathcal{O}_{E}(H)\cong \mathcal{O}_{E}(3F)$, where $S$ and $F$ are given in Example \ref{cohom-zero-F_1}.
Let $a$ be an integer. Then $X$ is also a projective bundle \cite{SW90}
\begin{equation}\label{eq:projective bundle 3}
X\cong \mathbb{P}(\mathcal{W}\otimes \mathcal{O}_{\mathbb{P}^2}(a)) \stackrel{\rho}\to \mathbb{P}^2,
\end{equation}
where the rank two vector bundle $\mathcal{W}$ over $\mathbb{P}^2$ is given by
\begin{equation*}
0\to \mathcal{O}_{\mathbb{P}^2}(-1)^{\oplus 2} \to \mathcal{O}_{\mathbb{P}^2}^{\oplus 4} \to \mathcal{W}(1)\to 0.
\end{equation*}
\subsection{Cohomologically zero line bundles}
\begin{lem}\label{Char-H0-H3-cubic}
$H^{0}(\mathcal{O}_{X}(aH+bE))=0$ if and only if $a<0$ or $a+2b<0$.
Consequently,
$H^{3}(\mathcal{O}_{X}(aH+bE))=0$ if and only if $a>-4$ or $a+2b>-2$.
\end{lem}
\begin{proof}
Similar to Lemma \ref{Char-H0-H3-point},
it suffices to show that if $aH+bE$ ($a, b\in \mathbb{Z}$) is an effective divisor,
then $a\geq 0$ and $a+2b\geq0$.
Since $H$ is a nef divisor,
we obtain the intersection number
$$
H^{2}(aH+bE)=a\geq 0.
$$
Since $2H-E$ is base-point free and hence are nef divisors,
we have the intersection number
$$
(2H-E)^2(aH+bE)=a+2b\geq 0.
$$
Next, by Serre duality, we have
$$
H^{3}(\mathcal{O}_{X}(aH+bE))\cong H^{0}(\mathcal{O}_{X}((-4-a)H-(b-1)E)).
$$
Note that $(-4-a)-2(b-1)=-2-a-2b$ and then the proposition follows.
\end{proof}
Let $Q$ be the proper transform of a smooth quadric surface containing $C$.
Then $\mathcal{O}_{X}(Q)\cong\mathcal{O}_{X}(2H-E)$.
Since $Q\cong \mathbb{P}^1\times \mathbb{P}^1$, we may assume $\mathrm{Pic}(Q)=\mathbb{Z}C_1\oplus \mathbb{Z}C_2$, where $C_1, C_2$ are smooth rational curves on $Q$ with $C_1^2=C_2^2=0$ and $C_1C_2=1$.
\begin{lem}\label{divisor-Q-lem}
$H|_{Q}\sim C_1+C_2$ and $(2H-E)|_{Q}\sim C_1$ or $(2H-E)|_{Q}\sim C_2$.
\end{lem}
\begin{proof}
Suppose $H|_{Q}\sim aC_1+bC_2$ are linear equivalent.
Since $H|_{Q}$ is a nef divisor, then $a, b\geq 0$.
So we get the intersection number
$$
2ab=(H|_{Q})^2=H^2(2H-E)=2H^3-H^2E=2,
$$
and thus $a=b=1$.
Suppose $(2H-E)|_{Q}\sim a'C_1+b'C_2$ are linear equivalent.
Since $(2H-E)|_{Q}$ is a nef divisor, then $a', b'\geq 0$.
We have the intersection number
$$
a'+b'=(H|_{Q})((2H-E)|_{Q})=H(2H-E)(2H-E)=1,
$$
and hence $a'=1,b'=0$ or $a'=0,b'=1$.
\end{proof}
\begin{rem}
From now on, we may assume $(2H-E)|_{Q}\sim C_2$.
\end{rem}
For the case of the blow-up of $\mathbb{P}^3$ at a twisted cubic curve, there is no similar result as Lemma \ref{H1H2-zero-lem-point} and Lemma \ref{H1H2-zero-lem-line}. In fact, this is the reason why we do not know whether $\mathcal{O}_X(aH+bE)$ is cohomologically zero in either of the two cases (10) and (11) in Proposition \ref{cohom-zero-lem-cubic}.
But there are still some characterizations of cohomologically zero line bundles.
In fact, we know that if a line bundles $\mathcal{O}_{X}(aH+bE)$ is cohomologically zero then $\chi(\mathcal{O}_{X}(aH+bE))=0$.
Combining with Lemma \ref{Char-H0-H3-cubic},
we obtain that a line bundle is cohomologically zero if it satisfy at least one of the conditions in the following proposition.
\begin{prop}\label{cohom-zero-lem-cubic}
A line bundle $\mathcal{O}_{X}(aH+bE)$ is cohomologically zero if it is one of the following:
\begin{enumerate}
\item[(1)] $a+2b=-1$;
\item[(2)] $a=-1, b=1$;
\item[(3)] $a=-2, b=0$;
\item[(4)] $a=-2, b=1$;
\item[(5)] $a=-3, b=0$;
\item[(6)] $a=-4, b=2$;
\item[(7)] $a=-7, b=4$;
\item[(8)] $a=0, b=-1$;
\item[(9)] $a=3, b=-3$.
\end{enumerate}
On the other hand, if $\mathcal{O}_{X}(aH+bE)$ is cohomologically zero and belongs to none of the nine cases above, then one of the following two statements is true:
\begin{enumerate}
\item[(10)] $a<-3, a+2b>3$ and $f(a,b):=a^2+5a+6-2ab-5b^2+b=0$;
\item[(11)] $a>-1, a+2b<-3$ and $f(a,b)=0$.
\end{enumerate}
\end{prop}
\begin{proof}
At first, by Lemma \ref{Char-H0-H3-cubic},
we have $H^{0}(\mathcal{O}_{X}(aH+bE))=H^{3}(\mathcal{O}_{X}(aH+bE))=0$
if and only if one of the following conditions hold:
\begin{enumerate}
\item[(i)] $-4<a<0$;
\item[(ii)] $a+2b>-2, a<0$;
\item[(iii)] $a>-4, a+2b<0$;
\item[(iv)] $-2<a+2b<0$.
\end{enumerate}
By Blow-up formula of Chern classes,
we have
$$
c_2(X)
=\pi^{\ast}(c_2(\mathbb{P}^3)+\mu_{C})-\pi^{\ast}c_{1}(\mathbb{P}^3)E
=9H^2-4HE,
$$
where $\mu_{C}=3H^2$.
By Riemann-Roch formula \eqref{RR-formula},
we have
\begin{eqnarray*}
\chi(\mathcal{O}_{X}(aH+bE))
&=& \frac{1}{6}(a^3+6a^2+11a+6-10b^3-3b^2+13b+9ab-9ab^2)\\
&=& \frac{1}{6}(a+2b+1) f(a,b).
\end{eqnarray*}
Therefore, a line bundle is cohomologically zero only if it lies in one of the $11$ cases.
Second, from the short exact sequence,
$$
0\to \mathcal{O}_{X}(-Q) \to \mathcal{O}_{X} \to \mathcal{O}_{Q} \to 0,
$$
we tensor with $\mathcal{O}_{X}(kQ-H)$ and $\mathcal{O}_{X}(-2H+E)$ to obtain two exact sequences,
\begin{equation}\label{exact-sequ-for-(1)}
0\to \mathcal{O}_{X}((k-1)Q-H)\to \mathcal{O}_{X}(kQ-H)\to \mathcal{O}_{Q}(kQ-H)\to 0, \;(\forall k\in \mathbb{Z}),
\end{equation}
and
\begin{equation}\label{exact-sequ-for-(6)}
0\to \mathcal{O}_{X}(-4H+2E)\to \mathcal{O}_{X}(-2H+E)\to \mathcal{O}_{Q}(-2H+E)\to 0.
\end{equation}
To show $(1)$, it is important to note that $ \mathcal{O}_{Q}(kQ-H)= \mathcal{O}_{Q}(-C_1+(k-1)C_2)$, and then $\mathcal{O}_{Q}(kQ-H)$ is cohomologically zero on $Q\cong \mathbb{P}^1\times \mathbb{P}^1$.
Therefore, by \eqref{exact-sequ-for-(1)},
we obtain
$$
h^{i}(\mathcal{O}_{X}(kQ-H))=h^{i}(\mathcal{O}_{X}((k-1)Q-H))=\cdots=h^{i}(\mathcal{O}_{X}(-H))=0, \forall k, i\in \mathbb{Z}.
$$
Hence, if $a+2b=-1$ then $H^{i}(\mathcal{O}_{X}(aH+bE))=H^{i}(\mathcal{O}_{X}(-b(2H-E)-H))=0$
for all $i \in \mathbb{Z}$, that is, $\mathcal{O}_{X}(aH+bE)$ is cohomologically zero.
To prove $(6)$, from Lemma \ref{divisor-Q-lem} and Example \ref{cohom-zero-F_1},
we first notice that the line bundle $\mathcal{O}_{Q}(-2H+E)\cong \mathcal{O}_{Q}(-C_2)$ is cohomologically zero.
From $(4)$, we will see that the line bundle $\mathcal{O}_{X}(-2H+E)$ is cohomologically zero.
By \eqref{exact-sequ-for-(6)}, we get that $\mathcal{O}_{X}(-4H+2E)$ is cohomologically zero.
Third, it is not difficult to see that if $(3)$, $(5)$ and $(8)$ hold
then $\mathcal{O}_{X}(aH+bE)$ is cohomologically zero.
At last, from the short exact sequence,
$$
0\to \mathcal{O}_{X}(-E) \to \mathcal{O}_{X} \to \mathcal{O}_{E} \to 0,
$$
we tensor with $\mathcal{O}_{X}(-H+E)$, $\mathcal{O}_{X}(-2H+E)$, $\mathcal{O}_{X}(-7H+4E)$ and $\mathcal{O}_{X}(3H-2E)$ respectively to obtain the following exact sequences,
\begin{equation}\label{exact-sequ-for-(2)}
0\to \mathcal{O}_{X}(-H)\to \mathcal{O}_{X}(-H+E)\to \mathcal{O}_{E}(-H+E)\to 0,
\end{equation}
\begin{equation}\label{exact-sequ-for-(4)}
0\to \mathcal{O}_{X}(-2H)\to \mathcal{O}_{X}(-2H+E)\to \mathcal{O}_{E}(-2H+E)\to 0,
\end{equation}
\begin{equation}\label{exact-sequ-for-(7)}
0\to \mathcal{O}_{X}(-7H+3E)\to \mathcal{O}_{X}(-7H+4E)\to \mathcal{O}_{E}(-7H+4E)\to 0,
\end{equation}
\begin{equation}\label{exact-sequ-for-(9)}
0\to \mathcal{O}_{X}(3H-3E)\to \mathcal{O}_{X}(3H-2E)\to \mathcal{O}_{E}(3H-2E)\to 0.
\end{equation}
From Example \ref{cohom-zero-F_1}, we obtain that the line bundles,
$\mathcal{O}_{E}(-H+E)\cong \mathcal{O}_{E}(-S+2F)$,
$\mathcal{O}_{E}(-2H+E)\cong \mathcal{O}_{E}(-S-F)$,
$\mathcal{O}_{E}(-7H+4E)\cong \mathcal{O}_{E}(-4S-F)$
and
$\mathcal{O}_{E}(3H-2E)\cong \mathcal{O}_{E}(2S-F)$
are cohomologically zero on $E\cong \mathbb{P}^1\times \mathbb{P}^1$.
Since $3+2\times (-2)=-1$ and $-7+2\times 3=-1$,
from $(1)$, we get that $\mathcal{O}_{X}(3H-2E)$ and $\mathcal{O}_{X}(-7H+3E)$ are cohomologically zero.
Then $(2)$, $(4)$, $(7)$ and $(9)$ hold following the exact sequences \eqref{exact-sequ-for-(2)} \eqref{exact-sequ-for-(4)}, \eqref{exact-sequ-for-(7)} and \eqref{exact-sequ-for-(9)} respectively.
\end{proof}
\subsection{Classification results}
\begin{thm}\label{Classify-Blowup-twisted-cubic-P3}
Let $X$ be the blow-up of $\mathbb{P}^{3}$ at a twisted cubic curve.
Then the normalized sequence
\begin{equation}\label{basic-type-of-EFC-linebundle3}
\{\mathcal{O}_{X},\mathcal{O}_{X}(D_1), \mathcal{O}_{X}(D_2),\mathcal{O}_{X}(D_3), \mathcal{O}_{X}(D_4), \mathcal{O}_{X}(D_5)\}
\end{equation}
is an exceptional collection of line bundles
if and only if the ordered set of divisors $\{D_1, D_2, D_3, D_4, D_5\}$ is one of the following types:
\begin{enumerate}
\item[$(1)$] $\{H, 3H-E,
E, 2H, 3H \}$;
\item[$(2)$] $\{2H-E, -H+E,
H, 2H, 3H-E\}$;
\item[$(3)$] $\{-3H+2E, -H+E,
E, H, 2H\}$;
\item[$(4)$] $\{2H-E, 3H-E,
4H-2E, 5H-2E, 7H-3E\}$;
\item[$(5)$] $\{H, 2H-E,
3H-E, 5H-2E, 2H \}$;
\item[$(6)$] $\{H-E, 2H-E,
4H-2E, H, 3H-E \}$;
\item[$(7)$] $\{2H-E, -3H+2E,
4H-2E, -H+E, H \}$;
\item[$(8)$] $\{-5H+3E, 2H-E,
-3H+2E, -H+E, 2H \}$;
\item[$(9)$] $\{7H-4E, 2H-E,
4H-2E, 7H-3E, 9H-4E \}$;
\item[$(10)$] $\{-5H+3E, -3H+2E,
E, 2H, -3H+3E \}$;
\item[$(11)$] $\{2H-E, 5H-2E,
7H-3E, 2H, 9H-4E\}$;
\item[$(12)$] $\{3H-E, 5H-2E,
E, 7H-3E, 2H\}$;
\item[$(13)_b$] $\{2H-E, 4H-2E,
(2b-3)H-(b-2)E,
(2b-1)H-(b-1)E, (2b+1)H-bE \}$;
\item[$(14)_b$] $\{2H-E,
(2b-3)H-(b-2)E, (2b-1)H-(b-1)E,
(2b+1)H-bE, 2H \}$;
\item[$(15)_b$] $\{(2b-3)H-(b-2)E,
(2b-1)H-(b-1)E, (2b+1)H-bE,
E, 2H \}$,
\end{enumerate}
where $b\in \mathbb{Z}$. Moreover, by mutations and normalizations, they are related as:
\begin{eqnarray*}
& & (1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)\Rightarrow(5)\Rightarrow (6) \Rightarrow (1); \\
& & (7)\Rightarrow(8)\Rightarrow(9)\Rightarrow(10)\Rightarrow(11)\Rightarrow (12)
\Rightarrow (7);\\
& & (13)_b\Rightarrow(14)_{b-1}\Rightarrow(15)_{b-2}\Rightarrow (13)_{5-b}.
\end{eqnarray*}
\end{thm}
\begin{proof}
The idea of the proof is the same as that of Theorem \ref{Classify-Blowup-p-P3}. Write $D_0=0$. By Lemma \ref{lem:exlb},
the sequence \eqref{basic-type-of-EFC-linebundle3}
is an exceptional collection if and only if for any integers $0\leq j<i\leq 5$ the line bundles $\mathcal{O}_{X}(D_j-D_i)$ are cohomologically zero.
Suppose the sequence \eqref{basic-type-of-EFC-linebundle3} is an exceptional collection.
By Proposition \ref{cohom-zero-lem-cubic},
the line bundle $\mathcal{O}_{X}(D_{i})$ must be one of the following line bundles:
$B_0=\mathcal{O}_{X}((2b+1)H-bE)$,
$B_1=\mathcal{O}_{X}(H-E)$,
$B_2=\mathcal{O}_{X}(2H)$,
$B_3=\mathcal{O}_{X}(2H-E)$,
$B_4=\mathcal{O}_{X}(3H)$,
$B_5=\mathcal{O}_{X}(4H-2E)$,
$B_6=\mathcal{O}_{X}(7H-4E)$,
$B_7=\mathcal{O}_{X}(E)$
$B_8=\mathcal{O}_{X}(-3H+3E)$,
$B_9=\mathcal{O}_{X}(aH+bE)$ ($a, b$ satisfy condition (10) in Proposition \ref{cohom-zero-lem-cubic})
and $B_{10}=\mathcal{O}_{X}(aH+bE)$ ($a, b$ satisfy condition (11) in Proposition \ref{cohom-zero-lem-cubic}).
Furthermore,
to find out all the exceptional collections \eqref{basic-type-of-EFC-linebundle3},
it suffices to pick up $\{B_{l_1},\cdots ,B_{l_5}\}$ such that each pair $(B_{l_j},B_{l_i})$ ($j<i$) is an exceptional pair.
To attain this, we shall build up a table of all exceptional pairs which consists of $B_i$ ($i=0,1, \cdots,10$).
Since a pair $(B_s, B_t)$ is an exceptional pair if and only if the line bundle $B_s \otimes B_t^{\vee}$ is cohomologically zero, by Proposition \ref{cohom-zero-lem-cubic} we get the following table:
\begin{table}[ht]
\caption{Exceptional pairs $(B_s, B_t)$ for blow-up of $\mathbb{P}^3$ at a twisted cubic curve}\label{tab3:g}
\begin{center}
{\tiny
\begin{tabular}{|p{3mm}|p{15mm}|p{3mm}|p{3mm}|p{12mm}|p{12mm}|p{12mm}|p{3mm}|p{3mm}|p{12mm}|p{3mm}|p{3mm}|}
\hline
& $B_0^{\prime}$ & $B_1$ & $B_2$ & $B_3$ & $B_4$ & $B_5$ & $B_6$ & $B_7$ & $B_8 $& $B_9$ & $B_{10}$\\
\hline
$B_0$& $b=b^{\prime}+1$, $b^{\prime}+2$& &$\forall b$&$b=0$, 3& $b=0$, $-1$&$b=-1$, $2$& &$\forall b$ &$b=2$, 3& & \\
\hline
$B_1$& $b^{\prime}=0$, $-1$& & &$\surd$& &$\surd$& & & & & \\
\hline
$B_2$& $b^{\prime}=-1$, $-4$& & & & $\surd$& & & &$\surd$& & \\
\hline
$B_3$&$\forall b^{\prime}$& &$\surd$ & & &$\surd$& & & & & \\
\hline
$B_4$& & & & & & & & & & & \\
\hline
$B_5$& $\forall b^{\prime}$& & & & & & & & & & \\
\hline
$B_6$& $b^{\prime}=-3$, $-4$& & &$\surd$& &$\surd$& & & & & \\
\hline
$B_7$& $b^{\prime}=0$, $-3$& & $\surd$ & &$\surd$ & & & &$\surd$ & & \\
\hline
$B_8$& & & & & & & & & & & \\
\hline
$B_9$& & & & & & & & & &? &? \\
\hline
$B_{10}$& & & & & & & & & & ?& ?\\
\hline
\end{tabular}
}
\end{center}
\;
\;
\small{
The mark ``?" means that we do not know whether the corresponding pair could be an exceptional pair or not.}
\end{table}
\begin{claim}\label{small-lem2}
$\{\mathcal{O}_{X}, \mathcal{O}_{X}((2b_1+1)H-b_1E), \cdots, \mathcal{O}_{X}((2b_i+1)H-b_iE)\}$ is an exceptional collection if and only if it is one of the following conditions hold:
\begin{enumerate}
\item[(1)] $i=1$, $b_1\in \mathbb{Z}$;
\item[(2)] $i=2$, $b_1=b_2-1$ or $b_1=b_2-2$;
\item[(3)] $i=3$, $b_1+1=b_2=b_3-1$.
\end{enumerate}
\end{claim}
\begin{proof}[Proof of Claim \ref{small-lem2} ]
The pair $\{\mathcal{O}_{X}((2a+1)H-aE), \mathcal{O}_{X}((2b+1)H-bE)\}$ is an exception collection if and only if $\mathcal{O}_{X}(2(a-b)H-(a-b)E)$ is cohomologically zero.
Then, by Proposition \ref{cohom-zero-lem-cubic} (4) and (6),
we get $a-b=-1$ or $a-b=-2$ and the claim follows.
\end{proof}
\begin{claim}\label{no-(10-11)}
Let $D_i=a_i H+b_i E$, $i=1,2,3$, be three divisors on $X$. We assume, for any $1\le i\le 3$, $-D_i$ satisfies either (10) or (11) in Proposition \ref{cohom-zero-lem-cubic}. Then $(D_1, D_2, D_3)$ cannot be an exceptional collection of length 3.
\end{claim}
\begin{proof}[Proof of Claim \ref{no-(10-11)}]
Denote $g(a,b):=\frac{1}{6}(a+2b+1) f(a,b)$, which is the Euler characteristic function in the proof of Proposition \ref{cohom-zero-lem-cubic}. Here $f(a,b):=a^2+5a+6-2ab-5b^2+b$.
The following system of equations
\begin{eqnarray*}
f(-a_1, -b_1) = f(-a_2, -b_2) = f(-a_3, -b_3) =0,\\
g(a_1 - a_2, b_1 - b_2) =g(a_1 - a_3, b_1 - b_3)=g(a_2 - a_3, b_2 - b_3) =0,
\end{eqnarray*}
has exactly four solutions (e.g., by computer software \texttt{Mathematica}):
$$
(a_1, b_1, a_2, b_2, a_3, b_3)=(0,1,2,0,-3,3), (0,1,2,0,3,0), (1,-1,2,-1,4,-2), \text{ or }(7,-4,2,-1,4,-2).
$$
Then the claim follows.
\end{proof}
From Claim \ref{small-lem2}, Claim \ref{no-(10-11)} and Table \ref{tab3:g},
we observe that the line bundle $\mathcal{O}_{X}(D_1)$ in the sequence \eqref{basic-type-of-EFC-linebundle3} may be only one of $B_0$, $B_1$, $B_3$ and $B_6$.
Similar to the proof of Theorem \ref{Classify-Blowup-p-P3},
one may discuss $\mathcal{O}_{X}(D_1)$ case-by-case to obtain Theorem \ref{Classify-Blowup-twisted-cubic-P3}.
Since this is totally the same as the proof of Theorem \ref{Classify-Blowup-twisted-cubic-P3}, we leave it to the interested readers.
\end{proof}
Now we are in the position to prove the main result of the current section.
\begin{thm}\label{main-thm-cubic}
Let $X$ be the blow-up of $\mathbb{P}^{3}$ at a twisted cubic curve.
Then any exceptional collection of line bundles of length $6$
on $X$ is full.
\end{thm}
\begin{proof}
By Lemma \ref{normalization-lem} and Lemma \ref{mutation-fullness-lem},
it suffices to show the exceptional collection of type $(1)$, $(7)$ and $(13)$ in Theorem \ref{Classify-Blowup-twisted-cubic-P3} are full. Then the proof is divided into three parts.
\textbf{Part 1}: Type $(1)$, i.e. $\{\mathcal{O}_{X}, \mathcal{O}_{X}(H), \mathcal{O}_{X}(3H-E),
\mathcal{O}_{X}(E), \mathcal{O}_{X}(2H), \mathcal{O}_{X}(3H) \}$ is a full exceptional collection.
Applying Orlov's blow-up formula (Theorem \ref{Orlov-blowup-formula}) to the blow-up $X$,
we obtain a semiorthogonal decomposition of $\mathrm{D}(X)$,
\begin{equation*}
\langle
j_{\ast}\rho^{\ast}\mathrm{D}(C)\otimes \mathcal{O}_{E}(-1),
\mathcal{O}_{X}, \mathcal{O}_{X}(H), \mathcal{O}_{X}(2H), \mathcal{O}_{X}(3H)
\rangle.
\end{equation*}
Since $\mathcal{O}_{E}(E)\cong \mathcal{O}_{E}(-1,5)$,
this turns to be
$$
\langle
\mathcal{O}_{E}(-1,4), \mathcal{O}_{E}(-1,5),
\mathcal{O}_{X}, \mathcal{O}_{X}(H), \mathcal{O}_{X}(2H), \mathcal{O}_{X}(3H)
\rangle.
$$
From the short exact sequence,
$$
0\to \mathcal{O}_{X}(-E) \to \mathcal{O}_{X} \to \mathcal{O}_{E} \to 0,
$$
we tensor with $\mathcal{O}_{X}(E)$ to gain an exact sequence,
$$
0\to \mathcal{O}_{X}\to \mathcal{O}_{X}(E)\to \mathcal{O}_{E}(E)\to 0.
$$
Hence
$\mathcal{O}_{E}(-1,5)\cong \mathcal{O}_{E}(E)\in\langle
\mathcal{O}_{X}, \mathcal{O}_{X}(H), \mathcal{O}_{X}(3H-E),
\mathcal{O}_{X}(E), \mathcal{O}_{X}(2H), \mathcal{O}_{X}(3H)
\rangle.
$
Since
$$
\mathrm{Hom}(\mathcal{O}_{X}(3H-E), \mathcal{O}_{E}(-1,4)[k])
=H^{k}(\mathcal{O}_{E}(-2,0))
=
\left\{
\begin{array}{ll}
0, k\neq 0; \\
\mathbb{C}, k=1;
\end{array},
\right.
$$
hence the pair $(\mathcal{O}_{E}(-1,4), \mathcal{O}_{X}(3H-E))$ is not an exceptional pair,
and then the result follows from the same idea as the proof of Lemma \ref{usefull-fullness-lem}.
\;
\;
\;
\textbf{Part 2}:
We start with the full exceptional collection of type $(6)$ in Theorem \ref{Classify-Blowup-twisted-cubic-P3},
$$
\mathrm{D}(X)
=\langle\mathcal{O}_{X}, \mathcal{O}_{X}(H-E), \mathcal{O}_{X}(2H-E),
\mathcal{O}_{X}(4H-2E), \mathcal{O}_{X}(H), \mathcal{O}_{X}(3H-E)\rangle.
$$
Since $\mathcal{O}_{X}(H-E)$ is cohomologically zero,
mutating $\mathcal{O}_{X}(H-E)$ and $\mathcal{O}_{X}$ turns this into
\begin{equation}\label{completion-mutation-sod}
\mathrm{D}(X)
=\langle\mathcal{O}_{X}(H-E), \mathcal{O}_{X}, \mathcal{O}_{X}(2H-E),
\mathcal{O}_{X}(4H-2E), \mathcal{O}_{X}(H), \mathcal{O}_{X}(3H-E)\rangle.
\end{equation}
Then, by Lemma \ref{mutation-fullness-lem}, we left mutate the last term to the front of the exceptional collection,
hence tensoring it with $\omega_X=\mathcal{O}_{X}(-4H+E)$ to obtain a semiorthogonal decomposition of $\mathrm{D}(X)$,
$$
\langle\mathcal{O}_{X}(-H), \mathcal{O}_{X}(H-E), \mathcal{O}_{X}, \mathcal{O}_{X}(2H-E),
\mathcal{O}_{X}(4H-2E), \mathcal{O}_{X}(H)\rangle.
$$
Next we show that type $(7)$, i.e. $\{\mathcal{O}_{X}, \mathcal{O}_{X}(2H-E), \mathcal{O}_{X}(-3H+2E),
\mathcal{O}_{X}(4H-2E), \mathcal{O}_{X}(-H+E), \mathcal{O}_{X}(H) \}$
is a full exceptional collection.
We assume
\begin{equation}\label{assumption-cubic-part2}
A\in \langle \mathcal{O}_{X}, \mathcal{O}_{X}(2H-E), \mathcal{O}_{X}(-3H+2E),
\mathcal{O}_{X}(4H-2E), \mathcal{O}_{X}(-H+E), \mathcal{O}_{X}(H) \rangle^{\bot}
\end{equation}
and $A\neq 0$.
Then $A\in \langle \mathcal{O}_{X}, \mathcal{O}_{X}(2H-E),
\mathcal{O}_{X}(4H-2E), \mathcal{O}_{X}(H)\rangle^{\bot}$.
Since $\langle \mathcal{O}_{X}(-H), \mathcal{O}_{X}(H-E) \rangle
\cong \langle \mathcal{O}_{X}, \mathcal{O}_{X}(2H-E),
\mathcal{O}_{X}(4H-2E), \mathcal{O}_{X}(H) \rangle^{\bot}$,
thus $A\in \langle \mathcal{O}_{X}(-H), \mathcal{O}_{X}(H-E)\rangle$.
Then there is a distinguished triangle
\begin{equation}\label{cubic-exact-triangle}
A_2 \to A \to A_1 \to A_2[1],
\end{equation}
where $A_2 \in \langle \mathcal{O}_{X}(H-E)\rangle$
and $A_1 \in \langle \mathcal{O}_{X}(-H)\rangle$.
Here we may assume $A_1=\bigoplus \mathcal{O}_{X}(-H)[i]^{\oplus j_i}$
and $A_2=\bigoplus \mathcal{O}_{X}(H-E)[s]^{\oplus t_s}$.
By Proposition \ref{cohom-zero-lem-cubic},
$\mathcal{O}_{X}(2H-2E)$ is not cohomologically zero,
i.e., for some $k_0$, $H^{k_0}(\mathcal{O}_{X}(2H-2E))\neq0$.
Then we have following:
\begin{enumerate}
\item[(i)] If $A_2=0$ , then $A\cong A_1\in \langle \mathcal{O}_{X}(-H) \rangle$ and
$$
\mathrm{Hom}(\mathcal{O}_{X}(-3H+2E), \mathcal{O}_{X}(-H)[k_0])=
H^{k_0}(\mathcal{O}_{X}(2H-2E))\neq 0.
$$
This is contradicting to the assumption \eqref{assumption-cubic-part2}.
\item[(ii)] If $A_1=0$ , then $A\cong A_2\in \langle \mathcal{O}_{X}(H-E) \rangle$, and
$$
\mathrm{Hom}(\mathcal{O}_{X}(-H+E), \mathcal{O}_{X}(H-E)[k_0])=
H^{k_0}(\mathcal{O}_{X}(2H-2E))\neq 0.
$$
This is also contradicting to the assumption \eqref{assumption-cubic-part2}.
\item[(iii)]
Assume $A_1\neq 0$ and $A_2\neq0$, and
pick up an object $B=\bigoplus \mathcal{O}_{X}(-H+E)[s]^{\oplus t_s}$.
Applying the functor $\mathrm{Hom}(B, -)$ to distinguished triangle \eqref{cubic-exact-triangle}, we gain the following long exact sequence
$$
\cdots \to \mathrm{Hom}(B, A[k]) \to \mathrm{Hom}(B, A_1[k]) \to \mathrm{Hom}(B, A_2[k+1]) \to \mathrm{Hom}(B, A[k+1])\to \cdots .
$$
By assumption \eqref{assumption-cubic-part2},
this exact sequence implies that
\begin{equation}\label{hom-exact-seq}
\mathrm{Hom}(B, A_1[k]) \cong \mathrm{Hom}(B, A_2[k+1])
\end{equation}
for any $k\in \mathbb{Z}$.
However,
since $\mathcal{O}_{X}(-E)$ is cohomologically zero,
for any $k\in \mathbb{Z}$, we have
$$
\mathrm{Hom}(B, A_1[k])
=\mathrm{Hom}(\bigoplus \mathcal{O}_{X}(-H+E)[s]^{\oplus t_s}, \bigoplus \mathcal{O}_{X}(-H)[i]^{\oplus j_i}[k])
=0,
$$
but
$$
\mathrm{Hom}(B, A_2[k_0-1])=
\mathrm{Hom}(\bigoplus \mathcal{O}_{X}(-H+E)[s]^{\oplus t_s}, \bigoplus \mathcal{O}_{X}(H-E)[s]^{\oplus t_s}[k_0])\neq 0
$$
gives a contradiction to \eqref{hom-exact-seq}.
\end{enumerate}
\;
\;
\;
\textbf{Part 3}: Type $(13)$, i.e. $\{\mathcal{O}_{X}, \mathcal{O}_{X}(2H-E), \mathcal{O}_{X}(4H-2E),
\mathcal{O}_{X}((2b-3)H-(b-2)E), \mathcal{O}_{X}((2b-1)H-(b-1)E),
\mathcal{O}_{X}((2b+1)H-bE) \}$ is a full exceptional collection.
First, by Lemma \ref{mutation-fullness-lem}, we right mutate the first term to the end of the semiorthogonal decomposition \eqref{completion-mutation-sod},
hence tensoring it with $\omega_X^{\vee}=\mathcal{O}_{X}(4H-E)$ to obtain a new semiorthogonal decomposition of $\mathrm{D}(X)$,
$$
\langle
\mathcal{O}_{X}, \mathcal{O}_{X}(2H-E), \mathcal{O}_{X}(4H-2E), \mathcal{O}_{X}(H), \mathcal{O}_{X}(3H-E), \mathcal{O}_{X}(5H-2E)\rangle.
$$
Therefore, for any given $a\in \mathbb{Z}$,
inductively on $b\in \mathbb{Z}$,
we may assume that for $b=k$ the exceptional collection
\begin{eqnarray}\label{main-thm-cubic-equ1}
&& \{\mathcal{O}_{X}, \mathcal{O}_{X}(2H-E), \mathcal{O}_{X}(4H-2E),
\mathcal{O}_{X}((2k-3)H-(k-2)E), \nonumber\\
&& \;\;\;\; \mathcal{O}_{X}((2k-1)H-(k-1)E), \mathcal{O}_{X}((2k+1)H-kE) \}
\end{eqnarray}
is full.
Then for $b=k-1$ we have the following exceptional collection
\begin{eqnarray}\label{main-thm-cubic-equ2}
&& \{\mathcal{O}_{X}, \mathcal{O}_{X}(2H-E), \mathcal{O}_{X}(4H-2E),
\mathcal{O}_{X}((2k-5)H-(k-3)E), \nonumber\\
&& \;\;\;\; \mathcal{O}_{X}((2k-3)H-(k-2)E), \mathcal{O}_{X}((2k-1)H-(k-1)E) \},
\end{eqnarray}
%
and for $b=k+1$ we obtain the exceptional collection
\begin{eqnarray}\label{main-thm-cubic-equ3}
&& \{\mathcal{O}_{X}, \mathcal{O}_{X}(2H-E), \mathcal{O}_{X}(4H-2E),
\mathcal{O}_{X}((2k-1)H-(k-1)E), \nonumber\\
&& \;\;\;\; \mathcal{O}_{X}((2k+1)H-kE), \mathcal{O}_{X}((2k+3)H-(k+1)E) \},
\end{eqnarray}
By comparing two exceptional collections \eqref{main-thm-cubic-equ2} and \eqref{main-thm-cubic-equ3} with \eqref{main-thm-cubic-equ1},
Lemma \ref{usefull-fullness-lem} implies that the exceptional collections \eqref{main-thm-cubic-equ2} and \eqref{main-thm-cubic-equ3} are full.
This completes the proof of Theorem \ref{main-thm-cubic}.
\end{proof}
\begin{rem}
It is possible to show the fullness of type $(13)$ by using the projective bundle structure (\ref{eq:projective bundle 3}) and projective bundle formula.
\end{rem}
\begin{rem}
Since the blow-up a point (or a line, a twisted cubic curve) of $\mathbb{P}^3$ is a Fano variety, theoretically one may use Bondal \cite[Theorem 4.1]{Bon90} to show the fullness of the exceptional collections in Theorem \ref{Classify-Blowup-p-P3}, Theorem \ref{Classify-Blowup-line-P3} and Theorem \ref{Classify-Blowup-twisted-cubic-P3}.
\end{rem}
\section{Final remarks}
In this section, we collect some interesting problems which are related to Kuznetsov's fullness conjecture and which we are interested in.
(1)
Recall that all smooth toric Fano $3$-folds and $4$-folds have full exceptional collections of line bundles.
Inspired by main theorem, one may hope the following to be true.
\begin{conj}
All exceptional collections of line bundles of maximal length on smooth toric Fano $3$-folds and $4$-folds are full.
\end{conj}
Of course, it is also very interesting to give an explicit classification of full exceptional collections of line bundles on smooth toric Fano $3$-folds and $4$-folds.
In dimensional $4$, there are some relatively easy examples of smooth toric Fano $4$-folds on which cohomologically zero line bundles are clear:
\begin{enumerate}
\item[(i)] $\mathcal{O}_{\mathbb{P}^2\times \mathbb{P}^2}(a, b)$ is cohomologically zero if and only if $a=-1, -2$, or $b=-1, -2$;
\item[(ii)] $\mathcal{O}_{\mathbb{P}^3\times \mathbb{P}^1}(a, b)$ is cohomologically zero if and only if $a=-1, -2, -3$, or $b=-1$;
\item[(iii)] $\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1\times \mathbb{P}^1}(a, b, c)$ is cohomologically zero if and only if $a=-1$, or $b=-1$, or $c=-1$.
\end{enumerate}
In general, the classification of cohomologically zero line bundles on smooth toric Fano 4-folds may be much more complicated.
For example, it is interesting to give the classification of cohomologically zero line bundles on the blow-up of $\mathbb{P}^{4}$ at a point or a line.
(2)
The existence of (quasi-)phantom categories on smooth projective varieties is becoming very important. There is an interesting conjecture on the existence of phantom categories on Barlow surfaces (see \cite[Conjecture 4.1]{DKK13} and \cite[Conjecture 4.9]{CKP13}), which is strongly connected to Kuznetsov's fullness conjecture.
\begin{conj}\label{Barlow-conj}
Given any Barlow surface $X$,
the derived category $\mathrm{D}(X)$ has an
exceptional collection of length $11$ with orthogonal complement a (non-trivial) phantom category,
but has no full exceptional collection.
\end{conj}
We see that Kuznetsov's fullness conjecture implies Conjecture \ref{Barlow-conj} for some Barlow surfaces.
As a matter of fact, in \cite{BGvBKS15}, B\"{a}hning-Graf von Bothmer-Katzarkov-Sosna
have exhibited a determinantal Barlow surface $S$
with an exceptional collection of line bundles of length $11$ whose orthogonal complement is a (non-trivial) phantom category.
Certainly, this does not mean that there is no full exceptional collection on the Barlow surface $S$.
If $S$ admits a full exceptional collection then its length must be $11$,
and then Kuznetsov's fullness conjecture give a contradiction with B\"{a}hning-Graf von Bothmer-Katzarkov-Sosna's result.
(3)
Since any smooth projective rational surface is a series of blow-up points of $\mathbb{P}^2$ or Hirzebruch surfaces, hence Orlov's blow-up formula implies that any smooth projective rational surface admits a full exceptional collection. Moreover, the augmentation implies that any smooth projective rational surface admits a full exceptional collection of line bundles.
Conversely, there is an open problem:
\begin{conj}[Orlov]\label{surf-rational-excep-collect}
Any smooth projective surface with a full exceptional collection (of line bundles) is rational.
\end{conj}
As a byproduct, Conjecture \ref{surf-rational-excep-collect} implies that any smooth projective surface with a full exceptional collection of length $4$ must be a Hirzebruch surface.
Recently, in \cite[Theorem 4.3]{BS15}, Brown-Shipman show that Conjecture \ref{surf-rational-excep-collect} holds for a smooth projective surface which admits a full strong exceptional collection of line bundles.
However, it is still widely open in general.
Also, there is a high dimensional analogous to Conjecture \ref{surf-rational-excep-collect} (see \cite[Conjecture 1.2]{EL15}).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
A presentation lemma proved by Gabber in \cite{gabber} (see also \cite{chk}) plays a foundational role in ${\mathbb A}^1$-algebraic topology as developed by F. Morel in \cite{Morel}. This lemma can be thought of as an algebro-geometric analogue of tubular neighbourhood theorem in differential geometry. The current published proof of Gabber's presentation lemma works only over infinite fields. In a private communication to F. Morel, Gabber has pointed out that the proof of this theorem also holds for finite fields. Unfortunately there is no published proof for this case.
The goal of this paper is to prove the following version of Gabber's presentation lemma over finite fields.
\begin{theorem}\label{gabberfinite}
Let $X$ be a smooth variety of dimension $d\geq 1$ over a finite field $F$ and $Z\subset X$ be a closed subvariety. Let $p\in Z$ be a point. Let ${\mathbb A}^d_F \xrightarrow{\pi} {\mathbb A}^{d-1}_F$ denote the projection onto the first $d-1$ coordinates. Then there exists
\begin{enumerate}
\item[(i)] an open neighbourhood $ U\subset X$ of $p$,
\item[(ii)] a map $\Phi:U\to {\mathbb A}_F^d$,
\item[(iii)] an open neighbourhood $V\subset {\mathbb A}^{d-1}_F$ of $\Psi(p)$ where
$\Psi:U\to {\mathbb A}_F^{d-1}$ denotes the composition
$$ U\xrightarrow{\Phi}{\mathbb A}_F^d \xrightarrow{\pi} {\mathbb A}_F^{d-1}$$
\end{enumerate}
such that
\begin{enumerate}
\item $\Phi$ is \'{e}tale.
\item $\Psi_{|Z_V}:Z_V \to V$ is finite where $Z_V:=Z\cap \Psi^{-1}(V)$.
\item $\Phi_{|Z_V}:Z_V \to {\mathbb A}^1_V= \pi^{-1}(V)$ is a closed immersion.
\end{enumerate}
\end{theorem}
\begin{remark}\label{simplify}
Without loss of generality, we may (and will) assume henceforth that $X$ is affine. Moreover, by \cite[3.2]{chk}, we may also assume that $Z$ is a principal divisor defined by $f\in {\mathcal O}(X)$ and $p$ is a closed point.
\end{remark}
\begin{remark}
If one finds $U,\Phi,V,\Psi$ satisfying conditions (1),(2),(3) of \eqref{gabberfinite}, one can also arrange, by shrinking $U$ if necessary, the following additional condition
\begin{enumerate}
\item[(4)] $Z_V=\Phi^{-1}\Phi(Z_V)$.
\end{enumerate}
To see this, let $\widetilde{Z}$ denote the image of $Z_V$ in ${\mathbb A}^1_V$. The morphism $\Phi^{-1}\Phi(Z_V)\to \widetilde{Z}$ is \'{e}tale, and it admits a section as $Z_V$ maps isomorphically onto $\widetilde{Z}$. Thus $\Phi^{-1}\Phi(Z_V)$ is a disjoint union of $Z_V$ and a closed subset $T$ of $\Psi^{-1}(V)$. Replacing $U$ by $U\backslash T$, one sees that the additional condition $(4)$ is satisfied.
\end{remark}
The proof of Gabber's presentation lemma for infinite fields (see \cite[3.1]{gabber} or \cite[3.1]{chk}) shows that for the maps $\Phi,\Psi$ appearing in the statement of the lemma, suitable generic choices work. The problem in making this proof work over a finite field is very similar to the problem of making Bertini's theorem work over a finite field. Bertini's theorem for finite fields was proved by Poonen in \cite{poonen} using an extremely clever counting argument. Because of the broad similarities of the issues involved, it is natural to try to use Poonen's argument to prove Gabber's presentation lemma over finite fields. However, Poonen's counting argument, in our opinion, is easier to apply in the case of subvarieties of an affine space. Thus, the first step of the proof of Theorem \ref{gabberfinite} is reduction to the case where $X$ is an open subset of ${\mathbb A}^d_F$. This is done in section \ref{firstred}. Unfortunately, we found even this case too complex to directly apply Poonen's ideas from \cite{poonen}. Fortunately, we are able to reduce this complexity by using induction on $d$ to reduce to the case where $d=2$, i.e. $X$ is an open subset of ${\mathbb A}^2_F$. This is done in section \ref{inductiond}. This induction argument, although short, was one of the most time-taking tasks for us in proving Theorem \ref{gabberfinite}. An important ingredient of this induction is a slightly modified version of Noether normalization trick (see \ref{nntrick}).
The case of open subsets of ${\mathbb A}^2_F$ is now ideal for using Poonen's counting argument. Indeed, the handling of points of small degree is very similar to that of \cite{poonen}. However, we are unable to handle the error term for `high degree points' as is done in \cite{poonen}. We fix this with a small trick (\ref{hexists}). \\
\noindent {\bf Acknowledgements}: We thank F. Morel for his comments and for answering our questions on the current status of this result. We thank A. Asok, F. D\'eglise, M. Levine and J. Riou for their comments during the early stage of this project. We thank Anand Sawant and Charanya Ravi for pointing out a mistake in the previous version of the paper. We also thank the referee for numerous suggestions.
\section{Reduction to open subsets of ${\mathbb A}^d_F$}\label{firstred}
The goal of this section is to prove Lemma \ref{redopen}, which reduces Theorem \ref{gabberfinite} to the case where $X$ is an open subset of ${\mathbb A}^d_F$ and $p\in Z\subset X$ is a closed point with first $d-1$ coordinates equal to $0$.
\begin{notation}\label{notation} Throughout this paper we work over a fixed finite field $F$. We further fix the following notation.
\begin{enumerate}
\item Let $Y$ be a subset of a scheme $X/F$. We let $Y_{ \scriptscriptstyle \leq r}:= \{x\in Y\ |\ \syn{deg}(x) \leq r\}$ and similarly $Y_{ \scriptscriptstyle < r}:= \{x\in Y\ |\ \syn{deg}(x) < r\}$ and $Y_{ \scriptscriptstyle = r}:= \{x\in Y\ |\ \syn{deg}(x) = r\}$.
\item For $f_1,...,f_i\in F[X_1,...,X_n]$ we let $\syn{Z}(f_1,...,f_i)$ denote the closed subscheme of ${\mathbb A}^n_F$ defined by the ideal $(f_1,...,f_i)$.
\end{enumerate}
\end{notation}
We first start by recalling the following standard trick (see \cite{mum}) used in the proof of Noether's normalization lemma.
\begin{lemma}\cite[page 2]{mum}\label{nntrick1}
Let $k$ be any field and $n\geq 1$ be any integer. Let $Z/k$ be a finitely generated affine scheme of dimension at most $n-1$. Let $$Z\xrightarrow{(\phi_1,...,\phi_n)}{\mathbb A}^{n}_k$$ be a finite map. Let $Q (T)\in k[T]$ be a non constant monic polynomial and $Q=Q(\phi_n)$. Then for $\ell>>0$, the map
$$ Z\xrightarrow{(\phi_{\scriptscriptstyle 1}-Q^{\ell^{n-1}},\ldots,\phi_{\scriptscriptstyle n-1}-Q^{\ell})} {\mathbb A}^{n-1}_k$$ is finite.
\end{lemma}
\begin{remark}\label{q=phin}
We claim that finiteness of $Z\xrightarrow{(\phi_1,...,\phi_n)}{\mathbb A}^{n}_k$ implies that of $ Z \xrightarrow{(\phi_1,...,Q)}{\mathbb A}^{n}_k$. This is because the later map is a composition of the following two finite maps
$$Z\xrightarrow{(\phi_1,...,\phi_n)}{\mathbb A}^{n}_k \xrightarrow{(Y_1,\ldots ,Q(Y_n))}{\mathbb A}^{n}_k.$$
One can thus easily reduce the proof of the above general case to the case where $Q(T)=T$. Unless explicitly mentioned, we will usually assume $Q(T)=T$ while applying the lemma. As in the proof of Noether normalization, the above lemma is usually applied repeatedly until one gets a map from $Z$ to ${\mathbb A}^{\dim(Z)}_k$.
\end{remark}
\begin{lemma}\label{redopen}
Let $p\in Z\subset X$ be as in Theorem \ref{gabberfinite}. Further, assume that $X$ is affine, $Z$ is a principal divisor and $p$ is a closed point (see Remark \ref{simplify}). Then there exists a map $\varphi:X\to {\mathbb A}^d_F$ and an open neighbourhood $W$ of $\varphi(p)$ such that
\begin{enumerate}
\item $\varphi^{-1}(W) \to W$ is \'{e}tale.
\item $Z_W:= Z\cap \varphi^{-1}(W) \to W$ is a closed immersion.
\item The first $d-1$ coordinates of $\varphi(p)$ are $0$.
\end{enumerate}
In particular, it suffices to prove Theorem \ref{gabberfinite} where $X$ is an open subset of ${\mathbb A}^d_F$ and the first $d-1$ coordinates of $p$ are zero.
\end{lemma}
\begin{proof} Let
\begin{enumerate}
\item[-] $X={\rm Spec \,}(A)$.
\item[-] $Z= {\rm Spec \,}(A/(f))$ and let $\overline{A}:= A/(f)$.
\item[-] $\mathfrak{m}\subset A$ be the maximal ideal of the closed point $p$.
\item[-] $F(p)$ denote the residue field of $p$.
\end{enumerate}
\noindent $\Step{1}$:
Since $X/F$ is smooth, $\dim_{F(p)}(\mathfrak{m}/\mathfrak{m}^2)=d$. Choose $\{x_{\scriptscriptstyle 1},...,x_{\scriptscriptstyle d-1}\} \subset {\mathfrak m}$ such that they span a $d-1$ dimensional $F(p)$-subspace of $\mathfrak{m}/\mathfrak{m}^2$. In this step we claim that there exists $y \in A$ such that
\begin{enumerate}
\item $y\ {\rm mod}\ \mathfrak{m}$ is a primitive element of $ F(p)/F$.
\item The set $\{x_1,...,x_{d-1},h(y)\}$ (modulo $\mathfrak{m}^2$) gives a $F(p)$-basis of ${\mathfrak m}/{\mathfrak m}^2$, where $h$ is the minimal polynomial of $y\ {\rm mod}\ \mathfrak{m}$.
\item The map $(x_1,\ldots,x_{d-1},{y}):X \xrightarrow{\eta} {\mathbb A}^d$ is \'{e}tale at $p$.
\item The map $\eta$ induces an isomorphism on residue fields $F(\eta(p)) \to F(p)$.
\end{enumerate}
Now let $w\in \mathfrak{m}$ be an element such that $\{x_{\scriptscriptstyle 1},...,x_{\scriptscriptstyle d-1},w\}$ span $\mathfrak{m}/\mathfrak{m}^2$ as a $F(p)$-vector space. Let $c$ be a primitive element of $F(p)/F$ and $h$ be its minimal polynomial. Choose $\hat{y}\in A$ such that
$$\hat{y} \equiv c \mod \mathfrak{m}.$$ Since $c$ is separable over $F$, $h'(c)\neq 0$. Thus $h'(\hat{y}) \notin \mathfrak{m}$ or equivalently $h'(\hat{y})$ is a unit in the ring $A/\mathfrak{m}^2$.
Choose $\epsilon\in \mathfrak{m}$ such that
$$ \epsilon \equiv \frac{w-h(\hat{y})}{h'(\hat{y})} \ \syn{mod} \ \mathfrak{m}^2.$$
Thus the $F(p)$-span of $\{x_{\scriptscriptstyle 1},...,x_{\scriptscriptstyle d-1},h(\hat{y})+\epsilon h'(\hat{y})\}$ is $\mathfrak{m}/\mathfrak{m}^2$.
Let
$${y}=\hat{y}+\epsilon.$$
We note that $$h({y})=h(\hat{y}+\epsilon) \equiv h(\hat{y})+\epsilon h'(\hat{y}) \mod
\mathfrak{m}^2.$$
Hence $\{x_1,\ldots,x_{d-1},h(y)\}$ gives a $F(p)$-basis for $\mathfrak{m}/\mathfrak{m}^2$.
Now let $\eta$ be the map $(x_1,\ldots,x_{d-1},{y}):X \xrightarrow{} {\mathbb A}^d_F$. Since $y \ \syn{mod} \ \mathfrak{m}$ is a primitive element of $F(p)$, one observes that $F(\eta(p)) \to F(p)$ is an isomorphism. It remains to show that $\eta$ is \'{e}tale at $p$. The maximal ideal of $\eta(p)$ in $F[X_1,...,X_d]$ is $\mathfrak{n}=(X_1,\ldots,X_{d-1},h(X_d))$.
As $\{x_1,\ldots,x_{d-1},h(y)\}$ is a $F(p)$-basis for $\mathfrak{m}/\mathfrak{m}^2$, {that $\eta$ is \'{e}tale at $p$} follows from the surjectivity of
$$\mathfrak{n}/\mathfrak{n}^2\xrightarrow{\eta^*}\mathfrak{m}/\mathfrak{m}^2 .$$
\noindent$\Step {2}$:
Let $U$ be an open neighbourhood of $p$ in $X$ such that $\eta_{\scriptscriptstyle{|U}}$ is \'{e}tale. Let $$ B= (X\setminus U) \ \sqcup Z.$$
In this step we modify $x_1,\ldots,x_{d-1}$ to $z_1,\ldots,z_{d-1}$ so that
\begin{enumerate}
\item The map $\tilde{\eta}=(z_1,\ldots,z_{d-1},{y}):X \rightarrow {\mathbb A}^d_F$ is \'{e}tale on $U$.
\item The set $\{z_1,\ldots,z_{d-1},h({y})\}$ is a $F(p)$ basis for $\mathfrak{m}/\mathfrak{m}^2$.
\item The map $B \xrightarrow{(z_1,\ldots,z_{d-1})}{\mathbb A}^{d-1}_F$ is finite.
\end{enumerate}
Let $\tilde{A}:= A/I(B)$ and $\tilde{\mathfrak{m}}$ denote the image of $\mathfrak{m}$ in $\tilde{A}$. For any element $\alpha \in A$, let $\tilde{\alpha}$ denote its image in $\tilde{A}$. Choose $y_1,...,y_m \in A$ which generate $A$ as an $F$ algebra. We expand this generating set to include the $x_i$'s. In particular
\begin{align*}
A & = F[x_{\scriptscriptstyle 1},...,x_{\scriptscriptstyle d-1},y_{\scriptscriptstyle 1},...,y_{\scriptscriptstyle m}],\\
\tilde{A} & = F[\tilde{x}_{\scriptscriptstyle 1},...,\tilde{x}_{\scriptscriptstyle d-1},\tilde{y}_{\scriptscriptstyle 1},...,\tilde{y}_{\scriptscriptstyle m}].
\end{align*}
The image of $y_{\scriptscriptstyle i}$ in $A/\mathfrak{m}$ satisfies a non-constant monic polynomial, say $f_{\scriptscriptstyle i}$, over $F$. Let
\begin{align*}
y_{\scriptscriptstyle i,0} & \ :=\ f_i(y_{\scriptscriptstyle i}) \in \mathfrak{m}.\\
x_{\scriptscriptstyle i,0} & \ := \ x_{\scriptscriptstyle i} \\
A_{\scriptscriptstyle 0} & \ := \ F[x_{\scriptscriptstyle 1,0},..,x_{\scriptscriptstyle d-1,0},y_{\scriptscriptstyle 1,0},...,y_{\scriptscriptstyle m,0}] \\
\tilde{A}_{\scriptscriptstyle 0} & \ := \ F[\tilde{x}_{\scriptscriptstyle 1,0},..,\tilde{x}_{\scriptscriptstyle d-1,0},\tilde{y}_{\scriptscriptstyle 1,0},...,\tilde{y}_{\scriptscriptstyle m,0}]
\end{align*}
Clearly, $\tilde{A}$ is finite over $\tilde{A}_{\scriptscriptstyle 0}$. \\
For $0 \leq r \leq m-1$, we inductively define $A_{\scriptscriptstyle r+1}$ and elements $x_{\scriptscriptstyle i,r+1}, y_{\scriptscriptstyle i, r+1}$ as follows. By \ref{nntrick1}, we choose an integer $\ell_{\scriptscriptstyle r}>1$ such that the following definitions make $\tilde{A}_{\scriptscriptstyle r}$ a finite $\tilde{A}_{\scriptscriptstyle r+1}$-algebra. Since any sufficiently large choice of $\ell_r$ works, we assume that $\ell_r$ is a multiple of the ${\rm char}(F)$. Let
\begin{align*}
x_{\scriptscriptstyle i,r+1}:= & \ \ x_{\scriptscriptstyle i,r}-(y_{\scriptscriptstyle m-r,r})^{\ell_{\scriptscriptstyle r}^i} & \ \forall \ 1\leq i \leq d-1\\
y_{\scriptscriptstyle i,r+1}:= & \ \ y_{\scriptscriptstyle i,r}-(y_{\scriptscriptstyle m-r,r})^{\ell_{\scriptscriptstyle r}^{d-1+i}} & \ \forall \ 1\leq i \leq m-r-1 \\
A_{\scriptscriptstyle r+1} := & \ \ F[x_{\scriptscriptstyle 1,r+1},..,x_{\scriptscriptstyle d-1,r+1},y_{\scriptscriptstyle 1,r+1},...,y_{\scriptscriptstyle m-r-1,r+1}] \\
\tilde{A}_{\scriptscriptstyle r+1} := & \ \ F[\tilde{x}_{\scriptscriptstyle 1,r+1},..,\tilde{x}_{\scriptscriptstyle d-1,r+1},\tilde{y}_{\scriptscriptstyle 1,r+1},...,\tilde{y}_{\scriptscriptstyle m-r-1,r+1}]
\end{align*}
Since, $x_{\scriptscriptstyle i,0}$ and $y_{\scriptscriptstyle i,0}$ belong to $ \mathfrak{m}$, inductively one can observe
\begin{align*}
& y_{\scriptscriptstyle i,r} \in \mathfrak{m} \\
& x_{\scriptscriptstyle i,r} \in \mathfrak{m} \\
& x_{\scriptscriptstyle i,r+1} \equiv x_{\scriptscriptstyle i,r} \ {\rm mod} \ \mathfrak{m}^2
\end{align*}
For ease of notation, let us rename
$$ z_{\scriptscriptstyle i}:= x_{\scriptscriptstyle i,m}.$$
Note that for all $i\leq d-1$, $ z_i- x_i$ is of the form $\beta_i^{k_i}$ for $\beta_i\in \mathfrak{m}$ and an integer $k_i$ divisible by ${\rm char}(F)$. This ensures requirements (1) and (2) of Step 2.
Recall that $m$ is an integer such that $\{y_1,...,y_m\}$ are the chosen generators of $A$ as an $F$ algebra.
It is now straightforward to see that $\{z_{\scriptscriptstyle 1},...,z_{\scriptscriptstyle d-1}\}\subset \mathfrak{m}$ such that $\tilde{A}$ is a finite algebra over $\tilde{A}_{\scriptscriptstyle m}=F[\tilde{z}_{\scriptscriptstyle 1},...,\tilde{z}_{\scriptscriptstyle d-1}]$. \\
\noindent $\Step{3}$: In this step we will further modify $y$ while ensuring that $(1)$ and $(2)$ of the above step continue to hold. Since the map $\tilde{\eta}_{\scriptscriptstyle |B}:B\to {\mathbb A}^{d-1}_F$ is finite, there exists finitely many points $\{p,p_1,...,p_t\}\subset B$ which are contained in the zero locus $\syn{Z}(z_{\scriptscriptstyle 1},...,z_{\scriptscriptstyle d-1})$. Let $\mathfrak{m}_i$ be the maximal ideal corresponding to $p_i$ for $1\leq i \leq t$.
By Chinese remainder theorem, choose $\delta \in A$ such that
\begin{align*}
\delta \equiv 0 & \ \syn{mod}\ \mathfrak{m} & \\
\delta^{\scriptscriptstyle {\rm char}(F)} \equiv {-y} & \ \syn{mod} \ \mathfrak{m}_i \ \ \forall \ 1 \leq i \leq t & ...(\text{note that } A/\mathfrak{m}_i \ \text{is perfect})
\end{align*}
Let $$ z= y+\delta^{\scriptscriptstyle {{\rm char}( F)}}.$$
For later use, we note that
$$ z\equiv 0 \ \syn{mod} \ \mathfrak{m}_i \ \ \ \forall \ 1\leq i \leq t.$$
Using the fact that $z-y$ is ${\rm char}(F)$-th power of an element of $\mathfrak{m}$, it is straightforward to deduce the following from (1) and (2) of the above step.
\begin{enumerate}
\item The map $\varphi: X\to {\mathbb A}^d_F$ defined by $(z_{\scriptscriptstyle 1},...,z_{\scriptscriptstyle d-1},z)$ is \'{e}tale at $p$.
\item $\{z_{\scriptscriptstyle 1},...,z_{\scriptscriptstyle d-1},h(z)\}$ is an $F(p)$-basis of $\mathfrak{m}/\mathfrak{m}^2$.
\item $z \ \syn{mod}\ \mathfrak{m}$ is the primitive element $c$ of $F(p)/F$.
\end{enumerate}
We further claim that we have the following equality of ideals of $\tilde{A}=A/(I(B))$ :
$$\sqrt{\left(\tilde{z}_{\scriptscriptstyle 1},...,\tilde{z}_{\scriptscriptstyle d-1},h(\tilde{z})\right)} = \tilde{\mathfrak{m}}.$$ To see the claim, we first observe
\begin{align*}
h(z) & \in \tilde{\mathfrak{m}} \\
h(z) & \notin \tilde{\mathfrak{m}}_i \ \ \ \forall \ 1\leq i \leq t.
\end{align*}
The first containment follows as $h$ is the irreducible polynomial of $z \ \syn{mod} \ \mathfrak{m}$.
Moreover, since $h(0)\neq 0$, the second statement follows from the fact that $z\equiv 0 \ \syn{mod} \ \mathfrak{m}_i.$
As $\{\tilde{\mathfrak{m}},\tilde{\mathfrak{m}}_1,...,\tilde{\mathfrak{m}}_t\}$ are the only prime ideals of $\tilde{A}$ containing the ideal $(\tilde{z}_{\scriptscriptstyle 1},...,\tilde{z}_{\scriptscriptstyle d-1})$, and $h(z)\notin \mathfrak{m}_i \ \forall \ i$, we conclude that $\tilde{\mathfrak{m}}$ is the unique prime ideal of $\tilde{A}$ containing the ideal
$\left(\tilde{z}_{\scriptscriptstyle 1},...,\tilde{z}_{\scriptscriptstyle d-1},h(\tilde{z})\right)$. Therefore
$$\sqrt{\left(\tilde{z}_{\scriptscriptstyle 1},...,\tilde{z}_{\scriptscriptstyle d-1},h(\tilde{z})\right)} = \tilde{\mathfrak{m}}.$$
\noindent $\Step{4}$: We claim that in fact $$\big(\tilde{z}_{\scriptscriptstyle 1},...,\tilde{z}_{\scriptscriptstyle d-1},h(\tilde{z})\big) = \tilde{\mathfrak{m}}.$$
Note that both are $\tilde{\mathfrak{m}}$-primary ideals and hence it is enough to show the equality in the localization $\tilde{A}_{\tilde{\mathfrak{m}}}.$ But the equality holds in this local ring by Nakayama's Lemma since it holds modulo $\tilde{\mathfrak{m}}^2$ as $\{ z_{\scriptscriptstyle 1},...,z_{\scriptscriptstyle d-1},h(z)\}$ $\syn{mod} \ \mathfrak{m}^2$ gives a basis of $\mathfrak{m}/\mathfrak{m}^2$ (see condition (2) of the the above Step). \\
\noindent $\Step{5}$: Recall that $\varphi: X\to {\mathbb A}^d_F$ is the map defined by $(z_{\scriptscriptstyle 1},...,z_{\scriptscriptstyle d-1},z)$. We claim that $p$ is the unique point in $\varphi^{-1}\varphi(p)\cap Z$. In fact we have that $p$ is the unique point of $\varphi^{-1}\varphi(p)\cap B$. This is a direct consequence of Step 3, since the ideal defining $\varphi^{-1}\varphi(p)\cap B$ in $B={\rm Spec \,}(\tilde{A})$ is equal to $(\tilde{z}_{\scriptscriptstyle 1},...,\tilde{z}_{\scriptscriptstyle d-1},h(\tilde{z}))=\tilde{\mathfrak m}$. Indeed, what we have observed is that the scheme $\varphi^{-1}\varphi(p)\cap B$ is reduced and has $p$ as the only underlying point. Thus the same holds for $\varphi^{-1}\varphi(p)\cap Z$. If $\mathfrak{n}$ denotes the maximal ideal in the coordinate ring of ${\mathbb A}^d_F$ of the point $\varphi(p)$, then $\mathfrak{n}\overline{A} = \overline{\mathfrak{m}}$. Recall that $\overline{A}:=A/(f)$ and $Z={\rm Spec \,}(\overline{A})$. \\
\noindent $\Step{6}$: In this step we prove the rest of the theorem using a trick used in the proof of \cite[3.5.1]{chk}.
In fact, the argument in this step has been directly taken from {\it loc. cit.} The map $\varphi:Z\to {\mathbb A}^d_F$ is finite. Let $\mathfrak{n}$ be the maximal ideal of $\varphi(p)$ in $F[X_1,...,X_d]$. By Step 5, $\mathfrak{n}\overline{A}=\overline{\mathfrak{m}}$ and the map
$$ \frac{F[X_1,...,X_d]}{\mathfrak{n}} \xrightarrow{\varphi^*} \frac{\overline{A}}{\mathfrak{n}{\overline{A}}} $$ is an isomorphism, in particular surjective. By Nakayama's lemma, there exists a $g\in F[X_1,...,X_d]\backslash \mathfrak{n}$ such that the map
$$ F[X_1,...,X_d]_g \to \overline{A}_g $$ is surjective. In particular, if $V={\mathbb A}^d_F\backslash \syn{Z}(g)$, then
$$Z \cap \varphi^{-1}(V) \to V$$ is a closed immersion. Note that $V$ is an open neighbourhood of $\varphi(p)$. \\
\noindent Let $D\subset X$ be the maximal closed subset on which the map $\varphi$ is not \'{e}tale. Clearly $p\notin D$. Also, since $D$ is a subset of $B$ (see Step 2) and $p$ is the only point in $\varphi^{-1}\varphi(p)\cap B$, we must have $\varphi(p)\notin \varphi(D)$. However, the map $\varphi_{|B}$ is finite, we have that $\varphi(D)$ is a closed subset of ${\mathbb A}^d_F$. Let
$$ W:= \big( {\mathbb A}^d_F\backslash \varphi(D) \big) \cap \big( {\mathbb A}^d_F\backslash \syn{Z}(g)\big). $$
Thus $\varphi^{-1}(W)\to W$ is \'{e}tale. Moreover, $\varphi^{-1}(W)$ is an open neighbourhood of $p$. It is now clear that $\varphi$ and $W$ satisfy conditions (1) and (2) of the Lemma. Condition (3) is also immediate since the map $\varphi$ is defined by $(z_{\scriptscriptstyle 1},...,z_{\scriptscriptstyle d-1},z)$ and $z_{\scriptscriptstyle i}$ vanish on $p$ for $1\leq i\leq d-1$.
\end{proof}
\section{Reduction to open subsets of ${\mathbb A}^2_F$} \label{inductiond}
The previous section reduces the general case of Theorem \ref{gabberfinite} to the case where $X$ is an open subset of ${\mathbb A}^d_F$. The goal of this section is to further reduce to the case where $d=2$ (see Lemma \ref{dto2}). This reduction, which is an induction argument, is an important step in the proof of Theorem \ref{gabberfinite}. One of the ingredients required for this induction argument to work is the following variation of the standard Noether normalization trick (see \eqref{nntrick1}).
\begin{lemma}\label{nntrick} Let $n\geq 2$ be any integer, $k$ be any field and
$Z/k$ be an affine variety of dimension $n-1$. Let $$Z\xrightarrow{(\phi_1,...,\phi_n)}{\mathbb A}^{n}_k$$ be a finite map. Let $Q \in k[\phi_{n}]$ be a non constant monic polynomial. Then for an integer $\ell>>0$, the map
$$ Z\xrightarrow{(\phi_{\scriptscriptstyle 1}-Q^\ell_{\scriptscriptstyle 1},\ldots,\phi_{\scriptscriptstyle n-1}-{Q^\ell_{\scriptscriptstyle n-1}})} {\mathbb A}^{n-1}_k$$ is finite, where $Q_i$'s are inductively defined by
$$ Q_{n-1}:= Q.$$
$$ Q_{i}:= \phi_{i+1}-Q_{i+1}^\ell \ \ \ \forall \ 1\leq i\leq n-2.$$
\end{lemma}
\begin{proof}The proof is similar to that of \ref{nntrick1} (see \cite[page 2]{mum}) and hence we only give a sketch. Since $\dim(Z)= n-1$, $\phi_1, \phi_2,\ldots,\phi_n$ cannot be algebraically independent. Thus there exists a non-zero polynomial $f \in k[Y_1,...,Y_n]$ such that $f(\phi_1, \phi_2,\ldots,\phi_n)=0$.
Let $\ell$ be any integer greater than $n\syn{deg}(f)$ where $\syn{deg}(f)$ is the total degree of $f$. Let $\tilde{Q}\in k[Y_n]$ be a polynomial such that $Q=\tilde{Q}(\phi_n)$. Inductively define $\tilde{Q}_i$ for $1\leq i \leq n-1$ as follows:
\begin{align*}
& \tilde{Q}_{n-1} : = \tilde{Q} \\
& \tilde{Q}_i := Y_{i+1} - \tilde{Q}_{i+1}^\ell & \hspace{-10cm} \forall \ 1 \leq i \ \leq n-2.
\end{align*}
Notice that the polynomials $\tilde{Q}_i$'s are defined such that
$$ \tilde{Q}_i(\phi_1,...,\phi_n) = Q_i.$$
Moreover, we note that if $d=Y_n$-degree of $\tilde{Q}$ then each $\tilde{Q}_i$ is monic in $Y_n$ of degree ${\ell}^{n-i-1}d$.
Consider the elements $Z_1,...,Z_{n-1}\in k[Y_1,...,Y_n]$ defined as follows:
$$ Z_i := Y_i - \tilde{Q}_i^\ell \ \ \ \forall 1 \leq i \leq n-1.$$
We leave it to the reader to check that
$$ k[Z_1,...,Z_{n-1},Y_n] = k[Y_1,...,Y_n].$$ For future reference, we note that the map $$\eta:{\mathbb A}^n_F\xrightarrow {(Y_1-\tilde{Q}_1^\ell,\ldots,Y_{n-1}-\tilde{Q}_{n-1}^\ell,Y_n)}{\mathbb A}^n_F $$ is an automorphism.
It is enough to show that the polynomial $f$, expressed in the variables $Z_1,...,Z_{n-1},Y_n$ is monic in $Y_n$. Let us write $f$ as $$f=\Sigma_{I=(i_1,\ldots i_n)} \alpha_I \cdot m_{I}$$ where $m_I$'s are monomials in $Y_1,...,Y_n$ and $\alpha_I \in k$.
We leave it to the reader to verify that when expressed in new coordinates $Z_1,\ldots Z_{n-1},Y_n$, each monomial $m_I$ becomes a polynomial which is monic in $Y_n$ of $Y_n$-degree equal to $i_n+\Sigma_{k=1}^{n-1} i_k\cdot {\ell}^{n-k}\cdot d$. Since $\ell > n \syn{deg}(f)$, one can show that these $Y_n$-degrees are distinct. Thus in the coordinates $Z_1,...,Z_{n-1},Y_n$, $f$ remains monic in $Y_n$.
\end{proof}
\begin{notation} \label{not:openad}
Let $d\geq 2$ be an integer and $f,g\in F[X_1,...,X_d]$ be nonzero polynomials with no common irreducible factors (see Remark \ref{fgfactors}). Let $X:={\mathbb A}^d_F\backslash \syn{Z}(g)$ and $Z:=\syn{Z}(f)\cap X$. Let $p\in Z$ be a closed point (see Remark \ref{simplify}) whose first $d-1$ coordinates are $0$.
\end{notation}
Recall that by \ref{redopen} it is enough to prove Theorem \ref{gabberfinite} for $(X,Z,p)$ as above. In order to prove this, we have to first come up with a map from $\Phi:X \to {\mathbb A}^d_F$. Indeed, we will look for maps $\Phi$ which are defined on the whole of ${\mathbb A}^d_F$. In other words, we will look for suitable polynomials $\{\phi_1,...,\phi_d\} \subset F[X_1,...,X_d]$. The goal of the following definition is to list necessary conditions on these polynomials which will ensure (see Lemma \ref{nakarg}) that the resulting map $\Phi$ is as desired in \eqref{gabberfinite}.
\begin{definition} \label{defpresents} Let $f,g, X, Z, p$ be as in Notation \ref{not:openad}.
For $\{\phi_1,...,\phi_d\}\subset F[X_1,...,X_d]$, let
\begin{enumerate}
\item[(i)] $\Phi:{\mathbb A}^d_F\xrightarrow{(\phi_1,...,\phi_d)} {\mathbb A}^d_F$.
\item[(ii)] $\Psi:{\mathbb A}^d_F\xrightarrow{(\phi_1,...,\phi_{d-1})} {\mathbb A}^{d-1}_F$.
\end{enumerate}
We say that $(\phi_1,...,\phi_d)$ presents $(X,\syn{Z}(f),p)$ if
\begin{enumerate}
\item $\Psi_{|\syn{Z}(f)}$ is finite and $\Psi(p)= (0,...,0)$.
\item $\Psi^{-1}\Psi(p)\cap \syn{Z}(f) \subset Z$
\item $\Phi$ is \'{e}tale at $S:=\Psi^{-1}\Psi(p)\cap Z$.
\item $\Phi$ is radicial at $S$.
\end{enumerate}
\end{definition}
Recall that $\Phi$ is said to be {\em radicial} \cite[Tag 01S2]{stacks} if $\Phi_{|S}$ is injective and for all $x \in S$ the residue field extension $F(x)/F(\Phi( x))$ is trivial.\\
The following lemma shows that in order to prove Theorem \ref{gabberfinite} for $X,Z,p$ as in Notation \ref{not:openad}, it is enough to find $\phi_1,...,\phi_d$ which presents $(X,\syn{Z}(f),p)$.
\begin{lemma}\label{nakarg} Let $X,Z,p$ be as above. Assume there exists $\{\phi_1,...,\phi_d\}$ which presents
$(X,\syn{Z}(f),p)$ and $\Phi,\Psi$ be as in Definition \ref{defpresents}. Then there exist open neighborhoods $V\subset {\mathbb A}^{d-1}_F$ of $\Psi(p)$ and $U\subset X$ of $p$, such that $\Phi_{|U},\Psi_{|U},U, V$ satisfy conditions (1),(2),(3) of Theorem \ref{gabberfinite}. Moreover, $\Psi^{-1}(V) \cap \syn{Z}(f)\subset U$.
\end{lemma}
\begin{proof} The argument here is directly taken from \cite[3.5.1]{chk}. We construct an open neighbourhood $V$ of $\Psi(p)$ in ${\mathbb A}^{d-1}_F$, such that
if $Z_V:=\Psi^{-1}(V) \cap \syn{Z}(f)$ then
\begin{enumerate}
\item[(i)] $Z_V \subset Z$
\item[(ii)] $\Phi$ is \'{e}tale at all points in $Z_V$
\item[(iii)] $\Phi|_{Z_V}: Z_V \rightarrow {\mathbb A}^ 1_V $ is closed immersion
\end{enumerate}
Let $B$ be the smallest closed subset of $\syn{Z}(f)$ containing all points of $\syn{Z}(f)$ at which $\Phi$ is not \'{e}tale and also containing $\syn{Z}(f)\backslash Z$. Since $\Psi_{|\syn{Z}(f)}$ is a finite map, $\Psi(B)$ is closed in ${\mathbb A}^{d-1}_F$. Moreover, because of conditions $(2)$ and $(3)$ of Definition \ref{defpresents}, we have $\Psi(p) \notin \Psi(B)$. Thus, we can choose affine open subset $W \subset {\mathbb A}^{d-1}_F$ such that $\Psi(p) \in W \subset {\mathbb A}^{d-1}_F \backslash \Psi(B)$. Let $Z_W= Z\cap \Psi^{-1}(W)$. We have following commutative diagram of affine schemes and consequently their coordinate rings.
\begin{center}
\begin{minipage}{.4\textwidth}
\begin{tikzpicture}
\node (E) at (0,0){$Z_W$};
\node at (2,1) (F) {${\mathbb A}^1_W$};
\node at (2,-1) (A) {$W$};
\draw[->] (E)--(F) node [midway,above] {$\Phi$};
\draw[->] (F)--(A) node [midway,right] {$\pi$} ;
\draw[->] (E)--(A) node [midway, left,below] {$\Psi$} ;
\end{tikzpicture}
\end{minipage}
\begin{minipage}{.2\textwidth}
\begin{tikzpicture}
\node (E) at (0,0){$ F[Z_W]$};
\node at (2,1) (F) {$F[{\mathbb A}^1_W]$};
\node at (2,-1) (A) {$F[W]$};
\draw[->] (F)--(E) node [midway,above] {$\Phi^{*}$};
\draw[->] (A)--(F) node [midway,right] {} ;
\draw[->] (A)--(E) node [midway, left,below] {$\Psi^{*}$} ;
\end{tikzpicture}
\end{minipage}
\end{center}
Let $\Psi(p)=q$ and ${\mathfrak m}_q$ be the maximal ideal in $F[W]$ corresponding to $q$. Thus the ideal corresponding to $S=\Psi^{-1}(q)\cap Z$ in $F[Z_W]$ is ${\mathfrak m}_q\cdot F[Z_W]$. Since $\Phi$ is radicial as well as \'{e}tale at $S$, $$\Phi_{|S}:S\hookrightarrow {\mathbb A}^1_W$$ is a closed immersion. Thus the map on the coordinate rings
$$F[{\mathbb A}^1_W] \twoheadrightarrow \frac{F[Z_W]}{{\mathfrak m}_q F[Z_W]}$$
is surjective. The surjectivity of the above map is equivalent to
$$ C\otimes_{F[W]} \frac{F[W]}{{\mathfrak m}_q}=0 $$
where $$C:={\rm Coker}\big(F[{\mathbb A}^1_W] \rightarrow F[Z_W]\big).$$
But $C$ is a finite $F[W]$ module. Hence by Nakayama's lemma $C_{{\mathfrak m}_q}=0$. Thus there exists $h \in F[W]\backslash {\mathfrak m}_q$ such that $C_h=0$ or equivalently
$$ {F[{\mathbb A}^1_W]}_h \twoheadrightarrow {F[Z_W]}_h $$
is surjective.
Let $V := W\backslash \syn{Z}(h)$. The coordinate ring of $Z_V:=\Psi^{-1}(V)\cap \syn{Z}(f)$ is $F[Z_W]_h$ and that of $\pi^{-1}V$ is $F[{\mathbb A}^1_W]_h$. Thus the surjectivity of the above map implies that
$$ Z_V \hookrightarrow {\mathbb A}^1_V$$ is a closed immersion as required. \\
Let $U\subset X$ be the maximal open subset containing points at which $\Phi$ is \'{e}tale. To finish the proof, we need to show that $U,V, \Phi_{|U}, \Psi_{|U}$ satisfy conditions $(1),(2),(3)$ of Theorem \ref{gabberfinite}. (1) is clearly satisfied by the definition of $U$. To see (2), note that $\Psi_{|Z(f)}$ is finite, and hence, as $Z_V= \Psi^{-1}(V) \cap Z(f)$, $\Psi_{|Z_V}:Z_V \to V$ is finite. (3) is precisely the condition (iii) mentioned at the beginning of the proof. By the construction of $W$, subsequently $V$, it follows that $\Psi^{-1}(V) \cap \syn{Z}(f) \subset U$.
\end{proof}
\begin{remark}\label{fgfactors} Since our main goal is to prove Theorem \ref{gabberfinite} for $(X,Z,p)$, we may change $\syn{Z}(f)$ as long as it does not change $Z$. If $f$ and $g$ have common irreducible factors, dividing $f$ by the g.c.d. of $f$ and $g$ does not change $\syn{Z}(f) \backslash \syn{Z}(g)$. This justifies our assumption in Definition \ref{not:openad} that $f,g$ have no common irreducible factors.
\end{remark}
The following lemma is proved using a simple coordinate change argument. It will be used in the proof of \eqref{dto2}, which is the main result of this section.
\begin{lemma}\label{normalizev}
Let $(\phi_1,...,\phi_d)$ present $(X,\syn{Z}(f),p)$. as in Lemma \ref{nakarg}. Then there exist $(\tilde{\phi}_1, ... ,\tilde{\phi}_d)$ which present $(X,\syn{Z}(f),p)$ such that there exists an open subset $V \subset {\mathbb A}^{d-1}_F$ satisfying the conclusion of Lemma \ref{nakarg} for $(\tilde{\phi}_1, ... ,\tilde{\phi}_d)$ and which satisfies the following additional condition:
$$ \dim\bigg( \syn{Z}(\tilde{\phi}_1, ... ,\tilde{\phi}_{d-2}) \cap \Psi^{-1}_{|Z(f)}\big( {\mathbb A}^{d-1}_F \backslash V \big)\bigg)=0.$$
\end{lemma}
\begin{proof} We note that if $d=2$, by convention,
$$\syn{Z}(\tilde{\phi}_1, ... ,\tilde{\phi}_{d-2}) \cap {\Psi^{-1}_{|Z(f)}}({\mathbb A}^{1}_F \backslash V) = {\Psi^{-1}_{|Z(f)}}({\mathbb A}^1_F\backslash V) $$
which is of zero dimension since $V$ is non-empty. Thus we may assume $d\geq 3$.
For an integer $\ell$, consider the automorphism $\rho: {\mathbb A}^{d-1}_F\to {\mathbb A}^{d-1}_F$ induced by
$$ (X_1,...,X_{d-1}) \mapsto (X_1-X_{d-1}^{\ell^{(d-1)-1}}, X_2-X_{d-1}^{\ell^{(d-1)-2}}, ..., X_{d-2}-X_{d-1}^{\ell^{1}}, X_{d-1}). $$
We choose $\ell >>0$, such that by \eqref{nntrick1}, $(X_1,...,X_{d-2})_{| \rho({\mathbb A}^{d-1}\backslash V)}$ is a finite map.
Let
\begin{align*}
\tilde{\phi}_i & : = {\phi}_i - \tilde{\phi}_{d-1}^{\ell^{d-1-i}} \ \ \ \ \text{for } \ i\leq d-2 \\
\tilde{\phi}_i & := \phi_i \ \ \ \ \ \ \text{for } i=d-1,d
\end{align*}
It is then straightforward to check that $(\tilde{\phi}_1,...,\tilde{\phi}_d)$ presents $(X,\syn{Z}(f),p)$ (since it is obtained by a coordinate change from the original $\phi_i$'s) and moreover
$$ \dim\bigg( \syn{Z}(\tilde{\phi}_1, ... ,\tilde{\phi}_{d-2}) \cap {\Psi^{-1}_{|Z(f)}} ({\mathbb A}^{d-1}_F \backslash \rho(V)) \bigg)=0.$$
\end{proof}
\begin{lemma}\label{fgcommon} Let $d\geq 3$, and $f,g\in F[X_1,...,X_d]$ be two non-zero polynomials with no common factors. Let $p$ be a closed point of ${\mathbb A}^d_F$ such that $X_i(p)=0$ for all $i\leq d-1$. Then there exists a coordinate change of $F[X_1,...,X_d]$, i.e. elements $Y_i\in F[X_1,...,X_d]$ with $$ F[X_1,...,X_d] = F[Y_1,...,Y_d]$$
such that $f(0,Y_2,...,Y_d)$ and $g(0,Y_2,...,Y_d)$ are nonzero polynomials with no common irreducible factors and $Y_i(p)=0$ for all $i\leq d-1$.
\end{lemma}
\begin{proof}
The condition that $f(0,Y_2,...,Y_d)$ and $g(0,Y_2,...,Y_d)$ are nonzero polynomials with no common irreducible factors is equivalent to the condition that no irreducible component of $\syn{Z}(f)\cap \syn{Z}(g)$ is contained in $\syn{Z}(Y_1)$.
By Noether normalization trick \ref{nntrick1}, we may assume, by a suitable coordinate change, that the projection
$$ (X_2,...,X_{d}) : \syn{Z}(f)\cap \syn{Z}(g) \xrightarrow{\eta} {\mathbb A}^{d-1}_F$$ is finite.
Note that since $d\geq 3$, the image of every irreducible component of $\syn{Z}(f)\cap \syn{Z}(g)$ under $\eta$ is of dimension at least one. Thus we may choose closed points $z_1,...,z_\tau$, one in each irreducible component of $\syn{Z}(f)\cap \syn{Z}(g)$ such that $\eta(z_i)$ are pairwise distinct and also different from $\eta(p)$. For every closed point $x$ of ${\mathbb A}^d_F$, either $X_1$ or $X_1+1$ is non-vanishing on $x$. Thus for each $z_i$, we choose $\epsilon_i=0$ or $1$, such that $X_1+\epsilon_i$ does not vanish on $z_i$. By Chinese remainder theorem, there exists a polynomial $\gamma\in F[X_2,...,X_{d}]$ such that
$$ \gamma(\eta(z_i))= \epsilon_i \ \ \ \text{and} \ \ \ \gamma(p)=0.$$
It is now straightforward to check that $$Y_1:= X_1-\gamma \ \ \ \text{and} \ \ \ Y_i:=X_i \ \ \forall \ 2\leq i\leq d$$
satisfies our requirement.
\end{proof}
\begin{lemma}\label{dto2} [Reduction to $d=2$] Assume that for $d=2$ and every $f,g,X,Z,p$ as in Notation \ref{not:openad}, there exists $\phi_1,\phi_2\in F[X_1,X_2]$ which presents $(X,\syn{Z}(f),p)$. Then the same holds for every $d\geq 2$.
\end{lemma}
\begin{proof}We prove this lemma by induction on $d$. Assume $d\geq 3$. \\
\noindent \underline{Step 0}:
As before, we let $F[X_1,...,X_d]$ be the coordinate ring of ${\mathbb A}^d_F$.
Let $\overline{f}(X_2,...,X_d):=f(0,X_2,...,X_d)$ and $\overline{g}(X_2,...,X_d):=g(0,X_2,...,X_d)$. By Lemma \ref{fgcommon}, we may assume that
then $\overline{f}$ and $\overline{g}$ are non-zero and have no common factors.
We let
\begin{enumerate}
\item[-] $\overline{X}:= X\cap \syn{Z}(X_1)$.
\item[-] $\overline{Z}:=Z\cap \overline{X}$.
\end{enumerate}
Note that $p\in \overline{Z}$ and $\overline{X}=\syn{Z}(X_1) \backslash \syn{Z}(g)$ where $\syn{Z}(X_1)\cong {\mathbb A}^{d-1}_F$ with coordinate ring $F[X_2,...,X_d]$. By induction, there exist $\{\overline{\phi}_2,...,\overline{\phi}_d\}\subset F[X_2,...,X_d]$ which presents $(\overline{X},\syn{Z}(\overline{f}),p)$. Let
$$\overline{\Phi}:=(\overline{\phi}_2,...,\overline{\phi}_d) \ \ \text{and} \ \ \overline{\Psi}:=(\overline{\phi}_2,...,\overline{\phi}_{d-1}).$$
By Lemma \ref{nakarg}, there exist neighbourhoods $\overline{V}\subset {\mathbb A}^{d-1}_F$ and $\overline{U}\subset \overline{X}$ of $\overline{\Psi}(p)$ and $p$ respectively such that if
$$ \overline{Z}_{\overline{V}}:= \overline{Z}\cap \overline{\Psi}^{-1}(\overline{V})$$
then the following conditions of Theorem \ref{gabberfinite}
\begin{enumerate}
\item $\overline{\Phi}_{|\overline{U}}$ is \'{e}tale
\item $\overline{\Psi}_{|\overline{Z}_{\overline{V}}} : \overline{Z}_{\overline{V}} \to \overline{V}$ is finite
\item $\overline{\Phi}_{|\overline{Z}_{\overline{V}} } : \overline{Z}_{\overline{V}} \to {\mathbb A}^1_{\overline{V}}$ is a closed immersion
\end{enumerate}
{are satisfied.}\\
Further, by Lemma \ref{normalizev}, we also assume (without loss of generality) that if $E$ is the closed subset of ${Z(\bar{f})}$ defined by
$$ {E:=Z(\bar{f})\backslash \overline{\Psi}^{-1}(\overline{V})} $$
then
\begin{enumerate}
\item[(4)] $ \dim( E \cap \syn{Z}(\overline{\phi}_2,...,\overline{\phi}_{d-2}) ) = 0.$
\end{enumerate}
Note that (4) is vacuously satisfied unless $d\geq 4$. Indeed for $d = 3$, ${\mathbb A}^{d-2}_{F}\setminus V$ is a finite set, and since $\Psi_{|Z(f)}:Z(\bar{f})\rightarrow {\mathbb A}^{d-2}_F$ is finite, $E$ is thus a
finite set.
\noindent $\Step{1}$: Since $\syn{Z}(\overline{f}) \xrightarrow{(\overline{\phi}_2,...,\overline{\phi}_{d-1})} {\mathbb A}^{d-2}_F$ is finite (see \ref{defpresents}(1)), for $2\leq i \leq d$, the image of $X_i$ in $F[X_2,...,X_d]/(\overline{f})$ satisfies a monic polynomial
$$ P_i(T) := T^{m_i}+a_{m_i-1,i}T^{m_i-1} + \cdots + a_{0,i} $$
where each $a_{i,j} \in F[\overline{\phi}_2,...,\overline{\phi}_{d-1}]$. So $P_i(X_i)$ is zero in $F[X_2,...,X_d]/(\bar{f})$. Note that each $\overline{\phi}_i$ is an element of $F[X_2,...,X_d]$. Thus we have a map of algebras
$$F[\overline{\phi}_2,...,\overline{\phi}_{d-1}][T] \rightarrow F[X_1,...,X_d][T]/(f).$$
We let $\tilde{P}_i(T)$ be the image of the polynomial $P_i(T)$ under this map.
Since $P_i(X_i)$ is zero in $F[X_2,...,X_d]/(\bar{f})$, $\tilde{P}_i(X_i)$ maps to zero via the map
$$ F[X_1,...,X_d]/(f) \xrightarrow{X_1\mapsto 0} F[X_2,...,X_d]/(\overline{f}).$$
Therefore
$$ \tilde{P}_i(X_i) = X_1g_i$$ for some $g_i\in F[X_1,...,X_d]/(f)$. We claim that the map
$$ \syn{Z}(f) \xrightarrow{(\overline{\phi}_2,...,\overline{\phi}_{d},X_1,X_1g_2,...,X_1g_d)} {\mathbb A}^{2d-1}_F $$
is finite. This is clear because for $i\geq 2$, each $X_i$ satisfies the monic polynomial $\tilde{P}_i(T)-X_1g_i$ with coefficients which are polynomial expressions in the functions defining the above map.
Applying \ref{nntrick1} repeatedly to this map (see Remark \ref{q=phin}), we get $\phi_2,...,\phi_d\in F[X_1,...,X_d]$ such that
$$ \phi_i \equiv \overline{\phi}_i \ \ \syn{mod} \ X_1$$
and the map $(\phi_2,...,\phi_{d})_{|\syn{Z}({f})}$ is finite.
\noindent $\Step{2}$: Consider the maps
\begin{enumerate}
\item[] $\widetilde{\Phi} :{\mathbb A}^d_F \xrightarrow{(X_1,\phi_2,...,\phi_d)} {\mathbb A}^{d}_F$
\item[] $\widetilde{\Psi}:{\mathbb A}^d_F \xrightarrow{(X_1,\phi_2,...,\phi_{d-1})} {\mathbb A}^{d-1}_F$.
\end{enumerate}
Note that for all points $x\in \syn{Z}(X_1)$, $\widetilde{\Phi}$ is \'{e}tale at $x$ iff $\syn{Z}(X_1)\xrightarrow{\overline{\phi}_2,...,\overline{\phi}_{d}} {\mathbb A}^{d-1}_F$ is \'{e}tale at $x$.
Let $E$ be the closed subset of $\syn{Z}(\overline{f})\subset \syn{Z}(f)$ defined in Step 0. We have the following:
\begin{enumerate}
\item $\widetilde{\Phi}_{|\syn{Z}(f)}$ is finite. In fact, the map $(\phi_2,...,\phi_d)_{|\syn{Z}(f)}$ is finite.
\item $\widetilde{\Psi}(p) \notin \widetilde{\Psi}(E)$ (this follows from the definition of $E$)
\item $\widetilde{\Phi}$ restricted to $\syn{Z}(\overline{f})\backslash E$ is a locally closed immersion.
\item $\widetilde{\Phi}$ is \'{e}tale at all points in $\syn{Z}(\overline{f})\backslash E$.
\end{enumerate}
By condition (4) of Step 0, $$E\cap \syn{Z}(\overline{\phi}_2,...,\overline{\phi}_{d-2})= E\cap \syn{Z}(\phi_2,...,\phi_{d-2}) \footnote{where by convention $\syn{Z}(\phi_2,...,\phi_{d-3})$ is the whole of ${\mathbb A}^d_F$ if $d\leq 3$.} $$ is finite.
Let $Q$ be any non-constant polynomial expression in $\phi_d$ which vanishes on the finite set
$$\bigg(E\cap \syn{Z}(\phi_2,...,\phi_{d-2})\bigg) \cup \big\{p\big\}.$$
Let $\ell$ be a large enough integer which is divisible by ${\rm char}(F)$. Let $\phi_1=X_1$ and
as in Lemma \ref{nntrick}, let $ Q_{d-1}:= Q$ and
$$ Q_i : = \phi_{i+1}-Q_{i+1}^\ell \ \ \forall \ i\leq d-2 . $$
Let
$$ \Phi:=(\phi_1-Q_1^\ell,\ldots, \phi_{d-1}-Q_{d-1}^\ell,\phi_d): {\mathbb A}^d_F \longrightarrow {\mathbb A}^d_F$$
$$ \Psi:=(\phi_1-Q_1^\ell,\ldots, \phi_{d-1}-Q_{d-1}^\ell): {\mathbb A}^d_F \to {\mathbb A}^{d-1}_F.$$
By Lemma \ref{nntrick} $\Psi_{|\syn{Z}(f)}$ is finite. We let
$S$ be the finite set of points in $\Psi^{-1}\Psi(p)\cap \syn{Z}(f)$.
To finish the proof, it suffices to verify the conditions (2)-(4) of Definition \eqref{defpresents}.
We first note that $S\subset \syn{Z}(\phi_1,...,\phi_{d-2})$. This is because if $x\in S$, then by definition of $S$,
$$ \phi_{i+1}-Q_{i+1}^\ell (x) = Q_{i}(x) = 0 \ \ \ \forall \ \ i\leq d-2.$$
And thus
$$ \phi_i-Q_i^{\ell}(x) = \phi_i(x) = 0 \ \ \ \forall \ \ i\leq d-2.$$
We now show that $S$ is disjoint from $E$. First note that
$S \subset \syn{Z}(\phi_1)=\syn{Z}(X_1)$. Also {$\Psi(p)=0$ since $Q(p)=0$ and $\phi_i(p)=0$ for $1\leq i\leq d-1$}. Let $x\in S\cap E$ if possible. Hence $x$ is necessarily in $E\cap \syn{Z}(\phi_2,...,\phi_{d-2})$ by the above argument. In particular we note that $\phi_{d-2}(x)=0$. Now we claim that
$$ \phi_{d-1}(x)=0.$$
Since $\Psi(x)=0$ we have $(\phi_{d-2}-Q_{d-2}^\ell)(x)=0$. But as $\phi_{d-2}(x)=0$, we conclude that $$ Q_{d-2}(x)=0.$$
Thus $$\phi_{d-1}(x)=(Q_{d-2}-Q^{\ell})(x)=0.$$
This proves the claim.
Consequently, $x\in Z(\phi_2,\ldots\phi_{d-1})$.
By definition of $E$, $x \in E$ implies $\bar{\Psi}(x)\notin \bar{V}$ where $\bar{V}$ is as defined in Step 0. As $\bar{V}$ is a neighborhood of $0=\Psi(p)$, we have $\bar{\Psi}(x) \neq0$. But as $x\in E \subset Z(X_1)$, we have $$ \bar{\Psi}(x)= (\phi_2,...,\phi_{d-1})(x)=(\bar{\phi}_2,...,\bar{\phi}_{d-1})(x).$$ Hence $\phi_i(x)\neq 0 $ for some $ i$ with $2\leq i \leq d-1.$ This is a contradiction to the fact that $x\in Z(\phi_2,\ldots\phi_{d-1})$.
Hence $S$ must be disjoint from $E$. Hence $ \tilde{\Phi}$ is a locally closed immersion on $S$ by property (3) of Step 2.
As in the proof of Lemma \ref{nntrick}, we let $${\mathbb A}^{d-2}_F\xrightarrow{\ \ \eta\ \ } {\mathbb A}^{d-2}_F$$ be the automorphism defined by
$$\eta={(Y_1-\tilde{Q}_1^\ell,\ldots,Y_{d-1}-\tilde{Q}_{d-1}^\ell,Y_d)}$$ where $\tilde{Q}_i\in F[Y_1,...,Y_d]$ are polynomials satisfying $Q_i= \tilde{Q}_i(\phi_1,...,\phi_{d})$.
It is straightforward to check that
$$ \Phi = \eta \circ \tilde{\Phi}. $$
Hence $\Phi$ is a locally closed immersion on $S$, this proves condition (4) of Definition \ref{defpresents}.
From Lemma \ref{nakarg} we have $Z(f) \cap \Psi^{-1}(V) \subset X$. This with the fact that $Z=Z(f)\cap X$ implies conditions (2) of Definition \ref{defpresents}. For checking condition (3), i.e. to check $\Phi$ is \'{e}tale at all points in $S$, we note that
since $\ell$ is divisible by ${\rm char}(F)$, $\Phi$ is \'{e}tale precisely at those points where $\widetilde{\Phi}$ is \'{e}tale. In particular $\Phi$ is \'{e}tale at all points of $\syn{Z}(\overline{f})\backslash E$.
\end{proof}
\section{Open subsets of ${\mathbb A}^2_F$}
In this section, we finish the proof of Theorem \ref{gabberfinite}. By Lemmas \ref{redopen}, \ref{dto2} we only have to deal with the case of open subsets of ${\mathbb A}^2_F$. While the handling of low degree points is similar, in spirit, to that of \cite{poonen}, for high degree points we proceed differently (see Lemma \ref{hexists}).
\begin{lemma}\label{prime}
Let $F$ be a finite field as before, and $C \subset {\mathbb A}^2_F$ be a closed curve such that the projection
onto the $Y$-coordinate $Y_{|C} : C\to {\mathbb A}^1_F$ is finite. Let $C^{(1)}$ denote the set of closed points of $C$. Then the following set of points is dense in $C$
$$ \big\{ x \in C^{(1)} \ | \ \syn{deg}_{{F}}(Y(x)) = \syn{deg}_{{F}}(x) \big\}.$$
\end{lemma}
\begin{proof} Without loss of generality, we may assume $C$ is irreducible and hence we simply have to show that the set $$ \big\{ x \in C^{(1)} \ | \ \syn{deg}_{{F}}(Y(x)) = \syn{deg}_{{F}}(x) \big\}$$ is infinite. Let $x_1,...,x_q$ be the $F$-rational points of ${\mathbb A}^1_F$. Let $C' : = C\backslash Y^{-1}(\{x_1,...,x_q\})$. $C'$ is a dense open subset of $C$ as $Y_{|C}$ is finite. Now, any point $x \in C'$ of prime degree satisfies $\syn{deg}_{{F}}(Y(x))=\syn{deg}_{{F}}(x)$. By Lang-Weil estimates \cite{lw}, for all large enough prime number $\ell$, there is a point $x\in C'^{(1)}$ of degree $\ell$. Hence, since $\ell$ is a prime, we must have $\syn{deg}_{{F}}(Y(x))=\syn{deg}_{{F}}(x)$. This proves the lemma.
\end{proof}
\begin{notation}\label{notationa2} Let
\begin{enumerate}
\item $A =F[X,Y]$ and for $d\geq 0$ let $A_{\scriptscriptstyle \leq d} = \{h\in A\ |\ \syn{deg}(h)\leq d \}$. Here $\syn{deg}(h)$ denotes the total degree.
\item $f,g\in A$ be two non-constant polynomials, with no common irreducible factors. By performing a change of coordinates if necessary, we will assume that $f$ is monic in $X$ of degree $m$.
\item $W:= {\mathbb A}^2_F \backslash \syn{Z}(g)$. In this section, we call our variety $W$ instead of $X$, since the later will denote a coordinate function on ${\mathbb A}^2_F$.
\item $Z:= \syn{Z}(f)\cap W$. Note that $\syn{Z}(f)\backslash Z$ is finite as $f,g$ have no common irreducible components.
\item $p\in Z$ be a closed point such that its $X$-coordinate is $0$. We also choose a set of closed points $\{p_1,...,p_t\}$ in $Z$ such that the set $T:= \{p,p_1,...,p_t\}$ satisfies
\begin{enumerate}
\item $T$ contains at least one point from each irreducible component of $Z$.
\item No two points in $T$ have same degrees and for all $p_i\in T$, $\syn{deg}(Y(p_i))= \syn{deg}(p_i)$. This can be ensured by Lemma \ref{prime}. Note that since $X$-coordinate of $p$ is $0$, we also have $\syn{deg}(Y(p))=\syn{deg}(p)$.
\end{enumerate}
\item Let $D= \{q_1,\ldots, q_s\}$ be a finite set of closed points in $\syn{Z}(f)$ satisfying:
\begin{enumerate}
\item $D$ contains all points in $\syn{Z}(f)\backslash Z$.
\item $D$ contains at least one point from each irreducible component of $\syn{Z}(f)$.
\item $D$ does not contain any point of $\{p,p_1,...,p_t\}$.
\end{enumerate}
\end{enumerate}
Moreover, for a point $x$ in $\syn{Z}(f)$, the notation ${\mathcal O}_x$ (resp. ${\mathfrak m}_x$) will denote ${\mathcal O}_{{\mathbb A}^2_F,x}$ (resp. ${\mathfrak m}_{{\mathbb A}^2_F,x}$) i.e. the local ring (resp. maximal ideal) of $x$ as a point of ${\mathbb A}^2_F$.
\end{notation}
The main result of this section is the following.
\begin{theorem}\label{dtwo}
There exists $(\phi_1,\phi_2)\in F[X,Y]$ which presents $(W,\syn{Z}(f),p).$
\end{theorem}
This is enough to prove Theorem \ref{gabberfinite}.
\begin{proof}[Proof of \ref{gabberfinite}]
This follows from Lemmas \ref{redopen}, \ref{nakarg} and \ref{dto2} and Theorem \ref{dtwo}.
\end{proof}
To prove \ref{dtwo}, we will find $\phi_1$ by using Lemma \ref{phiexists} and $\phi_2$ by Lemma \ref{hexists}. We heavily use the counting techniques by Poonen \cite{poonen} to prove these lemmas.
Recall from \ref{notation}, for $Y$ a subset of a scheme $X/F$, $Y_{ \scriptscriptstyle \leq r}:= \{x\in Y\ |\ \syn{deg}(x) \leq r\}$.
\begin{lemma}\label{phiexists}
Let the notation be as in \ref{notationa2}. There exists $c\in {\mathbb N}$, such that for every $d>>0$, there exists a $\phi \in A_{\scriptscriptstyle \leq d}$ satisfying
\begin{enumerate}
\item $\phi(p)=\phi(p_i)=0$ for all $i=1,...,t$ and $ \phi(q_i) \neq 0 $ for all $i=1,\ldots, s$.
\item $ (\phi,Y)$ is \'{e}tale at all $x\in S$ where $S:=\syn{Z}(\phi) \cap Z$.
\item The projection $Y:{\mathbb A}^2_F \to {\mathbb A}^1_F$ is radicial at $S_{\scriptscriptstyle \leq {(d-c)}/{3}}$.
\end{enumerate}
\end{lemma}
\begin{remark}\label{strategy} The above lemma is motivated by writing down conditions for $\phi$ such that $(\phi,Y)$ presents $(W,\syn{Z}(f),p)$, and then keeping only those which we can prove. Indeed, if $\phi_{\scriptscriptstyle | Z(f)}$ is a finite map and $Y$ is radicial at whole of $S$ (as opposed to $S_{\scriptscriptstyle (d-c)/{3}}$ above), then $(\phi,Y)$ would present $(W,Z,p)$ thereby proving \eqref{dtwo}.
\end{remark}
\begin{remark}\label{sfinite}
The set $S=\syn{Z}(\phi) \cap Z$ appearing in the statement of the above Lemma is necessarily finite. This is because, in each irreducible component of $Z$, there is at least one $q_i$ (see \eqref{notationa2}(6)(b)) on which $\phi$ does not vanish. Since $T$ intersects each irreducible component of $\syn{Z}(f)$ (see \eqref{notationa2}(5)(a)), we know that any open neighbourhood of $S$ is dense in $\syn{Z}(f)$.
\end{remark}
Following \cite{poonen} define the density of a subset ${\mathcal C} \subset A$ by $$\mu({\mathcal C}) := \displaystyle \lim_{d \rightarrow \infty} \frac{\raisebox{.3ex}{\footnotesize \# } ({\mathcal C}\cap A_{\scriptscriptstyle \leq d})}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}}$$ provided the limit exists. Similarly, the upper and lower densities of ${\mathcal C}$, denoted by $\overline{\mu}({\mathcal C})$ and $\underline{\mu}({\mathcal C})$, are defined by replacing limit in the above expression by lim sup and lim inf respectively. \\
To prove the existence of $\phi$ in \ref{phiexists}, we will show that the density of such $\phi$ is positive.
We prove Lemma \ref{phiexists} in two steps. First, we show (Lemma \ref{findg}) that $\phi$ satisfying conditions $(1),(3)$ and condition $(2)$ for points upto certain degree, exists. Next, we show (Lemma \ref{findh}) that the set of $\phi$ which does not satisfy condition (2) for points of higher degrees has zero density. \\
Let $\phi\in A$ and $r\geq 1$ be an integer. Consider the following conditions on $\phi$, which are closely related to the conditions (1), (2) and (3) of Lemma \ref{phiexists}.
\begin{enumerate}
\item[(a)] $\phi(p)=\phi(p_i)=0$ for all $1 \leq i\leq t$ and $\phi(q_i)=1$ for all $1\leq i\leq s$.
\item[($b_r$)] For all $x\in \syn{Z}(f)_{\scriptscriptstyle \leq r}$ such that $\phi(x)=0$, $\frac{\partial \phi}{\partial X}(x)\neq 0$.
\item[($c_r$)] For all points $x_1,x_2 \in \syn{Z}(f)_{\scriptscriptstyle \leq r}$, such that
$ \syn{deg}(x_1)=\syn{deg}(x_2)=\syn{deg}(Y(x_1))=\syn{deg}(Y(x_2))$ and $\phi(x_1)=\phi(x_2)=0$, we have $Y(x_1)\neq Y(x_2)$.
\item[($d_r$)] For all $x \in \syn{Z}(f)_{\scriptscriptstyle \leq r}$ such that $\phi(x)=0$, $\syn{deg}(Y(x))=\syn{deg}(x)$.
\end{enumerate}
It is easily seen that
\begin{remark} \label{conditions} The main motivation for introducing the above conditions, are the following straightforward implications between them and the conditions of \ref{phiexists}
\begin{enumerate}
\item[-] $\phi$ satisfies \eqref{phiexists}(1) if $\phi$ satisfies (a).
\item[-] $\phi$ satisfies \eqref{phiexists}(2) iff $\phi$ satisfies ($b_r$) for all $r\geq 1$.
\item[-] $\phi$ satisfies \eqref{phiexists}(3) iff $\phi$ satisfies ($c_r$) and ($d_r$) for all $r\leq (d-c)/{3}$.
\end{enumerate}
\end{remark}
\begin{lemma}\label{findg}
There exists integers $r_0, c\in {\mathbb N}$, with $$r_0> {\rm max} \big\{ \syn{deg}(p), \syn{deg}(p_1),...,\syn{deg}(p_t), \syn{deg}(q_1),..., \syn{deg}(q_s)\big\} $$ such that the lower density of the set
$$
{\mathcal P}:=\bigcup_{d>c+2r_0} \Big\{\phi \in A_{\scriptscriptstyle \leq d}\ |\ \phi \ {\rm satisfies}\ (a), (b_{\scriptscriptstyle (d-c)/{3}}), (c_{\scriptscriptstyle (d-c)/{3}}), (d_{\scriptscriptstyle (d-c)/{3}}) \ {\rm and} \ \phi(x)=1 \ \forall \ x\in \syn{Z}(f)_{\scriptscriptstyle \leq r_0} \backslash T \Big\} \\
$$
is positive.
\end{lemma}
\begin{proof}
By Lang-Weil estimates \cite{lw} there exists $c'\in {\mathbb N}$ such that for all $n\geq 1$,
$$ \raisebox{.3ex}{\footnotesize \# } \big( \syn{Z}(f)_{\scriptscriptstyle = n} \big) \leq c'\cdot q^{n}.$$
For reasons which be clear during the course of the calculations below, we choose $r_0$ and $c$ as follows. {Recall that $m$ is the $X$-degree of $f$.} Let $r_0$ be any integer satisfying
\begin{enumerate}
\item[(i)] $r_0 > {\rm max} \big\{ \syn{deg}(p), \syn{deg}(p_1),...,\syn{deg}(p_t), \syn{deg}(q_1),..., \syn{deg}(q_s)\big\}$.
\item[(ii)] $\displaystyle{\Bigg( \sum _{i>r_0/m} \frac{1}{q^{i}} \Bigg)\cdot \Bigg( c'+ \binom{m}{2} + \frac{m}{2} \Bigg) <1-\sum_{x\in T}q^{- \syn{deg}(x)}.}$
\end{enumerate}
Note that it is always possible to ensure (ii) as $$ \Bigg(\sum _{i>r_0/m} \frac{1}{q^{i}}\Bigg) \to 0 \ \ \ \text{as} \ \ \ r_0\to \infty $$
and as degrees of points in $T$ are distinct we have
$$ \sum_{x\in T}q^{-\syn{deg}(x)} < \sum_{i=1}^\infty q^{-i} \leq 1.$$
Let
$$ c = \sum_{x\in \syn{Z}(f)_{\scriptscriptstyle \leq r_0}} \syn{deg}(x). $$
Let $d\geq c+2r_0$ be any integer and $r:=(d-c)/{3}$. Let
\begin{align*}
{\mathcal T} & := \big\{ \phi \in A_{\scriptscriptstyle \leq d} \ | \ \phi \mbox{ satisfies (a) } \ \text{and} \ \phi(x)=1 \ \forall \ x\in \syn{Z}(f)_{\scriptscriptstyle \leq r_0} \backslash T \big\}. \\
{\mathcal T}_b & := \big\{\phi \in {\mathcal T} \ | \ \phi \ \text{does not satisfy} (b_{r}) \big\} \\
{\mathcal T}_c & := \big\{\phi \in {\mathcal T} \ | \ \phi \ \text{does not satisfy} (c_{r}) \big\} \\
{\mathcal T}_d & := \big\{\phi \in {\mathcal T} \ | \ \phi \ \text{does not satisfy} (d_{r}) \big\}
\end{align*}
Let
$$\delta := \frac{\raisebox{.3ex}{\footnotesize \# } {\mathcal T} }{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}}, \ \
\delta_b := \frac{\raisebox{.3ex}{\footnotesize \# } {\mathcal T}_b}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}}, \ \
\delta_c := \frac{\raisebox{.3ex}{\footnotesize \# } {\mathcal T}_c}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}}, \ \
\delta_d := \frac{\raisebox{.3ex}{\footnotesize \# } {\mathcal T}_d}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}}.
$$
\\ \\
In the following steps we will estimate $\delta,\delta_b,\delta_c,\delta_d$. \\ \\
\noindent $\Step{1}:$ (Estimation for $\delta$) :
Note that the condition that $\phi$ belongs to ${\mathcal T}$ depends solely on the image of $\phi$ in the zero dimensional ring
$$ \prod_{x\in \syn{Z}(f)_{\leq{r_0}}} ({\mathcal O}_x/m_x). $$
Since the {dimension over $F$} of the above ring is $c$ and since $d\geq c$, by \cite[Lemma 2.1]{poonen} the map
$$ A_{\scriptscriptstyle \leq d} \stackrel{\rho}{\longrightarrow} \prod_{x\in \syn{Z}(f)\leq{r_0}} ({\mathcal O}_x/m_x)$$
is surjective. One can easily see that ${\mathcal T}$ is a coset of ${\rm Ker }(\rho)$. Therefore
$$ \delta = \prod_{x\in \syn{Z}(f)_{\scriptscriptstyle \leq r_0}}q^{-\syn{deg}(x)}.$$ \\
\noindent $\Step{2}:$ (Estimation for $\delta_b$) :
Let $x \in \syn{Z}(f)_{\scriptscriptstyle \scriptscriptstyle \leq r}$ where recall that $r=(d-c)/{3}$. The following are equivalent :
\begin{enumerate}
\item[(i)] $\phi\in {\mathcal T}$ and $\phi(x)=0$ and $\frac{\partial \phi}{\partial X}(x)=0$.
\item[(ii)] $\phi \in {\mathcal T}$ and $\phi \ \syn{mod} \ {\mathfrak m}_x^2$ lies in the kernel of the linear map $\frac{\partial}{\partial X}: \frac{{\mathfrak m}_x}{{\mathfrak m}_x^2} \to F(x).$
\end{enumerate}
Let us first consider the case when $\syn{deg}(x)>r_0$. In this case,
each of the above condition for $\phi$ depends only on its image in the zero dimensional ring
$$ \Bigg(\prod_{\substack{ q\in \syn{Z}(f)_{\scriptscriptstyle \leq{r_0}}} } ({\mathcal O}_q/m_q) \Bigg)\times ({\mathcal O}_x/m_{x}^2) .$$
The cardinality of the above ring is $$\bigg(\prod_{y\in \syn{Z}(f)_{\scriptscriptstyle \leq r_0}} q^{\syn{deg}(y)}\bigg) \cdot q^{3 \syn{deg} (x)}.$$
Let us call an element $\xi$ in the above ring as a favorable value iff all $\phi$ mapping to $\xi$ satisfy the above conditions. It is an easy exercise to check that the set of all favorable values has cardinality $q^{\syn{deg}(x)}$.
Thus the ratio of the number of favorable values to the cardinality of the ring is nothing but $\delta q^{-2 \syn{deg}(x)}$.
The {dimension over $F$} of this ring is $c+3 {\cdot \syn{deg}(x)}$. Since $d\geq c+3{\cdot\syn{deg}(x)}$, by \cite[Lemma 2.1]{poonen}, $A_{\scriptscriptstyle \leq d}$ surjects onto this ring. Due to this, the ratio of $\phi \in A_{\scriptscriptstyle \leq d}$ satisfying the above two conditions to the $\# A_{\scriptscriptstyle \leq d}$ is nothing but $\delta q^{-2 \syn{deg}(x)}$.
In other words,
$$ \frac{ \raisebox{.3ex}{\footnotesize \# } \big\{ \phi \in {\mathcal T} \ | \ \phi(x)=0, \ \frac{\partial \phi}{\partial X}(x)=0 \big\}}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} = \delta \cdot q^{-2 \syn{deg}(x)}.$$
Now let us consider the case where $\syn{deg}(x)\leq r_0$. We claim that unless $x\in T$, there is no $\phi \in {\mathcal T}$ which vanishes on $x$. This follows from the definition of ${\mathcal T}$. So let us assume $ x\in T$. In this case, the above two conditions for $\phi$ depend solely on the image of $\phi$ in the ring
$$ \Bigg(\prod_{\substack{ q\in \syn{Z}(f)_{\scriptscriptstyle \leq{r_0}}\\ q\neq x } } ({\mathcal O}_q/m_q) \Bigg)\times ({\mathcal O}_x/m_{x}^2) .$$
Proceeding in a manner similar to the case where $\syn{deg}(x)>r_0$, we find that for $x\in T$,
$$ \frac{ \raisebox{.3ex}{\footnotesize \# } \big\{ \phi \in {\mathcal T} \ | \ \phi(x)=0, \ \frac{\partial \phi}{\partial X}(x)=0 \big\}}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} = \delta \cdot q^{- \syn{deg}(x)}.$$
Since
$$ {\mathcal T}_b = \bigcup_{\substack{x \in \syn{Z}(f)_{\scriptscriptstyle \leq r} \text{ such that} \\ \ x\in T \ {\text or} \ \syn{deg}(x)> r_0}} \hspace{-5mm} \big\{ \phi \in {\mathcal T} \ | \ \phi(x)=0, \ \frac{\partial \phi}{\partial X}(x)=0 \big\}$$
we get an estimate
\begin{align*} \delta_b & \leq \sum_{x\in T} \delta q^{- \syn{deg}(x)} + \sum_{\substack{x\in \syn{Z}(f)_{\scriptscriptstyle \leq r} \text{ such that} \\\syn{deg}(x)>r_0}} \delta \cdot q^{-2 \syn{deg}(x)} \\
& \leq \delta \bigg( \sum_{x\in T} q^{- \syn{deg}(x)}+ \sum_{r_0<i\leq r} c'q^{-i}\bigg)
\end{align*}\\
where recall that $c'$ was the constant in Lang-Weil estimates such that $\# \syn{Z}(f)_{=n} \leq c' q^n$. \\
\noindent $\Step{3}:$ (Estimation for $\delta_c$): Let $y\in {\mathbb A}^1_F$ with $i:=\syn{deg}(y)\leq r$. Let
$$ {\mathcal T}_c^y := \Big\{ \phi\in {\mathcal T} \ | \ \exists \ \text{distinct} \ x_1, x_2 \in \syn{Z}(f)_{=i} \ \text{with}\ Y(x_1)=Y(x_2)=y \ \text{and} \ \phi(x_1)=\phi(x_2)=0\Big\}.$$
First, note that ${\mathcal T}_c^y$ is empty unless $i>r_0$. This is because the only points of degree $\leq r_0$ on which a $\phi \in {\mathcal T}$ vanishes are the points in $T$. However, by choice, all points in $ x \in T$ have different degrees and satisfy $\deg(x)=\deg(Y(x))$. Thus, let us assume $i>r_0$.
In this case, we claim that
$$ \frac{ \raisebox{.3ex}{\footnotesize \# } {\mathcal T}_c^y } {\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} \leq \delta \cdot {m\choose2}\cdot q^{-2i}.$$
For fixed $x_1,x_2$ with $Y(x_1)=Y(x_2)=y$,
$$ \Big\{ \phi\in {\mathcal T} \ | \ \phi(x_1)=\phi(x_2)=0 \Big\}$$
is a coset of the kernel of the following map
$$ A_{\scriptscriptstyle \leq d} \longrightarrow \bigg( \prod_{q\in \syn{Z}(f)_{\scriptscriptstyle \leq{r_0}}} ({\mathcal O}_q/m_q) \bigg) \times ({\mathcal O}_{x_1}/m_{{x_1}}) \times ({\mathcal O}_{x_2}/m_{{x_2}}) $$
which is surjective by \cite[2.1]{poonen}. Thus
$$ \frac{ \raisebox{.3ex}{\footnotesize \# } \Big\{ \phi\in {\mathcal T} \ | \ \phi(x_1)=\phi(x_2)=0.\Big\}} {\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} \leq \delta \cdot q^{-2i}.$$
To prove the claim we now simply observe that
since $f$ is monic in $X$ of degree $m$ there are atmost $m\choose{2}$ possible choices for a pair $\{x_1,x_2\}$ as above. \\
\noindent As discussed above, since ${\mathcal T}_c^y$ is empty unless $i>r_0$, we have
$$ {\mathcal T}_c = \bigcup_{\substack{y\in {\mathbb A}^1_F \\ r_0< \syn{deg}(y)\leq r}} {\mathcal T}_c^y.$$
For a fixed $i$,
$$ \raisebox{.3ex}{\footnotesize \# } \big\{ y\in {\mathbb A}^1_F \ | \ \syn{deg}(y)=i\big\} \leq q^i.$$
From this, it is elementary to deduce
$$ \delta_c=\frac{\raisebox{.3ex}{\footnotesize \# } {\mathcal T}_c}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} \leq \delta \Big( \sum_{r_0<i\leq r} {m \choose 2} q^{-i}\Big).$$\\
\noindent $\Step{4}:$ (Estimation for $\delta_d$): As in the above step, let $y\in {\mathbb A}^1_F$ with $i:=\syn{deg}(y)\leq r$. Let
$$ {\mathcal T}_d^y := \Big\{ \phi\in {\mathcal T} \ | \ \exists \ x\in \syn{Z}(f)_{\scriptscriptstyle \leq r} \ \text{with} \ \phi(x)=0 \ , \ Y(x)=y \ \text{and}\ \syn{deg}(x)> i .\Big\}.$$
We first claim that ${\mathcal T}_d^y$ is empty unless $\syn{deg}(y)>r_0/m$. Otherwise, there would exist a $\phi\in {\mathcal T}$ and an $x\in \syn{Z}(f)_{\scriptscriptstyle \leq r}$ with $Y(x)=y$, $\phi(x)=0$ and $\syn{deg}(x)> \syn{deg}(y)$. But as $f$ is monic in $X$ of degree $m$, the maximum degree of a point $x$ lying over $y$ is $m \cdot \syn{deg}(y)\leq r_0$. Which means $x\in \syn{Z}(f)_{\scriptscriptstyle \leq r_0}$. However as $\phi \in {\mathcal T}$, the only points in $\syn{Z}(f)_{\scriptscriptstyle \leq r_0}$ on which $\phi$ vanishes are those in $T$. Thus $x\in T$. But by \eqref{notationa2}(5)(c), for such $x$, $\syn{deg}(Y(x))=\syn{deg}(y)=\syn{deg}(x)$ which is a contradiction. \\
We will now estimate
$$ \frac{ \raisebox{.3ex}{\footnotesize \# } {\mathcal T}_d^y } {\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}}.$$
Fix a point $x\in \syn{Z}(f)_{\leq r}$ with $\syn{deg}(x)>i$ and $Y(x)=y$. For this $x$, we first note that
because of \eqref{notationa2}(5)(c), $x\notin T$.
For $\syn{deg}(x)>r_0$ we note that
$$ \frac{ \raisebox{.3ex}{\footnotesize \# } \Big\{ \phi\in {\mathcal T} \ | \ \phi(x)=0.\Big\}} {\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} \leq \delta \cdot q^{-\syn{deg}(x)}\leq \delta \cdot q^{-2i}.$$
This is deduced, as before, from the surjectivity of
$$ A_{\scriptscriptstyle \leq d} \longrightarrow \bigg(\prod_{q\in \syn{Z}(f)_{\scriptscriptstyle \leq{r_0}}} ({\mathcal O}_q/m_q) \bigg)\times ({\mathcal O}_{x}/m_{{x}})$$
If $\syn{deg}(x)\leq r_0$, $\Big\{ \phi\in {\mathcal T} \ | \ \phi(x)=0\Big\}$ is empty as there is no points in $\syn{Z}(f)\backslash T$ on which a $\phi \in {\mathcal T}$ vanishes. And hence, the above estimate trivially holds in this case also.
As $f$ is monic in $X$ of degree $m$, and $\syn{deg}(x)\geq 2i$, there are at most $\frac{m}{2}$ possible choices for $x\in \syn{Z}(f)$ such that $Y(x)=y$. This shows that
$$ \frac{ \raisebox{.3ex}{\footnotesize \# } {\mathcal T}_d^y } {\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} \leq \delta \cdot \frac{m}{2}\cdot q^{-2i}.$$
Since, as discussed above, ${\mathcal T}_d^y$ is empty unless $\syn{deg}(y)>r_0/m$, we have
$$ {\mathcal T}_d = \bigcup_{y\in ({\mathbb A}^1_F)_{\scriptscriptstyle \geq r_0/m}} {\mathcal T}_d^y.$$
For a fixed $i$,
$$ \raisebox{.3ex}{\footnotesize \# } \big\{ y\in {\mathbb A}^1_F \ | \ \syn{deg}(y)=i\big\} \leq q^i.$$
Thus
$$ \delta_d=\frac{\raisebox{.3ex}{\footnotesize \# } {\mathcal T}_d}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} \leq \delta \Big( \sum_{r_0/m<i\leq r} \frac{m}{2}\ q^{-i}\Big).$$
\\
\noindent $\Step{5}:$ (Estimation for ${\mathcal P}$):
If we let
$$ {\mathcal P}_d := \Big\{ \phi \in A_{ \scriptscriptstyle \leq d} \ | \ \phi \ {\rm satisfies}\ (a), (b_{\scriptscriptstyle (d-c)/{3}}), (c_{\scriptscriptstyle (d-c)/{3}}), (d_{\scriptscriptstyle (d-c)/{3}}) \ {\rm and} \ \phi(x)=1 \ \forall \ x\in \syn{Z}(f)_{\scriptscriptstyle \leq r_0} \backslash T \Big\},$$
then $${\mathcal P}_d = {\mathcal T} \backslash({\mathcal T}_b\cup {\mathcal T}_c \cup {\mathcal T}_d). $$
Therefore
\begin{align*}
\frac{\raisebox{.3ex}{\footnotesize \# } {\mathcal P}_d}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} & \geq \delta - \delta_b- \delta_c -\delta_d \\
& \geq \delta \Bigg[1 - \sum_{x\in T}q^{- \syn{deg}(x)} - \sum_{r_0<i\leq r} c'q^{-i}- \sum_{r_0<i\leq r} {m \choose 2} q^{-i} - \sum_{r_0/m<i\leq r} \frac{m}{2}\ q^{-i} \Bigg] \\
& \geq \delta \Bigg[ 1- \sum_{x\in T}q^{- \syn{deg}(x)} - \Bigg( \sum _{r_0/m<i \leq r} \frac{1}{q^{i}} \Bigg)\cdot \Bigg( c'+ \binom{m}{2} + \frac{m}{2} \Bigg) \Bigg]
\end{align*}
Note that in the above expression $r=(d-c)/{3}$. As $d\to \infty$, so does $r$. Hence we observe that
$$ \syn{inf} \Big( \frac{\raisebox{.3ex}{\footnotesize \# } {\mathcal P}_d}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \leq d}} \Big) \geq \delta \Bigg[ 1- \sum_{x\in T}q^{- \syn{deg}(x)}- \Bigg( \sum _{i>r_0/m} \frac{1}{q^{i}} \Bigg)\cdot \Bigg( c'+ \binom{m}{2} + \frac{m}{2} \Bigg) \Bigg] $$
which is positive, thanks to the definition of $r_0$. Thus the lower density of
$${\mathcal P}= \bigcup_d {\mathcal P}_d$$ is positive as required.
\end{proof}
\begin{lemma} \label{findh} Let $c$ be as in Lemma \ref{findg} and let $${\mathcal Q}:=\bigcup_{d\geq 0} \Big\{\phi\in A_{\scriptscriptstyle \leq d} \ |\ \exists\ x\in \syn{Z}(f)_{\scriptscriptstyle > (d-c)/{3}} \ {\rm such \ that } \ \phi(x)= \frac{\partial \phi}{\partial X}(x)= 0\Big\}.$$ Then ${\mu}({\mathcal Q})=0.$
\end{lemma}
\begin{proof} The proof is identical to that of \cite[2.6]{poonen}. We reproduce the argument verbatim here for the convenience of the reader. We will bound the probability of $\phi$ constructed as
$$\phi=\phi_0+g^p X+h^p$$ and for which there is a point $x\in \syn{Z}(f)_{\scriptscriptstyle > (d-c)/{3}}$ with $\phi(x)= \frac{\partial \phi}{\partial X}(x)= 0$. Note that if $\phi$ is of the above form, then $$ \frac{\partial \phi}{\partial X}=\frac{\partial \phi_0}{\partial X}+g^p.$$ Further, if
$\phi_0\in A_{\scriptscriptstyle \leq d}$, $g \in A_{\scriptscriptstyle \leq d-1/p}$
and $h\in A_{\scriptscriptstyle \leq d/p}$, then $\phi\in A_{\scriptscriptstyle \leq d}$.
Define $$W_0:=\syn{Z}(f) \ \ \ \text{and} \ \ \ W_1:=\syn{Z}\Big(f,\frac{\partial \phi}{\partial X}\Big).$$
Note that $\syn{dim}(W_0)=1$. \\
Let $$ \gamma:= \rdown{\frac{d-1}{p}} \ \ \ \text{and} \ \ \ \eta= \rdown{\frac{d}{p}}.$$
\noindent Claim 1: The probability (as a function of $d$) of choosing $\phi_0 \in A_{\scriptscriptstyle \leq d}$ and $g\in A_{\scriptscriptstyle \leq (d-1)/p}$ such that $\syn{dim}(W_1)=0$ is $1-o(1)$ as $d\to \infty$.\\ \\
Let $V_1,...,V_{\ell}$ be $F$ irreducible components of $W_0$. Clearly $\ell\leq \syn{deg}(f)$ (where $\syn{deg}(f)$ is the total degree). Since the projection onto the $Y$ coordinate is finite on $\syn{Z}(f)$ (by \eqref{notationa2}(2)), we know that $Y(V_k)$ is of dimension one for all $k$.
We will now bound the set
$$ G_k^{\rm bad} := \Big\{ g \in A_{\scriptscriptstyle \leq \gamma} \ | \ \frac{\partial \phi}{\partial X}= \frac{\partial \phi_0}{\partial X}+g^p \ \text{vanishes identically on}\ V_k \Big\}.$$
If $g,g'\in G_k^{\rm bad}$, then $g-g'$ vanishes on $V_k$. Thus if $G_k^{\rm bad}$ is non-empty, it is a coset of the subspace of functions in $A_{\scriptscriptstyle \gamma}$ which vanish identically on $V_k$. The codimension of that subspace, or equivalently the dimension of the image of $A_{\scriptscriptstyle \gamma}$ in the regular functions on $V_k$ is at least $\gamma+1$, since no polynomial in $Y$ vanishes on $V_k$. Thus the probability that $\frac{\partial \phi}{\partial X}$ vanishes on $V_k$ is at most $q^{-\gamma-1}$. Thus, the probability that $\frac{\partial \phi}{\partial X}$ vanishes on some $V_k$ is at most $\ell q^{-\gamma-1}=o(1)$. Since $\dim(W_1)=0$ iff $\frac{\partial \phi}{\partial X}$ does not identically vanish on any component $V_k$, the claim follows. \\
We will now estimate the probability of choosing $h$ such that there is no bad point in $Z(f)$, i.e. a point in $\syn{Z}(f)_{\scriptscriptstyle > (d-c)/{3}}$ where both $\phi$ and $\frac{\partial \phi}{\partial X}$ vanish. Note that the set of such bad points is precisely
$$ \syn{Z}(\phi)\cap W_1 \cap \syn{Z}(f)_{\scriptscriptstyle > (d-c)/{3}}.$$
\noindent Claim 2: Conditioned on the choice of $\phi_0$ and $g$ such that $W_1$ is finite, the probability of choosing $h$ such that $$ \syn{Z}(\phi)\cap W_1 \cap \syn{Z}(f)_{\scriptscriptstyle > (d-c)/{3}}=\emptyset$$ is
$1-o(1)$ as $d\to \infty$. \\ \\ It is clear by the Bezout theorem that $\raisebox{.3ex}{\footnotesize \# } W_1 = O(d)$. For a given $x\in W_1$, the set $$H^{\rm bad} = \big\{ h\in A_{\eta} \ | \ \phi=\phi_0+g^pX+h^p \text{ vanishes on } x \big\} $$
is either $\emptyset$ or a coset of ${\rm Ker }\big( A_{\scriptscriptstyle \eta} \xrightarrow{ev_x} F(x)\big)$ where $F(x)$ is the residue field of $x$. For the purpose of this claim, we only need to consider $x$ such that $\syn{deg}(x)>(d-c)/{3}$. In this case, \cite[Lemma 2.5]{poonen} implies that
$$ \frac{\raisebox{.3ex}{\footnotesize \# } H^{\rm bad}}{\raisebox{.3ex}{\footnotesize \# } A_{\scriptscriptstyle \eta}} \leq q^{-\nu} \ \ \ \text{where } \ \nu=\syn{min}(\eta+1,(d-c)/{3}).$$ Thus, the probability that both $\phi$ and $\frac{\partial \phi}{\partial X}$ vanish at such $x$ is at most $q^{-\nu}$. There are at most $\raisebox{.3ex}{\footnotesize \# } W_1$ many possibilities for $x$. Thus the probability that there exists a 'bad point', i.e. point in $x\in W_1$ with $\syn{deg}(x)> (d-c)/{3}$ such that both $\phi$ and $\frac{\partial \phi}{\partial X}$ vanish at such $x$ is at most $\big(\raisebox{.3ex}{\footnotesize \# } W_1\big)q^{-\nu} = O(dq^{-\nu})$. Since as $d\to \infty$, $\nu$ grows linearly in $d$,
$O(dq^{-\nu})=o(1)$. In other words, the probability of choosing $h$ such that there is no bad point is $1-o(1)$. \\
Combining the above two claims, it follows that the probability of choosing $\phi=\phi_0+g^pX+h^p$ such that $$\syn{Z}(\phi)\cap W_1 \cap \syn{Z}(f)_{\scriptscriptstyle > (d-c)/{3}}=\emptyset$$ is equal to
$(1-o(1))(1-o(1))=1-o(1)$. This shows that $\mu(Q)=0$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{phiexists}]
Let $\overline{Q}$ denote the complement of $Q$ in $A$. Let ${\mathcal P}$ be as in Lemma \ref{findg}. To prove Lemma \ref{phiexists} we need to show that ${\mathcal P}\cap \overline{Q}$ is non-empty. However, combining the above two lemma's, we in fact get that
$\mu({\mathcal P}\cap \overline{Q})>0$. This finishes the proof.
\end{proof}
Condition (3) of Lemma \ref{phiexists} ensures that $Y:{\mathbb A}^2_F\to {\mathbb A}^1_F$ is radicial at $S_{\scriptscriptstyle \leq (d-c)/{3}}$. We would have ideally liked to have $S$ instead of $S_{\scriptscriptstyle \leq (d-c)/{3}}$ here. If this was the case, and if $\phi_{ |Z(f)}$ was finite, as mentioned in Remark \ref{strategy}, we would be able to deduce that $(\phi,Y)$ presents $(W,\syn{Z}(f),p)$. However we are unable to handle points in $S$ of degree greater than $ (d-c)/{3}$. In order to rectify that, we replace the map $(\phi,Y)$ with a map $(\phi, h)$ for a suitable $h$ as found by the following lemma. Finiteness of $\phi$ will be handled later using a Noether normalization argument.
\begin{lemma}\label{hexists} Let $c\in {\mathbb N}$ be as in Lemma \ref{phiexists}. Let $d>>0$ be an integer such that for every $i>(d-c)/{3}$,
$$ \raisebox{.3ex}{\footnotesize \# } ({\mathbb A}^1_F)_{\scriptscriptstyle =i} > dm. $$
Let $\phi\in A_{\scriptscriptstyle \leq d}$ be as given by \eqref{phiexists} and $S:=\syn{Z}(\phi)\cap Z$ . Then, there exists $h \in F[X,Y]$ such that
\begin{enumerate}
\item $h_{|S} :S \rightarrow {\mathbb A}^1$ is radicial, i.e. injective and preserves the degree.
\item The map ${\mathbb A}^2_F \xrightarrow{(\phi,h)} {\mathbb A}^2_F$ is \'{e}tale at all $x\in S$.
\item $h_{|\syn{Z}(f)}: \syn{Z}(f) \to {\mathbb A}^1_F$ is a finite map.
\end{enumerate}
\end{lemma}
\begin{proof} \underline{Step(1)}: In this step we will show that with the given choice of $d$, it is possible to choose $h_1$ which satisfies condition (1) of the Lemma. \\
We claim that
$$ \raisebox{.3ex}{\footnotesize \# } S_{=i} \leq \raisebox{.3ex}{\footnotesize \# } ({\mathbb A}^1_F)_{\scriptscriptstyle =i} \ \ \forall \ i\geq 1.$$
As explained in Remark \ref{sfinite}, $\syn{Z}(\phi)\cap \syn{Z}(f)$ is finite. By Bezout theorem, $\raisebox{.3ex}{\footnotesize \# } S\leq \syn{deg}(\phi)\syn{deg}(f) = dm$. Thus the above claim holds for all $i> (d-c)/{3}$ by the choice of $d$. On the other hand, the claim also holds for $i\leq (d-c)/{3}$, since by Lemma \ref{phiexists}, $Y$ is radicial at $S_{\scriptscriptstyle \leq (d-c)/{3}}$. Thus we can choose a set theoretic map $S \xrightarrow{\tilde{h}} {\mathbb A}^1_F$ which is injective and preserves degree of points. By Chinese remainder theorem, there exists an $h_1\in F[X,Y]$ such that for all $x\in S$
$$ h_1(x)= \tilde{h}(x).$$\\
\noindent \underline{Step(2)}: Now, using the $h_1$ from above step, we will find a $h_2\in F[X,Y]$ which satisfies conditions (1) and (2) of the Lemma.
It is sufficient to find an $h_2\in F[X,Y]$ such that
\begin{align*}
(i) \ \ h_2 \equiv h_1 &\ \ \ \ \syn{mod} \ {\mathfrak m}_x \ \ \forall \ x\in S \\
(ii) \ \ \frac{\partial h_2}{\partial X}(x) & = 0 \ \ \ \forall\ x\in S \\
(iii) \ \ \frac{\partial h_2}{\partial Y}(x) & = 1 \ \ \ \forall \ x\in S
\end{align*}
First, we claim that for any closed point $x\in {\mathbb A}^2_F$, there exists an $h_x \in F[X,Y]$ such that
\begin{align*}
\ \ h_x \equiv h_1 &\ \ \ \ \syn{mod} \ {\mathfrak m}_x \\
\ \ \frac{\partial h_x}{\partial X}(x) & = 0 \\
\ \ \frac{\partial h_x}{\partial Y}(x) & = 1
\end{align*}
We choose a polynomial $f_1\in F[X]$ such that $f_1(x)=0$ and $\partial f_1/\partial X(x)\neq 0$. To see that such a choice is possible, let $\pi_1:{\mathbb A}^2_F\to {\mathbb A}^1_F$ be the projection on to the $X$-coordinate. The minimal polynomial of any primitive element of the residue field of $\pi_1(x)$ satisfies our requirement. Similarly, we choose $f_2\in F[Y]$ such that $f_2(x)=0$ and $\partial f_2/\partial Y(x)\neq 0.$ Using Chinese remainder theorem and the fact that the residue field $F(x)$ is perfect, we choose $g_1,g_2 \in F[X,Y]$ such that
$$g_1^p(x) = - \frac{ \partial h_1/ \partial X(x)}{\partial f_1/ \partial X(x)},$$
$$g_2^p(x) = \frac{ \big(1-\partial h_1/ \partial Y(x)\big)}{\partial f_2/ \partial Y(x)}.$$
We leave it to the reader that
$$ h_x = h_1 + g_1^p f_1 + g_2^p f_2 $$
satisfies the requirement of our claim.
Now, by Chinese remainder theorem, there exists $h_2\in F[X,Y]$ such that
$$ h_2 \equiv h_x \ \ \ \syn{mod} \ {\mathfrak m}_x^2 \ \ \forall \ x\in S.$$
It is straightforward to see that $h_2$ satisfies conditions (1) and (2) of the Lemma. \\
\noindent \underline{Step (3)}: Choose a non-constant polynomial $\beta \in F[Y]$ such that
$\beta(x)=0$ for all $x\in S$. Since $f$ is monic in $X$, $\syn{Z}(f)\xrightarrow{Y} {\mathbb A}^1_F$ is a finite map. Thus $\beta :\syn{Z}(f) \to {\mathbb A}^1_F$ is also a finite map. As $\syn{dim}(\syn{Z}(f))=1$, for a sufficiently large integer $\ell$, $$h:=h_2-\beta^{p\ell}$$
defines a finite map $\syn{Z}(f) \xrightarrow{h} {\mathbb A}^1_F$ by Noether normalization trick (see \eqref{nntrick1}). Clearly $h$ continues to satisfy conditions (1) and (2) of the Lemma since $\beta^{p\ell} \in {\mathfrak m}_x^2$ for all $x\in S$.
\end{proof}
\begin{proof}[Proof of theorem \ref{dtwo}]
Let $\phi,h$ be as in Lemmas \ref{phiexists} and \ref{hexists} respectively. Let $\widetilde{\Phi}$ be the map ${\mathbb A}^2_F \xrightarrow{(\phi,h)} {\mathbb A}^{2}_F$, and $\widetilde{\Psi}:=\phi$. Recall that $\tilde{S}:= \phi^{-1}(0)\cap \syn{Z}(f)$ (with reduced scheme structure). By Remark \ref{sfinite} it is finite.
\noindent \underline{Step 1}: We claim that there exists a $g\in F[X,Y]$ such that if $W_g:={\mathbb A}^2_F\backslash \syn{Z}(g)$, then $\widetilde{\Phi}(\tilde{S})\subset W_g$ and
$$ \widetilde{\Phi}_{|\widetilde{\Phi}^{-1}(W_g)\cap \syn{Z}(f)} : \widetilde{\Phi}^{-1}(W_g)\cap \syn{Z}(f) \longrightarrow W_g$$
is a closed immersion. The proof of this claim is a repetition of the argument in \cite[3.5.1]{chk} (see also \eqref{nakarg}). Let $\{p, x_1,...,x_n\}$ be the set of points in $\tilde{S}$. Since $\widetilde{\Phi}$ is \'{e}tale and radicial at all points of $\tilde{S}$ (see \eqref{phiexists}(2) and \eqref{hexists}(1) ) we have
$ \widetilde{\Phi}^{-1}(\widetilde{\Phi}(\tilde{S})) \to {\mathbb A}^2_F$ is a closed immersion. Let $y_0,...,y_n$ be the points in $\widetilde{\Phi}(\tilde{S})$. Let $\eta_i$ be the maximal ideal in $F[X,Y]$ corresponding to the closed point $y_i$. Thus the above closed immersion gives us a surjective map
$$ F[X,Y] \twoheadrightarrow \frac{ F[\syn{Z}(f)]}{ \eta_0\cdots \eta_n}$$
where $F[\syn{Z}(f)]$ is the coordinate ring of $\syn{Z}(f)$.
If $C$ denotes the cokernel of $F[X,Y] \to F[\syn{Z}(f)]$ (as $F[X,Y]$ modules), then the above surjective map implies that $$C \otimes \frac{F[X,Y]}{\eta_0\cdots \eta_n} = 0 .$$
Note that $\widetilde{\Phi}_{| \syn{Z}(f)}$ is a finite map, since $h$ is a finite map (\eqref{hexists}(3)). Thus $F[\syn{Z}(f)]$ is a finite $F[X,Y]$ module. Thus, by Nakayama's lemma, there exists an element $g \in F[X,Y]$ such that
$g \notin \eta_0\cdots \eta_n $ and $C_g=0$. In other words, the map
$$ F[X,Y]_g \twoheadrightarrow F[\syn{Z}(f)]_g$$
is surjective. This proves the claim since if $W_g:= {\mathbb A}^2_F \backslash \syn{Z}(g)$, the above surjectivity is equivalent to the following being a closed immersion
$$ \widetilde{\Phi} : \syn{Z}(f) \cap \widetilde{\Phi}^{-1}(W_g) \to W_g.$$ \\
\noindent \underline{Step 2}: Let $E$ be the smallest closed subset of $\syn{Z}(f)$ satisfying the following three conditions
\begin{enumerate}
\item[(i)] $x\in E$ if $x\in \syn{Z}(f)$ and $\widetilde{\Phi}$ is not \'{e}tale at $x$.
\item[(ii)] $\syn{Z}(f)\backslash Z\subset E$.
\item[(iii)] $\syn{Z}(f) \backslash \big(\widetilde{\Phi}^{-1}(W_g) \cap \syn{Z}(f)\big) \subset E$.
\end{enumerate}
Since $\tilde{S}$ contains at least one point in each irreducible component of $\syn{Z}(f)$, (iii) implies that $E$ is finite (see also Remark \ref{sfinite}). Moreover, by the above step and condition (iii) we have
$$ \syn{Z}(f) \backslash E \longrightarrow {\mathbb A}^2_F$$
is a locally closed immersion. Moreover, $\tilde{S}$ and $E$ are disjoint, and hence $\phi(p) \notin \phi(E)$. Since $E$ is finite, we choose a non-constant polynomial expression $Q$ in $h$ which vanishes on $p$ as well as $E$. For an integer $\ell >>0$ and divisible by ${\rm char}(F)$, we claim that $(\phi-Q^\ell, h)$ presents $(W,\syn{Z}(f),p)$. Let
$$ \Phi:= (\phi-Q^\ell, h) \ \ \ \ \text{and} \ \ \ \Psi:= \phi-Q^{\ell}.$$
To prove the claim we need to verify the conditions of the Definition (1)-(4) \ref{defpresents}.
Condition (1), i.e. finiteness of $\Psi_{|\syn{Z}(f)}$, follows by \eqref{nntrick1} since
$\ell$ is large, and $h_{|\syn{Z}(f)}$ is finite. As $Q$ vanishes on $p$ and $E$, $\Psi(p)\notin \Psi(E)$ follows from $\phi(p)\notin \phi(E)$. Thus if $S:= \Psi^{-1}\Psi(p)\cap Z$, then $S \subset \syn{Z}(f)\backslash E$. Conditions (2) to (4) of \eqref{defpresents} follow from the conditions (i) to (iii) of $E$ in the beginning of this step.
\end{proof}
|
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| 4,077
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Description The custom QUICKUP is a bottle opener that you wear on your hand as ring. It is a very functional, durable and unique (QCS patent) bottle opener and bar tool. It can also come with a key ring attachment and be used as a keychain.
The QUICKUP is a trademark and WorldWide patented innovative promotional products and gift.
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"redpajama_set_name": "RedPajamaC4"
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| 1,441
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16-year old Daniela Moroz Does It A ...
16-year old Daniela Moroz Does It Again
December 7, 2017 Michelle Slade News
Daniela Moroz crowned World Champion
Two-time World Champion Daniela Moroz talks about her recent sweep of the Oman Formula Kite World Championships. From race 1, Moroz had the outcome dialed, winning with a 27-point lead over Elena Kalinina (Russia) in second. Per the International Kiteboarding Association point system, Moroz was ranked third going into the Worlds because she didn't attend as many IKA events as the other women competitors and so received more points than those women. Unofficially however, she was ranked first since she'd won every foiling event she competed in this year.
Was there anything specific you were focused on to reach the level you needed to be for the Worlds?
Daniela Moroz takes her second World Championship title
DM: It's hard to point out one specific strength since I feel like I've become pretty well-rounded. I trained a lot over the summer, and tried to get some time in light wind conditions, which were my weakness last year. Improving my light wind skills was definitely a focus for me this year. Once school started at the end of August, I didn't have as many opportunities to really train on the water outside of events, so I tried to make up for that by going to the gym and swim practice, and really paying attention to what I was eating.
Going into this event, I knew there would be A LOT of expectations of me. I knew that everyone was expecting me to win. Personally, I hate having expectations for myself because it is so disappointing if you don't do what you expected to. But I knew that everyone expected me to win. It was difficult to tune all of that out at first, especially traveling to Oman for the event. On the training day, I was pretty stressed out but as soon as racing started, something just kind of clicked and I didn't worry about the different possibilities. I just focused on what I had to do, race by race. It worked out pretty well in the end – haha!
What did you feel you were particularly strong at?
DM: One of my strengths was being able to mentally tune out all those expectations and just focus on myself and my race, one by one. Physically, racing in strong wind is a big strength for me, being from San Francisco. However, I also think something that I really improved on this year was understanding my gear and especially knowing how to adjust my kites. Last year, I was brand new to it and was honestly afraid of doing it on my own. I always had one of my teammates help me out with it. But this time I felt a lot more confident about it. For example, on the last training day before racing began, I went out on a kite that I had not spent too much time on, and it did not feel the way I normally like. I felt much more confident about coming in and adjusting the knots to the way I felt would be better. It's something minor, but it made a big difference – that kite, an 11m, ended up being PERFECT on the windy day and it definitely made me trust my own judgement more.
How were the conditions and were they favorable for you?
DM: We had a variety of conditions throughout the week. I think that's the best kind of event because that way only the true best people can win because they have to be good in everything. We raced in everything from 6 knots on the morning of the first day and last day to around 22-25 on the second day to 10-13 on the days in between. I used almost every kite I registered.
How did the competition compare this year to last? Who were you looking out for and why?
DM: I think the overall level of the women's fleet has improved a lot. Elena Kalinina, from Russia, is really strong in light wind, as I saw at the Worlds last year. I was definitely looking out for her when it was light. Alexia Fancelli from France has improved a lot over this season, and I was surprised at her speed when it was windier. I felt like I kind of knew most of the girls' strengths and weaknesses, but I mostly just tried to focus on myself and my own race.
It seems like for the most part you were strong from the get-go – what were the defining moments of the competition?
DM: I felt pretty good after the first day, winning all but one race. I just tried to be consistent and sail clean. The second day was the windy day when we were on 11m's, and I loved those conditions. I think my defining moment was after the second day. I had won 10 out of the 12 races we'd done so far and it gave me a lot of confidence going into the rest of the week.
What equipment were you on?
DM: Ozone R1 V2's and Mike'sLab board and foil.
What's next for a two-time World Champion????
DM: I HAVE to focus on finishing junior year! It's a tough year, so I really want to just keep my grades up. Other than that, I love what I'm doing and I'm gonna go for #3 next year!
Images: Courtesy Toby Bromwich
Results: http://formulakite.com/images/documents/2017_FK_Worlds_Results_Women.htm
Michelle Slade16-year old Daniela Moroz Does It Again 12.07.2017
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"redpajama_set_name": "RedPajamaCommonCrawl"
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<!DOCTYPE html>
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<meta charset="utf-8">
<title>JSDoc: Source: controllers/FacilityTableWindow.js</title>
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<h1 class="page-title">Source: controllers/FacilityTableWindow.js</h1>
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<pre class="prettyprint source"><code>/**
* @fileOverview TableView window to view facilities with selected amenities
* @name FacilityTableWindow
* @namespace FacilityTableWindow
* @author Chris Golding
*/
Alloy.Collections.Facility.trigger('change');
/**
* Filters facilities to only show facilities that have amenities selected
* @memberOf FacilityTableWindow
* @param {Facility} collection to be filtered
* @return {Facility} filtered collection
*/
function facilityFilter(collection) {
return Alloy.Collections.Facility.filterByIds(
Alloy.Collections.FacilityAmenity.filterByAmenityIdsAsArray(
Alloy.Collections.Amenity.getSelectedIds()));
}
/**
* Closes the current window
* @memberOf FacilityTableWindow
*/
function closeWindow(e) {
$.navGroupWin.close();
}
/**
* Opens the facility details window for a specified facility
* @memberOf FacilityTableWindow
* @param {Titanium.Event} e - event containing the facilityId to display
*/
function showFacilityDetailsWindow(e) {
var arg = {
facilityId: e.source.facilityId
};
if (OS_ANDROID) {
Alloy.createController('FacilityDetails', arg).getView().open({modal:false});
} else {
$.navgroup.open(Alloy.createController('FacilityDetails', arg).getView());
}
}
/**
* Calls showFacilityDetailsWindow from the window and not an event
* @memberOf FacilityTableWindow
* @param {JSON} e facilityId of the target facility
*/
$.navGroupWin.addEventListener("showFacilityDetailsForTarget",function(e){
showFacilityDetailsWindow({source:{facilityId:e.facilityId}});
});
$.navGroupWin.addEventListener("close", function(){
//TODO - Appcelerator recommended, throws an error??
// $.destroy();
});
Alloy.Globals.parent = $.navGroupWin;</code></pre>
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<h2><a href="index.html">Index</a></h2><h3>Classes</h3><ul><li><a href="Amenity.html">Amenity</a></li><li><a href="Facility.html">Facility</a></li><li><a href="FacilityAmenity.html">FacilityAmenity</a></li></ul><h3>Namespaces</h3><ul><li><a href="AmenityFilter.html">AmenityFilter</a></li><li><a href="FacilityDetails.html">FacilityDetails</a></li><li><a href="FacilityTableWindow.html">FacilityTableWindow</a></li><li><a href="index_.html">index</a></li><li><a href="mapView.html">mapView</a></li></ul>
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Download Small GTPases and Their Regulators, Part B: Rho Family by John N. Abelson, Melvin I. Simon, W. E. Balch, Channing J. PDF
By John N. Abelson, Melvin I. Simon, W. E. Balch, Channing J. Der, Alan Hall
Normal Description of the Volume:Small GTPases play a key function in lots of elements of latest mobilephone biology: regulate of telephone progress and differentiation; rules of phone adhesion and mobilephone flow; the association of the actin cytoskeleton; and the legislation of intracellular vesicular delivery. This quantity plus its significant other Volumes 255 and 257 hide all biochemical and organic assays at the moment in use for reading the function of small GTPases in those facets of telephone biology on the molecular point. it's the first compendium of functional options for operating with small GTPases of the Rho group.General Description of the Series:The severely acclaimed laboratory usual for greater than 40 years, tools in Enzymology is likely one of the so much hugely revered guides within the box of biochemistry. in view that 1955, every one quantity has been eagerly awaited, usually consulted, and praised through researchers and reviewers alike. Now with greater than three hundred volumes (all of them nonetheless in print), the sequence comprises a lot fabric nonetheless appropriate today--truly a necessary book for researchers in all fields of existence sciences.
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Biopsy Interpretation of the Central Nervous System
Extra info for Small GTPases and Their Regulators, Part B: Rho Family
Segal, Biochem. J. 298, 585 (1994). 26T. Sasaki, M. Kato, and Y. Takai, J. Biol. Chem. 268, 23959 (1993). [4] P u r i f i c a t i o n of Rac2 from Human Neutrophils By ULLA G. KNAUS and GARY M. BOKOCH The Rac2 protein belongs to the Rho family of GTP-binding proteins and is closely related to the Racl protein (92% identity). Whereas Racl is ubiquitously expressed, Rac2 is only found in cells of myeloid origin. Studies with a human leukemia cell line (HL-60) showed a seven- to ninefold increase of Rac2 expression on differentiation to a neutrophiMike type, whereas R a c l levels rose only slightly.
After completion of the gradient, the column is further eluted with 150 ml of 1 M NaC1 in TEDMPM buffer. 5 ml are collected and assayed for GTPySbinding activity. The flow through and two overlapping peaks in the salt gradient (100-170 mM NaC1) show GTPyS binding (Fig. 1). When assayed for N A D P H oxidase activity in a cell-free system, as mentioned earlier, the first part of the GTPyS-binding peak (100-130 mM NaC1) correlates with oxidase stimulatory activity. The second part of the biphasic GTPySbinding peak represents the Rac-related proteins Rho (detected by ADPribosylation with botulinum toxin C3 ADP-ribosyltransferase) and Cdc42Hs (detected by Western blotting).
Kotani, K. Hirata, M. Katayama, and Y. Takai, J. Biol. Chem. 267, 14611 (1992). 15S. Kuroda, A. Kikuchi, K. Hirata, T. Masuda, K. Kishi, T. Sasaki, and Y. Takai, Biochem. Biophys. Res. Commun. 185, 473 (1992). 16H. Yaku, T. Sasaki, and Y. Takai, Biochem. Biophys. Res. Commun. 198, 811 (1994). 17M. Isomura, A. Kikuchi, N. Ohga, and Y. Takai, Oncogene 6, 119 (1991). 18T. Sasaki, M. Kato, and Y. Takai, Z Biol. Chem. 268, 23959 (1993). a9y. Miura, A. Kikuchi, T. Musha, S. Kuroda, H. Yaku, T. Sasaki, and Y.
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Q: Log a clicked (external) link to webserver log or a file I have a website written in php. I have a lots of external links like:
<a href="http://example.com" target="_blank">
What i want is that if the user clicks on these kind of links it should log this somewhere. Preferred is the apache log. If not possible then log in a file.
How can I do this with (probably with js)?
A: A simple solution to get it in apache log is to replace external links with something like:
<a href="tracker.php?url=http://example.com" target="_blank>
where tracker.php is on your server and simply redirects to the external site:
header('Location: ' . $_GET['url']);
This could be done manually, or you could replace the links client side using jQuery like:
$("a[target='_blank']").each(function(e){
var href = $(this).attr("href");
$(this).attr("href", "tracker.php?url=" + encodeURIComponent(href));
});
Fiddle of the above
Alternatively you can capture the click event using jQuery and make a request to your tracker.php file with the target URL as a query param and then redirect:
<a href="http://example.com" target="_blank">test</a>
// Click event for a tags with target='_blank' attribute
$("a[target='_blank']").click(function(e){
e.preventDefault(); // Prevent the default behaviour (the link opening)
var href = $(this).attr("href"); // Get the href attribute of the clicked element
// Make a get request to tracker.php passing the encoded href
$.get("tracker.php?url=" + encodeURIComponent(href), function(data) {
window.open(href); // Open the link in a new window once http request is complete
});
});
The downsides to this method are:
*
*Right click + open in new tab won't be captured (nor middle mouse button clicks)
*There will be a slight delay for the user during the http request.
Keyboard controls (focus + return key) will be captured.
For either of these to work you will need to ensure apache is configured to log GET requests, which it most likely will be by default, something like:
0.0.0.0 - - [20/Feb/2020:13:15:48 +0100] "GET /tracker.php?url=http://example.com HTTP/1.1" 200
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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What Innovators Can Learn About Communication from a Teenage Girl
by Mitch Ditkoff
There are 16,593,242 teenage girls living in America. One of them lives in my house. That would be my daughter, Mimi, an extraordinary 17-year old who, shall we say, has been quite an education for me.
If you have a teenage daughter, you know what I mean. If you've been a teenage daughter, you know what I mean. If you have a friend with a teenage daughter (and spent hours chanting "It's just a phase she's going through, it too shall pass") you know what I mean. Everyone else — oh ye of no teenagers in your life — please give me the benefit of the doubt for a moment while I shed some major light on the little understood emerging science of how to communicate to a teenage girl.
Most people who know me would assume I'd have no trouble communicating to my teenage daughter. I'm smart. I'm likable. I'm laid back and usually thought of as "cool". I am also a professional communicator — my work taking me all over the world to speak with all kinds of people: rocket scientists, MTV programmers, actuaries, college students, polymer chemists, PR wizards, cultural creatives, Hollywood executives, video game makers, and everybody else in between — a percentage for whom English is their second language.
Compared to communicating to my teenage daughter, these people are a piece of cake.
Usually, my attempts to engage my daughter in meaningful conversation are perceived of as lame. I ask what I consider to be authentic, thoughtful, caring questions and, more often than not, get only inscrutable, one word answers – "Fine", "Good", and "OK" being the three most popular, as she mounts the carpeted staircase to her room.
If I try to get clever in my conversation-opening mode, I succeed only in getting "the look" — the non-verbal equivalent of "Yo, dude, I see through your game of trying to have a conversation with me and, God, why would I want to talk with anyone as old as you when, in fact, I have some serious texting and Netflix watching to do?"
But today… ah, today… driving Ms. Mimi to school was a Red Sea parting experience — a glorious epiphany, free parking in Monopoly — one of those Archimedes-in-the-bathtub moments we've all heard about.
Are you ready for the the secret to communicating to a teenage girl? STORY!
Yes, story! Today, instead of my pitiful, Socratically-infused, semi-desparate attempt to engage my still-not-yet-fully-formed-frontal-cortex-challenged daughter, I completely shifted gears. I took a left turn, instead of a right, segueing from something she said to the spontaneous telling of a personal story — the passionate, no holds barred sharing of a life-changing moment, for me, that happened five years ago in Australia — a moment when the eternal adolescent in me made a quantum leap.
I was not probing. I was not teaching. I was not "looking for an opening" to establish more rapport. I was merely recounting a story that mattered to me — one, it turns out, that mattered to her, she being an edgy, aspiring artist who, like me, sometimes wrestles with doubt.
The vibe in the car? Totally transformed from the kind of teenage black hole moment where only a father's bald spot is visible to the sudden brilliance of a Christmas morning.
When my story was over, my daughter was not only fully present, engaged, and responsive… she asked ME questions. Here in this space, Mimi and I were one, two members of the same tribe sitting around the same fire, the light in each others' eyes all we needed to find our way home to ourselves and each other.
While there probably aren't a whole of teenage girls in your life right now, you, as a human being, innovator, entrepreneur, manager, team leader, worker bee, or business owner, are faced with the same challenge every single day that millions of parents of teenage girls are faced with — and that is how to how to BRIDGE THE GAP between you and "that other person"… how to connect… how to engage in a way that works.
May I suggest that STORY is the way to go — the convertible, low-carbon emission vehicle that allows you to travel vast distances between others who may be very different from yourself. Story, quite simply, is the BRIDGE, the universally understood medium that makes it profoundly easy to deliver and receive a message in the least amount of time and in a way that is empowering, inspiring, and memorable.
What story will YOU tell this week? And who will you tell it to?
image credit: Tammy McGary
Mitch Ditkoff is the Co-Founder and President of Idea Champions, an innovation consulting and training company, headquartered in Woodstock NY. He is also a big believer in the inspired words of Margaret Mead: "Never doubt that a handful of concerned citizens can change the world. Indeed, that's all that ever has." Follow him @mitchditkoff
Communication Leadership Management 2015-03-18
Mitch Ditkoff
Pingback: What Innovators Can Learn About Communication from a Teenage Girl | Innovation Revealed
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{
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}
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Q: From Keyword Not Found Where Expected Error in Self Join I have a data table Employees, I want to show the employee name, the employee number, the manager number and the manager name of the employee who has the least salary in the company. I decide to perform a self join, and here's my code:
select worker.employee_id, worker.last_name "Worker Last Name",
worker.salary manager.last_name "Manager Last Name", manager.manager_id
from employees worker join employees manager
on worker.manager_id = manager.employee_id
having worker.salary = (select min(salary)
from employees);
However, when I run this, the error "from keyword not found where expected" pops up. What should I do?
A: Oops, realized my own mistakes. I forgot to place a comma between worker.salary and manager.last_name, and I should not have WHERE instead of HAVING.
select worker.employee_id, worker.last_name "Worker Last Name",
worker.salary, manager.last_name "Manager Last Name", manager.manager_id
from employees worker join employees manager
on worker.manager_id = manager.employee_id
where worker.salary = (select min(salary)
from employees);
After fixing those two mistakes, the code runs fine.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
In the broad spectrum of cognitive states, vigilance plays an essential role in most human activities. Many working occupations require high levels of alertness to prevent and avoid potential dangers to the labourer.
An example of such activities is driving: drivers, in any circumstance, are required to be active to maintain a correct vehicle trajectory, avert casualties, and react to random dangerous episodes. Research studies show that in a considerable percentage of road accidents, sleep and fatigue have been stated to be contributing factors \cite{horne1995sleep, sagberg1999road}.
A system capable of detecting when the driver is getting tired, or even falling asleep, and of giving real-time feedback could be a huge benefit in the prevention of both non-fatal and fatal crashes.
\\
Electroencephalography (EEG) is a monitoring method based on the principles of electrophysiology used to record brain activity.
In the transition between wakefulness and sleep, the EEG is one of the most indicative bio-signals that reflect the event, for the very reason of being a direct indication of those events \cite{berka2007eeg}. Its core characteristic makes it a promising indicator of vigilance and fatigue levels.
Starting from Ref. \cite{zheng2017multimodal}, which developed a multimodal approach for vigilance estimation regarding temporal dependency and combining EEG and forehead EOG in a simulated driving environment, we wanted to exploit the temporal evolution of the mental state during the transition from an \textit{awake} to a \textit{drowsy} state, to detect the event and alert the driver.
We hypothesize that these transitions can be represented by a sequence of a set of bit-codes computed by the one-dimensional version of the Local Binary Pattern operator (1D-LBP, \cite{ojala2002multiresolution}). This work's novelty lies in the use of the 1D-LBP for feature extraction from the EEG signal in a driver's state monitoring system. The analysis is done over a given time-window, named epoch. The goal is to identify the epochs where the subject is getting tired or falling asleep and send a signal (e.g. an audio alarm) to wake the driver up.
The second point is to investigate if this representation can generalize the drowsiness detection to any user or requires user-specific settings.
The white-box nature of the proposed method, unlike deep-learning approaches, allows to provide an explanation of the BCI's behavior. Moreover, to make the system response more explainable, we propose new performance metrics that can help standardizing comparisons and evaluate the temporal response of the classifier.
We focused on three basic properties: the ability to detect a certain class transition, the percentage of cases where the transition is detected at the time reported by the ground truth, and the average delay time in detecting the transition occurrence. The comparison of the reported performance with one of the most effective methods at the state-of-the-art is also performed. Experiments are carried out on the sole dataset publicly available and specifically collected for drowsiness detection, namely, the SEED-VIG data set \cite{zheng2017multimodal}. This data set is very recent. Consequently, the majority of published works used home-made data, thus it is not easy to make the exact point on current achievements. For this reason, we believe that the SEED-VIG data set should deserve more attention from the scientific community.
The rest of this paper is organized in the following sections. Section \ref{chap:state-of-the-art} provides an overview of the state of the art. Section\ \ref{proposed} describes the purpose of the experimentation. The experimental methodology is presented in Section \ref{exprotocol}. Experiments are reported in Section \ref{results}, while Section \ref{conclusion} summarizes the main results of this paper and offers concluding remarks.
\section{State of the art}
\label{chap:state-of-the-art}
Many methods for the analysis of vigilance levels have been proposed in the past. Early studies focused on image processing techniques by analyzing driver's physical changes, such as eye-closure degree, eyelid movements, head pose, and gaze direction \cite{ueno1994development,boverie1998intelligent,ji2002real,matsumoto2000algorithm}.
These visual-behaviors are usually recorded by one or more video cameras or thermographic cameras to avoid problems in poor or very bright lighting conditions \cite{bergasa2006real}. Another line in active safety system development employs indirect vehicle characteristics like speed, lateral position, and steering angle as driver's alertness state. Although these methods can achieve a satisfactory level of accuracy, their performance may vary in different environmental situations and driving conditions, and it also may require a significant amount of time to analyze user behaviors.
In this respect, the most accurate indicators are physiological signals such as EEG, electrooculography (EOG), electrocardiography (ECG), and electromyography (EMG). Among them, EEG-based methods appear to be promising in detecting sleep onset while driving. EEG recordings contain information that varies with drowsiness, arousal, sleep, and attention \cite{santamaria1987eeg}.
Wireless and wearable EEG systems have been tested in virtual driving environments to evaluate and get feedback on driving performance \cite{lin2014wireless, lin2013can}. In fact, through the analysis of EEG signals, it is possible to discover a lapse of disengagement of sustained attention \cite{davidson2007eeg}.
This detection is based on the examination of the temporal dynamics of the signal that depicts specific and recognizable patterns up to 20 seconds before the actual error\cite{o2009uncovering}.
Many previous studies focus on the best way to discriminate the transition between wakefulness and sleep.
Yeo \textit{et al. }\cite{yeo2009can} demonstrated that Support Vector Machines are the best classifiers for this task. They used samples of EEG signals from both states to train the model. In \cite{mardi2011eeg}, chaotic feature extraction is employed to create a significant level of difference between sleepiness and alertness in each EEG channel.
One more approach consists of analysis in the frequency domain, dividing the transformed signal in different EEG rhythms: the main widely adopted intervals are delta (1-\SI{4}{\hertz}), theta (4-\SI{8}{\hertz}), alpha (8-\SI{14}{\hertz}), beta (14-\SI{31}{\hertz}), and gamma (31-\SI{50}{\hertz}) bands \cite{kang}. Each of these bands has a functional meaning: a dense or particular activity in any of these frequency intervals generally corresponds to peculiar cognitive states, physiological processes, and pathologies. In the last decades, various works \cite{LAL2001173, SUBASI2005701} showed that the energy of the feature waves changes with different degrees of fatigue. In particular, an increase in delta, theta, and alpha activity characterizes the EEG signal during driver fatigue.
Therefore, many studies base their fatigue detection systems on the energy analysis of these waves \cite{JAP20092352,bachao}.
In particular, Zheng \emph{et al.} \cite{zheng2017multimodal} adopted the differential entropy (DE) for features extraction. The ability of DE to discriminate EEG pattern between low and high frequency energy made it suitable to perform both emotion recognition \cite{duan2013differential}, and vigilance estimation. \cite{shi2013differential}.
Other handcrafted features can be computed starting from DE, as differential
asymmetry (DASM) and rational asymmetry (RASM). Nevertheless, as Zheng \emph{et al.} showed in \cite{zheng2015investigating} making a comparison also with the conventional power spectral density (PSD), the DE features are more accurate for this task.
Relying on these results, we used differential entropy and \cite{zheng2017multimodal}'s work as a benchmark in our investigation: firstly, because they created a shareable dataset for vigilance estimation (SEED-VIG); secondly, we both investigated the changes in neural patterns associated with vigilance. To the best of our knowledge, this is the most recent and promising state-of-the-art method proposed for this task, whose results can be comparable by a unified experimental protocol.
\section{A system for drowsiness
detection}
\label{proposed}
In this work, we proposed and analysed a hypothetical BCI system to detect drowsiness states and feedback in the driving environment. The BCI is composed of an EEG device capable of recording, a computational unit capable of running the signal processing and
the classification process and a feedback system in real-time to alert the person behind the wheel and wake him up (Fig.\ref{fig:bci}).
\begin{figure}[!ht]
\centering
\includegraphics[width=0.5\linewidth]{figures/bci.jpeg}
\caption{Operating diagram of the BCI system for monitoring the driver's state.}
\label{fig:bci}
\end{figure}
Standard measures for BCI efficiency are the accuracy of the classifier installed on the device and temporal analysis of the response of the test classification. In the experiments, we carried out both these criteria.
The classification of the samples was made in three vigilance classes: awake, tired, and drowsy.
The BCI should identify these states with the smallest latency and, secondly, it should generate the least number of false positives (considering the tired/drowsy classes as the positive class), in order to avoid the alarm when the driver is entirely awake.
The novelty of our approach is in the feature extraction algorithm. This is based on the one-dimensional local binary pattern computation for extracting the quantitative histograms from EEG signals. Local Binary Pattern (LBP) \cite{ojala2002multiresolution} is a visual descriptor, primarily used in 2-D image processing for texture segmentation and feature detection \cite{ojala1999unsupervised,he2009quantitative}, whose characteristics can be adapted to work in the one-dimensional (1-D) case.
The 1D-LBP operator can be defined as a function that takes a signal $x$ as input and examines the neighbour points of $x[i]$ sample, assigning a LBP code to it. The 1D-LBP code is calculated in this form:
\begin{equation}
\resizebox{0.8\textwidth}{!}{
$LBP^{1D}\left(x[i]\right)= \sum ^{\frac{P}{2}-1} _{r=0}\left\lbrace sgn\left(x\left[i+r-\frac{P}{2}\right]-x[i]\right)2^r+ sgn\left(x[i+r+1]-x[i]\right)2^{r+\frac{P}{2}}\right\rbrace$
}
\label{lbp:eq}
\end{equation}
where $sgn(x)$ is the sign function, defined as
\begin{equation}
sgn(x) = \begin{cases}
0 & \text{if } x < 0, \\
1 & \text{if } x \geq 0. \end{cases},
\label{lbp:dec}
\end{equation}
and $P$ is a parameter arbitrarily chosen, that represents the number of neighbouring samples thresholded against the centre one. If the neighbour sample is greater or equal to $x[i]$, its binary code would be 1, if it's lesser, the code is 0. Each binary number is multiplied by a specific binomial weight, relative to the position of the corresponding sample, and summed to obtain the 1D-LBP code that locally describes data. The 1D-LBP operator produces $ 2^P $ different configurations, through whose frequency the analyzed signal is characterized within a given window.
The use of 1D-LBP for signal processing was introduced by \cite{chatlani2010local} in signal segmentation and voice activity detection. To the best of our knowledge, this method has never been applied to the driver's state monitoring.
We have chosen 1D-LBP because, given the properties of the EEG measurements in the drowsiness and sleep states, it could show similarities between windows of signals representing the same transition from a vigilance state to another. As stated by Eqs. \ref{lbp:eq}-\ref{lbp:dec}, this method represents the signal by a sequence of binary strings of $P$ size at each sampling time. This leads to $2^P$ possible configurations of the signal. For a given time window, the histogram of each configuration's frequency represents a specific 'signal state'. We believe that the alteration of the histogram may reflect the alteration of the driver's vigilance from a time window to another. The more the histogram changes, the more abrupt the vigilance state transition.
Therefore, our aim is to investigate whether the configurations where a state transition is present, have different characteristics from those in which there is a state of rest or other types of switching.
In order to verify this claim, we codified each channel of the EEG signal by 1D-LBP over a time window of pre-set size, thus obtaining a sequence of histograms put under classification by a machine learning-based method. We trained two classification models: the linear Support Vector Machine (SVM) and the SVM with a gaussian kernel (RBF).
Being the SVM a binary discrimination model and our classification task a multiclass problem, we adopted the one-vs.-one strategy, where each combination of two classes is used for a distinct model training. Hence, we obtained a system made up of $K(K-1)/ 2$ classifiers, where $K$ is the number of vigilance classes in the set $\{ drowsy, tired, awake\}$. $K=3$ in our case, thus:
\[
N_{\text{classifiers}} = \frac{K(K-1)}{2} = \frac{3(2-1)}{2} = 3.
\]
The kernel scale adopted for every Gaussian SVM model was $\sqrt{n}$, being $n$ the size of the feature vector.
Finally, for the sake of comparison with our method, we implemented the method proposed by \cite{zheng2017multimodal}, which consists in extracting the differential entropy (DE) from the EEG signal. Their claim is similar to ours: by the DE extraction, they quantify the amount of uncertainty or randomness of the pattern time: in a state of rest, the signal should be more predictable or repetitive.
\section{Dataset and pre-processing}
\label{exprotocol}
In order to evaluate the effectiveness of the proposed system, SEED-VIG dataset was used.
SEED-VIG is a freely available multimodal dataset for vigilance estimation published by Zheng et al. in 2017 \cite{zheng2017multimodal}. It consists of EEG and EOG data collected from 23 subjects. All participants had a normal or corrected-to-normal vision and were asked to abstain from assuming caffeine, tobacco, and alcohol before the experiment. Most experiments were performed around 1:30 PM, to induce fatigue easily, and the duration of an entire experiment was approximately 2 hours.
The environment was a simulated driving system: a four-lane highway scene was shown on a LCD screen in front of a real vehicle deprived of unnecessary components (e.g. the engine) and controlled with steering wheel and pedals under the participants' actions. The route was monotonous and primarily straight to induce fatigue more easily in the subjects.
EEG and forehead EOG signals were recorded using Neuroscan system at a \SI{1}{\kilo \hertz} sampling rate (\SI{500}{Hz} bandwidth). The EEG setup recorded 12-channel EEG signals from the posterior site (CP1,
CPZ, CP2, P1, PZ, P2, PO3, POZ, PO4, O1, OZ, and O2)
and 6-channel EEG signals from the temporal site (FT7,
FT8, T7, T8, TP7, and TP8) using the 10-20 system.
Like \cite{zheng2017multimodal}, we split the EEG data into three categories based on the PERCLOS labels, which represents the percentage of eyelid closure over the pupil over time \cite{dinges1998perclos}.
Three vigilance classes, based on the PERCLOS index, were defined:\begin{itemize}
\item \emph{Awake} class: ${PERCLOS} < 0.35$;
\item \emph{Tired} class: $ 0.35 \leq {PERCLOS} < 0.7$
\item \emph{Drowsy} class: $ {PERCLOS} \geq 0.7$
\end{itemize}
A preprocessing was done to filter out any artefact from the EEG data of every experiment: a zero-phase bandpass filtering from \SI{1}{Hz} to \SI{75}{Hz} was applied to the raw data, using a 4th-order Butterworth bandpass filter. The same procedure was done to obtain the frequency bands of interest (\emph{delta}, \emph{theta}, \emph{alpha}, \emph{beta}, \emph{gamma}).
After the pre-processing phase,the 1-D-LPB method \cite{chatlani2010local} and DE method \cite{zheng2017multimodal} were applied for the feature extraction.
The 1D-LBP feature was extracted with parameters $P=2$ and $P=4$ ($P$ is a parameter representing the number of neighbour samples thresholded against the centre one) as a window function of 8 seconds non-overlapping intervals over the EEG signal of every 17 channels for each band. The values of P were chosen small to keep the dimensions of the feature vector contained. Concatenating the feature vector of every channel, we obtained a feature vector for every frame composed of 68 components (4 1D-LBP bins times 17 channels) in the $P=2$ case and 272 (16 1D-LBP bins times 17 channels) in the $P=4$ case.
\subsection{New metrics for temporal response}
To evaluate the classifier's temporal response, we defined new metrics to determine the system's performance.
\textbf{\textit{Hit rate:}}
The \emph{hit rate} is the ratio of successfully recognized class transitions (hits) to the total count of state changes: only one sample is needed to identify a transition and to send an alarm to the driver. We also defined \emph{0-delay hit rate} as the percentage of hits that happen on the first sample of the state change. These values are estimated for two distinct class transitions: \emph{awake to tired} (AT) and \emph{tired to drowsy} (TD). The \emph{awake to drowsy} case was not considered, because it is a very rare event in real applications and this is confirmed by the insignificant number of occurrences in the dataset (the participants go through the intermediate tired state before getting drowsy).
The \emph{hit rate} formula is:
\begin{equation}
\text{Hit rate}=\frac{H_{AT}}{n_{AT}} + \frac{H_{TD}}{n_{TD}}
\end{equation}
where $H_{\textit{xy}}$ is the \emph{hit rate} for the transition from class \textit{x} to class \textit{y} and $n_{xy}$ the number of transitions.
\textbf{\textit{Mean hit delay:}}
The \emph{hit delay} is defined as the mean of seconds of delay with which the response of the classifier correctly identifies a state change. \emph{Awake to tired} (AT) and \emph{tired to drowsy} (TD) class transitions were considered to obtain this measure.
The \emph{mean hit delay }can be formulated as:
\begin{equation}
\text{Mean hit delay}=\frac{\sum \limits^{n_{AT}} _{i=0} \Delta^{AT}_{i}+\sum\limits ^{n_{TD}} _{i=0}\Delta^{TD}_{i}}{n_{AT} + n_{TD}}
\end{equation}
where $\Delta^{\textit{xy}}= t(hit)_{xy}-t(change)_{xy}$ is the difference between the algorithm detection time of the transition from the class \textit{x} to the class \textit{y} and the actual time of that transition.
\textbf{\textit{False hit rate:}}
The \emph{false hit rate} is the rate of misclassified ``awake'' samples. This value is calculated separately for the \emph{tired instead of awake}, \emph{drowsy instead of awake}, \emph{tired or drowsy instead of awake} cases:
\begin{equation}
\text{False hit rate}=\frac{\text{FN}_{(awake)}}{\text{FN}_{(awake)}+\text{TP}_{(awake)}}
\end{equation}
where $FN_{(awake)}$ is the rate of false negative samples and $TP_{(awake)}$ is the rate of true positive samples for the awake class.
In addition to these new metrics, the accuracy was calculated as:
\begin{equation}
\text{Accuracy}=\frac{\sum \limits^{K} _{i=0} \frac{TP_i+TN_i}{TP_i+FP_i+TN_i+FN_i}}{K}
\end{equation}
where $K$ is the number of classes.
\subsection{Settings optimization}
\renewcommand{\arraystretch}{1.3}
\begin{table*}[!ht]
\centering
\caption{Assessment of the impact of filtering band on system accuracy [\%] using two different SVM classifiers and 1D-LBP features with $P = 2$ and $P = 4$. The accuracies are written as: `mean $\pm$ SD'.
}
{
\begin{tabular}{cc|c|c|c|c|}
\cline{3-6}
& & \multicolumn{2}{c|}{\textbf{Linear}} & \multicolumn{2}{c|}{\textbf{RBF}} \\\cline{3-6}
& & \textbf{ P=2} &\textbf{ P=4}&\textbf{ P=2} &\textbf{ P=4 }\\ \hline
\multicolumn{1}{|c|}{\multirow{6}{*}{\rotatebox[origin=c]{90}{\textbf{ EEG band}}}} &\textbf{ Delta} & $48.75\pm 0.09$ & $48.31\pm $0.10 & $51.74\pm 0.15$ & $49.30\pm 0.20$ \\ \cline{2-6}
\multicolumn{1}{|c|}{} & \textbf{ Theta} & $43.73\pm 0.08$ & $44.08\pm 0.09$ & $46.98\pm 0.21$ & $45.21\pm 0.29$ \\ \cline{2-6}
\multicolumn{1}{|c|}{} &\textbf{ Alpha} & $47.12\pm 0.06$ & $46.58\pm 0.08$ & $52.57\pm 0.10$ & $50.26\pm 0.06$ \\ \cline{2-6}
\multicolumn{1}{|c|}{} & \textbf{ Beta} & $54.65\pm 0.12$ & $54.42\pm 0.19$ & $60.94\pm 0.17$ & $58.11\pm 0.16$ \\ \cline{2-6}
\multicolumn{1}{|c|}{} &\textbf{ Gamma }& $51.93\pm 0.08$ & $52.15\pm 0.11$ & $57.80\pm 0.10$ & $55.81\pm 0.18$ \\ \cline{2-6}
\multicolumn{1}{|c|}{} & \textbf{ Total} & $63.81\pm 0.08$ & $65.02\pm 0.10$ & \boldmath $73.78\pm 0.05$ & $72.95\pm 0.09$ \\ \hline
\end{tabular}%
}
\label{tab:r1}
\end{table*}
We conduct a 5-fold-cross validation process for every EEG band, to gain a general perspective on how each band could perform in a test process.
Table \ref{tab:r1} shows the results of the different bands analyzed using two different classifiers, a linear SVM model and a SVM model with a Gaussian (RBF) kernel. Features were extracted using 1D-LBP with $P = 2$ and $P = 4$.
The reported results show that the system performs better using the whole band (1-50 Hz) than using single bands. The SVM model with the Gaussian kernel performed better in every case. Moreover, accuracy is higher for the 1D-LBP feature with $P=2$.
This first evidence allows us to point out that:
\begin{itemize}
\item The vigilance state is not provided by a specific sub-band of the EEG signal. It is a holistic process that requires the contributions of all bands.
\item The contribution of all bands cannot be described with a linear function, as it can be seen by comparing the accuracy of linear SVM and RBF SVM. As a matter of fact, if the performance difference between these classifiers is not significant when inspecting each EEG sub-band, it is noteworthy when we consider the broadband signal.
\end{itemize}
Therefore, the broadband signal is kept for the rest of the experiments. In order to assess the contribution of each channel, a 5-fold cross-validation was carried out, using $P=2$ for the 1D-LBP extraction and the SVM with RBF kernel ($k = 2$). Results are reported in Table \ref{tab:valchn}.
\begin{table}[!ht]
\centering
\caption{Channels' contribution in terms of mean accuracy [\%] using the 1D-LBP feature extraction ($P=2$) and a SVM classifier with RBF kernel ($k = 2$).}{
\begin{tabular}{lc|c|c|}
\cline{3-3}
& & \textbf{5-fold cross-validation accuracy}\\ \hline
\multicolumn{1}{|l|}{\multirow{17}{*}{\rotatebox[origin=c]{90}{\textbf{EEG channel}}}} & FT7 & \boldmath $52.14\pm 0.07$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & FT8 & $49.75\pm 0.11$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & T7 & $51.54\pm 0.08$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & T8 & $50.25\pm 0.06$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & TP7 & $50.07\pm 0.08$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & TP8 & $48.87\pm 0.13$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & CP1 & $47.10\pm 0.07$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & CP2 & $46.31\pm 0.14$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & P1 & $49.22\pm 0.02$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & PZ & $48.87\pm 0.11$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & P2 & $47.24\pm 0.10$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & PO3 & $49.44\pm 0.06$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & POZ & $50.18\pm 0.06$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & PO4 & $50.28\pm 0.04$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & O1 & $50.87\pm 0.17$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & OZ & $51.79\pm 0.14$ \\ \cline{2-3}
\multicolumn{1}{|l|}{} & O2 & $51.28\pm 0.03$ \\ \hline
\end{tabular}
}
\label{tab:valchn}
\end{table}
We observe that a channel alone cannot achieve the performance of all 17 channels. However, the performance increase from $52\%$ of the $FT7$ channel to $74\%$ of all channels suggests a weak correlation among 1D-LBP features per channel.
On the basis of these observations, we keep the whole channels for the final BCI design and implementation.
\section{BCI design and test}
\subsection{Experimental protocol}
To test the BCI performance, we carried out two types of tests:
\begin{itemize}
\item In a first attempt, we train and test the system on the same group of users, a common approach when designing BCI interfaces. We refer to this as ``user-specific'' test.
\item Secondly, we train the system on a user population that is totally different from that involved for the final use, in order to investigate the system's generalization capability. We indicate this test as ``generic-users'' test.
\end{itemize}
The dataset was randomly split into two parts, a first part containing $\sim80\%$ of users (18 users) and a second part containing the remaining $\sim20\%$ of users (5 users).
The first part was divided to make a 5-fold-cross validation, and for each iteration after training, the accuracy was calculated on both the validation set (user-specific) and the unknown users from the second part (generic-user).
The experiments were done both with features extracted using the proposed 1D-LBP, and DE features \cite{zheng2017multimodal}.
Each experiment was repeated 20 times and the results are averaged over those runs.
\subsection{Results}
\label{results}
In this section, we reported the results of the experiments carried out to evaluate the applicability of the 1D-LBP method for the feature extraction in a system for drivers' vigilance estimation through the analysis of the EEG signal.
At first, the experiments to evaluate the feasibility of the generic-users vigilance estimation system were made.
We then evaluated how the system responds in the temporal domain, using the metrics introduced in section \ref{exprotocol}. Finally, we evaluate the statistical significance of the experiments performed.
\subsubsection{User-specific and generic-users systems and comparison with the state of the art}
Tab. \ref{tab:r2} shows the generic-user and user-specific system's accuracy. The comparison with the state of the art shows that the best general performance is obtained by the SVM model with RBF kernel trained with 1D-LBP features. The improvement we obtained is relevant by considering the amount of data and the compactness of the 1D-LBP features set.
\begin{table*}[!ht]
\centering
\caption{User-specific and generic user test accuracies [\%]. 1D-LBP is compared to DE in terms of accuracy in classification. Results for generic user tests show how a generic user system could have very low performance.}
{
\begin{tabular}{c|c|c|c|}
\cline{2-4}
& 1D-LBP (linear) & 1D-LBP (RBF) & DE (RBF) \\ \hline
\multicolumn{1}{|c|}{User-specific} & $65.78 \pm 1.68$ & \boldmath $77.89 \pm 1.17$ & $70.36 \pm 1.32$ \\ \hline
\multicolumn{1}{|c|}{Generic user} & $43.27 \pm 8.46$ & \boldmath $44.50 \pm 8.55$ & $42.36 \pm 9.24$ \\ \hline
\end{tabular}%
}
\label{tab:r2}
\end{table*}
Concerning the generic-users results, both 1D-LBP and DE features achieve a low, unacceptable performance. Actually, the EEG related to fatigue and drowsiness appears stable within an individual across different sessions, but it varies between users because of the individuality degree embedded in any EEG signal \cite{lin2005eeg}.
Indeed, the neurophysiological signals are characterized by strong inter-subjects variations. If we suppose that both 1D-LBP and DE are able to extract partly such individuality, the configurations of 1D-LBP and DE is consequently different even when observing the awake class. This led to a bad parametrization of the decision surface of the adopted classifier. This also explain why EEG is also proposed for personal identification, and in particular the differential entropy as measurement of individuality by \cite{Phung2014UsingSE}. The confirmation of this is given by the correspondent accuracies achieved by the user-specific system.
However, another point of view must be observed, before going in depth of the user-specific system performance. Fig. \ref{fig:confmats2} shows the confusion matrices of the three SVM models for the generic-users system. We are not interested in `tired' samples being confused with the drowsy class, because the corresponding feedback's alarm of the hypothetical BCI could be used as a precautionary measure in real-world applications. What we are more concerned about is the misclassification of `awake' samples. From Fig. \ref{fig:confmats2}, it is possible to notice that all the classifiers have a high accuracy in the classification of `drowsy' samples while generating confusion in the 'awake' and 'tired' ones. We read this result as an indirect confirmation that each user has its own way of representing, by EEG, a brain state in which he/she is still ''conscious'', that is, he/she has full or partial control of his/her actions. This makes very difficult to classify this cases. Where getting drowsy, this ability is partially lost, and the system becomes more accurate.
\begin{figure}[!ht]
\centering
\includegraphics[width=16cm]{figures/confmats2.eps}
\caption{Confusion matrices related to the comparison with the state of the art (DE features) for ``generic-users'' experiments. Each matrix was calculated as the `mean matrix' of all sessions' matrices and is colour coded (greater values, darker colours). For all three analyzed classifiers, the system is perfectly able to distinguish the ``drowsy'' class while the ``tired'' and ``awake'' classes are confused.}
\label{fig:confmats2}
\end{figure}
\subsubsection{Response dynamics} \label{dyncs}
In this section, we refer to the user-specific system, which show the best accuracy. Therefore, this design approach is the most promising candidate for a real BCI for the driver's vigilance detection.
We go in depth of its performance analysis, because it is not only important how accurate the system is, but also how the system responds in the temporal domain. In fact, what the hypothetical BCI should do is to identify the abrupt changes in the driver's cognitive state with the minimum delay.
Therefore, we defined new metrics to determine the performance of the system in the time domain. These metrics are the hit rate, the 0-delay hit rate, the mean hit delay and the false hit rate (presented in the Section \ref{exprotocol}). Tables \ref{tab:rates1}, \ref{tab:rates2} and \ref{tab:falserates} report the evaluation of these metrics.
First of all, we observe that the LBP-based classifier detects the $96.23\%$ of \emph{awake to tired} transitions with a mean delay of $6.52$ seconds and $86.75\%$ of \emph{tired to drowsy} transitions with a mean delay of $70.96$ seconds. The \emph{awake to tired} state change is detected with zero delay in the $7.36\%$ of hits. These data suggest that the extracted feature could be used in the case that BCI system has to alarm the driver when he or she is getting tired. However, the false rate relative to the tired class discourages this kind of detection in real-world scenarios, because the $52\%$ of alarms are given when the driver is in the awake state. Consequently, 1 over 2 times the driver would be bored by a useless advice of stop driving because of getting tired. With such a behaviour of the vigilance system, the risk is that the driver would be lead to turn off it.
Beside this, the reported tables motivates the following observations:
\begin{itemize}
\item the \emph{awake to tired} transition is detected better than the \emph{tired to drowsy} one. This difference could be caused by a higher confusion between the drowsy and tired classes. Due to the natural smoothness of such classes, this could be avoided by the assessment of different PERCLOS thresholds for labeling;
\item the 1D-LBP features perform better than the DE feature. This means that the micro-textural information extracted through the 1D-LBP is more discriminating than the signal entropy, but their performance leads to an indirect confirmation of the respective claims;
\item the \emph{tired to drowsy} state change is recognized with a substantial delay, over a minute, that is unacceptable in real-world applications. In a real application the BCI should therefore rely its operation on the \emph{awake to tired} transition, the only one approaching a reasonable speed to alert the driver;
\item `awake' samples are wrongly marked as `tired' the 52\% of the time. This means that the BCI could alarm the awake driver in a significant number of occurrences. Anyway, this kind of misclassification is much less crucial than the \textit{awake-to-drowsy} or the \textit{tired-to-drowsy} transitions, therefore the system could be based on the latter to generate the alarm and avoid false alarms.
Unfortunately, as mentioned before, the \textit{awake-to-drowsy} transition is infrequent and the \textit{awake-to-drowsy} transition has a mean hit delay above the minute. For this reason, it is possible to state that these limitations make this kind of systems immature for a real application.
\end{itemize}
\begin{table*}[!ht]
\centering
\caption{Hit rate [\%], 0-delay hit rate [\%] and mean hit delay [s] for the \emph{Awake to tired} transition. The detection of this transition is very accurate, especially for the 1D-LPB method.}{
\begin{tabular}{c|c|c|c|}
\cline{2-4}
\emph{Awake $\rightarrow$ tired} & 1D-LBP (linear) & 1D-LBP (RBF) & DE (RBF) \\ \hline
\multicolumn{1}{|c|}{Hit rate [\%]} & $96.20\pm 6.30$ & \boldmath $98.10\pm 2.70$ & $92.40\pm 8.70$ \\ \hline
\multicolumn{1}{|c|}{0-delay hit rate [\%]} & $72.80 \pm 15.10$ & \boldmath $73.40\pm 13.20$ & $70.70\pm 16.80$ \\ \hline
\multicolumn{1}{|c|}{Mean hit delay [s]} & $16.55 \pm 19.97$ & \boldmath $6.52 \pm 5.66$ & $10.35 \pm 8.72$ \\ \hline
\end{tabular}%
}
\label{tab:rates1}
\end{table*}
\begin{table*}[!ht]
\centering
\caption{Hit rate [\%], 0-delay hit rate [\%] and mean hit delay [s] for the \emph{Tired to drowsy} transition. The comparison with the previous table shows that this status change is detected less than the \emph{awake to tired} transition. However the use of the 1D-LBP features with respect to the state of the art considerably increases the detection of this transition.}
{
\begin{tabular}{c|c|c|c|}
\cline{2-4}
\emph{Tired $\rightarrow$ drowsy} & 1D-LBP (linear) & 1D-LBP (RBF) & DE (RBF) \\ \hline
\multicolumn{1}{|c|}{Hit rate [\%]} & \boldmath $86.80\pm 13.40$ & $80.10\pm 18.80$ & $48.80\pm 24.90$ \\ \hline
\multicolumn{1}{|c|}{0-delay hit rate [\%]} & \boldmath $31.10\pm 17.60$ & $20.00\pm 14.10$ & $13.80\pm 16.08$ \\ \hline
\multicolumn{1}{|c|}{Mean hit delay [s]} & $70.96\pm 58.65$ & \boldmath $66.40\pm 43.47$ & $118.64 \pm 98.21$ \\ \hline
\end{tabular}%
}
\label{tab:rates2}
\end{table*}
\begin{table*}[!ht]
\centering
\caption{False hit rates [\%]: the values were calculated for three misclassification cases. The evaluation of this metric confirms what is shown by the confusion matrices. The awake class is more confused with the tired class. The low confusion between awake and drowsy classes is a positive result for the BCI operation.}
{
\begin{tabular}{c|c|c|c|}
\cline{2-4}
\emph{Awake misclassified as} & 1D-LBP (linear) & 1D-LBP (RBF) & DE (RBF) \\ \hline
\multicolumn{1}{|c|}{Tired} & \boldmath $52.25\pm 15.66$ & $56.21\pm 14.10$ & $74.12\pm 15.96$ \\ \hline
\multicolumn{1}{|c|}{Drowsy} & \boldmath $6.52\pm 10.59$ & $8.79\pm 11.55$ & $5.45\pm 7.03$ \\ \hline
\multicolumn{1}{|c|}{Tired or drowsy} & \boldmath $58.77\pm 15.76$ & $65.00\pm 15.63$ & $79.57 \pm 17.14$ \\ \hline
\end{tabular}%
}
\label{tab:falserates}
\end{table*}
\subsubsection{Statistical significance analysis}
In this section, we analyze if the hit rate values calculated for the \emph{awake to tired} and the \emph{tired to drowsy} transitions and the value of false hit rate calculated for the awake class have a statistical significance concerning the sample size. By obtaining a positive answer, our results could be taken into account by the scientific community for further investigations or to confirm or deny our claims by adopting the same data, experimental protocols and evaluation metrics.
In Table \ref{table:samples}, the number of samples of the awake class with respect to the total number of samples and the number of transitions between the awake and tired classes and between the tired and drowsy classes have been reported.
Their statistical significance is assessed by evaluating the \textit{margin of error}, that is, the interval estimation expressing the range of values above and below a confidence interval \cite{stat}.
It is possible to calculate the margin of error by the following:
\begin{equation}
\centering
margin \thinspace of \thinspace error = critical\thinspace value * standard\thinspace error
\end{equation}
When the sampling distribution of a statistic is normal or nearly gaussian, the critical value can be expressed as a t-score or as a z-score. The critical value as a z-score at 95\% level of confidence is $\pm1.96$.
The standard error ($SE$) can be calculated as:
\begin{equation}
\centering
SE=\sqrt{\dfrac{p(1-p)}{n}}
\end{equation}
in which $n$ is the sample size and $p$ is the population proportion.
The goal was to calculate the margin of error on the hit rate calculated for the \emph{awake to tired} and the \emph{tired to drowsy} transitions and on the false hit rate calculated for the awake class.
By considering the samples distribution shown in Table \ref{table:samples}, we have:
$p=7405/20355=0.36$, being the sample size equal to 20355 (23 experiments * 885 samples).
From Eq. (8), we get a standard error equal to $0.0033$ for the awake class.
\begin{table}
\centering
\caption{SEED-VIG dataset composition based on the awake classes and the \emph{Awake $\rightarrow$ tired} and \emph{Tired $\rightarrow$ drowsy} transitions.}{
\begin{tabular}{|l|l|}
\hline
\textbf{Total samples} & 20355 \\ \hline
\textbf{Awake samples} & 7405 \\ \hline
\textbf{Total transitions} & 333 \\ \hline
\textbf{\emph{Awake $\rightarrow$ tired}} & 114 \\ \hline
\textbf{\emph{Tired $\rightarrow$ drowsy}} & 60 \\ \hline
\end{tabular}
}
\label{table:samples}
\end{table}
\begin{table}
\centering
\caption{Margins of error referred to the false hit rate calculated for the awake class and to the hit rate calculated for the \emph{awake to tired} and the \emph{tired to drowsy} transitions.}
{
\begin{tabular}{|l|c|}
\hline
\textbf{} & \textbf{Margin of error} \\ \hline
\textbf{Awake samples} & 0.66\% \\ \hline
\textbf{\emph{Awake $\rightarrow$ tired}} & 5.10\% \\ \hline
\textbf{\emph{Tired $\rightarrow$ drowsy}} & 4.13\% \\ \hline
\end{tabular}
}
\label{table:margin}
\end{table}
The margin of error is 0.66\%, which corresponds to the maximum estimation error under the assumption of gaussian distribution of measurements. Performing the same steps but considering $n=333$, for the \emph{awake to tired} and the \emph{tired to drowsy} transitions, the margin of error is, respectively, 5.10\% and 4.13\% (Table \ref{table:margin}).
In conclusion, the hit rate results for the \emph{awake to tired} and the \emph{tired to drowsy} transitions are accurate at the 95\% confidence level respectively plus or minus 5 and 4 percentage points. The false hit rate (Table \ref{tab:falserates}) is accurate plus or minus one percentage point. These reported values allow us to consider reliable the performance parameters values obtained in our experiments.
\section{Conclusions}
\label{conclusion}
In this work, we proposed the 1D-LBP algorithm for assessing the driver's vigilance by the acquisition of the electroencephalography signal (EEG). We dealt with a three-classes classification problem where the assigned classes are awake, tired and drowsy. The novelty of the approach relies on a textural descriptor with the aim of characterizing the transitions from a class to another one by a set of bit-codes over a time window. It was expected that, whilst a transition awake-to-drowsy appeared, the electrodes might point out a decrease in the neural activity devoted to the scene observation, and this variation impact on the frequencies of the bit-codes from a time window to the next one. The one-dimensional LBP (1D-LBP) is a instrument designed to investigate if this is true.
For confirming our claim, we assessed well-known performance parameters and introduced novel ones for the transitions detection and the related time delay. These new temporal metrics allowed us to evaluate accurately the limitations of the proposed method. We compared our results with those of a reference method, based on a similar hypothesis, by adopting the same publicly available data set and the same experimental protocol. Instead, the majority of published works on this topic used home-made data, thus making impossible to assess the level of current achievements.
The proposed method appeared to exhibit strong effectiveness in detecting the awake-to-tired transitions (only 6 sec of delay and the best hit rate, namely $98\%$). Therefore, the claim of differentiating this relevant changes in the vigilance state by
extracting a set of bit-codes frequencies appeared to be confirmed.
On the other hand, the proposed method did not exhibit an user-independent performance and still needs user-specific settings to be effective.
Future works will be devoted to improve the performance, especially by reducing the false detection rate, in order to make the implementation of a BCI based on 1D-LBP feasible in real applications.
\bibliographystyle{abbrv}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: A problem concerning real functions What are the all possible values of $r$ such that there is a function $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any $x$, $$f''(x) > f'(x) + r$$ and $$f'(x) > f(x) + r$$ ??
A: Leaving it to you to fill in the details, but (barring a mistake) the following approach seems to work.
*
*Show that the choice $f(x)=e^{2x}$ works, whenever $r\le0$.
*If $r>0$, then, assuming that a suitable function $f(x)$ exists, consider the function $$g(x)=f'(x)-f(x).$$ Show that $g'(x)>r$ for all $x$. With the aid of mean value theorem show that $$\lim_{x\to-\infty}g(x)=-\infty.$$ Then find a contradiction.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: Ordenar filas iguais Eu tenho 4 colunas no excel: A,B,C,D certo?
Só que eu preciso ordenar para realizar uma comparação pois está toda a informação desorganizada me explico:
Arquivo Original em .CSV (Informação desorganizada )
A(Inglês) B (Tradução) C(Inglês) D (Trad.)
"xxx.yyy.Hello" "Oi" "xxx.yyy.Text" Texto
"xxx.yyy.Green" "Verde" "xxx.yyy.Hello" Hello
Como gostaria que ficasse assim:
A(Inglês) B (Tradução) C(Inglês) D (Trad.)
"xxx.yyy.Hello" "Oi" "xxx.yyy.Hello" "Hello"
"xxx.yyy.Green" "Verde" "xxx.yyy.Text" "Texto"
Resumindo devem ser iguais os valores dos campos A e C, pra ficar mais fácil de realizar a comparação das traduções.
Observação: O arquivo contém aprox 4000 linhas em cada fila (A,B,C,D).
Existe forma de realizar esse tipo de ordenação no excel, plugin, etc...
Desde já obrigado!!!
A: É possível realizar o que foi pedido por fórmulas do Excel.
Dados de Entrada
Resultado
Como fazer?
Foram inseridas novas fórmulas nas colunas F,G,H e I para comparar as duas colunas e retornar na mesma linha quando houver palavras iguais
Coluna F
=SE($H1="";"";A1)
Coluna G
=SE($H1="";"";B1)
Coluna H
=SE(É.NÃO.DISP(CORRESP(A1;C:C;0));"";ÍNDICE(C:C;CORRESP(A1;C:C;0)))
Coluna I
=SE($H1="";"";ÍNDICE(D:D;CORRESP(A1;C:C;0)))
Problema
Quando houver duplicatas no segundo arquivo, não funcionará do modo que está descrito.
Portanto a sugestão para corrigir este problema (se houver) é olhar este link
A: Sugiro utilizar a formula PROCV. Pode construir uma folha com as seguintes colunas:
Referencia | Tradução PT | Tradução EN
Algo como:
Em que a Tradução PT e Tradução EN são obtidas através de uma procura na folha/documento onde estão. As referências sendo coincidentes nas duas folhas bastaria copiar de uma delas e colar nesta nova folha.
Para a Tradução PT utilizaria a formula:
=PROCV(A2;Folha2!$A$1:$B$4;2;FALSO)
Em que A2 é a referencia e Folha2!$A$1:$B$4 é a folha com as traduções em Português. Visualmente a folha de traduções em português ficaria assim:
Note que a ordem das referências nesta folha é irrelevante pois os valores são obtidos por procura.
Para a Tradução EN a formula seria igual mas noutra folha:
=PROCV(A2;Folha3!$A$1:$B$4;2;FALSO)
Com isto consegue rapidamente ver quais estão em falta. Pode inclusivamente fazer uma formula ao lado para lhe indicar sempre que uma tradução está em falta assim:
Em que o campo Em falta utiliza a formula:
SE(B2=C2;"FALTA";"")
Que apenas compara se ambas as traduções são iguais e se forem diz que a tradução está em falta.
Com isto consegue filtrar pelos que faltam simplificando consideravelmente o manuseamento da folha. Pode até saber quantas traduções estão em falta no total, com uma formula como =CONTAR.SE(D2:D5;"FALTA") especificando como intervalo os valores todos para a coluna Em falta.
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"redpajama_set_name": "RedPajamaStackExchange"
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The dragon flying over Tiadun bay is the only thing that Darna loves in the provinces, and she's also the only person she knows of who can see it. There's nothing else she likes about life at Tiadun keep. When she learns that she might be the daughter of the prince, she's afraid she'll be trapped there forever so she flees to the city of Anamat. In the city, there will be others who can see the dragons, or so the minstrels say.
Along the way, she meets Myril, an older girl with frequent premonitions and an eerie sense of hearing. At the walls, they find Iola, so dragon-struck that she wants to be a priestess, and Thorat, her devoted champion.
Despite these newfound friends, life in the city isn't easy. Darna scavenges for scraps and just about gets by, but when she's offered a sack of gold beads for a small bit of thieving, she takes her chances… and ends up angering the city's patron dragon.
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"redpajama_set_name": "RedPajamaC4"
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Lycosa iranii är en spindelart som beskrevs av Pocock 1901. Lycosa iranii ingår i släktet Lycosa och familjen vargspindlar. Inga underarter finns listade i Catalogue of Life.
Källor
Externa länkar
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iranii
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,358
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\section{Introduction}
Interference effects under multiple light scattering in optically thick disordered media
have been subject to intense investigation for almost three decades.
Coherent backscattering (CBS), which is closely related to weak localization, is one of the striking
examples of the types of effects which can occur. CBS manifest itself in an enhancement in the
intensity of light scattered in the nearly backwards direction. Wave
scattering in this direction along reciprocal, or time-reversed, multiple
scattering paths preserves the relative phase, which results in constructive
interference. The first detailed observations and analysis of this effect for light were
made in \cite{1}-\cite{3} and by now CBS in solids and liquids has
been investigated in detail \cite{4}-\cite{8}.
Observation of CBS of light in atomic gases is complicated because atomic motion
causes random phase shifts of scattering waves which are different for
reciprocal paths. For this reason, the weak localization has been observed only for a cold
atomic ensemble. The first experiments on CBS in ultracold atomic samples
prepared in a magneto-optical trap \cite{9}-\cite{11} shown that there are a
range of interesting features of this phenomena compared with CBS observed earlier
for solids and liquids \cite{4}-\cite{8}. These features, which have their origin in the atomic nature of the scatterers, could not be understood and quantitatively described by approaches developed previously
for classical scatterers such as powders and suspensions.
Detailed treatments of CBS, taking into account the quantum nature of atomic
scatterers, was developed by several groups \cite{12}-\cite{19.0}.
In these papers different aspects of weak localization were considered.
Particularly it was shown that one can have a strong influence on the observed interferences by
reinforcing those scattering channels which lead to interference and suppressing
those which do not. Such desirable effects can be realized for example by
polarization of atomic angular moments. By means of optical orientation
effects it is possible to collect all atoms in one Zeeman sublevels
ensuring the fulfillment of optimal interference. Similar effects can be
achieved by applying an auxiliary static magnetic field and tuning the frequency of
the light in such a way that light would interact only with the desire Zeeman
sublevel \cite{19}.
Experiment \cite{19} has shown that a static external field
influences the process of multiple scattering and the associated interferences. At the same time it is well known that control of optical properties of matter by means of auxiliary quasiresonant electromagnetic fields is
much more effective than by a static one. Electromagnetically induced atomic
coherence changes the optical properties of atomic samples in sometimes
dramatic ways, and is responsible for such effects as population trapping,
electromagnetically induced transparency (EIT), "slow light"\,, "stopped
light," to name a few \cite{20}-\cite{22}.
In this paper we are going to analyze how a resonant control field influences
coherent backscattering of light. In particular, we consider CBS effects under conditions of electro-magnetically
induced transparency. Our attention is focused on the spectral
dependence of the relative amplitude of the backscattering cone in a steady-state
regime, i.e. on the dependence of dthe enhancement factor on frequency of scattered
light which assumed to be monochromatic. We show that a control field can
essentially change the type of interference and even can cause destructive
interference. For some scattering polarization channels and for some
detunings of the probe light, the enhancement factor can be less than one. With
the developed knowledge of the spectrum we also consider the dynamics of CBS in the case of
pulsed probe radiation.
\section{Basic assumptions}
One of main quantitative characteristics of CBS is the enhancement factor, which
determines the relative contribution of interference effects to the total
scattered light intensity. In experiment it is measured as the relative amount of light
intensity scattered into a given direction inside CBS cone to the background
intensity registered outside the cone. In theory it is more convenient to
evaluate the differential cross section of scattering from the input to outgoing
mode and calculate the enhancement factor as the ratio of this cross section to
its non-interfering part.
Our theoretical approach allowing calculation of the differential cross section of
light scattered from optically thick ultracold atomic ensembles is described
in details in a series of papers \cite{13}, \cite{16}-\cite{18}. This
approach is based on a diagrammatic technique for nonequilibrium systems and allows
us to obtain separately both interfering and noninterfering parts as
a series over a number of incoherent scatterings.
The generalization of this approach to the case of the presence of a coherent
control field was made in \cite{23} and \cite{24}. In Appendix A we show, as
an example, the double scattering contributions to the differential cross section.
On the basis of these and similar expressions for higher order scattering
contributions, we calculate here the spectral dependence of the enhancement
factor. We consider probe light scattering from ultracold clouds of $\phantom{a}^{87}$Rb
atoms, prepared in a magneto-optical trap after the trapping and repumping
lasers and the quadrupole magnetic field are switched off. All atoms populate
the F=1 hyperfine sublevel of the ground state, while the distribution over Zeeman sublevels
is uniform. The spatial distribution of atoms is assumed to be spherically
symmetric and Gaussian. For the typical conditions of the trap, the Doppler
width is many times smaller than the natural line width of the excited state
and the interatomic distances on average are much larger than the optical
wavelength (dilute medium). This allows us to neglect all effects
associated with atomic motion, and atomic collisions.
Probe radiation is quasi resonant with the $F=1\rightarrow F^{\prime }=1$
transition of the $D_{1}$ line (see Fig. 1) and its polarization can be arbitrary. However, for
definiteness we will consider right or left-handed circularly polarized
light. This light is assumed to be weak; all nonlinear effects connected
with the probe radiation will be neglected. In our calculations this field will
be taken into account only in the first non-vanishing order. Besides the probe
light, the atomic ensemble interacts with a coupling, control field. In this paper
we will consider this field tuned to exact resonance with $F=2\rightarrow
F^{\prime }=1$ transition. Its amplitude is determined by the Rabi frequency
$\Omega _{c}=2|V_{nm^{\prime }}|,$ $V_{nm^{\prime }}$ are the transition
matrix elements for the coupling mode between states $|n\rangle $ and $%
|m^{\prime }\rangle \equiv |F,m^{\prime }\rangle $, which we define with
respect to the $|m^{\prime }\rangle $ $=|{F=2,m^{\prime }=-1>}\rightarrow
|n\rangle =|{F^{\prime }=1,n=0>}$ hyperfine Zeeman transition. Other
transition matrix elements are proportional to $\Omega _{c}$ and algebraic
factors depending on corresponding Clebsch-Gordon coefficients.
\begin{figure}[th]
\begin{center}
{$\scalebox{0.5}{\includegraphics*{fig1.eps}}$ }
\caption{Excitation scheme for observation of the EIT effect in
the system of hyperfine and Zeeman sublevels of the $D_1$-line of
${}^{87}$Rb. The coupling field is applied with right-handed
circular polarization to the $F=2\to F'=1$ transition. The probe
mode is applied to $F=1\to F'=1$ transition and can cause
different excitations depending on its polarization and
propagation direction. }
\end{center}
\par
\label{f1}
\end{figure}
The polarization of the control field is assumed to be right handed and circular. The
detuning of the probe laser frequencies from the corresponding atomic
resonances is assumed to be much less than the hyperfine splitting. This
circumstance, along with the relatively large hyperfine splitting in the excited
state, allows us take into account only one hyperfine sublevel $F' = 1$ of
this state.
In this paper we will not focus our attention on the angular distribution and shape of the CBS
cone, but will instead restrict ourselves by consideration of exact backscattering only.
The ground state hyperfine splitting of ${}^{87} Rb$ is about 6.83 GHz, so Rayleigh
and elastic Raman scattering (associated with the $F=1\rightarrow F^{\prime
}=1\rightarrow F=1$ transition) is well spectrally separated from inelastically Raman
scattered light $(F=1\rightarrow F^{\prime }=1\rightarrow F=2)$. We will
assume spectral selection under photo detection. The inelastic Raman
component is assumed to be not registered. Depending on the type of
polarization analyzer used for photodetection, four main polarization schemes
can be considered $H_{+}\rightarrow H_{+},\,H_{+}\rightarrow
H_{-},\,H_{-}\rightarrow H_{+},\,H_{-}\rightarrow H_{-}$. Here $H_{\pm }$
represents the helicity of the input and outgoing light. Note that, despite homogeneous
population of the Zeeman sublevels, the susceptibility tensor becomes
essentially anisotropic due to the presence of the coupling field \cite{23},%
\cite{24}. In such a case the enhancement factor depends not only on the relative
polarizations of the input and output light but also on their specific types.
\section{Results}
The results of calculations of the spectrum of the enhancement factor for a weak
control field are shown in Fig. 2a. The calculations were performed for
an atomic cloud with a Gaussian radius equal to $r_{0}=0.5\,cm$; maximal density is $%
n_{0}=3.2\cdot 10^{10}cm^{-3}$. The Rabi frequency of the control field is $\Omega
_{c}=0.5\Gamma $. For a weak control field we observe a not particularly surprising behavior of
the spectrum. Against a background typical for the spectral dependence of CBS
effect we see narrow spectral gap which has its origin in the decreasing of
the optical depth of the cloud caused by the EIT phenomena. Under the EIT effect the
probability of higher order scattering relatively decreases compared with
single scattering and all interference effects which connect with multiple
scattering are reduced. The width of the gap in the spectrum is about the width
of the transparency window determined by the EIT phenomenon. We point out only that
there is a certain difference in the width for different polarization
channels caused by the above-mentioned optical anisotropy of the atomic ensemble.
The situation changes dramatically when the control Rabi frequency becomes comparable to or
larger than the spontaneous decay rate. In Fig. 2b, which is calculated for $%
\Omega _{c}=3\Gamma $, in addition to the more noticeable anisotropy, an essential
transformation of the spectrum takes place. The range of the structure of this spectrum
connects with the difference in the Autler-Townes splitting for different Zeeman
transitions. The latter is caused by different dipole moments of
the corresponding transitions and consequently with different Rabi frequencies for
them. The maximal value of the enhancement factor for polarization channels with
changing helicity is almost the same as for weak control field but for the
case with preserving helicity we see qualitative modifications. The main one of these is
that, instead of constructive interference for some spectral regions, we
observe destructive interference. In these regions, for channels $H_{-}\rightarrow H_{-}
$ and $H_{+}\rightarrow H_{+}$, the enhancement factor becomes less than unity. That is, in
place of a CBS cone we have a CBS gap, or anticone. In spite of the relatively small value of the
gap it seems physically important because CBS or weak localization itself in
its \textquotedblleft traditional\textquotedblright\ interpretation connects
with time-reversal and always causes enhancement in back scattering.
\begin{figure}[th]
\begin{center}
{$\scalebox{0.5}{\includegraphics*{Figure2acolor.eps}}$ }{$\scalebox{0.5}{%
\includegraphics*{Figure2bcolor.eps}}$ }
\caption{Spectrum of the enhancement factor. (a) $\Omega_{c}=0.5\Gamma $, (b) $\Omega_{c}=3\Gamma $}
\end{center}
\par
\label{fig2}
\end{figure}
A similar effect for scattering of polarized electrons in a solid was shown in
\cite{25}. Following this, antilocalization in electron transport has been widely studied in
a range of physical systems where electrons interact directly with magnetic
impurities and where the spin-orbit interaction is important \cite{26,27}. The
possibility to observe destructive interference in the case of light
scattering from ultracold atomic ensemble was predicted for the first time in \cite{16}.
This effect was explained in \cite{16} by the hyperfine interaction in atoms and by possible
interference of transitions through different hyperfine sublevels of excited
atomic states. In the case considered now, the mechanism of antilocalization is
different. This is emphasized by Fig. 2b, in which the results are obtained for only one excited state sublevel $F'=1$
without taking into account possible hyperfine interactions.
The fundamental possibility of destructive interference under multiple
scattering connected with the coupling field is illustrated in Fig. 3. Here
we show double backscattering of a positive helicity incoming photon from the probe light beam on a
system consisting of two ${}^{87}$Rb atoms; the exit channel consists of detection
of light also of positive helicity. Here the double scattering is a
combination of the Raleigh-type and Raman-type transitions. In the direct path the scattering
consists of a sequence of Rayleigh-type scattering in the first step and of
Raman-type scattering in the second one. In the reciprocal path, Raman-type
scattering occurs first, and the positive helicity photon undergoes
Rayleigh-type scattering in the second step. There is an
important difference in the transition amplitudes associated with Raleigh
process for these two interfering channels. Indeed, in the direct and
reciprocal path the scattered mode is coupled with different Zeeman
transitions. In the absence of a coupling field these transitions have the
same amplitude and we always have constructive interference. The coupling field
essentially modifies the scattering process and this modification is different
for different Zeeman transitions. For definite frequencies of probe light, the
scattering amplitudes connecting the direct and reciprocal scattering
channels can be comparable in absolute value but can have phases shifted by
an angle close to $\pi$. In this case these two channels suppress each other.
The considered example is not the only one, and appears only in the double scattering channel. There are
some others which lead to constructive interference. In the higher
scattering order the situation is similar. That is why in our results we have
only partial destructive interference and for the considered parameters only a small suppression of
backscattering.
\begin{figure}[th]
\begin{center}
{$\scalebox{0.9}{\includegraphics*{fig3.eps}}$ }
\caption{Diagram explaining the antilocalization phenomenon in the example of double scattering in the helicity
preserving scattering channel. Only the interfering transitions initiated by the probe are shown and
the presence of the control mode is specified in Fig. 1. }
\end{center}
\par
\label{f3}
\end{figure}
Note that the antilocalization effect discussed in \cite{16} takes place for
essentially nonresonant light and consequently is difficult for observation,
because it is challenging to prepare clouds with large optical depth for
such radiation. The effect considered here takes place for radiation which is
not far from resonance. It makes this case more promising for experimental
verification. Note also that similar antilocalization phenomena can be
observed in the case when the control field is absent but the probe radiation
is strong enough to make the Autler-Townes splitting noticeable \cite{28}.
Consider now how these peculiarities in the spectrum manifest themselves in time
dependent CBS in the case of a pulsed probe light. Here the most interesting
effects are observed for a weak control field when peculiarities in the
spectrum are sharp. Different spectral components of the input pulse scatter
differently and it is the reason for the essential transformation of the spectrum and
consequently the time profile of the pulse. Scattered light has a deficit of
near resonant photons for which the EIT mechanism works best. The gap in
the spectrum causes light beating effects which are different for scattering of
different orders. In Fig. 4 we show shapes of pulses which undergo single
(4a) and double (4b) scattering for the $H_{+}\rightarrow H_{-}$ polarization
channel. These graphs are calculated for a cloud with $n_{0}=3.2\cdot
10^{10}cm^{-3}$ and $r_{0}=0.5\,cm$. The input pulse has a Gaussian shape $%
I=I_{0}\exp (-t^{2}/\tau ^{2})$, the length of the pulse is $\tau ^{=}200\Gamma
^{-1}$.
\begin{figure}[th]
\begin{center}
{$\scalebox{0.5}{\includegraphics*{fig4a.eps}}$ }{$\scalebox{0.5}{%
\includegraphics*{fig4b.eps}}$ }
\caption{Shapes of the outgoing pulse. (a) single scattering, (b) double scattering}
\end{center}
\par
\label{f4}
\end{figure}
The shapes of single and doubly scattered pulses are essentially
different. Most important is that the intensity peaks for them are separated in time. When single
scattering intensity is maximal double scattering is very small and vice
versa. This effect arises through the relative time scales associated with single scattering in comparison with double scattering, as illustrated in Fig. 5. For comparison the input pulse shape is also shown.
\begin{figure}[th]
\begin{center}
{$\scalebox{0.5}{\includegraphics*{fig5.eps}}$ }
\caption{Time dependence of the enhancement factor in the case of pulsed probe radiation. Input pulse shape (in arbitrary
intensity units) is shown in gray}
\end{center}
\par
\label{f5}
\end{figure}
\section*{Acknowledgments}
This work was supported by the Russian Foundation for Basic Research (Grants
08-02-91355 and 10-02-00103). M.H. acknowledges support from the National
Science Foundation (NSF-PHY-0654226).
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"redpajama_set_name": "RedPajamaArXiv"
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One of the goals of the NLASW is to advance health and social policy to ensure the well-being of the citizens of Newfoundland and Labrador. This is achieved through the analysis of health and social policies through a social determinants of health framework and the production of policy documents highlighting the social work perspective. Members can access all of NLASW policy submissions by visiting the following links. While these policy documents are organized by category, there is overlap between all of these health and social policy issues.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}
Interleaved texts, e.g., multi-author entries for activity reports, and social media conversations, such as Slack are increasingly common.
However, getting a quick sense of different threads in interleaved texts is often difficult due to entanglement of threads, i.e, posts belonging to different threads occurring in one sequence; see a hypothetical example in \figref{interlv_summ}.
\begin{figure}[t!]
\footnotesize
\begin{center}
\resizebox{0.85\linewidth}{!}{
\includegraphics[width=\textwidth]{multiauthor_part5.png}
}
\end{center}
\caption{\label{fig:interlv_summ}In the upper part, 7 interleaved posts belonging to different threads occur in a sequence. In the background at the bottom, posts are disentangled (clustered) into 3 threads (posts are outlined with colors corresponding to threads), and in the foreground, single sentence abstractive summaries are generated for each thread.
}
\end{figure}
In conversation disentanglement, interleaved posts are grouped by the thread. However, a reader still has to read all posts in all clustered threads to get the insights. To address this shortcoming, \newcite{P18-1062} proposed a system that takes an interleaved text as input and provides the reader with its summaries. Their system is an unsupervised two-step system, first, a conversation disentanglement component clusters the posts thread-wise, and second, a multi-sentence compression component compresses the thread-wise posts to single-sentence summaries. However, this system has two major disadvantages: first, the disentanglement obtained through either supervised \cite{N18-1164} or unsupervised \cite{wang2009context} methods propagate its errors to the downstream summarization task, and therefore, degrades the overall performance, and second, the compression component is restricted to formulate summaries out of disentangled threads, and therefore, cannot bring new words to improve the fluency.
We aim to tackle these issues
through an end-to-end trainable encoder-decoder system that takes a variable length input, e.g., interleaved texts, processes it and generates a variable length output, e.g., a multi-sentence summary.
An end-to-end system eliminates the disentanglement component, and thus, the error propagation. Furthermore, the corpus-level vocabulary of the decoder provided it with greater selection of words, and thus, a possibility to improve language fluency.
In the domain of text summarization, hierarchical encoder, encoding words in a sentence (post) followed by the encoding of sentences in a document (channel), is a very commonly used method \cite{DBLP:conf/conll/NallapatiZSGX16,P18-1013}. However, hierarchical decoding is rare, as many works in the domain aim to comprehend an important fact from single or multiple documents. Summarizing interleaved texts provides us a unique opportunity to employ hierarchical decoding as such texts comprise several facts from several threads.
We also propose novel hierarchical attention, which assists the decoder in its summary generation process with 3-levels of information from the interleaved text; posts, phrases, and words, rather than traditional two levels; post and word \cite{AAAI1714636,DBLP:conf/conll/NallapatiZSGX16,tan2017neural,cheng2016neural}.
As labeling of interleaved texts is a difficult and expensive tas
, we devised an algorithm that synthesizes interleaved text-summary pairs corpora of different difficulty levels (in terms of entanglement) from a regular document-summary pairs corpus. Using these corpora, we show the encoder-decoder system not only obviates disentanglement component, but also enhances performance. Further, our hierarchical encoder-decoder system consistently outperforms traditional sequential ones.
To summarize, our contributions are
\begin{itemize}
\item We propose an end-to-end encoder-decoder system over pipeline to obtain a quick overview of interleaved texts.
\item To the best of our knowledge, we are first to use a hierarchical decoder to obtain multi-sentence abstractive summaries from texts.
\item We propose a novel hierarchical attention that integrates information from 3 levels; posts, phrases and words, and is trained end-to-end.
\item We devise an algorithm that synthesizes interleaved text-summary corpora, on which we verify pipeline system vs. encoder-decoder, sequential vs. hierarchical decoding, 2- vs. 3-level hierarchical attention. Overall, the proposed system attains 20-40\% performance gains on both real-world (AMI) and synthetic datasets.
\end{itemize}
\input{tsum_relatedwork}
\input{tsum_model}
\input{tsum_experiments}
\section{Conclusion}
We presented an end-to-end trainable hierarchical encoder-decoder architecture with novel hierarchical attention which implicitly disentangles interleaved texts and generates abstractive summaries covering the text threads. The architecture addresses the error propagation and fluency issues that occur in the two-step architectures, and thereby, adding performance gains of 20-40\% on both real-world and synthetic datasets.
\section*{Appendix}
\section{Interleave Algorithm}
In \algref{interleave_algo}, $\textsc{Interleave}$ takes a set of concatenated abstracts and titles, $\textit{C} = \langle\textit{A}_1;\textit{T}_1,\ldots,\textit{A}_{|C|};\textit{T}_{|C|}\rangle$, minimum, $a$, and maximum, $b$, number of abstracts to interleave, and minimum, $m$, and maximum, $n$, number of sentences in a source, and then returns a set of concatenated interleaved texts and summaries. $\textsc{window}$ takes a sequence of texts, $\textit{X}$, and returns a window iterator of size $\frac{|\mathit{X}|-\textit{w}}{t}+1$, where $\textit{w}$ and $\textit{t}$ are window size and sliding step respectively. $\textit{window}$ reuses elements of $\textit{X}$, and therefore, enlarges the corpus size. Notations $\mathcal{U}$ refers to a uniform sampling, $\left[\cdot\right]$ to array indexing, and $\textsc{Reverse}$ to reversing an array.
\label{app:algo}
\begin{algorithm}[!h]
\footnotesize
\caption{Interleaving Algorithm}\label{alg:interleave_algo}
\begin{algorithmic}[1]
\Procedure{Interleave}{$\textit{C}, \textit{a}, \textit{b}, \textit{m}, \textit{n}$}
\State $\hat{\textit{C}},\textit{Z}\gets \textsc{window}(\textit{C}, w=b, t=1)$, Array()
\While {$\hat{\textit{C}} \neq \emptyset$}
\State $\textit{C}^\prime, \textit{A}^\prime, \textit{T}^\prime, \textit{S} \gets \hat{\textit{C}}.\textsc{Next}()$, Array(), Array(), $\{\}$
\State ${\textit{r}}$ $\sim$ $\mathcal{U}$($\textit{a}, \textit{b}$)
\For {$\textit{j}$ = 1 to $\textit{r}$} \Comment{Selection}
\State $\textit{A}, \textit{T} \gets \hat{{\textit{C}}}[\textit{j}]$
\State $\textit{T}^\prime.\textsc{Add}(\textit{T}$)
\State ${\textit{q}}$ $\sim$ $\mathcal{U}$($\textit{m}, \textit{n}$)
\State $\textit{A}^\prime.\textsc{Add}(\textit{A}$[1:$\textit{q}$])
\State $\textit{S} \gets \textit{S}\cup\lbrace j_{\times q}\rbrace$
\EndFor
\State $\hat{\textit{A}}, \hat{\textit{T}} \gets$ Array(), Array()
\For {1 to $|{\textit{S}}|$} \Comment{Interleaving}
\State $\textit{k} \gets \mathcal{U}(\textit{S})$
\State $\textit{S} \gets \textit{S}\backslash{\textit{k}}$
\State $\textit{I} \gets \textsc{Reverse}(\textit{A}^\prime$[$\textit{k}]).\textsc{pop}()$
\State $\hat{\textit{A}}.\textsc{Add}(\textit{I}$)
\State $\textit{J} \gets \textit{T}^{\prime}$[$\textit{k}$]
\If {$\textit{J} \not\in \hat{\textit{T}}$}:
\State $\hat{\textit{T}}.\textsc{Add}(\textit{J}$)
\EndIf
\EndFor
\State $\textit{Z}.\textsc{Add}(\hat{\textit{A}}$;$\hat{\textit{T}}$)
\EndWhile
\State \Return $\textit{Z}$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{Parameters}
For the word-to-word encoder, the steps are limited to 20, while the steps in the word-to-word decoder are limited to 15. The steps in the post-to-post encoder and thread-to-thread decoder depend on the corpus type, e.g., Medium has 15 steps in post-to-post and 3 steps in thread-to-thread. In seq2seq experiments, the source is flattened, and therefore, the number of steps in the source is limited to 300. We initialized all weights, including word embeddings, with a random normal distribution with mean 0 and standard deviation 0.1. The embedding vectors and hidden states of the encoder and decoder in the models are set to dimension 100. Texts are lowercased. The vocabulary size is limited to 8000 and 15000 for Pubmed and Stack Exchange corpora respectively. We pad short sequences with a special token, $\langle PAD\rangle$. We use Adam (Kingma et al. 2014
with an initial learning rate of .0001 and batch size of 64 for training.
\section{Training Loss}
\begin{figure}[h!]
\centering
\resizebox{0.65\linewidth}{!}{
\includegraphics[width=0.99\textwidth]{running_avg_loss.PNG}
}
\caption{Running average training loss between seq2seq (pink) and hier2hier (gray) for Stack Exchange Hard corpus.}
\figlabel{hier_perf_comp2}
\end{figure}
\section{Examples}
\begin{table*}[htbp!]
\begin{center}
\resizebox{0.99\linewidth}{!}{
\footnotesize
\begin{tabular}{p{0.54\linewidth}|p{0.45\linewidth}}
\hline
\ding{51} botulinum toxin a is effective for treatment\ldots&\multirow{4}{5.5cm}{\ding{51} prospective randomised controlled trial comparing trigone-sparing versus trigone-including intradetrusor injection of abobotulinumtoxina for refractory idiopathic detrusor overactivity.}\\
\ding{51} the trigone is generally spared because of the theoretical\ldots&\\
\ding{51} evaluate efficacy and safety of trigone-including .\ldots&\\
\ding{70} most methadone-maintained injection drug users \ldots&\\
\ding{81} gender-related differences in the incidence of bleeding\ldots&\\
\ding{81} we studied patients with stemi receiving fibrinolysis\ldots&\\
\ding{70} physicians may be reluctant to treat hcv in idus because \ldots&\multirow{3}{5.8cm}{\ding{70} rationale and design of a randomized controlled trial of directly observed hepatitis c treatment delivered in methadone clinics.}\\
\ding{81} outcomes included moderate or severe bleeding defined
\ldots&\\
\ding{70} optimal hcv management approaches for idus remain
\ldots&\\
\ding{81} moderate or severe bleeding was 1.9-fold higher
\ldots&\multirow{2}{5cm}{\ding{81} comparison of incidence of bleeding and mortality of men versus women with st-elevation myocardial infarction treated with fibrinolysis.}\\
\ding{70} we are conducting a randomized controlled trial in a network\ldots&\\
\ding{81} bleeding remained higher in women even after adjustment
\ldots&\\
\hline
\end{tabular}}
\end{center}
\caption{\tablabel{example_dataset_a1}The left rows contain interleaving of 3 articles with 2 to 5 sentences and the right rows contain their interleaved titles. Associated sentences and titles are depicted by similar symbols.}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\resizebox{0.99\linewidth}{!}{
\footnotesize
\begin{tabular}{p{0.55\linewidth}|p{0.45\linewidth}}
\hline
\ding{51} the effects of short-course antiretrovirals given to\ldots&\multirow{4}{5.5cm}{\ding{51} hiv-1 persists in breast milk cells despite antiretroviral treatment to prevent mother-to-child transmission.}\\
\ding{81} good adherence is essential for successful antiretroviral\ldots&\\
\ding{51} women in kenya received short-course zidovudine ( zdv )\ldots&\\
\ding{51} breast milk samples were collected two to three times weekly.\ldots&\\
\ding{70} the present primary analysis of antiretroviral therapy with\ldots&\multirow{3}{5.8cm}{\ding{81} patterns of individual and population-level adherence to antiretroviral therapy and risk factors for poor adherence in the first year of the dart trial in uganda and zimbabwe.}\\
\ding{81} this was an observational analysis of an open multicenter\ldots&\\
\ding{70} patients with hiv-1 rna at least 5000 copies/ml were\ldots&\\
\ding{81} at 4-weekly clinic visits , art drugs were provided and \ldots&\\
\ding{70} the primary objective was to demonstrate non-inferiority\ldots&\\
\ding{81} viral load response was assessed in a subset of patients\ldots&\multirow{2}{5cm}{\ding{70} efficacy and safety of once-daily darunavir/ritonavir versus lopinavir/ritonavir in treatment-naive hiv-1-infected patients at week 48.}\\
\ding{168} we explored the link between serum alpha-fetoprotein levels\ldots&\\
\ding{81} drug possession ratio ( percentage of drugs taken between\ldots&\\
\ding{168} a low alpha-fetoprotein level ( $<$ 5.0 ng/ml ) was an\ldots&\\
\ding{70} six hundred and eighty-nine patients were randomized\ldots&\\
\ding{70} at 48 weeks , 84 \% of drv/r and 78 \% of lpv/r patients\ldots&\multirow{2}{5cm}{\ding{168} serum alpha-fetoprotein predicts virologic response to hepatitis c treatment in hiv coinfected patients.}\\
\ding{51} hiv-1 dna was quantified by real-time pcr .\ldots&\\
\ding{168} serum alpha-fetoprote in measurement should be integrated \ldots&\\
\hline
\end{tabular}
}
\end{center}
\caption{\tablabel{example_dataset_a2}The left rows contain interleaving of 4 articles with 2 to 5 sentences and the right rows contain their interleaved titles. Associated sentences and titles are depicted by similar symbols.}
\end{table*}
\begin{table}[!h]
\begin{center}
\resizebox{0.95\linewidth}{!}{
\begin{tabular}{m{0.02\linewidth}|m{0.95\linewidth}}
\hline
&\multicolumn{1}{c}{Interleaved Texts}\\
\Tstrut $0$& \boldblue{botulinum} \boldblue{toxin} a is \boldblue{effective} for \boldblue{treatment} of \boldblue{idiopathic} \boldblue{detrusor} overactivity ( [UNK] )\\
$1$& the [UNK] is generally [UNK] because of the theoretical risk of [UNK] reflux ( [UNK] ) , although studies assessing\ldots\\
$\dots$ & \multicolumn{1}{c}{\dots}\\
$3$& \boldgreen{most} \boldgreen{[UNK]} \boldgreen{injection} \boldgreen{drug} \boldgreen{users} ( \boldgreen{idus} ) have been infected with hepatitis c virus ( hcv ) , but\ldots\\
$4$&\boldred{[UNK]} \boldred{differences} in the \boldred{incidence} of \boldred{bleeding} and its relation to subsequent mortality in patients with st-segment elevation myocardial infarction\ldots\\
$\dots$ & \multicolumn{1}{c}{\dots}\\
$8$& \boldred{optimal} hcv management approaches for idus remain unknown .\ldots\\
$\dots$ & \multicolumn{1}{c}{\dots}\\
\hline
&\multicolumn{1}{c}{GroundTruth/Generation}\\
& prospective randomised controlled trial comparing trigone-sparing versus trigone-including intradetrusor injection of abobotulinumtoxina for refractory idiopathic detrusor overactivity.\\
0,1& \boldblue{efficacy of [UNK] [UNK] in patients with idiopathic detrusor overactivity : rationale , design}\\
& \\
& rationale and design of a randomized controlled trial of directly observed hepatitis c treatment delivered in methadone clinics.\\
3,4& \boldgreen{validation of a point-of-care hepatitis injection drug injection drug , hcv medication , and}\\
& \\
& comparison of incidence of bleeding and mortality of men versus women with st-elevation myocardial infarction treated with fibrinolysis .\\
4,8& \boldred{subgroup analysis of patients with st-elevation myocardial infarction with st-elevation myocardial infarction .}\\
\hline
\end{tabular}
}
\end{center}
\caption{\tablabel{result_example_a1}Interleaved sentences of 3 articles, and corresponding ground-truth and hier2hier generated summaries. The top 2 sentences that were attended ($\boldsymbol{\gamma}$) for the generation are on the left. Additionally, top words ($\boldsymbol{\beta}$) attended for the generation are colored accordingly.}
\end{table}
\begin{table}[!h]
\begin{center}
\resizebox{0.95\linewidth}{!}{
\begin{tabular}{m{0.04\linewidth}|m{0.95\linewidth}}
\hline
&\multicolumn{1}{c}{Interleaved Texts}\\
\Tstrut $0$& the \boldblue{effects} of \boldblue{short-course} \boldblue{antiretrovirals} \boldblue{given} to \boldblue{reduce} \boldblue{mother-to-child} \boldblue{transmission} ( [UNK] ) on temporal patterns of [UNK] hiv-1 rna\\
$1$& \boldgreen{good} \boldgreen{adherence} \boldgreen{is} \boldgreen{essential} \boldgreen{for} \boldgreen{successful} \boldgreen{antiretroviral} \boldgreen{therapy} ( \boldgreen{art} ) provision , but simple measures have rarely been validated\ldots\\
$2$& women in kenya received short-course zidovudine ( zdv ) , single-dose nevirapine ( [UNK] ) , combination [UNK] or short-course\ldots\\
$3$& breast milk samples were collected two to three times weekly for 4-6 weeks .\ldots\\
$4$&the \boldred{present} \boldred{primary} \boldred{analysis} of \boldred{antiretroviral} \boldred{therapy} \boldred{with} \boldred{
[UNK]} \boldred{examined} \boldred{in} naive subjects ( [UNK] ) compares the efficacy and\ldots\\
$\dots$ & \multicolumn{1}{c}{\dots}\\
$10$& we \boldyellow{explored} \boldyellow{the} \boldyellow{link} \boldyellow{between} \boldyellow{serum} \boldyellow{[UNK]} levels and virologic response in [UNK] [UNK] c virus coinfected patients .\ldots\\
$\dots$ & \multicolumn{1}{c}{\dots}\\
\hline
&\multicolumn{1}{c}{GroundTruth/Generation}\\
& hiv-1 persists in breast milk cells despite antiretroviral treatment to prevent mother-to-child transmission .\\
0,2& \boldblue{impact of hiv-1 persists on hiv-1 rna in human immunodeficiency virus-infected individuals with hiv-1}\\
& \\
& patterns of individual and population-level adherence to antiretroviral therapy and risk factors for poor adherence in the first year of the dart trial in uganda and zimbabwe .\\
1,3& \boldgreen{impact of a antiretroviral treatment algorithm on adherence to antiretroviral therapy in [UNK] ,}\\
& \\
& efficacy and safety of once-daily darunavir/ritonavir versus lopinavir/ritonavir in treatment-naive hiv-1-infected patients at week 48 .\\
4,2& \boldred{a randomized trial of [UNK] versus [UNK] in treatment-naive hiv-1-infected patients with hiv-1 infection}\\
& \\
& serum alpha-fetoprotein predicts virologic response to hepatitis c treatment in hiv coinfected patients .\\
\multirow{2}{1cm}{10,12}& \boldyellow{predicting virologic response in [UNK] coinfected patients coinfected with hiv-1 : a [UNK] randomized}\\
\hline
\end{tabular}
}
\end{center}
\caption{\tablabel{result_example_a2}Interleaved sentences of 4 articles, and corresponding ground-truth and hier2hier generated summaries. The top 2 sentences that were attended ($\boldsymbol{\gamma}$) for the generation are on the left. Additionally, top words ($\boldsymbol{\beta}$) attended for the generation are colored accordingly.}
\end{table}
\end{document}
\section{Dataset}
Obtaining labeled training data for conversation summarization is challenging. The available ones are either extractive \cite{verberne2018creating} or too small \cite{barker2016sensei,anguera2012speaker} to train a neural model. To get around this issue and thoroughly verify the proposed architecture, we synthesized a dataset by utilizing a corpus of conventional texts for which summaries are available.
We create two corpora of interleaved texts: one from the abstracts and titles of articles from the PubMed corpu
and one from the questions and titles of Stack Exchange questions.
A random interleaving of sentences from a few PubMed abstracts or Stack Exchange questions roughly resembles interleaved texts, and correspondingly interleaving of titles resembles its multi-sentence summary.
The algorithm that we devised for creating synthetic interleaved texts is defined in detail in the Appendix.
The number of abstracts to include in the interleaved texts is given as a range (from $a$ to $b$) and the number of sentences per abstract to include is given as a second range (from $m$ to $n$). We vary the
parameters as below and create three different corpora for experiments: \textbf{Easy} ($a$=2, $b$=2, $m$=5 and $n$=5), \textbf{Medium} ($a$=2, $b$=3, $m$=2 and $n$=5) and \textbf{Hard} ($a$=2, $b$=5, $m$=2 and $n$=5).
\tabref{example_dataset1} shows an example of a data instance in the Hard Interleaved RCT corpus.
\def0.20cm{0.20cm}
\def0.125cm{0.125cm}
\def0.08cm{0.08cm}
\begin{table*}[t!]
\begin{center}
\footnotesize
\resizebox{1.0\linewidth}{!}{
\begin{tabular}
{l@{\hspace{0.08cm}}|c|
c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|}
\multirow{2}{1cm}{Input Text} &\multirow{2}*{Model} &\multicolumn{3}{c|}{\bf Easy} &\multicolumn{3}{c|}{\bf Medium}&\multicolumn{3}{c|}{\bf Hard}\\
& & Rouge-1 & Rouge-2 & Rouge-L & Rouge-1 & Rouge-2 & Rouge-L& Rouge-1 & Rouge-2 & Rouge-L\\
\hline
\tikzmark{top left 1}ind& seq2seq & 35.09& 28.72& 13.16& 36.31& 28.78& 13.45& 37.74& 28.72& 13.76\\
dis& seq2seq& \bf36.38& \bf29.90& \bf14.78& 35.63& 28.45& 13.98& 37.87& 28.85& 14.77\\
dis& hier2hier & 35.30& 28.93& 13.35& \bf37.30& \bf29.83& \bf14.90& \bf39.09& \bf30.11& \bf15.22\\
\hline
kmn& seq2seq(dis) & 34.48& 27.51& 13.31& 34.05& 26.58& 13.14& 35.54& 26.36& 13.65\tikzmark{bottom right 1}\\
\tikzmark{top left 2}kmn& seq2seq & 34.28& 27.84& 13.86& 34.89& 27.42& 13.68& 31.22& 23.37& 11.77\\
\hline
kmn& compress & 30.04& 19.83& 10.75& 29.37& 17.54& 10.43& 29.11& 15.76& 10.13\\
ent& seq2seq & 35.78& \bf28.89& \bf14.62& 35.20& 27.44& 13.54& 32.46& 24.17& 12.17\\
ent& hier2hier & \bf35.88& 28.47& 13.33& \bf37.29& \bf29.63& \bf 14.95& \bf37.11& \bf27.97& \bf14.26\tikzmark{bottom right 2}\\
\end{tabular}
\DrawBox[ thick, draw=blue, dotted]{top left 1}{bottom right 1}
\DrawBox[ thick, draw=green, dashed]{top left 2}{bottom right 2}
}
\end{center}
\caption{\tablabel{base_perf_comp} Summarization performance (Rouge Recall-Scores) comparing models when the threads are disentangled (top blue dotted section, upper bounds) and when the threads are entangled (bottom green dashed section, real-world) on the Easy, Medium and Hard Pubmed Corpora. \textbf{ind} = individual, \textbf{dis} = disentangled (ground-truth), \textbf{kmn} = K-means disentangled and \textbf{ent} = entangled.
In the middle, the first row shows a seq2seq model trained on ground-truth disentangled texts and tested on unsupervised disentangled texts, and the second row shows a seq2seq model trained and tested on unsupervised disentangled texts. The best performance for the entangled threads and for the disentangled threads are in bold.
}
\end{table*}
\section{Experiments}
We report ROUGE-1, ROUGE-2, and ROUGE-L as the quantitative evaluation of the models.
The hyper-parameters for experiments are described in detail in the Appendix and remain the same unless otherwise noted.
\subsection{Upper-bound}
In upper-bound experiments, we check the impact of disentanglement on the abstractive summarization models, e.g., seq2seq and hier2hier. In order to do this, firstly, we provide the ground-truth disentanglement (cluster) information and evaluate the performance of these models. Secondly, we let the models to perform either end-to-end or two-step summarization. In order to perform these experiments, we compiled three corpora of different entanglement difficulty using Pubmed corpus of MeSH type Disease and Chemical\footnote{interleaving is performed within a MeSH type}. The training, evaluation and test sets are of sizes of 170k, 4k and 4k respectively.
The seq2seq model can use ground-truth disentanglement information in two ways, i.e., summarize threads individually or summarize concatenated threads. The first two rows in \tabref{base_perf_comp} compares performance of those two sets of experiments. Clearly, seq2seq model can easily detect thread boundary in concatenated threads and perform as good as individual model. However, hier2hier is better than seq2seq in detecting thread boundaries as indicated by its performance gain on Medium and Hard corpora (see row 3 in \tabref{base_perf_comp}), and therefore, sets the upper bound for interleaved text summarization.
Additionally, we also utilize \newcite{P18-1062}'s unsupervised disentanglement component and cluster the entangled threads. Importantly, their disentanglement component requires a fixed cluster size as an input; however, our Medium and Hard corpora have a varying cluster size, and therefore, we give their system benefit of the doubt and input the maximum cluster size, i.e., 3 and 5 respectively. We sort the clusters by their association to a sequence of summary, where the association is measured by Rouge-L between them. We then take the seq2seq trained on ground-truth disentanglement and test it on these unsupervised-disentangled texts to understand the strength of unsupervised clustering. The performance of the pre-trained model remains somewhat similar (see row 2 and 4), indicating a strong disentanglement component. We also train and test a seq2seq on unsupervised-disentangled texts; however, its performance lowers slightly (see row 5), which we believe is due to noise inserted by heuristic sorting of clusters.
In real-world scenarios, i.e., without ground-truth disentanglement, \cite{P18-1062}'s unsupervised two-step system performs worse than seq2seq on unsupervised disentanglement (see row 5 and 6), the reason being a seq2seq model trained on a sufficiently large dataset is better at summarization than the unsupervised sentence compression (extractive) method. At the same time, a seq2seq model trained on entangled texts performs similar to a seq2seq trained on unsupervised disentangled texts (see row 5 and 7), and thereby, showing that the disentanglement component is not necessary. Finally, a hier2hier trained on entangled texts is the only model that reaches closest to the upper-bound set by hier2hier on disentangled texts (see row 3 and 8).
\begin{table*}[t!]
\begin{center}
\footnotesize
\begin{tabular}
{l|l|
c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|}
Corpus& \multirow{2}*{Model} &\multicolumn{3}{c|}{\bf Pubmed} &\multicolumn{3}{c|}{\bf Stack Exchange}\\
Difficulty & & Rouge-1 & Rouge-2 & Rouge-L & Rouge-1 & Rouge-2 & Rouge-L\\
\hline
\multirow{2}*{Medium} & seq2seq & 30.67 & 11.71 & 23.80 & 18.78 & 03.52 & 14.73\\
& hier2hier & \bf32.78 & \bf12.36 & \bf25.33 & \bf24.34 & \bf05.07 & \bf18.63\\
\hline
\multirow{2}*{Hard} & seq2seq & 29.07 & 10.96 & 21.76 & 20.21 & 04.03 & 14.93\\
& hier2hier & \bf 33.36 & \bf 12.69 & \bf 24.72 & \bf 24.96 & \bf 05.56 & \bf 17.95\\
\end{tabular}
\end{center}
\caption{\tablabel{hier_perf_comp} Rouge Recall-Scores on the Medium and Hard Corpora. The base Pubmed has abstract-summary pairs of 10 MeSH types, while base Stack Exchange has posts-question pairs from 12 topics.}
\end{table*}
\section{Seq2seq vs. hier2hier models}
Further, we compare the proposed hierarchical approach against the seq-to-seq approach in summarizing the interleaved texts by experimenting on the Medium and Hard corpora obtained from much-varied base document-summary pairs. We interleave Pubmed corpus of 10 MeSH types, e.g., anatomy and organism. Similarly, we interleave Stack Exchange posts-question pairs of 12 different categories with regular vocabularies, e.g., science fiction and travel. As before, the interleaving is performed within a type or category. The training, evaluation and test sets of Pubmed are of sizes 280k, 5k and 5k and Stack Exchange are of sizes 140k, 4k and 4k respectively. Results in \tabref{hier_perf_comp} shows that a noticeable improvement is observed on changing the decoder to hierarchical, i.e., 1.5-3 Rouge points in Pubmed and 2-4.5 points in Stack Exchange.
Additionally, we evaluated models strength in recognizing threads where summaries are ordered by the location of each thread's greatest density. Here, density refers to smallest window of posts with over 50\% of posts belonging to a thread; e.g.,
post1-thread1, post1-thread2, post-2-thread2, post2-thread1, post3-thread1, post4-thread1 $\rightarrow$ thread2-summary, thread1-summary. In this example, although thread1 occurs early, as the majority of posts on thread1 occurs latter, therefore, its summary also occurs later.
So, we create Medium and Hard corpora of the Stack Exchange with summaries sorted by thread density and perform abstractive summarization studies. As seen in \tabref{hier_dnsty_comp}, both the seq2seq and hier2hier models perform similar to the corpora with summaries sorted by thread occurrence (see \tabref{hier_perf_comp}), which indicates a strong disentanglement in such abstractive models irrespective of summary arrangement.
In addition, the hier2hier model is still consistently better than the seq2seq model.
\begin{table}[h!]
\footnotesize
\begin{center}
\begin{tabular}
{l|c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|}
&\multicolumn{3}{c|}{\bf Medium Corpus}\\
Model & Rouge-1 & Rouge-2 & Rouge-L \\
\hline
seq2seq & 19.67 & 03.88 & 15.37 \\
hier2hier & \bf23.97 & \bf05.63 & \bf18.75 \\
& \multicolumn{3}{c|}{\bf Hard Corpus}\\
seq2seq & 19.62 & 03.71 & 14.90\\
hier2hier & \bf 24.14 & \bf 05.00 & \bf 17.25\\
\end{tabular}
\end{center}
\caption{\tablabel{hier_dnsty_comp} Rouge Recall-Scores of models on the Stack Exchange Medium and Hard Corpus.}
\end{table}
To understand the impact of hierarchy on the hier2hier model, we perform an ablation study and use the Hard Pubmed corpus for the experiments, and \tabref{hier_ablt_comp} shows the results. Clearly, adding hierarchical decoding already provides a boost in the performance. Hierarchical encoding also adds some improvements to the performance; however, the enhancement attained in training and inference speed by the hierarchical encoding is much more valuable (see Figure 1 in Appendix C
\footnote{hier2hier takes $\approx$1.5 days for training on a Tesla V100 GPU, while seq2seq takes $\approx$4.5 days}. Thus, hier2hier model not only achieves greater accuracy but also reduces training and inference time.
\begin{table}[h!]
\footnotesize
\begin{center}
\begin{tabular}
{l|
c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|}
\Tstrut &\multicolumn{3}{c|}{\bf Pubmed Hard Corpus}\\
Model & Rouge-1 & Rouge-2 & Rouge-L\\
\hline
seq2seq & 29.07 & 10.96 & 21.76\\
seq2hier & 32.92 & 11.87 & 24.43\\
hier2seq & 31.86 & 11.9 & 23.57\\
hier2hier & \bf 33.36 & \bf 12.69 & \bf 24.72\\
\end{tabular}
\end{center}
\caption{\tablabel{hier_ablt_comp} Rouge Recall-Scores of ablated models (encoder-decoder) on the Pubmed Hard Corpus.}
\end{table}
\section{Hierarchical attention}
To understand the impact of hierarchical attention on the hier2hier model, we perform an ablation study of post-level attentions ($\boldsymbol{\gamma}$) and phrase-level attentions ($\boldsymbol{\beta}$), using the Pubmed Hard corpus
\def0.2cm{0.2cm}
\begin{table}[h!]
\begin{center}
\footnotesize
\resizebox{1.0\linewidth}{!}{
\begin{tabular}
{l@{\hspace{0.2cm}}|
c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|}
Model & Rouge-1 & Rouge-2 & Rouge-L\\
\hline
hier2hier$(+\boldsymbol{\gamma}+\boldsymbol{\beta})$ & \bf 33.36 & \bf 12.69 & \bf 24.72\\
hier2hier$(-\boldsymbol{\gamma}+\boldsymbol{\beta})$ & 32.65 & 12.21 & 24.23\\
hier2hier$(+\boldsymbol{\gamma}-\boldsymbol{\beta})$ & 31.28 & 10.20 & 23.49\\
hier2hier(Li et al.) & 29.83 & 09.80 & 22.17\\
hier2hier$(-\boldsymbol{\gamma}-\boldsymbol{\beta})$ & 30.58 & 10.00 & 22.96\\
seq2seq & 29.07 & 10.96 & 21.76\\
\end{tabular}
}
\end{center}
\caption{\tablabel{hier_attn_ablt} Rouge Recall-Scores of ablated models (attentions) on the Hard Pubmed Corpus.}
\end{table}
\tabref{hier_attn_ablt} shows the performance comparison. $\boldsymbol{\gamma}$ attention improves the performance (0.5-1) of hierarchical decoding but not a lot. The phrase-level attention, i.e., $\boldsymbol{\beta}$ is very important as without it the model performance is noticeably reduced (Rouge values decrease from 2-3).
The closest hierarchical attentions to ours, i.e., \cite{AAAI1714636,DBLP:conf/conll/NallapatiZSGX16,tan2017neural,cheng2016neural} do not use $\boldsymbol{\beta}$, and therefore, is equivalent to hier2hier$(+\boldsymbol{\gamma}-\boldsymbol{\beta})$, whose performs worse than hier2hier$(-\boldsymbol{\gamma}+\boldsymbol{\beta})$ and hier2hier$(+\boldsymbol{\gamma}+\boldsymbol{\beta})$, thus signifying importance of $\beta$. We also include \newcite{P15-1107} type post-level attention technique in the comparison, where a softmax $\gamma$ instead of $\sigma(\cdot)$ based $\gamma$ and $\beta$ is used to compute thread representation. Results indicate $\sigma(\cdot)$ fits better in this case.
Lastly, removing both the $\boldsymbol{\gamma}$ and $\boldsymbol{\beta})$ makes the hier2hier similar to seq2seq, except a few more parameters, i.e., two additional LSTM, and the performance is also very similar.
\section{AMI Experiments}
We also experimented both abstractive models; seq2seq and hier2hier, on the popular meeting AMI corpus \cite{7529878bc1a143dbad4fa019e742fdb8}, and compare them against \newcite{P18-1062} two-step system. We follow the standard train, eval and test split. Results in \tabref{abst_ami} show hier2hier outperforms both systems by a large margin.
\begin{table}[h!]
\begin{center}
\footnotesize
\resizebox{1.0\linewidth}{!}{
\begin{tabular}
{l@{\hspace{0.2cm}}|
c@{\hspace{0.125cm}}c@{\hspace{0.125cm}}c|}
Model & Rouge-1 & Rouge-2 & Rouge-L\\
\hline
Shang et al.& 29.00 & -&-\\
seq2seq & 31.60 & 10.60 & 25.03\\
hier2hier & \bf39.75 & \bf12.75 & \bf25.41\\
\end{tabular}
}
\end{center}
\caption{\tablabel{abst_ami} Rouge F1 Scores of models on AMI Corpus with summary size 150.}
\end{table}
\section{Discussion}
\tabref{result_example} shows an output of our hierarchical abstractive system, in which interleaved texts are in the top, and ground-truth and generated summaries in the bottom. \tabref{result_example} also shows the top two post indexes attended by the post-level attention ($\boldsymbol{\gamma}$) while generating those summaries, and they coincide with relevant posts. Similarly, the top 10 indexes (words) of the phrase-level attention ($\boldsymbol{\beta}$) is directly visualized in the table through the color coding matching the generation. The system not only manages to disentangle the interleaved texts but also to generate appropriate abstractive summaries. Meanwhile, $\boldsymbol{\beta}$ provides explainability of the output.
The next step in this research is transfer learning of the hierarchical system trained on the synthetic corpus to real-world examples.
Further, we aim to modify hier2hier to include some of the recent additions of seq2seq models, e.g., \newcite{P17-1099} pointer mechanism.
\begin{table}[!t]
\begin{center}
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{m{0.04\linewidth}|m{0.96\linewidth}}
\hline
&\multicolumn{1}{c}{Interleaved Texts}\\
\Tstrut $0$& this study was conducted \boldblue{to evaluate the influence of} excessive \boldblue{sweating} during \boldblue{long-distance} running \boldblue{on} the urinary concentration of \boldblue{caffeine}\ldots\\
$1$& \boldgreen{to assess the effect of} a \boldgreen{program of} supervised \boldgreen{fitness} walking and patient education on functional status , pain , and\ldots\\
$\dots$ & \multicolumn{1}{c}{\dots}\\
$5$& a total of 102 patients with a documented diagnosis of primary osteoarthritis of one or both knees participated\ldots\\
$6$& we \boldred{examined the effects of intensity of training} on ratings of perceived exertion \boldred{(}\ldots\\
$\dots$ & \multicolumn{1}{c}{\dots}\\
\hline
&\multicolumn{1}{c}{GroundTruth/Generation}\\
& caffeine in sport . influence of endurance exercise on the urinary caffeine concentration .\\
0,2\vspace{0.2cm}& \boldblue{effect of excessive [UNK] during [UNK] running on the urinary concentration of caffeine .}\vspace{0.2cm}\\
& supervised fitness walking in patients with osteoarthritis of the knee . a randomized , controlled trial .\\
1,4\vspace{0.2cm}& \boldgreen{effect of a physical fitness walking on functional status , pain , and pain}\vspace{0.2cm}\\
& the effect of training intensity on ratings of perceived exertion .\\
6,8& \boldred{effects of intensity of training on perceived [UNK] in [UNK] athletes .}\\
\hline
\end{tabular}
}
\end{center}
\caption{\tablabel{result_example}Interleaved sentences of 3 articles, and corresponding ground-truth and hier2hier generated summaries. The top 2 sentences that were attended ($\boldsymbol{\gamma}$) for the generation are on the left. Additionally, top words ($\boldsymbol{\beta}$) attended for the generation are colored accordingly.}
\end{table}
\section{Model}
\subsection*{Problem Statement}
We aim to design a system that when given a sequence of posts, $\mathit{C} = \langle\mathit{P}_1,\ldots,\mathit{P}_{|\mathit{C}|}\rangle$, produces a sequence of summaries, $\mathit{T} = \langle\mathit{S}_1,\ldots,\mathit{S}_{|\mathit{T}|}\rangle$. For simplicity and clarity, unless otherwise noted, we will use lowercase italics for variables, uppercase italics for sequences, lowercase bold for vectors and uppercase bold for matrices.
\subsection{Encoder}
The hierarchical encoder (see \figref{architecture_joint} left hand section) is based on \newcite{AAAI1714636}, where word-to-word and post-to-post encoders are bi-directional LSTMs.
The word-to-word BiLSTM encoder ($E_{w2w}$) runs over word embeddings of post $\mathit{P}_i$ and generates a set of hidden representations, $\langle\mathbf{h}^{{E_{w2w}}}_{i,0},\ldots,\mathbf{h}^{{E_{w2w}}}_{i,p}\rangle$, of $d$ dimensions.
The average pooled value of the word-to-word representations of post $\mathit{P}_i$ ($\frac{1}{p}\sum_{j=0}^{p} \mathbf{h}^{{E_{w2w}}}_{i,j}$) is input to the post-to-post BiLSTM encoder ($E_{t2t}$), which then generates a set of representations, $\langle\mathbf{h}^{E_{p2p}}_{0},\ldots,\mathbf{h}^{E_{p2p}}_{n}\rangle$, corresponding to the posts.
Overall, for a given channel $\mathit{C}$, output representations of word-to-word, $\mathbf{W}$, and post-to-post, $\mathbf{P}$, has $n\times p\times 2d$ and $n\times 2d$ dimensions respectively.
\subsection{Decoder}
Our hierarchical decoder structure and arrangement is similar to \newcite{P15-1107} hierarchical auto encoder, with two uni-directional LSTM decoders, thread-to-thread and word-to-word (see right-hand side in \figref{architecture_joint}), however, in terms of inputs, initial states and attentions it differs a lot, which we explain in the next two sections.
The initial state $\mathbf{h}^{D_{t2t}}_{0}$ of the thread-to-thread LSTM decoder ($f^{D_{t2t}}$) is set with a feedforward-mapped representation of an average pooled post representations ($\mathbf{c}^\prime = \frac{1}{n}\sum_{i=0}^{n} \mathbf{h}^{p2p}_{i}$).
At each step $k$ of the $f^{D_{t2t}}$, a sequence of attention weights, $\langle\mathit{\hat{\beta}}^{k}_{0,0},\ldots,\mathit{\hat{\beta}}^{k}_{n,p}\rangle$, corresponding to the set of encoded word representations, $\langle\mathbf{h}^{w2w}_{0,0},\ldots,\mathbf{h}^{w2w}_{n,p}\rangle$ are computed utilizing the previous state, $\mathbf{h}^{D_{t2t}}_{k-1}$. We will elaborate the attention computation in the next section.
A weighted representation of the words (crossed blue circle) is then computed: $\sum_{i=1}^{n}\sum_{j=1}^{p}\hat{\beta}^{k}_{i,j}\mathbf{W}_{ij}$,
Additionally, we use the last hidden state $\mathbf{h}^{D_{w2w}}_{k-1,q}$ of the word-to-word decoder LSTM (${D_{w2w}}$) of the previously generated summary sentence as the second input to compute the next state of thread-to-thread decoder, i.e., $\mathbf{h}^{D_{t2t}}_{k}$. The motivation is to provide information about the previous sentence.
The current state $\mathbf{h}^{D_{t2t}}_{k}$ is passed through a single layer feedforward network and a distribution over STOP=1 and CONTINUE=0 is computed:
\begin{equation}
\mathit{p}_{k}^{STOP} = \sigma(\mbox{g}\left({\mathbf{h}^{D_{t2t}}_{k}}\right))
\eqlabel{stop_predict}
\end{equation} where $\mbox{g}$ is a feedforward network. In \figref{architecture_joint}, the process is depicted by a yellow circle. The thread-to-thread decoder keeps decoding until $\mathit{p}_{k}^{STOP}$ is greater than 0.5.
Additionally, the current state $\mathbf{h}^{D_{t2t}}_{k}$ and inputs to $D_{t2t}$ at that step are passed through a two-layer feedforward network \text{r} followed by a dropout layer
to compute the thread representation $\mathbf{s}_k =
\mbox{r}\left({\mathbf{h}^{D_{t2t}}_{k};\mathbf{h}^{D_{w2w}}_{k-1,q};\boldsymbol{\hat{\beta}}^k*\mathbf{W}}\right)$.
Given a thread representation $\mathbf{s}_k$, the word-to-word decoder generates a summary for the thread. Our word-to-word decoder is based on \newcite{DBLP:journals/corr/BahdanauCB14}. It is a unidirectional attentional LSTM ($f^{D_{w2w}}$); see the right-hand side of \figref{architecture_joint}.
We refer to \cite{DBLP:journals/corr/BahdanauCB14} for further details
\subsection{Hierarchical Attention}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.5\textwidth]{hierattn_acl20_1.pdf}
\caption{Hierarchical attention mechanism. Dotted lines indicate involvement in the mechanism.
}\figlabel{architecture_hierattn}
\end{figure}
\begin{table*}[t!]
\begin{center}
\resizebox{0.99\textwidth}{!}{
\footnotesize
\begin{tabular}{p{0.60\textwidth}|p{0.35\textwidth}}
\hline
\ding{51} this study was conducted to evaluate the influence of e\ldots&\multirow{4}{5cm}{\ding{51} caffeine in sport . influence of endurance exercise on the urinary caffeine concentration .}\\
\ding{70} to assess the effect of a program of supervised fitness\ldots&\\
\ding{70} an 8-week randomized , controlled trial .\ldots&\\
\ding{51} nine endurance-trained athletes participated in a randomised\ldots&\\
\multicolumn{1}{c|}{\ldots}&\multirow{3}{5.5cm}{\ding{70} supervised fitness walking in patients with osteoarthritis of the knee . a randomized , controlled trial .}\\
\ding{81} we examined the effects of intensity of training on ratings\ldots&\\
\ding{81} subjects were recruited as sedentary controls or were randomly\ldots&\\
\ding{81} the at lt group trained at velocity lt and the greater than\ldots&\multirow{2}{5cm}{\ding{81} the effect of training intensity on ratings of perceived exertion .}\\
\ding{51} data were obtained on 47 of 51 intervention patients and 45\ldots&\\
\hline
\end{tabular}
}
\end{center}
\caption{\tablabel{example_dataset1}The left rows contain interleaving of 3 articles with 2 to 5 sentences and the right rows contain their interleaved titles. Associated sentences and titles are depicted by similar symbols.}
\end{table*}
Our novel hierarchical attention works at 3 levels, the post level (corresponding to posts), i.e., $\boldsymbol{\gamma}$, and phrase level (corresponding to source tokens), i.e., $\boldsymbol{\beta}$, and are computed while obtaining a thread representation, $\mathbf{s}$. The word level attention (also corresponding to source tokens), i.e., $\boldsymbol{\alpha}$, is computed while generating a word, $y$, of a summary, $\mathit{S}$.; see \figref{architecture_hierattn}.
We draw inspiration for the hierarchical attention from some of the recent works in computer vision \cite{noh2017large,teichmann2019detect}, in which, they show a convolutional neural network (CNN)-based local descriptor with attention is better at obtaining key points from an image than CNN-based global descriptor. Phrases from posts of interleaved texts are equivalent to visual patterns in images, and thus, extracting phrases is more relevant for thread recognition than extracting posts. Thus, contrary to popular hierarchical attention \cite{DBLP:conf/conll/NallapatiZSGX16,cheng2016neural,tan2017neural}, we have additional phrase-level attention focusing again on words, but with a different responsibility. Further, the popularly held intuition of hierarchical attention, i.e., sentence attention scales word attention, is still intact as gamma (post-attention) scales beta.
At step $k$ of thread decoding, we compute elements of post-level attention, i.e., $\boldsymbol{\gamma}^{k,\cdot}$ as.
\begin{equation}
\gamma^{k}_{i} = \sigma(\mbox{attn}^{\gamma}(\mathbf{h}^{D_{t2t}}_{k-1}, \mathbf{P}_{i})\quad i \in \{1,\dotsc,n\}
\label{eqn:gamma_attn}
\end{equation}, where $\mbox{attn}^{\gamma}$ aligns the current thread decoder state vector $\mathbf{h}^{D_{t2t}}_{i-1}$ to vectors of matrix $\mathbf{P}_{i}$ and then maps aligned vectors to scalar values through a feed-forward network.
At the same step, we also compute elements of phrase-level attention, i.e, $\boldsymbol{\beta}^{k}_{i,j}$ as.
\begin{equation}
\begin{aligned}
\beta^{k}_{i,j} &= \sigma(\mbox{attn}^{\beta}(\mathbf{h}^{D_{t2t}}_{k-1}, \mathbf{a}_{i,j})) \\
& \text{where} \enskip \mathbf{a}_{i,j} = add(\mathbf{W}_{i,j}, \mathbf{P}_{i}),\\
& i \in \{1,\dotsc,n\},\quad j \in \{1,\dotsc,p\}
\label{eqn:beta_attn}
\end{aligned}
\end{equation},
${add}$ aligns a post representation to its constituting word representations and does element-wise addition, and $\mbox{attn}^{\beta}$ is a feedforward network that maps the current thread decoder state $\mathbf{h}^{D_{t2t}}_{k-1}$ and vector $\mathbf{a}_{i,j}$ to a scalar value.
Importantly, $\sigma(\cdot)$ in $\gamma$ and $\beta$ will allow a thread not to be associated with any relevant phrase, and thereby, indicating a halt in decoding.
Then, we use $\boldsymbol{\gamma}^{k}$ to rescale phrase-level attentions, $\boldsymbol{\beta}^{k}$ as $\hat{\beta}^{k}_{i,j} = \beta^{k}_{i,j}*\gamma^{k}_{i}$.
At step $l$ of word-to-word decoding of summary thread $k$, we compute elements of word level attention, i.e., $\boldsymbol{\alpha}^{k,l}_{i,\cdot}$ as below.
\begin{equation}
\begin{aligned}
\alpha^{k,l}_{i,j} &= \frac{\exp(\mathbf{e}^{k,l}_{i,j})}{\sum_{i=1}^{n}\sum_{j=1}^{p}\exp(\mathbf{e}^{k,l}_{i,j})}\\
&\text{where} \enskip
\mathbf{e}^{k,l}_{ij}=\mbox{attn}^{\alpha}(\mathbf{h}^{D_{w2w}}_{k,l-1}, \mathbf{a}_{i,j})
\end{aligned}
\label{eqn:alpha_attn}
\end{equation}, and $\mathbf{a}_{k}$ is same as in \eqref{beta_attn} and $\mbox{attn}^{\alpha}$ is a feedforward network that maps the current word decoder state $\mathbf{h}^{D_{w2w}}_{k,l-1}$ and vector $\mathbf{a}_{i,j}$ to a scalar value.
Finally, we use rescaled phrase-level word attentions, $\hat{\boldsymbol{\beta}^{k}}$, for rescaling word level attention, $\alpha^{k,l}$ as $\hat{\alpha}^{k,l}_{i,j} = \hat{\beta}^{k}_{i,j} \times \alpha^{k,l}_{ij}$
\subsection{Training Objective}
We train our hierarchical encoder-decoder network similarly to an attentive seq2seq model \cite{DBLP:journals/corr/BahdanauCB14}, but with an additional weighted sum of sigmoid cross-entropy loss on stopping
distribution; see \eqref{stop_predict}. Given a thread summary, $\mathit{Y}^k = \langle \mathit{w}^{k,0},\ldots, \mathit{w}^{k,q}\rangle$, our word-to-word decoder generates a target $\hat{\mathit{Y}}^k = \langle \mathit{y}^{k,0},\ldots, \mathit{y}^{k,q} \rangle$,
with words from a same vocabulary $\mathit{U}$.
We train our model end-to-end by minimizing the objective
given in \eqref{log_likelihood}.
\begin{equation}
\begin{aligned}
{\underset{k=1}{\overset{m}{\sum}}}{\underset{l=1}{\overset{q}{\sum}}}&\log{p}_\theta\left(y^{k,l}\vert\textit{w}_{k,\cdot<l}, \textbf{W}\right)
\eqlabel{log_likelihood}\\
&+\lambda {\underset{k=1}{\overset{m}{\sum}}}\textit{y}_{k}^{STOP}\log(p_{k}^{STOP})
\end{aligned}
\end{equation}
\section{Related Work}
\newcite{ma2012topic,aker2016automatic,P18-1062} designed earlier systems that summarize posts in multi-party conversations in order to provide readers with overview on the discussed matters. They broadly follow the same approach: cluster the posts and then extract a summary from each cluster.
There are two kinds of summarization: abstractive and extractive. In abstractive summarization, the model utilizes a corpus level vocabulary and generates novel sentences as the summary, while extractive models extract or rearrange the source words as the summary. Abstractive models based on neural sequence-to-sequence (seq2seq) \cite{DBLP:conf/emnlp/RushCW15} proved to generate summaries with higher ROUGE scores than the feature-based abstractive models. Integration of attention into seq2seq \cite{DBLP:journals/corr/BahdanauCB14} led to further advancement of abstractive summarization \cite{DBLP:conf/conll/NallapatiZSGX16,DBLP:conf/naacl/ChopraAR16}.
\newcite{P15-1107} proposed an encoder-decoder (auto-encoder) model that utilizes a hierarchy of networks: word-to-word followed by sentence-to-sentence. Their model is better at capturing the underlying structure than a vanilla sequential encoder-decoder model (seq2seq). \newcite{krause2016paragraphs,P18-1240} showed multi-sentence captioning of an image through
hierarchical Recurrent Neural Network (RNN), topic-to-topic followed by word-to-word, is better than seq2seq.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.99\textwidth]{joint_architecture_acl20.pdf}
\caption{Our hierarchical encoder-decoder architecture. On the left, interleaved posts are encoded hierarchically, i.e., word-to-word ($E_{w2w}$) followed by post-to-post ($E_{p2p}$). On the right, summaries are generated hierarchically, thread-to-thread ($D_{t2t}$) followed by word-to-word ($D_{t2t}$).
}\figlabel{architecture_joint}
\end{figure*}
These works suggest a hierarchical encoder, with word-to-word encoding followed by post-to-post, will better recognize the dispersed information in interleaved texts. Similarly, a hierarchical decoder, thread-to-thread followed by word-to-word, will intrinsically disentangle the posts, and therefore, generate more appropriate summaries.
\newcite{DBLP:conf/conll/NallapatiZSGX16} devised a hierarchical attention mechanism for a seq2seq model, where two levels of attention distributions over the source, i.e., sentence and word, are computed at every step of the word decoding. Based on the sentence attentions, the word attentions are rescaled. \newcite{P18-1013} slightly simplified this mechanism and computed the sentence attention only at the first step. Our hierarchical attention is more intuitive and computes new sentence attentions for every new summary sentence, and unlike \newcite{P18-1013}, is trained end-to-end.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,527
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Produced by Chris Curnow, Craig Kirkwood, and the Online
Distributed Proofreading Team at http://www.pgdp.net
Transcriber's Notes:
This book catalogue for W. & R. Chambers, Limited, was extracted from
Mary Louisa Molesworth, _Hoodie_, W. & R. Chambers, Limited, London and
Edinburgh, 1897.
Text enclosed by underscores is in italics (_italics_), and text
enclosed by equal signs is in bold (=bold=).
Additional Transcriber's Notes are at the end.
* * * * *
BOOKS SUITABLE FOR PRIZES AND PRESENTATION.
Price 5s.
=MEG LANGHOLME=, or the Day after To-morrow. By Mrs MOLESWORTH, author
of _Philippa_, _Olivia_, _Blanche_, _Carrots_, _Imogen_, &c. With eight
Illustrations by W. Rainey. =5/=
Mrs Molesworth with her usual charm of manner, and easy natural
grace, traces the development of Meg Langholme from early girlhood
to young womanhood, with her friends and companions in the home of
Bray Weald, where she is like an adopted daughter, until mysterious
warnings bode the disaster of her life; for certain reasons she is
kidnapped and concealed until cleverly rescued, and happily married
to a lifelong friend then home from India.
=VINCE THE REBEL=, or the Sanctuary in the Bog. By GEORGE MANVILLE
FENN, author of _The Black Tor_, _Roy Royland_, _Diamond Dyke_, _The
Rajah of Dah_, _Real Gold_, &c. With eight Illustrations by W. H. C.
Groome. =5/=
Relates the troubles at Mere Abbey, a fine South-of-England mansion,
surrounded by bogs and woodlands, during the reign of James II. of
England, and how Vince the Rebel lay in hiding here after Sedgemoor,
and escaped the soldiers sent in pursuit. The free and healthy
country life enjoyed by Walter Heron and his cousin Vince, along with
Sol Bogg, the man-servant, who aids in all the fishing, hunting, and
woodland adventures, form a fascinating and enjoyable narrative for
readers of all ages.
=WILD KITTY.= By L. T. MEADE, author of _Catalina_, &c. With eight
Illustrations by J. Ayton Symington. =5/=
Mrs Meade again gives a picture of school-girl life, in which many
varied characters play a part, the most interesting and original
being Kitty Malone from Castle Malone in Ireland, who earns
the nickname of Wild Kitty because of her love of mischief and
unconventional manners. Mrs Meade is herself a native of Ireland and
quite at home in sketching such a character, and she does not fail
to weave a fascinating narrative, and one which she herself believes
will rank amongst her best efforts.
=PHILIPPA.= By Mrs MOLESWORTH, author of _Olivia_, _Blanche_, _Robin
Redbreast_, _Carrots_, _Imogen_, &c. With eight Illustrations by J.
Finnemore. =5/=
'Very clever, very fantastic, and very enjoyable.'--_Spectator._
'One of Mrs Molesworth's best stories for girls.'--_The Queen._
'Fully maintains her charm of style and narration.'--_Leeds Mercury._
=THE GIRL AT THE DOWER HOUSE, AND AFTERWARD.= By AGNES GIBERNE,
author of _Sun, Moon, and Stars_; _A Lady of England_, &c. With eight
Illustrations by J. Finnemore. =5/=
'An absorbing story.'--_Daily Free Press._
'A charming love-tale.'--_Westminster Review._
=CATALINA=: Art Student. By L. T. MEADE, author of _Betty_, _Four on an
Island_, _Wilton Chase_, &c. With eight Illustrations, by W. Boucher.
=5/=
'The story is managed with great skills.'--_Daily Free Press._
'Unquestionably one of Mrs Meade's best books.'--_Evening News._
'Very brightly told.'--_Punch._
=THE BLACK TOR=: A Tale of the Reign of James I. By GEORGE MANVILLE
FENN, author of _Roy Royland_, _Diamond Dyke_, _The Rajah of Dah_,
_Real Gold_, &c. With eight Illustrations by W. S. Stacey. =5/=
'A capital story ... full of incident and adventure.'--_The Standard._
'There is a fine manly tone about the book, which makes it
particularly appropriate for youth.'--_Sheffield Daily Telegraph._
[Illustration: All my senses were now concentrating into the one
maddening desire to reach shelter and safety. _From_ MEG LANGHOLME, _by
Mrs Molesworth; price 5s._ PAGE 222.]
=ROY ROYLAND=, or the Young Castellan. By GEORGE MANVILLE FENN. With
eight Illustrations by W. Boucher. =5/=
'Fascinating from beginning to end ... is told with much spirit and
go.'--_Birmingham Gazette._
=THE BROTHERHOOD OF THE COAST.= By DAVID LAWSON JOHNSTONE. With
twenty-one Illustrations by W. Boucher. Large crown 8vo, antique cloth
gilt. =5/=
'There is fascination for every healthily-minded boy in the very name
of the Buccaneers.... Mr D. Lawson Johnstone's new story of adventure
is already sure of a warm welcome.'--_Manchester Guardian._
=GIRLS NEW AND OLD.= By L. T. MEADE. With eight Illustrations by J.
Williamson. =5/=
'A sound as well as entertaining romance.'--_Yorkshire Daily Post._
'It is a fine, bright, wholesome book, well bound and
illustrated.'--_Saturday Review._
=DON.= By the author of _Laddie_, &c. With eight Illustrations by J.
Finnemore. Large crown 8vo, antique cloth gilt. =5/=
'A fresh and happy story ... told with great spirit ... it is as pure
as spring air.'--_Glasgow Herald._
=OLIVIA.= By Mrs MOLESWORTH. With eight Illustrations by Robert Barnes.
=5/=
'A beautiful story, an ideal gift-book for girls.'--_British Weekly._
=BETTY=: a School Girl. By L. T. MEADE. With eight Illustrations by
Everard Hopkins. =5/=
'This is an admirable tale of school-girl life: her history involves
an excellent moral skilfully conveyed.'--_Glasgow Herald._
=WESTERN STORIES.= By WILLIAM ATKINSON. With Frontispiece. =5/=
'These stories touch a very high point of excellence. They are
natural, vivid, and thoroughly interesting.'--_Speaker._
=BLANCHE.= By Mrs MOLESWORTH, author of _Robin Redbreast_, _The
Next-Door House_, &c. With eight Illustrations by Robert Barnes. =5/=
'Eminently healthy ... pretty and interesting, free from
sentimentality.'--_Queen._
[Illustration: Sol sat staring straight at Wat with his mouth open.
_From_ VINCE THE REBEL, _by G. Manville Fenn; price 5s._ PAGE 167.]
=DIAMOND <DW18>=, or the Lone Farm on the Veldt: a Story of South African
Adventure. By GEORGE MANVILLE FENN, author of _The Rajah of Dah_,
_Dingo Boys_, &c. With eight Illustrations by W. Boucher. =5/=
'There is not a dull page in the book.'--_Aberdeen Free Press._
=REAL GOLD=: a Story of Adventure. By GEORGE MANVILLE FENN. With eight
Illustrations by W. S. Stacey. =5/=
'In the author's best style, and brimful of life and adventure....
Equal to any of the tales of adventure Mr Fenn has yet
written.'--_Standard._
=POMONA.= By the author of _Laddie_, _Rose and Lavender_, _Zoe_, _Baby
John_, &c. With eight Illustrations by Robert Barnes. =5/=
'A bright, healthy story for girls.'--_Bookseller._
=DOMESTIC ANNALS OF SCOTLAND=, from the Reformation to the Rebellion
of 1745. By ROBERT CHAMBERS, LL.D. Abridged from the original octavo
edition in three volumes. =5/=
=ALL ROUND THE YEAR.= A Monthly Garland by THOMAS MILLER, author of
_English Country Life_, &c. And Key to the Calendar. With Twelve
Allegorical Designs by John Leighton, F.S.A., and other Illustrations.
=5/=
Price 3s. 6d.
=HUNTED THROUGH FIJI=, or 'Twixt Convict and Cannibal. By REGINALD
HORSLEY, author of _The Yellow God_, _The Blue Balloon_, &c. With six
Illustrations by J. Ayton Symington. =3/6=
Dr Horsley is here at his best in following the fortunes of three
young lads pursued by convicts and natives through Fiji in the
cannibal days. The pages are crowded with adventures and hairbreadth
escapes, sufficient to carry any reader from beginning to close
without abatement of interest.
=HOODIE.= By Mrs MOLESWORTH. With seventeen Illustrations by Lewis
Baumer. =3/6=
The story, very simply and naturally told, is of a rather naughty
little girl who at first has a mistaken idea that she is out of
favour with everybody, but who gets brought to a better mind by an
illness. The little heroine displays great character.
=THE 'ROVER'S' QUEST=: a Story of Foam, Fire, and Fight. By HUGH ST
LEGER, author of _Sou'wester and Sword_, &c. With six Illustrations by
J. Ayton Symington. =3/6=
A tough yarn, which relates how Noel Hamilton is picked up from a
boat in the Channel by a passing merchant ship and carried into
eastern seas, where he encounters all the horrors of a mutiny, a
sea-quake, and shipwreck, his loneliness on a barren island being
shared by two fine old salts named Sam Port and Eli Grouse. How they
are rescued by the _Rover_, out on a strange quest, and how this
quest is accomplished, form the thread of an interesting narrative of
sea life.
=A DAUGHTER OF THE KLEPHTS=, or A Girl of Modern Greece. By ISABELLA
FYVIE MAYO (Edward Garrett), author of _Occupations of a Retired
Life_, _By Still Waters_, &c. Crown 8vo, art linen, gilt. With six
Illustrations by W. Boucher. =3/6=
'A well-written, sensible piece of work, likely to please educated
and thoughtful girls.'--_The Globe._
'The book is interesting as a dramatic representation of incidents
both tragical and heroic.'--_Inverness Courier._
'The numerous characters in the story are vivid portraitures, the
very humblest has nothing of the puppet in him or her, and the story
from the first page to the last is highly interesting, realistic, and
natural.'--_Scotsman._
=YOUNG DENYS=: a Story of the Days of Napoleon. By ELEANOR C. PRICE,
author of _In the Lion's Mouth_, _Miss Latimer of Bryans_, _The Little
One_, _A Lost Battle_, &c. With six Illustrations by G. Nicolet. =3/6=
'An interesting tale of the great Napoleon.'--_Punch._
'Children of any age can enjoy its quiet vigour and character
sketches.'--_Spectator._
=A SOLDIER OF THE LEGION=: a Romance. By DAVID LAWSON JOHNSTONE, author
of _The Brotherhood of the Coast_, _The Rebel Commodore_, &c. With
seventeen Illustrations by W. Boucher. =3/6=
'A spirited romance of adventure ... which follows the career of a
young Englishman in the Carlist wars.'--_Scotsman._
'Distinguished alike for accuracy in detail and for vivid
imagination.'--_The Standard._
=SWEPT OUT TO SEA.= By DAVID KER, author of _Prisoner among Pirates_,
_Cossack and Czar_, _Vanished_, _The Wizard King_, &c. With six
Illustrations by J. Ayton Symington. =3/6=
'A fine stirring story of adventure on sea and land.... The
local colour of the West Indies is laid on delicately and
truthfully.'--_Birmingham Gazette._
'Crowded with adventure and excitement.'--_Black and White._
=TWO BOY TRAMPS.= By J. MACDONALD OXLEY, author of _Bert Lloyd's
Boyhood_, _Fergus Mactavish_, &c. With six Illustrations by H. Sandham.
=3/6=
'An uncommonly good tale.'--_School Board Chronicle._
'There is plenty of incident, and the interest is throughout well
kept up.'--_Spectator._
=THE BLUE BALLOON=: a Tale of the Shenandoah Valley. By REGINALD
HORSLEY. With six Illustrations by W. S. Stacey. =3/6=
'We have seldom read a finer tale. It is a kind of
masterpiece.'--_Methodist Times._
=THE WIZARD KING=: a Story of the Last Moslem Invasion of Europe. By
DAVID KER. With six Illustrations by W. S. Stacey. =3/6=
'This volume ought to find an army of admiring readers.'--_Liverpool
Mercury._
=THE REBEL COMMODORE= (Paul Jones); being Memoirs of the Earlier
Adventures of Sir Ascott Dalrymple. By D. LAWSON JOHNSTONE. With six
Illustrations by W. Boucher. =3/6=
'It is a good story, full of hairbreadth escapes and perilous
adventures.'--_To-day._
[Illustration: 'My land, William, I've got the drop on you.' _From_
HUNTED THROUGH FIJI, _by Reginald Horsley: price 3s. 6d._]
=ROBIN REDBREAST.= By MRS MOLESWORTH, author of _Imogen_, _Next-Door
House_, _The Cuckoo Clock_, &c. With six original Illustrations by
Robert Barnes. =3/6=
'It is a long time since we read a story for girls more simple,
natural, or interesting.'--_Publishers' Circular._
=THE WHITE KAID OF THE ATLAS.= By J. MACLAREN COBBAN. With six
Illustrations by W. S. Stacey. =3/6=
'A well-told tale of adventure and daring in Morocco, in which the
late and the present Sultan both figure.... A very pleasant book to
read.'--_Imperial and Asiatic Quarterly Review._
=THE YELLOW GOD=: a Tale of some Strange Adventures. By REGINALD
HORSLEY. With six Illustrations by W. S. Stacey. =3/6=
'Admirably designed, and set forth with life-like force.... A
first-rate book for boys.'--_Saturday Review._
=PRISONER AMONG PIRATES.= By DAVID KER, author of _Cossack and Czar_,
_The Wild Horseman of the Pampas_, &c. With six Illustrations by W. S.
Stacey. =3/6=
'A singularly good story, calculated to encourage what is noble and
manly in boys.'--_Athenæum._
=JOSIAH MASON: A BIOGRAPHY.= With Sketches of the History of the Steel
Pen and Electroplating Trades. By JOHN THACKRAY BUNCE. With Portrait
and Illustrations. =3/6=
=FOUR ON AN ISLAND=: a Story of Adventure. By L. T. MEADE, author of
_Daddy's Boy_, _Scamp and I_, _Wilton Chase_, &c. With six original
Illustrations by W. Rainey. =3/6=
'This is a very bright description of modern Crusoes.'--_Graphic._
=IN THE LAND OF THE GOLDEN PLUME=: a Tale of Adventure. By DAVID
LAWSON JOHNSTONE, author of _The Paradise of the North_, _The Mountain
Kingdom_, &c. With six Illustrations by W. S. Stacey. =3/6=
'Most thrilling, and excellently worked out.'--_Graphic._
=THE DINGO BOYS=; or the Squatters of Wallaby Range. By GEORGE MANVILLE
FENN, author of _The Rajah of Dah_, _In the King's Name_, &c. With six
original Illustrations by W. S. Stacey. =3/6=
=THE CHILDREN OF WILTON CHASE.= By L. T. MEADE, author of _Four on an
Island_, _Scamp and I_, &c. With six Illustrations by Everard Hopkins.
=3/6=
'Both entertaining and instructive.'--_Spectator._
=THE PARADISE OF THE NORTH=: a Story of Discovery and Adventure around
the Pole. By D. LAWSON JOHNSTONE, author of _Richard Tregellas_, _The
Mountain Kingdom_, &c. With fifteen Illustrations by W. Boucher. =3/6=
'Marked by a Verne-like fertility of fancy.'--_Saturday Review._
=THE RAJAH OF DAH.= By GEORGE MANVILLE FENN, author of _In the King's
Name_, &c. With six Illustrations by W. S. Stacey. =3/6=
Price 2s. 6d.
=ANIMAL STORIES.= Selected and edited by ROBERT COCHRANE, editor of
_Great Thinkers and Workers_, _Romance of Industry and Invention_, &c.
Profusely Illustrated. =2/6=
A selection of varied true stories of animal life, illustrating
sagacity, instinct, the almost human traits of monkeys, speaking
powers of parrots, the usefulness and cleverness of many dogs,
horses, elephants, and hairbreadth escapes from lions, tigers,
bears, and snakes. The examples are drawn from a wide field, and the
narratives are brightly written.
=ELSIE'S MAGICIAN.= By FRED WHISHAW, author of _Boris the Bear Hunter_,
_A Tsar's Gratitude_, &c. With ten Illustrations by Lewis Baumer. =2/6=
A pretty story told with real humour and vivacity of how a little
London girl managed to provide for her mother a much-needed holiday
abroad, and brought together a father and daughter who had been
alienated for many years to the sorrow and misfortune of both.
=THE ROMANCE OF COMMERCE.= By J. MACDONALD OXLEY, LL.B., B.A. With
fifteen Illustrations. =2/6=
'Sure to fascinate young lads fond of tales of adventure and
daring.'--_Evening News._
=ABIGAIL TEMPLETON=; or Brave Efforts. A Story of To-day. By
EMMA MARSHALL, author of _Under Salisbury Spire_, &c. With four
Illustrations by J. Finnemore. =2/6=
'A bright and happy narrative.... Told with great
spirit.'--_Birmingham Gazette._
=THE ROMANCE OF INDUSTRY AND INVENTION.= Selected by ROBERT COCHRANE,
editor of _Great Thinkers and Workers_, _Beneficent and Useful Lives_,
_Adventure and Adventurers_, _Recent Travel and Adventure_, _Good
and Great Women_, _Heroic Lives_, &c. With 34 process and woodcut
Illustrations. =2/6=
'It is hard to say which chapter is the best, for each seems more
interesting than the last.'--_The Queen._
'A most interesting and inspiring book.'--_Colliery Guardian._
'We can recommend this work as at once instructive and
interesting.'--_New Age._
=THROUGH THICK AND THIN=: The Story of a School Campaign. By ANDREW
HOME, author of _From Fag to Monitor_, _Disturbers of the Peace_, &c.
With four Illustrations by W. Rainey. =2/6=
'This is just the kind of book for boys to rave over; it does not
cram moral axioms down their throats, the characters act them
instead.'--_Glasgow Daily Mail._
=PLAYMATES=: A Story for Boys and Girls. By L. T. MEADE. With six
Illustrations by G. Nicolet. =2/6=
'The charm of Mrs Meade's stories for children is well sustained in
this pretty and instructive tale.'--_Liverpool Mercury._
=OUTSKERRY=: The Story of an Island. By HELEN WATERS. With four
Illustrations by R. Burns. =2/6=
'The diversion provided is varied beyond expectation (and indeed
belief). We read of an "Arabian Night's Entertainment," but here is
enough for an Arctic night.'--_The Times._
[Illustration: 'There'll be more than one dead corpse amongst you afore
you can say knife, mark me!' _From_ THE 'ROVER'S' QUEST, _by Hugh St
Leger; price 3s. 6d._ Page 91.]
=WHITE TURRETS.= By Mrs MOLESWORTH, author of _Carrots_, _Olivia_, &c.
With four Illustrations by W. Rainey. =2/6=
'A charming story.... A capital antidote to the unrest that inspires
young folks that seek for some great thing to do, while the great
thing for them is at their hand and at their home.'--_Scotsman._
=HUGH MELVILLE'S QUEST=: a Boy's Adventures in the Days of the Armada.
By F. M. HOLMES. With four Illustrations by W. Boucher. =2/6=
'A refreshing, stirring story ... and one sure to delight young boys
and young girls too.'--_Spectator._
=ELOCUTION=, a Book for Reciters and Readers. Edited by R. C. H.
MORISON. =2/6=
'No elocutionist's library can be said to be complete without this
neatly bound volume of 500 pages.... An introduction on the art of
elocution is a gem of conciseness and intellectual teaching.'--_Era._
'One of the best books of its kind in the English
language.'--_Glasgow Citizen._
=VANISHED=, or the Strange Adventures of Arthur Hawkesleigh. By DAVID
KER. Illustrated by W. Boucher. =2/6=
'It must be ranked high amongst its kind.'--_Spectator._
'A quite entrancing tale of adventure.'--_Athenæum._
=THISTLE AND ROSE.= By AMY WALTON. Illustrated by Robert Barnes. =2/6=
'Is as desirable a present to make to a girl as any one could
wish.'--_Sheffield Daily Telegraph._
=ADVENTURE AND ADVENTURERS=; being True Tales of Daring, Peril, and
Heroism. With Illustrations. =2/6=
'The narratives are as fascinating as fiction.'--_British Weekly._
=BLACK, WHITE, AND GRAY=: a Story of Three Homes. By AMY WALTON, author
of _White Lilac_, _A Pair of Clogs_, &c. With four Illustrations by
Robert Barnes. =2/6=
=OUT OF REACH=: a Story. By ESMÈ STUART, author of _Through the Flood_,
_A Little Brown Girl_, &c. With four Illustrations by Robert Barnes.
=2/6=
'The story is a very good one, and the book can be recommended for
girls' reading.'--_Standard._
=IMOGEN=, or Only Eighteen. By Mrs MOLESWORTH. With four Illustrations
by H. A. Bone. =2/6=
'The book is an extremely clever one.'--_Daily Chronicle._
'A readable and very pretty story.'--_Black and White._
=THE LOST TRADER=, or the Mystery of the _Lombardy_. By HENRY FRITH,
author of _The Cruise of the 'Wasp,'_ _The Log of the 'Bombastes,'_ &c.
With four Illustrations by W. Boucher. =2/6=
'Mr Frith writes good sea-stories, and this is the best of them that
we have read.'--_Academy._
=BASIL WOOLLCOMBE, MIDSHIPMAN.= By ARTHUR LEE KNIGHT, author of _The
Adventures of a Midshipmite_, &c. With Frontispiece by W. S. Stacey,
and other Illustrations. =2/6=
=THE NEXT-DOOR HOUSE.= By Mrs MOLESWORTH. With six Illustrations by W.
Hatherell. =2/6=
'I venture to predict for it as loving a welcome as that received by
the inimitable _Carrots_.'--_Manchester Courier._
=COSSACK AND CZAR.= By DAVID KER, author of _The Boy Slave in Bokhara_,
_The Wild Horseman of the Pampas_, &c. With original Illustrations by
W. S. Stacey. =2/6=
'There is not an uninteresting line in it.'--_Spectator._
=THROUGH THE FLOOD=, the Story of an Out-of-the-way Place. By ESMÈ
STUART. With Illustrations. =2/6=
'A bright story of two girls, and shows how goodness rather than
beauty in a face can heal old strifes.'--_Friendly Leaves._
=WHEN WE WERE YOUNG.= By Mrs O'REILLY, author of _Joan and Jerry_,
_Phœbe's Fortunes_, &c. With four Illustrations by H. A. Bone. =2/6=
'A delightfully natural and attractive story.'--_Journal of
Education._
=ROSE AND LAVENDER.= By the author of _Laddie_, _Miss Toosey's
Mission_, &c. With four original Illustrations by Herbert A. Bone. =2/6=
'A brightly-written tale, the characters in which, taken from humble
life, are sketched with lifelike naturalness.'--_Manchester Examiner._
=JOAN AND JERRY.= By Mrs O'REILLY, author of _Sussex Stories_, &c. With
four original Illustrations by Herbert A. Bone. =2/6=
'An unusually satisfactory story for girls.'--_Manchester Guardian._
=THE YOUNG RANCHMEN=, or Perils of Pioneering in the Wild West. By
CHARLES R. KENYON. With four original Illustrations by W. S. Stacey,
and other Illustrations. =2/6=
=MEMOIR OF WILLIAM AND ROBERT CHAMBERS.= With Autobiographic
Reminiscences of William Chambers, and Supplemental Chapter. 15th
edition. With Portraits and Illustrations. 2/6
=POPULAR RHYMES OF SCOTLAND.= By ROBERT CHAMBERS. =2/6=
=TRADITIONS OF EDINBURGH.= By ROBERT CHAMBERS. _New Edition._ With
Illustrations. 2/6
=GOOD AND GREAT WOMEN=: a Book for Girls. Comprises brief lives of
Queen Victoria, Florence Nightingale, Baroness Burdett-Coutts, Mrs
Beecher-Stowe, Jenny Lind, Charlotte Brontë, Mrs Hemans, Dorothy
Pattison. Numerous Illustrations. =2/6=
'A brightly written volume, full to the brim of interesting and
instructive matter; and either as reader, reward, or library book, is
equally suitable.'--_Teachers' Aid._
=LIVES OF LEADING NATURALISTS.= By H. ALLEYNE NICHOLSON, Professor of
Natural History in the University of Aberdeen. Illustrated. =2/6=
'Popular and interesting by the skilful manner in which notices of
the lives of distinguished naturalists, from John Ray and Francis
Willoughby to Charles Darwin, are interwoven with the methodical
exposition of the progress of the science to which they are
devoted.'--_Scotsman._
=HISTORY OF THE REBELLION OF 1745-6.= By ROBERT CHAMBERS. _New
Edition_, with Index and Illustrations. 2/6
'There is not to be found anywhere a better account of the events of
'45 than that given here.'--_Newcastle Chronicle._
=BENEFICENT AND USEFUL LIVES.= Comprising Lord Shaftesbury, George
Peabody, Andrew Carnegie, Walter Besant, Samuel Morley, Sir James
Y. Simpson, Dr Arnold of Rugby, &c. By R. COCHRANE. With numerous
Illustrations. =2/6=
'Nothing could be better than the author's selection of facts setting
forth the beneficent lives of those generous men in the narrow
compass which the capacity of the volume allows.'--_School Board
Chronicle._
=GREAT THINKERS AND WORKERS=; being the Lives of Thomas Carlyle,
Lord Armstrong, Lord Tennyson, Charles Dickens, Sir Titus Salt,
W. M. Thackeray, Sir Henry Bessemer, John Ruskin, James Nasmyth,
Charles Kingsley, Builders of the Forth Bridge, &c. With numerous
Illustrations. =2/6=
'One of the most fitting presents for a thoughtful boy that we have
come across.'--_Review of Reviews._
=RECENT TRAVEL AND ADVENTURE.= Comprising Stanley and the Congo,
Lieutenant Greely, Joseph Thomson, Livingstone, Lady Brassey, Vambéry,
Burton, &c. Illustrated. Cloth. =2/6=
'It is wonderful how much that is of absorbing interest has been
packed into this small volume.'--_Scotsman._
=LITERARY CELEBRITIES=; being brief biographies of Wordsworth,
Campbell, Moore, Jeffrey, and Macaulay. Illustrated. =2/6=
=SONGS OF SCOTLAND= prior to Burns, with the Tunes, edited by ROBERT
CHAMBERS, LL.D. With Illustrations. =2/6=
This volume embodies the whole of the pre-Burnsian songs of Scotland
that possess merit and are presentable, along with the music; each
accompanied by its own history.
=GREAT HISTORIC EVENTS.= The Conquest of India, Indian Mutiny, French
Revolution, the Crusades, the Conquest of Mexico, Napoleon's Russian
Campaign. Illustrated. =2/6=
=HISTORICAL CELEBRITIES.= Comprising lives of Oliver Cromwell,
Washington, Napoleon Bonaparte, Duke of Wellington. Illustrated. 2/6
'The story of their life-work is told in such a way as to teach
important historical, as well as personal, lessons bearing upon the
political history of this country.'--_Schoolmaster._
=STORIES OF REMARKABLE PERSONS.= The Herschels, Mary Somerville, Sir
Walter Scott, A. T. Stewart, &c. By WILLIAM CHAMBERS, LL.D. =2/6=
Embraces about two dozen lives, and the biographical sketches are
freely interspersed with anecdotes, so as to make it popular and
stimulating reading for both young and old.
=STORIES OF OLD FAMILIES.= By W. CHAMBERS, LL.D. =2/6=
The Setons--Lady Jean Gordon--Countess of Nithsdale--Lady Grisell
Baillie--Grisell Cochrane--the Keiths--Lady Grange--Lady Jane
Douglas--Story of Wedderburn--Story of Erskine--Countess of
Eglintoun--Lady Forbes--the Dalrymples--Montrose--Buccleuch
Family--Argyll Family, &c.
=YOUTH'S COMPANION AND COUNSELLOR.= By WILLIAM CHAMBERS, LL.D. =2/6=
=TALES FOR TRAVELLERS.= Selected from Chambers's _Papers for the
People_. 2 volumes, each =2/6=
Containing twelve tales by the author of _John Halifax, Gentleman_,
George Cupples, and other well-known writers.
Price 2s.
=BUNYAN'S PILGRIM'S PROGRESS.= With Index; and Prefatory Memoir by Rev.
JOHN BROWN, D.D., Bedford. Illustrated by J. D. Watson. =2/=
A careful reprint, giving the best text of Bunyan's masterpiece, with
a useful index for ready reference.
=BRUCE'S TRAVELS.= Travels of James Bruce through part of Africa,
Syria, Egypt, and Arabia, into Abyssinia, to discover the source of the
Nile. Illustrated. =2/=
'An ideal volume for a school prize.'--_Publishers' Circular._
'The record of his journey in this volume is full of fascination and
freshness. Few travellers have followed in Bruce's footsteps; none
have seen with a clearer eye or left more vivid impressions of what
he saw.'--_Aberdeen Free Press._
'A healthier or more entertaining book it would be impossible to
place in the hands of any youth. When we look to the 358 pages of
clear letterpress, the capital illustrations, and the pretty binding,
the book seems a marvel of cheapness.'--_Perthshire Courier._
=THE HALF-CASTE=: an Old Governess's Story, and other Tales. By the
author of _John Halifax, Gentleman_. =2/=
'Cannot but edify, while it must of necessity gratify and please the
fortunate reader.'--_Liverpool Mercury._
'The volume contains six short stories, which may be unhesitatingly
recommended to such as relish fiction that is free from all
morbidness, and is at the same time interesting.'--_Publishers'
Circular._
=THE LIFE AND TRAVELS OF MUNGO PARK IN AFRICA.= With Illustrations,
Introduction, and concluding chapter on the Present Position of Affairs
in the Niger Territory. =2/=
'Few books of travel have acquired so speedy and extensive a
reputation as this of Park's.'--THOMAS CARLYLE.
'A notable work well worthy of recommendation.'--_Birmingham Gazette._
=TWO ROYAL LIVES=: Queen Victoria, William I., German Emperor. =2/=
=FOUR GREAT PHILANTHROPISTS=: Lord Shaftesbury, George Peabody, John
Howard, J. F. Oberlin. Illustrated. =2/=
Shows the good accomplished through the agency of the lives and
labours of a noble Earl, a millionaire, a prison reformer, and the
humble pastor of the remote Ban de la Roche.
=TWO GREAT AUTHORS.= Lives of Scott and Carlyle. =2/=
'Youthful readers will find these accounts of the boyhood
and youth of two of the three Scotch literary giants full of
interest.'--_Schoolmaster._
=EMINENT ENGINEERS.= Lives of Watt, Stephenson, Telford, and Brindley.
=2/=
'All young persons should read it, for it is in an excellent sense
educational. It were devoutly to be wished that young people would
take delight in such biographies.'--_Indian Engineer._
=TALES OF THE GREAT AND BRAVE.= By MARGARET FRASER TYTLER. =2/=
A collection of interesting biographies and anecdotes of great
men and women of history, in the style of Scott's _Tales of a
Grandfather_, written by a niece of the historian of Scotland.
=THROUGH STORM AND STRESS.= By J. S. FLETCHER. With Frontispiece by W.
S. Stacey. =2/=
'Full of excitement and incident.'--_Dundee Advertiser._
=GREAT WARRIORS=: Nelson, Wellington, Napoleon. =2/=
'One of the most instructive books published this
season.'--_Liverpool Mercury._
=HEROIC LIVES=: Livingstone, Stanley, General Gordon, Lord Dundonald.
=2/=
'It would be difficult to name four other lives in which we find
more enterprise, adventure, achievement.... The book is sure to
please.'--_Leeds Mercury._
=THE REMARKABLE ADVENTURES OF WALTER TRELAWNEY=, Parish 'Prentice of
Plymouth, in the year of the Great Armada. Re-told by J. S. FLETCHER,
author of _Through Storm and Stress_, &c. With Frontispiece by W. S.
Stacey. =2/=
'A wonderfully vivid story of the year of the Great Armada; far more
effective than the unwholesome trash which so often does duty for
boys' books nowadays.'--_Idler._
=FIVE VICTIMS=: a School-room Story. By M. BRAMSTON, author of _Boys
and Girls_, _Uncle Ivan_, &c. With Frontispiece by H. A. Bone. =2/=
'A delightful book for children. Miss Bramston has told her simple
story extremely well.'--_Associates' Journal._
=SOME BRAVE BOYS AND GIRLS.= By EDITH C. KENYON, author of _The Little
Knight_, _Wilfrid Clifford_, &c. =2/=
'A capital book: will be read with delight by both boys and
girls.'--_Manchester Examiner._
=ELIZABETH=, or Cloud and Sunshine. By HENLEY I. ARDEN, author of
_Leather Mill Farm_, _Aunt Bell_, &c. With Frontispiece by Herbert A.
Bone. =2/=
'This is a charming story, and in every way suitable as a gift-book
or prize for girls.'--_Schoolmaster._
=HEROES OF ROMANTIC ADVENTURE=, being Biographical Sketches of Lord
Clive, founder of British supremacy in India; Captain John Smith,
founder of the colony of Virginia; the Good Knight Bayard; and
Garibaldi, the Italian patriot. Illustrated. =2/=
=FAMOUS MEN.= Illustrated. =2/=
Biographical Sketches of Lord Dundonald, George Stephenson, Lord
Nelson, Louis Napoleon, Captain Cook, George Washington, Sir Walter
Scott, Peter the Great, &c.
=LIFE OF BENJAMIN FRANKLIN.= Illustrated. =2/=
'A fine example of attractive biographical writing.... A short
address, "The Way to Wealth," should be read by every young man in
the kingdom.'--_Teachers' Aid._
=EMINENT WOMEN=, and Tales for Girls. Illustrated. =2/=
'The lives include those of Grace Darling, Joan of Arc, Flora
Macdonald, Helen Gray, Madame Roland, and others.'--_Teachers' Aid._
=TALES FROM CHAMBERS'S JOURNAL.= 4 vols., each =2/=
Comprise interesting short stories by James Payn, Hugh Conway, D.
Christie Murray, Walter Thornbury, G. Manville Fenn, Dutton Cook, J.
B. Harwood, and other popular writers.
=BIOGRAPHY, EXEMPLARY AND INSTRUCTIVE.= Edited by W. CHAMBERS, LL.D.
=2/=
The Editor gives in this volume a selection of biographies of those
who, while exemplary in their private lives, became the benefactors
of their species by the still more exemplary efforts of their
intellect.
=OUR ANIMAL FRIENDS=--the Dog, Cat, Horse, and Elephant. With numerous
Illustrations. =2/=
=AILIE GILROY.= By W. CHAMBERS, LL.D. =2/=
'The life of a poor Scotch lassie ... a book that will
be highly esteemed for its goodness as well as for its
attractiveness.'--_Teachers' Aid._
=ESSAYS, FAMILIAR AND HUMOROUS.= By ROBERT CHAMBERS, LL.D. 2 vols.,
each =2/=
Contains some of the finest essays, tales, and social sketches of
the author of _Traditions of Edinburgh_, reprinted from _Chambers's
Journal_.
=MARITIME DISCOVERY AND ADVENTURE.= Illustrated. =2/=
Columbus--Balboa--Richard Falconer--North-east Passage--South
Sea Marauders--Alexander Selkirk--Crossing the Line--Genuine
Crusoes--Castaway--Scene with a Pirate, &c.
=SHIPWRECKS AND TALES OF THE SEA.= Illustrated. =2/=
'A collection of narratives of many famous shipwrecks, with other
tales of the sea.... The tales of fortitude under difficulties,
and in times of extreme peril, as well as the records of
adherence to duty, contained in this volume, cannot but be of
service.'--_Practical Teacher._
=SKETCHES, LIGHT AND DESCRIPTIVE.= By W. CHAMBERS, LL.D. =2/=
A selection from contributions to _Chambers's Journal_, ranging over
a period of thirty years.
=MISCELLANY OF INSTRUCTIVE AND ENTERTAINING TRACTS.= Each =2/=
These Tracts comprise Tales, Poetry, Ballads, Remarkable Episodes
in History, Papers on Social Economy, Domestic Management, Science,
Travel, &c. The articles contain wholesome and attractive reading for
Mechanics', Parish, School, and Cottage Libraries.
_s._ _d._
20 Vols. cloth 20 0
10 Vols. cloth 20 0
10 Vols. cloth, gilt edges 25 0
10 Vols, half-calf 45 0
160 Nos. each 0 1
Which may be had separately.
Price 1s. 6d.
With Illustrations.
=SWISS FAMILY ROBINSON.= Their Life and Adventures on a Desert Island.
=1/6=
=SKETCHES OF ANIMAL LIFE AND HABITS.= By ANDREW WILSON, Ph.D., &c =1/6=
A popular natural history text-book, and a guide to the use of the
observing powers. Compiled with a view of affording the young and the
general reader trustworthy ideas of the animal world.
=RAILWAYS AND RAILWAY MEN.= =1/6=
'A readable and entertaining book.'--_Manchester Guardian._
=EXPERIENCES OF A BARRISTER.= =1/6=
Eleven tales embracing experiences of a barrister and attorney.
=BEGUMBAGH=, a Tale of the Indian Mutiny. =1/6=
A thrilling tale by GEORGE MANVILLE FENN.
=THE BUFFALO HUNTERS=, and other Tales. =1/6=
Fourteen short stories reprinted from _Chambers's Journal_.
=TALES OF THE COASTGUARD=, and other Stories. =1/6=
Fifteen interesting stories from _Chambers's Journal_.
=THE CONSCRIPT=, and other Tales. =1/6=
Twenty-two short stories specially adapted for perusal by the young.
=THE DETECTIVE OFFICER=, by 'WATERS;' and other Tales. =1/6=
Nine entertaining detective stories, with three others.
=FIRESIDE TALES AND SKETCHES.= =1/6=
Contains eighteen tales and sketches by R. Chambers, LL.D., and
others by P. B. St John, A. M. Sargeant, &c.
=THE GOLD-SEEKERS=, and other Tales. =1/6=
Seventeen interesting tales from _Chambers's Journal_.
=THE HOPE OF LEASCOMBE=, and other Stories. =1/6=
The principal tale inculcates the lesson that we cannot have
everything our own way, and that passion and impulse are not reliable
counsellors.
=THE ITALIAN'S CHILD=, and other Tales. =1/6=
Fifteen short stories from _Chambers's Journal_.
=JURY-ROOM TALES.= =1/6=
Entertaining stories by James Payn, G. M. Fenn, and others.
=KINDNESS TO ANIMALS.= By W. CHAMBERS, LL.D. =1/6=
'Illustrates, by means of a series of anecdotes, the intelligence,
gentleness, and docility of the brute creation.'--_Sunday Times._
=THE MIDNIGHT JOURNEY.= By LEITCH RITCHIE; and other Tales. =1/6=
Sixteen short stories from _Chambers's Journal_.
=OLDEN STORIES.= =1/6=
Sixteen short stories from _Chambers's Journal_.
[Illustration: Patience was sitting idly crooning a monotonous wailing
sound to which she put no words. _From_ A DAUGHTER OF THE KLEPHTS, _by
Mrs Isabella Fyvie Mayo; price 3s. 6d._ P. 148]
=THE RIVAL CLERKS=, and other Tales. =1/6=
The first tale shows how dishonesty and roguery are punished, and
virtue triumphs in the end.
=ROBINSON CRUSOE.= By DANIEL DEFOE. =1/6=
A handy edition, profusely illustrated.
=PARLOUR TALES AND STORIES.= =1/6=
Seventeen short tales from the old series of _Chambers's Journal_, by
Anna Maria Sargeant, Mrs Crowe, Percy B. St John, Leitch Ritchie, &c.
=THE SQUIRE'S DAUGHTER=, and other Tales. =1/6=
Fifteen short stories from _Chambers's Journal_.
=TALES FOR HOME READING.= =1/6=
Sixteen short stories from the old series of _Chambers's Journal_,
by A. M. Sargeant, Frances Brown, Percy B. St John, Mrs Crowe, and
others.
=TALES FOR YOUNG AND OLD.= =1/6=
Fourteen short stories from _Chambers's Journal_, by Mrs Crowe, Miss
Sargeant, Percy B. St John, &c.
=TALES OF ADVENTURE.= =1/6=
Twenty-one tales, comprising wonderful escapes from wolves and bears,
American Indians, and pirates; life on a desert island; extraordinary
swimming adventures, &c.
=TALES OF THE SEA.= =1/6=
Five thrilling sea tales, by G. Manville Fenn, J. B. Harwood, and
others.
=TALES AND STORIES TO SHORTEN THE WAY.= =1/6=
Fifteen interesting tales from _Chambers's Journal_.
=TALES FOR TOWN AND COUNTRY.= =1/6=
Twenty-two tales and sketches, by R. CHAMBERS, LL.D., and other
writers.
=HOME-NURSING.= By RACHEL A. NEUMAN. Paper, =1/=; cloth, =1/6=
A work intended to help the inexperienced and those who in a sudden
emergency are called upon to do the work of home-nursing.
Price 1s.
=COOKERY FOR YOUNG HOUSEWIVES.= By ANNIE M. GRIGGS. =1/=
A book of practical utility, showing how tasteful and nutritious
dishes may be prepared at little expense.
NEW SERIES OF CHAMBERS'S LIBRARY FOR YOUNG PEOPLE.
ILLUSTRATED.
Price 1s.
'Excellent popular biographies.'--_British Weekly._
POPULAR BIOGRAPHIES.
=WALLACE AND BRUCE=: Heroes of Scotland. By MARY COCHRANE, L.L.A.
Illustrated. =1/=
This little book gives the main outlines of the lives of the
founders of Scottish political freedom. In its preparation the best
authorities have been consulted, and here is given in small bulk the
results of research only to be found in larger volumes more difficult
of access.
=WILLIAM SHAKESPEARE=: the Story of his Life and Times. By EVAN J.
CUTHBERTSON. With Portrait and numerous Illustrations. =1/=
Gives in brief and compact form what history, tradition, and research
are able to tell us of the life-story of the world's greatest
dramatist. An attempt is made to picture the England he lived in, the
scenes among which he moved, the people he associated with, and the
customs that bound him.
=QUEEN VICTORIA=: the Story of her Life and Reign. =1/=
'A sympathetic and popular sketch of the life and rule of our Queen
up to the present day.'--_Manchester Guardian._
=LORD SHAFTESBURY AND GEORGE PEABODY.= Being the Story of Two Great
Public Benefactors. With Portraits. =1/=
'Cheap, interesting, and readable biographies.'--_Methodist Times._
'May be recommended to young readers as being as inspiring as it is
interesting.'--_Scotsman._
=WILLIAM I., GERMAN EMPEROR, AND HIS SUCCESSORS.= By MARY COCHRANE,
L.L.A. Illustrated. =1/=
'Must take a prominent place among compilations on the same
subject.... Compact and comprehensive.'--_Daily Chronicle._
=THOMAS CARLYLE=: the Story of his Life and Writings. =1/=
'We don't know where to find a better biography of any man at the
price.'--_Methodist Times._
=THOMAS ALVA EDISON=: the Story of his Life and Inventions. By E. C.
KENYON. =1/=
'It will repay any one who is interested in Edison's various works to
read this little book.'--_Inventions._
=THE STORY OF WATT AND STEPHENSON.= =1/=
'As a gift-book for boys this is simply first-rate.'--_Schoolmaster._
=THE STORY OF NELSON AND WELLINGTON.= =1/=
'This book is cheap, artistic, and instructive. It should be in the
library of every home and school.'--_Schoolmaster._
=GENERAL GORDON AND LORD DUNDONALD=: the Story of Two Heroic Lives. =1/=
=THOMAS TELFORD AND JAMES BRINDLEY.= =1/=
'This is a capital book for boys of active and inquiring
mind.'--_Saturday Review._
=LIVINGSTONE AND STANLEY=: the Story of the opening up of the Dark
Continent. =1/=
=COLUMBUS AND COOK=: the Story of their Lives, Voyages, and
Discoveries. =1/=
'Models of compact biography.'--_Christian World._
'Is a fascinating and historical account of daring
adventure.'--_Bristol Mercury._
=THE STORY OF THE LIFE OF SIR WALTER SCOTT.= By ROBERT CHAMBERS, LL.D.
Revised, with additions, including the AUTOBIOGRAPHY. =1/=
Besides the AUTOBIOGRAPHY, many interesting and characteristic
anecdotes of the boyhood of Scott, which challenge the attention of
the young reader, have been added; while the whole has been revised
and brought up to date.
=THE STORY OF HOWARD AND OBERLIN.= =1/=
The book is equally divided between the lives of Howard the prison
reformer, and Oberlin the pastor and philanthropist, who worked such
a wonderful reformation amongst the dwellers in a valley of the
Vosges Mountains.
=THE STORY OF NAPOLEON BONAPARTE.= =1/=
A brief and graphic life of the first Napoleon, set in a history of
his own times: the battle of Waterloo, as of special interest to
English readers, being fully narrated.
=PERSEVERANCE AND SUCCESS=: the Life of William Hutton. =1/=
=STORY OF A LONG AND BUSY LIFE.= By W. CHAMBERS, LL.D. =1/=
STORIES FOR YOUNG PEOPLE.
=WONDERFUL STORIES FOR CHILDREN.= By HANS CHRISTIAN ANDERSEN.
Translated by Mary Howitt. Illustrated. =1/=
One of the first forms in which these ever-delightful stories of Hans
Andersen were given to the British public.
=A FAIRY GRANDMOTHER=; or, Madge Ridd, a Little London Waif. By L. E.
TIDDEMAN, author of _A Humble Heroine_. =1/=
A realistic story of a London waif, who runs off from a drunken
mother, and who after many adventures is adopted by a good old lady
in the country, who proves herself a fairy grandmother indeed.
=THE CHILDREN OF MELBY HALL.= By M. and J. M'KEAN. Illustrated. =1/=
These talks and stories of plant and animal life afford simple
lessons on the importance of 'Eyes and No Eyes,' and show what an
immense interest the study of natural history, even in its simplest
forms, will produce in the minds of young folks.
=MARK WESTCROFT, CORDWAINER=: a Village Story. By F. SCARLETT POTTER.
=1/=
=A HUMBLE HEROINE.= By L. E. TIDDEMAN. =1/=
=BABY JOHN.= By the author of _Laddie_, _Tip-Cat_, _Rose and Lavender_,
&c. With Frontispiece by H. A. Bone. =1/=
'Told with quite an unusual amount of pathos.'--_Spectator._
=THE GREEN CASKET=; =LEO'S POST-OFFICE=; =BRAVE LITTLE DENIS=. By Mrs
MOLESWORTH. =1/=
Three charming stories by the author of the _Cuckoo Clock_, each
teaching an important moral lesson.
=JOHN'S ADVENTURES=: a Tale of Old England. By THOMAS MILLER, author of
_Boy's Country Book_, &c. =1/=
=THE BEWITCHED LAMP.= By Mrs MOLESWORTH. With Frontispiece by Robert
Barnes. =1/=
=ERNEST'S GOLDEN THREAD.= =1/=
=LITTLE MARY=, and other Stories. By L. T. MEADE. =1/=
=THE LITTLE KNIGHT.= By EDITH C. KENYON. =1/=
'Has an admirable moral.... Natural, amusing, pathetic.'--_Manchester
Guardian._
=WILFRID CLIFFORD=, or The Little Knight Again. By EDITH C. KENYON.
With Frontispiece by W. S. Stacey. =1/=
=ZOE.= By the author of _Tip-Cat_, _Laddie_, &c. =1/=
'A charming and touching study of child life.'--_Scotsman._
=UNCLE SAM'S MONEY-BOX.= By Mrs S. C. Hall. =1/=
=THEIR HAPPIEST CHRISTMAS.= By EDNA LYALL, author of _Donovan_, &c. =1/=
=FIRESIDE AMUSEMENTS=; a Book of Indoor Games. =1/=
'A thoroughly useful work, which should be welcomed by all who have
the organisation of children's parties.'--_Review of Reviews._
=THE STEADFAST GABRIEL=: a Tale of Wichnor Wood. By MARY HOWITT. =1/=
=GRANDMAMMA'S POCKETS.= By Mrs S. C. HALL. =1/=
=THE SWAN'S EGG.= By Mrs S. C. HALL. =1/=
=MUTINY OF THE BOUNTY=, and =LIFE OF A SAILOR BOY=. =1/=
=DUTY AND AFFECTION=, or the Drummer-boy. =1/=
A thrilling narrative of the wars of the first Napoleon.
=FAMOUS POETRY.= Being a collection of the best English verse.
Illustrated. =1/=
Price 9d.
Cloth, Illustrated.
=YOUNG KING ARTHUR.=
=THE LITTLE CAPTIVE KING.=
=FOUND ON THE BATTLEFIELD.=
=ALICE ERROL=, and other Tales.
=THE WHISPERER.= By Mrs S. C. HALL.
=TRUE HEROISM=, and other Stories.
=PICCIOLA=, and other Tales.
=TWELFTH NIGHT KING.=
=JOE FULWOOD'S TRUST.=
=PAUL ARNOLD.=
=CLEVER BOYS.=
=THE LITTLE ROBINSON.=
=MIDSUMMER HOLIDAY.=
=MY BIRTHDAY BOOK.=
Price 6d.
Cloth, with Illustrations.
'For good literature at a cheap rate, commend us to a little series
published by W. & R. Chambers, which consists of a number of readable
stories by good writers.'--_Review of Reviews._
'One contains three little stories from the pen of Mrs Molesworth,
one of the most charming of writers for the little ones; and the
name of L. T. Meade is a guarantee of good reading of a kind which
children are sure to enjoy.'--_School Board Chronicle._
=CASSIE, and LITTLE MARY.= By L. T. MEADE.
=A LONELY PUPPY=, and =THE TAMBOURINE GIRL=. By L. T. MEADE.
=LEO'S POST-OFFICE=, and =BRAVE LITTLE DENIS=. By Mrs MOLESWORTH.
=GERALD AND DOT.= By Mrs FAIRBAIRN.
=KITTY AND HARRY.= By EMMA GELLIBRAND, author of _J. Cole_.
=DICKORY DOCK.= By L. T. MEADE, author of _Scamp and I_, &c.
=FRED STAMFORD'S START IN LIFE.= By Mrs FAIRBAIRN.
=NESTA=; or Fragments of a Little Life. By Mrs MOLESWORTH.
=NIGHT-HAWKS.= By the Hon. EVA KNATCHBULL-HUGESSEN.
=A FARTHINGFUL.= By L. T. MEADE.
=POOR MISS CAROLINA.= By L. T. MEADE.
=THE GOLDEN LADY.= By L. T. MEADE.
=MALCOLM AND DORIS=; or Learning to Help. By DAVINA WATERSON.
=WILLIE NICHOLLS=; or False Shame and True Shame.
=SELF-DENIAL.= By Miss EDGEWORTH.
_W. & R. Chambers, Limited, London and Edinburgh._
* * * * *
Transcriber's Notes:
Captions for illustrations have been made consistent.
Punctuation has been made consistent.
Variations in spelling and hyphenation were retained as they appear in
the original publication, except that obvious typographical errors have
been corrected.
End of the Project Gutenberg EBook of W. & R. Chambers' Catalogue. - 1897, by
W. & R. Chambers
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 8,482
|
A post shared by Keith Urban (@keithurban)
Keith Urban Honors Don Williams With Touching Video [WATCH]
Angela Stefano Published: September 9, 2017
Keith Urban honored Don Williams with a sweet post to social media shortly after Williams' death on Friday (Sept. 8). Urban counts Williams among his many influences, specifically because of Urban's father's love for the artist.
"I cannot put into words the depth of sadness I feel right now at hearing of Don's passing," Urban writes along with a short video, which is a clip from the pair's "Imagine That" music video. The song appears on Williams' 2012 album And So It Goes, and its music video shows the two performing the song together and goofing around in between takes.
"For me, all roads lead to Don Williams. And the reason I say that is because my dad loved country music ... and he was a fan of the Pozo Seco Singers, and there was a member in that group called Donald Williams. And when Donald became Don Williams and went solo, my dad followed him as a fan, a real fan," Urban recalled in 2015. "The records I grew up with were mostly Don Williams records. My dad went and bought every record Don made the day it went on sale."
Urban's recent single "Blue Ain't Your Color" is heavily influenced by Williams' style -- specifically, Urban notes, "that kind of strong downbeat, backbeat and little in-between rhythmic thing."
"Don also had that attitude, too, like, the song is the picture, and the record is the frame," Urban reflects. "You've got to find the right frame, not too much and not too little, to make the picture really work."
Williams was 78 years old when he died following a brief illness. After making a name for himself in folk music as a member of the Pozo Seco Singers, Williams found his way to Nashville. By 1971, he had earned a songwriting contract, and the following year, he signed with JMI Records. Williams debuted on the country charts in 1973, with "The Shelter of Your Eyes," the same year in which he released his first album, Don Williams Volume One. In 1974, Williams earned his first No. 1 song, "I Wouldn't Want to Live If You Didn't Love Me."
From 1974 through 1991, all of Williams' singles landed in the Top 40 on the Billboard country charts. He was named Male Vocalist of the Year at the CMA Awards in 1978, and in 2010, Williams became a member of the Country Music Hall of Fame.
In March of 2016, Williams announced his retirement. The 6-foot-1 singer was recently forced to cancel his 2016 tour due to an unexpected hip replacement surgery, which likely played a part in his decision to retire. Williams most recently released a new album, Reflections, in 2014 and spent much of 2015 on the road.
In 2017, Williams was the subject of a tribute album, Gentle Giants: The Songs of Don Williams. The disc features Lady Antebellum and Garth Brooks, among many others.
Funeral arrangements for Williams are pending.
Country Artists Who Have Died in 2017
Unforgettable Keith Urban Moments
NEXT: Is Traditional Country Music Dead?
Filed Under: Don Williams, Keith Urban
Categories: Country News, Legends, R.I.P.
Keith Urban Gives $250K to Various Nashville Charities for the Holidays
19 Years Ago: Keith Urban's Debut Solo Album Certified Platinum
Keith Urban Sells Master Recordings, Including 10 Studio Albums + a Greatest Hits Package, to Litmus
Top 10 Country Thanksgiving Songs
Keith Urban Shares Sweet Moment With Young Fan Born With Brain Condition [Watch]
Keith Urban Plots New Las Vegas Residency, Opening March 2023
18 Years Ago: Keith Urban's 'Be Here' Goes Gold, Platinum
Which Artists Have Won the Most CMA Awards?
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,783
|
using System;
using System.Collections.Generic;
using System.Linq;
using Kudu.Contracts.Settings;
using System.IO;
namespace Kudu.Core.Infrastructure
{
internal static class ExecutableExtensions
{
public static void PrependToPath(this Executable exe, IEnumerable<string> paths)
{
if (!paths.Any())
{
throw new ArgumentNullException("paths");
}
string pathEnv;
exe.EnvironmentVariables.TryGetValue("PATH", out pathEnv);
if (!String.IsNullOrEmpty(pathEnv))
{
paths = paths.Concat(new[] { pathEnv });
}
exe.EnvironmentVariables["PATH"] = String.Join(Path.PathSeparator.ToString(), paths);
}
public static void AddDeploymentSettingsAsEnvironmentVariables(this Executable exe, IDeploymentSettingsManager deploymentSettingsManager)
{
IEnumerable<KeyValuePair<string, string>> deploymentSettings = deploymentSettingsManager.GetValues();
foreach (var keyValuePair in deploymentSettings)
{
exe.EnvironmentVariables[keyValuePair.Key] = keyValuePair.Value;
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,452
|
There might be a number of factors regarding why you wish to know How To View A Private Facebook Page. Everybody has actually been in a scenario prior to where they wanted to see what people from their past depended on without absolutely making a connection with them. Maybe you want to see just what your crush from high school is doing since you're in your 20s and out of university, or you wonder just what ever took place to your middle school bully. We've all had individuals that we wonder about every so often when their names cross our minds, however it isn't really constantly as very easy as pulling up their Facebook account. Probably their profile is secured down, without a means to access their material, as well as you could just see their name and account photo. As well as while including an individual is always an option, adding some people simply isn't an alternative if you don't currently have a preexisting partnership with that individual. there must be a method to gain access to an exclusive account on Facebook, but just how? The adhering to article will certainly cover some pointers that will aid you learn ways to go about it without being friends.
It is undoubtedly a little bit difficult to watch a personal profiles without being a friend. You can do so by getting a public URL of the individual from the Facebook site. And also how do you protect a public LINK? It is very basic to do. Do not log into your account. Then, look for the customer account from Facebook search. You will discover a public URL for the customer and all you have to do is duplicate the URL web link. Then, paste the LINK in the address bar as well as you will certainly be able to watch a little the user's account. You can also Google the users name as well as find his/her account in results. When you click open the page, you will certainly be able to see their friends listing, some of the usual teams they participate in and also perhaps even some of their personal info.
Social engineering is a psychology concept, where one has the tendency to make someone conform to their desires. You make the individual begin talking with you and also subsequently enable you to access their account. All you have to do is just send out a simple message. You see, when you open a person's account, you can see their photo and on the contrary side you could see three alternatives. Initially one states, 'Include as Close friend', which is certainly not just what we want, second is 'Send out a Message', as well as the last is 'View Friends'. The 2nd options is what we need. All you should do is send out the person a message, 'Hello, I am Rob. I assume we had actually met at Camp New Rock last summer season. If yes, please message me back'. If you are lucky, the person could respond nicely, or rudely ask you to 'obtain lost'. Whatever possibly the reply, you will certainly currently be able to access their limited private account.
You may find the above technique a total waste, if the individual does not respond. Or perhaps the person knows you, and also clearly you do not want him/her to understand you are sneaking about. You can try one more strategy that will assist. All you should do is watch the person's friends checklist. You could find there are some friends without a picture. Open their accounts and also you could discover they are not really active on Facebook. See to it you make a list of these people as well as open a brand-new account under their name. Send brand-new pal demands to other members of the person's pal list in addition to the person you have an interest in with your new phony identity. You could send out a message along, claiming 'Hi, I have neglected my old password and have produced a new account. Please accept my friend demand'. Possibilities are the individual will approve the buddy request and you will certainly now obtain accessibility to their total profile.
You might assume you are doing no damage in aiming to see some private as well as personal details, but are you conscious, it is an intrusion of somebody's right to privacy. Facebook is a social networking site where personal details is revealed. However, all info published is copyrighted against each private customers. Every participant of Facebook can decide who can and that can not see their accounts. If you think of developing a phony ID, it could total up to identity burglary. This is a serious crime under the law court. If you pester them with messages, it may amount to harassment. And most of all, trying different methods might amount to tracking. You could undergo the pros and cons before attempting anything that turns out to be a serious offense under the law.
It is noticeable that you might intend to keep a tab on your kids communication on the social networking sites. Or possibly you want to catch a cheating partner openly. It may also happen that you want to ensure, a person you know is not succumbing to a serial killer! Whatever could be the reason, make certain you do not go across the limits. Keep in mind that someone else may understand how you can utilize Facebook unethically or attempt the exact same methods to take a look at your account. My friend gave me a wonderful pointer, make some buddies with men in CIA or the police. Or even better, if you are as well certain something is wrong somewhere, work with a private detective. They will show to be ideal resource to help you with private details.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,384
|
Gypona ileota är en insektsart som beskrevs av Freytag 2005. Gypona ileota ingår i släktet Gypona och familjen dvärgstritar. Inga underarter finns listade i Catalogue of Life.
Källor
Dvärgstritar
ileota
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,692
|
Arecibo, Puerto Rico
La Estatua de Colón
This giant bronze image of Christopher Columbus embarked on a voyage of its own.
Birth of the New World statue, completed. Prof.D.H.R./cc by-sa 4.0
The statue from a distance, 2016. CDVV86 (Atlas Obscura User)
The statue being assembled in 2016. CDVV86 (Atlas Obscura User)
The Genoese navigator Christopher Columbus is best known for "discovering" the New World in 1492. In commemoration of the 500-year anniversary of that historic day, a colossal monument in his honor would embark on its own voyage, in search for a place to settle.
Top Places in Arecibo
The world's second largest single-dish radio telescope, once used to contact E.T.
Added by Trevor
Cueva Ventana
This limestone cave is scenically perched above Puerto Rico's Río Grande de Arecibo valley.
See more things to do in Arecibo »
In 1991, Georgian sculptor Zurab Tsereteli built a 360-foot bronze statue depicting Columbus at the helm, with a backdrop of the three ships used during his first voyage. Titled "Birth of the New World" and commonly known as "La Estatua de Colón (the Statue of Columbus)", it is currently one of the tallest statues in the Americas, dwarfing famous monuments like the Statue of Liberty.
Tsereteli offered the statue to several cities in the United States, including Miami, Baltimore, Fort Lauderdale, New York, and Columbus, Ohio. However, it was considered an eyesore and too disruptive. In 1998, the statue's voyage brought it to Cataño, Puerto Rico. The town's mayor at the time was interested, and the monument was brought in. This, however, was the easy part.
Originally, the plan was to erect the statue by the entrance of Cataño Bay. Despite the Puerto Rican government funding the project, it was overwhelmed with controversy. Several houses needed to be demolished to make way for it, the assembly costs were astronomical ($150 million for the base alone, not adjusted for inflation), and unpaid import taxes prompted an investigation. The statue's possible interference with air traffic, missing permits, and other safety concerns brought everything to a screeching halt.
The statue was left by Cataño Bay, lingering and rusting for several years. In 2005, Tsereteli resumed his search for developers. There were no takers until 2008, when a plan emerged to move the pieces of Columbus to the Mayagüez coastline in time for the 2010 Central American and Caribbean Games. Yet that project was killed as well.
Other possible venues would follow, and Tsereteli chose the town of Arecibo in the end. Finally, the statue would be assembled without significant hurdles, and it was completed by mid-2017. Some activists protested the statue, as they considered Columbus an image of genocide and slavery. Still, the controversy did not hinder the project.
All that remains now is the grand opening to visitors, which has been rescheduled several times. Among the reasons for these delays was the devastation caused by Hurricane María. The statue survived the hurricane, and for the time being, at least, Columbus stands tall by the Arecibo coastline.
The statue is listed on Google Maps and is open daily from 9:00 a.m. to 5:00 p.m. The site is still under construction and is not fully accessible yet, but you can park your vehicle on the street and appreciate the monument from a distance. The area tends to be crowded on weekends.
christopher columbusexplorerscolonialismstrange statuesstatues
CDVV86
https://www.telegraph.co.uk/news/worldnews/centralamericaandthecaribbean/puertorico/7916692/Too-ugly-Christopher-Columbus-statue-finds-home-after-20-years.html
https://en.wikipedia.org/wiki/Birth_of_the_New_World
Calle Los Bajas
Quebrada, Puerto Rico
Cueva Clara
An enchanting natural wonder in Puerto Rico's Camuy River Park.
Added by AlexBarmi
Bundoora, Australia
Upside-Down Charles La Trobe Statue
This strange portrayal of the state's first lieutenant-governor literally stands on its head.
-37.7195, 145.0459
Peter the Great Statue
One of the world's tallest statues is also one of its most hated.
Added by ahvenas
Cercado de Lima, Peru
Francisco Pizarro Statue
A monument to the conquistador became increasingly controversial in the very city that he founded.
-12.0445, -77.0271
'Alec the Goose'
The statue honors a beloved bird that once roamed St. George's Market.
Added by katielou106
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,108
|
Q: Run JUnit test from Java I would like running my JUnit test from Java.
I use :
JUnitCore runner = new JUnitCore();
runner.addListener(new TextListener(System.out));
runner.run(AdditionTest.class);
But I would like the name of the test, the result (true or false), the failure
How I have it?
A: Read the documentation! JUnitCore returns a Result object when you call run(...). The result object has methods like getFailures(), getRunTime() and of course wasSuccessful().
JUnitCore runner = new JUnitCore();
runner.addListener(new TextListener(System.out));
Result result = runner.run(AdditionTest.class);
boolean wasSuccessful = result.wasSuccessful();
System.out.println("tests were successful: " + wasSuccessful);
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,892
|
\section{Basic integrals in terms of dilogarithms and
logarithms}\label{appF}
In presence of vanishing internal masses, specific integrals of the
following type
\[
H
=
\int^a_b du \,
\frac{\ln ( A \, u^2 + B)}{A \, u^2 + B}
\]
for the contour $[0,1]$ as
well as the two other contours $[0,+\infty[$ and $[1,+\infty[$ are involved.
\vspace{0.3cm}
\noindent
We hereby compute all the above types of integrals successively.
The presentation is ordered according to the integration contours $(a,b)$
considered.
We last provide an extra load of back-up integrals.
This appendix often makes use of the identity
\begin{align}
\ln(z) &= \ln(-z) + i \, \pi \, S(z)
\label{eqdeflnzlnmz}
\end{align}
where $S(z)$ is given by eq.~(\ref{eqdeffuncS0}).
\subsection{$H$-type integrals for the IR case}
\subsubsection{First kind}
\[
H_{0,1}(A,B) = \int^1_0 du \, \frac{\ln(A \, u^2 + B)}{A \, u^2 + B}
\]
The cases of real masses ($A$ real) and of complex masses ($\Im(A) \ne 0$) are
treated all at once considering $A$ and $B$ both complex yet such that
sign$(\Im(A \, u^2 + B)$ is kept constant when $u$ spans the range $[0,1]$,
as is always the case for all our needs (cf. sections \ref{3point_ir} and \ref{sectfourpointir}).
We write:
\begin{equation}
H_{0,1}(A,B)
=
\frac{1}{A} \, \int^1_0 du \, \frac{C_A + \ln(u^2 - \bar{u}^2)}{u^2-\bar{u}^2}
\label{eqdefh01}
\end{equation}
where
\begin{align}
C_A &= \ln(A - i \, \lambda \, S(-\bar{u}^2))
\quad \text{if} \; \Im(A)=0
\label{eqdefca1a}\\
C_A &= \ln(A) + \eta(A,-\bar{u}^2)
\quad \text{otherwise}
\label{eqdefca1b}
\end{align}
and $\bar{u}^2 = - B/A$.
The $\eta$ function is given by eq.~(\ref{P1-eqdefeta01}) of ref.\ \cite{paper1}.
The term $\ln(u^2-\bar{u}^2)$ can be split without $\eta$ function since
$\bar{u}$ and $-\bar{u}$ have imaginary parts of opposite signs. Performing a
partial fraction decomposition, we get:
\begin{align}
H_{0,1}(A,B)
&= \frac{1}{2 \, A \, \bar{u}}
\left[
C_A \, \int^1_0 du \, \left( \frac{1}{u-\bar{u}} - \frac{1}{u+\bar{u}} \right)
\right.
\notag \\
&\quad {}\quad {} \quad {} \quad {}\quad {}
+
\int^1_0 du \, \frac{\ln(u-\bar{u})}{u-\bar{u}}
-
\int^1_0 du \, \frac{\ln(u-\bar{u})}{u+\bar{u}}
\notag \\
&\quad {}\quad {} \quad {} \quad {}\quad {}
\left.
+
\int^1_0 du \, \frac{\ln(u+\bar{u})}{u-\bar{u}} -
\int^1_0 du \, \frac{\ln(u+\bar{u})}{u+\bar{u}}
\right]
\label{eqdefh02}
\end{align}
We can rearrange the terms of the r.h.s. of
eq. (\ref{eqdefh02}) in the following way:
\begin{align}
H_{0,1}(A,B)
&= \frac{1}{2 \, A \, \bar{u}}
\left\{
C_A \,
\left[
\ln \left( \frac{\bar{u}-1}{\bar{u}} \right) -
\ln \left( \frac{\bar{u}+1}{\bar{u}} \right)
\right]
\right.
\notag \\
&\quad {}\quad {}\quad {}\quad {} +
\frac{1}{2} \,
\left[
\ln^2(1-\bar{u}) - \ln^2(-\bar{u}) -\ln^2(1+\bar{u}) + \ln^2(\bar{u})
\right]
\notag \\
&\quad {} \quad {}\quad {}\quad {}+
\int^1_0 du \, \frac{\ln(u+\bar{u}) - \ln(2 \, \bar{u})}{u-\bar{u}}
\;\;\; + \;\;\;
\int^1_0 du \, \frac{\ln(2 \, \bar{u})}{u-\bar{u}}
\notag \\
&\quad {}\quad {}\quad {}\quad {} -
\left.
\int^1_0 du \, \frac{\ln(u-\bar{u}) - \ln(-2 \, \bar{u})}{u+\bar{u}} -
\int^1_0 du \, \frac{\ln(-2 \, \bar{u})}{u+\bar{u}}
\;\;
\right\}
\label{eqdefh03}
\end{align}
Using eq. (\ref{eqdeflnzlnmz}) we can write eq.
(\ref{eqdefh03}) in the following way:
\begin{align}
H_{0,1}(A,B)
&= \frac{1}{2 \, A \, \bar{u}}
\left\{
\ln \left( \frac{\bar{u}-1}{\bar{u}+1} \right) \,
\left[
C_A +
\frac{1}{2} \, \left[ \ln(\bar{u}-1) + \ln(\bar{u}+1) \right] +
i \, \pi \, S(-\bar{u}) + \ln(2 \, \bar{u})
\right]
\right.
\notag \\
&\quad {} \quad {}\quad {}\quad {}+
\left.
R^{\prime}(-\bar{u},\bar{u})
\vphantom{\ln \left( \frac{\Lambda - \bar{u}}{1 - \bar{u}} \right)}
\right\}
\label{eqdefh04}
\end{align}
where the function $R^{\prime}$ has been defined in eq. (\ref{P1-eqdeffctr10}) of \cite{paper1}\footnote{The subtlety discussed in \cite{paper1} does not appear in this case.}.
Using eq. (\ref{P1-eqdefrprime4}) of \cite{paper1} with
$y=-\bar{u}$ and $z=\bar{u}$ and rearranging the term in square brackets, we get:
\begin{align}
H_{0,1}(A,B)
&= \frac{1}{2 \, A \, \bar{u}}
\left\{
\ln \left( \frac{\bar{u}-1}{\bar{u}+1} \right) \,
\left[
C_A +
\frac{1}{2} \,
\left[
\ln \left( 1 - \bar{u}^2 \right) + \ln \left( - 4 \, \bar{u}^2 \right)
\right]
\right]
\right.
\notag \\
&\quad {} \quad {}\quad {}\quad {} +
\left.
\mbox{Li}_2 \left( \frac{\bar{u}+1}{2 \, \bar{u}} \right) -
\mbox{Li}_2 \left( \frac{\bar{u}-1}{2 \, \bar{u}} \right)
\right\}
\label{eqdefh05}
\end{align}
\subsubsection{Second kind}
With complex masses we need also to compute:
\begin{equation}
H_{1,\infty}(A,B) = \int^{\infty}_1 du \,
\frac{\ln(A \, u^2 + B)}{A \, u^2 + B}
\label{eqdefh11}
\end{equation}
where $A$ and $B$ are complex yet such that $\mbox{sign}(\Im(A \, u^2 + B))$ is kept
constant while $u$ spans $[1,+\infty[$. The quantity $H_{1,\infty}(A,B)$ can
be written as:
\begin{equation}
H_{1,\infty}(A,B)
=
\frac{1}{A} \, \int^{\infty}_1 du \,
\frac{C^{\prime}_A + \ln(u^2 - \bar{u}^2)}{u^2 - \bar{u}^2}
\label{eqdefh12}
\end{equation}
where $\bar{u}^2 = - B/A$ and $C^{\prime}_A$ is given by:
\begin{align}
C^{\prime}_A &= \ln(A) + \eta(A,1-\bar{u}^2)
\label{eqdefca1c}
\end{align}
We perform a partial fraction decomposition and, writing
$H_{1,\infty}(A,B)$ as a sum
of terms which are individually divergent when $u \rightarrow \infty$,
we face a
situation similar to the one met in subsec. \ref{P2-g2} of \cite{paper2}. We proceed likewise,
introducing a large $u$-cut-off $\Lambda$ and write $H_{1,\infty}(A,B)$ as:
\begin{align}
H_{1,\infty}(A,B)
&= \frac{1}{2 \, A \, \bar{u}} \lim_{\Lambda \rightarrow + \infty}
{\cal H}_{1,\infty}^{\Lambda}(A,B)
\notag
\end{align}
where
\begin{align}
{\cal H}_{1,\infty}^{\Lambda}(A,B)
&=
\left\{
C^{\prime}_A \,
\int^{\Lambda}_1 du \,
\left[ \frac{1}{u-\bar{u}} - \frac{1}{u+\bar{u}} \right]
\right.
\notag \\
& \quad {} \quad {} +
\int^{\Lambda}_1 du \, \frac{\ln(u-\bar{u})}{u-\bar{u}}
-
\int^{\Lambda}_1 du \, \frac{\ln(u+\bar{u})}{u+\bar{u}}
\notag \\
& \quad {} \quad {}
+
\int^{\Lambda}_1 du \, \frac{\ln(u+\bar{u})- \ln(2 \, \bar{u})}{u-\bar{u}}
\;\;\;+\;\;\;
\ln ( 2 \, \bar{u}) \, \int^{\Lambda}_1 \frac{du}{u - \bar{u}}
\notag \\
&\quad {} \quad {}
\left.
-
\int^{\Lambda}_1 du \, \frac{\ln(u-\bar{u}) - \ln(- 2 \, \bar{u})}{u+\bar{u}}
-
\; \ln ( - 2 \, \bar{u}) \, \int^{\Lambda}_1 \frac{du}{u + \bar{u}}
\right\}
\label{eqdefh24}
\end{align}
We express ${\cal H}_{1,\infty}^{\Lambda}(A,B)$ in term of the function
$R^{\Lambda}(y,z) $ defined by eq. (\ref{P2-eqdefrlamda1}) of \cite{paper2}:
\begin{align}
{\cal H}_{1,\infty}^{\Lambda}(A,B)
&=
\left\{
\left[ C^{\prime}_A + \ln ( 2 \, \bar{u}) \right] \,
\ln \left( \frac{\Lambda - \bar{u}}{1 - \bar{u}} \right) -
\left[ C^{\prime}_A + \ln ( - 2 \, \bar{u}) \right] \,\ln \left( \frac{\Lambda + \bar{u}}{1 + \bar{u}} \right)
\right.
\notag \\
&\quad {} +
\frac{1}{2} \, \left[ \ln^2 \left( \Lambda - \bar{u} \right) - \ln^2 \left( 1 - \bar{u} \right) \right] -
\frac{1}{2} \, \left[ \ln^2 \left( \Lambda + \bar{u} \right) - \ln^2 \left( 1 + \bar{u} \right) \right]
\notag \\
&\quad {} +
\left.
R^{\Lambda}(-\bar{u},\bar{u}) - R^{\Lambda}(\bar{u},-\bar{u})
\vphantom{\ln \left( \frac{\Lambda - \bar{u}}{1 - \bar{u}} \right)}
\right\}
\label{eqdefh25}
\end{align}
Using eq. (\ref{P2-eqcalcrlambda3}) of \cite{paper2} for the $R^{\Lambda}$ terms we take the
limit $\Lambda \rightarrow \infty$. The terms proportional to
$\ln^2(\Lambda)$ and those proportional to $\ln (\Lambda)$ drop out
and we get:
\begin{align}
H_{1,\infty}(A,B)
&= \frac{1}{2 \, A \, \bar{u}}
\left\{ \vphantom{\mbox{Li}_2 \left( \frac{\bar{u} + 1}{2 \, \bar{u}} \right)}
\left[ C^{\prime}_A + \ln ( - 2 \, \bar{u}) \right] \,\ln \left( 1 + \bar{u} \right) -
\left[ C^{\prime}_A + \ln ( 2 \, \bar{u}) \right] \, \ln \left( 1 - \bar{u} \right)
\right.
\notag \\
&\quad {} \quad {}\quad {}\quad {}+
\frac{1}{2} \,
\left[
\ln^2 \left( 1 + \bar{u} \right) - \ln^2 \left( 1 - \bar{u} \right) +
\ln^2 \left( 2 \, \bar{u} \right) - \ln^2 \left( - 2 \, \bar{u} \right)
\right]
\notag \\
&\quad {}\quad {}\quad {} \quad {}
\left.
- \mbox{Li}_2 \left( \frac{\bar{u} + 1}{2 \, \bar{u}} \right)
+ \mbox{Li}_2 \left( \frac{\bar{u} - 1}{2 \, \bar{u}} \right)
\label{eqdefh26}
\right\}
\end{align}
Noting that:
\[
\ln \left( \frac{1 + \bar{u}}{1 - \bar{u}} \right)
= \ln (1 + \bar{u}) - \ln (1 - \bar{u})
\]
eq. (\ref{eqdefh26}) becomes after some algebra:
\begin{align}
H_{1,\infty}(A,B)
&= \frac{1}{2 \, A \, \bar{u}}
\left\{
\ln \left( \frac{1 + \bar{u}}{1 - \bar{u}} \right) \,
\left[ C^{\prime}_A + \frac{1}{2} \, \ln ( 1 - \bar{u}^2) + \ln ( 2 \, \bar{u}) \right] +
\frac{\pi^2}{2}
\right.
\notag \\
&\quad {} \quad {}\quad {} \quad {} -
\left.
i \, \pi \, S(\bar{u}) \, \ln \left( \frac{\bar{u}+1}{2 \, \bar{u}} \right)
-
\mbox{Li}_2 \left( \frac{\bar{u} + 1}{2 \, \bar{u}} \right)
+
\mbox{Li}_2 \left( \frac{\bar{u} - 1}{2 \, \bar{u}} \right)
\right\}
\label{eqdefh27}
\end{align}
We remark that the same combination of dilogarithms - up to a sign -
appears in eq. (\ref{eqdefh27}) and in $H_{0,1}(A,B)$ so that we can rewrite
$H_{1,\infty}$ as:
\begin{align}
H_{1,\infty}(A,B)
&= - \, H_{0,1}(A,B) +
\frac{1}{2 \, A \, \bar{u}}
\left\{ \vphantom{\frac{1+\bar{u}}{1-\bar{u}}}
i \, \pi \, S(\bar{u}) \, \left[ 2 \, \ln ( 2 \, \bar{u}) + C_A \right] + \pi^2 \right.
\notag \\
&\qquad {} +
\left. \left[ \eta(A,-\bar{u}^2) - \eta(A,1-\bar{u}^2) \right] \, \ln \left( \frac{1+\bar{u}}{1-\bar{u}} \right)
\right\}
\label{eqdefh28}
\end{align}
\subsubsection{Third kind}
With complex masses a third kind of integrals has also to be considered:
\begin{equation}
H_{0,\infty}(A,B) = \int^{\infty}_0 du \, \frac{\ln(A \, u^2 + B)}{A \, u^2 + B}
\label{eqdefh21}
\end{equation}
where $A$ and $B$ are complex yet such that $\mbox{sign}(\Im(A \, u^2 + B))$ is kept
constant while $u$ spans $[0,+\infty[$. The quantity $H_{0,\infty}(A,B)$ can
be split as:
\begin{equation}
H_{0,\infty}(A,B) = H_{0,1}(A,B) + H_{1,\infty}(A,B)
\label{eqdefh22}
\end{equation}
From eq. (\ref{eqdefh28}) and reminding that the assumption on the sign of $\Im(A \, u^2 + B)$ implies that $\eta(A,-\bar{u}^2) = \eta(A,1-\bar{u}^2)$
, we immediately get:
\begin{align}
H_{0,\infty}(A,B)
&= \frac{1}{2 \, A \, \bar{u}}
\Bigl[
i \, \pi \, S(\bar{u}) \, \left[ 2 \, \ln ( 2 \, \bar{u}) + C_A \right] + \pi^2
\Bigr]
\label{eqdefh23}
\end{align}
As happened for $K^{C}_{0,\infty}(A,B)$ (cf.\ appendix \ref{P2-appF} of \cite{paper2}), $H_{0,\infty}(A,B)$ contains only logarithmic
terms.
\subsection{An extra load of back-up integrals}
We also need the following load of simpler integrals:
\begin{align}
W_1(u_0^2) &= \int^1_0 du \, \frac{\ln(1-u^2)}{u^2 - u_0^2} \\
W_2(u_0^2) &= \int^1_0 du \, \frac{\ln(u)}{u^2 - u_0^2} \\
W_3(u_0^2) &= \int^{\infty}_1 du \, \frac{\ln(u^2-1)}{u^2 - u_0^2} \\
W_4(u_0^2) &= \int^{\infty}_0 du \, \frac{\ln(u)}{u^2 - u_0^2} \\
W_5(u_0^2) &= \int^{\infty}_0 du \, \frac{\ln(u^2+1)}{u^2 + u_0^2}
\end{align}
For all these integrals, $u_0^2$ is assumed to be a complex number, this is indeed the
case because these integrals appear in the computation of the four-point function in
the IR case where $u_0^2$ is either a complex number with an imaginary part $\propto \lambda$
or a genuine complex number.
\vspace{0.3cm}
\noindent
One might compute these integrals using specific values for $A$ and $B$
in $K^R_{0,1}(A,B)$, $K^C_{0,1}(A,B)$, $K^C_{1,\infty}(A,B)$ and $K^C_{0,\infty}(A,B)$
given in appendices~\ref{P1-appF} of ref. \cite{paper1} and~\ref{P2-appF} of ref. \cite{paper2},
however these integrals are simple enough to be computed
directly (we verified that the results can be retrieved using the K-type
integrals after some transformation on the dilogarithms).
We give here the result of these integrals without any details:
\begin{align}
W_1(u_0^2)
&= \frac{1}{2 \, u_0} \,
\left[
\mbox{Li}_2 \left( \frac{2}{1+u_0} \right) -
\mbox{Li}_2 \left( \frac{2}{1-u_0} \right) -
2 \, \ln(2) \, \ln \left( \frac{u_0+1}{u_0-1} \right)
\right]
\label{eqresw1} \\
W_2(u_0^2)
&= \frac{1}{2 \, u_0} \,
\left[
\mbox{Li}_2 \left( \frac{1}{u_0} \right) -
\mbox{Li}_2 \left( - \, \frac{1}{u_0} \right)
\right]
\label{eqresw3} \\
W_3(u_0^2)
&= - W_1(u_0^2) + \frac{1}{2 \, u_0} \, i \, S(u_0) \, \pi \ln \left( 1 - u_0^2 \right)
\label{eqresw2} \\
W_4(u_0^2)
&= \frac{1}{4 \, u_0} \, i \, S(u_0) \, \pi \, \ln(-u^2_0)
\label{eqresw4} \\
W_5(u_0^2)
&= \frac{\pi}{u_0} \, \ln(1+u_0)
\label{eqresw5}
\end{align}
with $u_0 = \sqrt{u_0^2}$.
\section{Herbarium of utilitarian integrals \label{herba}}
This appendix collects a bunch of integrals appearing in the three- and four-point cases to make the reading easier.
\vspace{0.3cm}
\noindent
The first series appears under the following forms and can be computed in terms of the Euler Beta function:
\begin{align}
\int^{+\infty}_0 dz \left(1+z^2 \right)^{-1-\varepsilon} & = \frac{1}{2} \, \int^{+\infty}_0 \frac{dy}{\sqrt{y}} \, (1+y)^{-1-\varepsilon}
= \frac{1}{2} \, B \left( \frac{1}{2},\frac{1}{2} + \varepsilon \right)
\label{firstinty0} \\
\int^{+\infty}_1 dz \left(z^2-1 \right)^{-1-\varepsilon} & = \frac{1}{2} \, \int^{+\infty}_1 \frac{dy}{\sqrt{y}} \, (y-1)^{-1-\varepsilon}
=
\frac{1}{2} \,B \left( \frac{1}{2} + \varepsilon, - \, \varepsilon \right)
\label{secondinty0} \\
\int^1_0 dz \left(1-z^2 \right)^{-1-\varepsilon} & = \frac{1}{2} \,\int^{1}_0 \frac{dy}{\sqrt{y}} \, (1-y)^{-1-\varepsilon}
\quad {}
= \frac{1}{2} \, B \left( \frac{1}{2}, - \, \varepsilon \right)
\label{thirdinty0}
\end{align}
Using the duplication formula for the Gamma functions \cite{abramowitz}, The $z$ integrals computed in closed form read:
\begin{align}
\int^{+\infty}_0 dz \left(1+z^2 \right)^{-1-\varepsilon}
&= \frac{\tan(\pi \, \varepsilon)}{2\varepsilon} \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1- 2\, \varepsilon)} \,
2^{-2 \, \varepsilon} \,
\label{firstinty} \\
\int^{+\infty}_1 dz \left(z^2-1 \right)^{-1-\varepsilon}
&= - \frac{1}{2 \, \varepsilon} \, \frac{1}{\cos(\pi \, \varepsilon)}
\frac{\Gamma^{2}(1-\varepsilon)}{\Gamma(1-2 \, \varepsilon)} \,
2^{- 2 \, \varepsilon}
\label{secondinty} \\
\int^1_0 dz \left(1-z^2 \right)^{-1-\varepsilon}
&=
- \frac{1}{2 \, \varepsilon} \, 2^{-2 \, \varepsilon} \,
\frac{\Gamma(1-\varepsilon)^2}{\Gamma(1-2 \, \varepsilon)}
\label{thirdinty}
\end{align}
\vspace{0.3cm}
\noindent
For the four-point functions in the real mass case or in the complex mass case when $\mbox{sign}(\Im(R_{ij})) = \mbox{sign}(\Im(T))$ , the following integral needs to be evaluated:
\begin{align}
K_2(R_{ij},T)
&= \int^1_0 \frac{d v}{v} \,
\left[
\frac{1}{[ v \, R_{ij} + (1-v) \, T \, - i \, \lambda ]^{1+\varepsilon}}
-
\frac{1}{[(1-v) \, T \, - i \, \lambda]^{1+\varepsilon}}
\right]
\label{eqdefK2}
\end{align}
Note that in both cases, $\Im(v \, R_{ij} + (1-v) \, T \, - i \, \lambda)$ has a constant sign when $v$ spans $[0,1]$.
After a partial fraction decomposition w.r.t. the variable $v$, $K_2(R_{ij},T)$
can be written as :
\begin{align}
K_2(R_{ij},T)
&=
\frac{1}{T} \, \int^{1}_{0} dv \,
\Biggl\{
- (R_{ij}-T) \,
\left[ R_{ij} \, v + T \, (1-v) - i \, \lambda \right]^{-1-\varepsilon}
-
(T - i \, \lambda)^{-\varepsilon} \, (1-v)^{-1-\varepsilon}
\notag \\
&\quad {}\quad {}\quad {} \quad {} \quad {}\quad {}
+
\frac{1}{v} \,
\left[
\left( R_{ij} \,v + T \, (1-v) - i \, \lambda \right)^{- \varepsilon}
- (T - i \, \lambda)^{- \varepsilon}
\right]
\notag \\
&\quad {}\quad {}\quad {} \quad {}\quad {}\quad {}
+ \frac{(T - i \, \lambda)^{-\varepsilon}}{v} \,
\left[ 1 - (1-v)^{- \varepsilon} \right]
\Biggr\}
\label{eqsecondontv}
\end{align}
The first two terms of eq. (\ref{eqsecondontv}) which yield a $1/\varepsilon$
pole are integrated in closed form, whereas
the last three terms of eq. (\ref{eqsecondontv}) which are not divergent can be
expanded around $\varepsilon=0$ up to order $\varepsilon$:
\begin{align}
K_2(R_{ij},T)
&= \frac{1}{T} \,
\Biggl[
\;\;\;\; \frac{1}{\varepsilon} \, (R_{ij} - i \, \lambda)^{-\varepsilon}
\notag \\
&\quad {} \quad {}\quad {}
- \; \varepsilon \, \int^{1}_{0} \frac{dv}{v} \,
\left[
\ln \left( R_{ij} \,v + T \, (1-v) - i \, \lambda \right)
- \ln \left(T - i \, \lambda \right)
\right]
\notag \\
&\quad {} \quad {}\quad {}
+ \; \varepsilon \, (T - i \, \lambda)^{-\varepsilon} \,
\int^1_0 \frac{dv}{v} \, \ln(1-v)
\;
\Biggr]
\label{eqsecondontv1}
\end{align}
Since $\mbox{sign}(\Im(R_{ij} \, v + T \, (1-v) - i \, \lambda))
= \mbox{sign}(\Im(T - i \, \lambda))$ when $v \in [0,1]$ the logarithms in the first
integral can be combined together. The last integration is performed explicitly
and we end with :
\begin{align}
K_2(R_{ij},T)
&= \frac{1}{T} \,
\Biggl[
\frac{1}{\varepsilon} \, (R_{ij} - i \, \lambda)^{-\varepsilon}
+ \varepsilon \, \mbox{Li}_2 \left( \frac{T - R_{ij}}{T - i \, \lambda} \right)
- \varepsilon \, \frac{\pi^2}{6} \Biggr]
\label{eqsecondontv3}
\end{align}
\vspace{0.3cm}
\noindent
With respect to the preceding case, two new integrals show up when $\mbox{sign}(\Im(T)) \ne \mbox{sign}(\Im(R_{ij}))$:
\begin{align}
K_3(A,B)
&= \int^{+\infty}_0 \frac{dv}{v} \,
\left[ (A \, v + B)^{-1-\varepsilon} - (B \, (1+v))^{-1-\varepsilon} \right]
\label{eqdefk3text} \\
K_4(A^{\prime},B^{\prime})
&= \int^{+\infty}_1 \frac{dv}{v} \,
\left[ (A^{\prime} \, v + B^{\prime})^{-1-\varepsilon} - (B^{\prime} \, (1-v))^{-1-\varepsilon} \right]
\label{eqdefk4text}
\end{align}
where $\mbox{sign}(\Im(A \, v + B))$ (resp. $\mbox{sign}(\Im(A^{\prime} \, v + B^{\prime}))$) keeps a constant sign when $v$ spans $[0, + \infty[$ (resp. $[1, + \infty[$).
To compute $K_3(A,B)$, we expand the r.h.s. of eq. (\ref{eqdefk3text}) around
$\varepsilon=0$ and we get:
\begin{align}
K_3(A,B)
&= \frac{1}{B} \,
\ln \left( \frac{B}{A} \right) \, \left[ 1 - \varepsilon \, \ln(B) \right]
\label{eqresulK3}
\end{align}
To compute $K_4(A^{\prime},B^{\prime})$, we expand $(A^{\prime} \, v + B^{\prime})^{-1-\varepsilon}$ in eq.
(\ref{eqdefk4text}) around $\varepsilon=0$, keeping in mind that
$\mbox{sign}(\Im(A^{\prime}+B^{\prime})) = \mbox{sign}(\Im(A^{\prime}))$, and we get:
\begin{align}
K_4(A^{\prime},B^{\prime})
&=
\frac{1}{B^{\prime}}
\left\{
- \, \frac{1}{\varepsilon} (-B^{\prime})^{\varepsilon}
+ \left( \ln(A^{\prime}+B^{\prime}) - \ln(A^{\prime}) \right)
\right.
\notag\\
& \quad {}\quad {}\quad {}
+ \left. \varepsilon \,
\left[
\mbox{Li}_2 \left( - \, \frac{B^{\prime}}{A^{\prime}} \right) - \frac{\pi^2}{6}
-
\frac{1}{2}
\left( \ln^2(A^{\prime}+B^{\prime}) - \ln^2(A^{\prime}) \right)
\right]
\right\}
\label{eqresulK4proto}
\end{align}
With the additional assumption that $\mbox{sign}(\Im(A^{\prime})) = - \mbox{sign}(\Im(B^{\prime}))$ and after some algebra $K_4(A^{\prime},B^{\prime})$ can be recast in the
alternative more useful form:
\begin{align}
K_4(A^{\prime},B^{\prime})
&=
\frac{1}{B^{\prime}}
\left\{
- \, \frac{1}{\varepsilon} (A^{\prime}+B^{\prime})^{\varepsilon}
+ \left( \ln(-B^{\prime}) - \ln(A^{\prime}) \right)
\right.
\notag\\
& \quad {}\quad {}\quad {}
+ \left. \varepsilon \,
\left[
\mbox{Li}_2 \left( \frac{A^{\prime}+B^{\prime}}{B^{\prime}} \right) +
\ln \left( A^{\prime}+B^{\prime} \right) \,
\ln \left( - \, \frac{A^{\prime}}{B^{\prime}} \right)
\right]
\right\}
\label{eqresulK4}
\end{align}
Notice that this additional assumption is always fulfilled in the cases met.
\section{Detailed comparisons with the ``direct way"
( cf. sec. \ref{3point_ir})}\label{direcway3pIR}
Our present goal is to check that the ``indirect way'' leads to the same results for
infrared divergent three-point functions as the ``direct way''. By performing
explicitly the sum over the $j$ index in eq. (\ref{eqdef3n3}), we recover the
results of section (\ref{3point_ir}).
This part is not necessary for the understanding of the method proposed in this article and can be skipped in a first reading.
\subsubsection*{Real mass case}
Let us start by the real mass case. We successively revisit
the examples examined in subsection (\ref{exp_exemp_ir}).
\vspace{0.3cm}
\noindent
{\bf 1. Occurrence of a soft divergence}\\
\noindent
Let us recap the texture of the $\text{$\cal S$}$ matrix (cf.\ eq.~(\ref{eqcalssoft})):
\begin{equation}
\text{$\cal S$} =
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & -2 \, m_2^2 & s_3 - m_2^2 - m_3^2 \\
0 & s_3 - m_2^2 - m_3^2 & - 2 \, m_3^2
\end{array}
\right)
\label{eqcalssoft_ver}
\end{equation}
thus $\det(\text{$\cal S$}) = 0$. Let us single out row and column 1. We readily see that
the vector $V^{(1)}$ vanishes and so do the coefficients $\overline{b}_2$ and
$\overline{b}_3$; thus $\overline{b}_1 = \det{(G)}$ since the coefficients $\overline{b}_i$ fulfil
$\sum_{i \in S_3} \overline{b}_i = \det{(G)}$. The three-point function is thus given by (cf.\ eq.~(\ref{eqdef3n3})):
\begin{align}
I_3^n
&=
\frac{\bbj{2}{1}}{\detgj{1}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{1\}},\widetilde{D}_{12} \right)
+ \frac{\bbj{3}{1}}{\detgj{1}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{1\}},\widetilde{D}_{13} \right)
\label{eq_verif_ir1}
\end{align}
The only relevant reduced $\text{$\cal S$}$ matrix is:
\begin{equation}
\text{$\cal S$}^{\{1\}} =
\left(
\begin{array}{cc}
-2 \, m_2^2 & s_3 - m_2^2 - m_3^2 \\
s_3 - m_2^2 - m_3^2 & - 2 \, m_3^2
\end{array}
\right)
\label{eqcalssoft_ver1}
\end{equation}
whose determinant $\det (\text{$\cal S$}^{\{1\}}) = - {\cal K}(s_3,m_2^2,m_3^2)$
involves the K\"all\'en function given by eq.~(\ref{eqkallenfunc}).
The associated Gram ``matrix" degenerates into a single scalar:
\begin{equation}
G^{\{1\}(2)} = \detgj{1} = \left( 2 \, s_3 \right)
\label{eqgrammat_ver1}
\end{equation}
One easily reads the $\bbj{j}{1}$ coefficients from the reduced Gram matrix $G^{\{1\}(2)}$ and the vector $V^{\{1\}(2)}$ (cf.\ the group of eqs.~(\ref{P1-Rij}) of ref.~\cite{paper1}):
\begin{align}
\bbj{2}{1} &= m_2^2 - m_3^2 - s_3, \quad \bbj{3}{1} = m_3^2 - m_2^2 - s_3 \label{bbar21}
\end{align}
Since $\widetilde{D}_{12} = 2 \, m_3^2$ and $\widetilde{D}_{13} = 2 \, m_2^2$, the quantities
$L_3^n(0,\Delta_1^{\{1\}},\widetilde{D}_{12})$ and $L_3^n(0,\Delta_1^{\{1\}},\widetilde{D}_{13})$ are given by eq. (\ref{eqlijsoft41}). The
$\widetilde{D}_{12} +
\Delta_1^{\{1\}}$, $\widetilde{D}_{13} + \Delta_1^{\{1\}}$ and $\Delta_1^{\{1\}}$ terms are given by eqs.~(\ref{P1-eqtruc1}) and (\ref{P1-eqtruc3}) of \cite{paper1}:
\begin{align}
\widetilde{D}_{12} + \Delta_1^{\{1\}}
&= \frac{(s_3 + m_3^2 - m_2^2)^2}{2 \, s_3}
\label{eqdelta1pd12_ver} \\
\widetilde{D}_{13} + \Delta_1^{\{1\}}
&= \frac{(s_3 + m_2^2 - m_3^2)^2}{2 \, s_3}
\label{eqdelta1pd13_ver} \\
\Delta_1^{\{1\}}
&= \frac{{\cal K}(s_3,m_2^2,m_3^2)}{2 \, s_3}
\label{eqdelta1_ver}
\end{align}
The roots of the denominator of eq. (\ref{eqlijsoft41}) are such that:
\begin{align}
(\bar{z}^{12})^2
&= \frac{\Delta_1^{\{1\}} + i \, \lambda}{\widetilde{D}_{12} + \Delta_1^{\{1\}}}
=
\frac{{\cal K}(s_3,m_2^2,m_3^2) + i \, \lambda \, \sigma_s}
{(s_3 + m_3^2 - m_2^2)^2}
\label{eqroot_ver12} \\
(\bar{z}^{13})^2
&=
\frac{\Delta_1^{\{1\}} + i \, \lambda}{\widetilde{D}_{13} + \Delta_1^{\{1\}}}
= \frac{{\cal K}(s_3,m_2^2,m_3^2) + i \, \lambda \, \sigma_s}
{(s_3 + m_2^2 - m_3^2)^2}
\label{eqroot_ver13}
\end{align}
with $\sigma_s = \mbox{sign}(s_3)$. We specify:
\begin{align}
\bar{z}^{12}
= \frac{\sqrt{{\cal K}(s_3,m_2^2,m_3^2) + i \, \lambda \, \sigma_s}}
{s_3 + m_3^2 - m_2^2}
& \quad {} , \quad {}
\bar{z}^{13}
= \frac{\sqrt{{\cal K}(s_3,m_2^2,m_3^2) + i \, \lambda \, \sigma_s}}
{s_3 + m_2^2 - m_3^2}
\label{eqroot_ver13r}
\end{align}
Introducing two new quantities: $\tilde{x}_1$ and $\tilde{x}_2$ which are the two roots of the equation $D^{\{1\}\, (2)}(x) = 0$ appearing in the ``direct way'':
\begin{align}
\tilde{x}_1
&=
\frac{s_3 + m_2^2 - m_3^2 + \sqrt{{\cal K}(s_3,m_2^2,m_3^2)
+ i \, \lambda \, \sigma_s}}{2 \, s_3}
\label{eqxtilde1def} \\
\tilde{x}_2
&= \frac{s_3 + m_2^2 - m_3^2 - \sqrt{{\cal K}(s_3,m_2^2,m_3^2)
+ i \, \lambda \, \sigma_s}}{2 \, s_3}
\label{eqxtilde2def}
\end{align}
Notice that the quantities $1-\tilde{x}_1$ and $1-\tilde{x}_2$ are the
roots of the equation $D^{\{1\} \, (3)}(x) = 0$.
The quantities $\bar{z}^{12}$, one of the roots of the equation $(\widetilde{D}_{12} + \Delta_1^{\{1\}}) \, z^2 - \Delta_1^{\{1\}} -i \, \lambda = 0$,
and $\bar{z}^{13}$, a root of the equation $(\widetilde{D}_{13} + \Delta_1^{\{1\}}) \, z^2 - \Delta_1^{\{1\}} -i \, \lambda = 0$,
can be related
to the roots $\tilde{x}_1$ and $\tilde{x}_2$ by the following relations:
\begin{align}
\bar{z}^{12}
= \frac{\tilde{x}_1 - \tilde{x}_2}{2 - \tilde{x}_1 - \tilde{x}_2}
&\quad {} , \quad {}
\bar{z}^{13}
= \frac{\tilde{x}_1 - \tilde{x}_2}{\tilde{x}_1 + \tilde{x}_2} \label{eqroot_ver13r1}
\end{align}
Putting things together, the $I_3^n$ can be written:
\begin{align}
I_3^n
&=
- \frac{2^{\varepsilon} \, \Gamma(1 + \varepsilon)}
{2 \, \sqrt{{\cal K}(s_3,m_2^2,m_3^2) + i \, \lambda \, \sigma_s}} \,
\left\{
- \frac{1}{\varepsilon} \,
\left[
\ln \left( \frac{\bar{z}^{12}-1}{\bar{z}^{12}+1} \right) +
\ln \left( \frac{\bar{z}^{13}-1}{\bar{z}^{13}+1} \right)
\right]
\right.
\notag \\
&\quad
+
\left.
\bar{H}_{0,1}
\left( \widetilde{D}_{12}+\Delta_1^{\{1\}},-\Delta_1^{\{1\}}- i \, \lambda \right)
+
\bar{H}_{0,1}
\left( \widetilde{D}_{13}+\Delta_1^{\{1\}},-\Delta_1^{\{1\}}- i \, \lambda \right)
\vphantom{\frac{\bar{z}^{12}-1}{\bar{z}^{12}+1}} \right\}
\label{eq_verif_ir2}
\end{align}
with:
\begin{equation}
\bar{H}_{0,1}
\left( A,B \right)
=
2 \, A \, \sqrt{ - \frac{B}{A} } \,
H_{0,1}
\left( A,B \right)
\label{eqdefhbarij}
\end{equation}
which is effectively the content between the curly brackets of
eq. (\ref{eqdefh05}) in appendix \ref{appF}.
Expressing all the arguments of logarithms and dilogarithms in eq.
(\ref{eqdefh05}) in terms of $\tilde{x}_1$ and $\tilde{x}_2$, we get:
\begin{align}
I_3^n
&=
- \, \frac{2^{\varepsilon} \, \Gamma(1 + \varepsilon)}
{(\tilde{x}_1 - \tilde{x}_2) \, \detgj{1}} \,
\notag\\
& \quad {} \times
\left\{
- \, \frac{1}{\varepsilon} \,
\left[
\ln \left( - \, \frac{1 - \tilde{x}_1}{1 - \tilde{x}_2} \right)
+
\ln \left( - \, \frac{\tilde{x}_2}{\tilde{x}_1} \right)
\right]
\right.
\notag \\
&\qquad \quad {}
+ \ln \left( - \, \frac{1 - \tilde{x}_1}{1- \tilde{x}_2} \right) \,
\left[
\; \ln
\left(
\frac{(2 - \tilde{x}_1 - \tilde{x}_2)^2}{4} \, \detgj{1} + i \, \lambda \, \sigma_s
\right)
\right.
\notag \\
&\qquad \qquad \quad \quad \quad \quad \quad \quad \quad {}
\left.
+
\frac{1}{2}
\left(
\ln \left( \frac{4 \, (1-\tilde{x}_1) \, (1-\tilde{x}_2)}{(2-\tilde{x}_1-\tilde{x}_2)^2} \right)
+
\ln \left( - \, \frac{4 \, (\tilde{x}_1 - \tilde{x}_2)^2}{(2 - \tilde{x}_1 - \tilde{x}_2)^2} \right)
\right)
\right]
\notag \\
&\qquad \quad {}
+
\ln \left( - \, \frac{\tilde{x}_2}{\tilde{x}_1} \right) \,
\left[
\ln
\left(
\frac{(\tilde{x}_1 + \tilde{x}_2)^2}{4} \, \detgj{1} + i \, \lambda \, \sigma_s
\right)
\right.
\notag \\
&\qquad \qquad \quad \quad \quad \quad \quad \quad {}
\left.
+ \frac{1}{2} \,
\left( \ln \left( \frac{4 \, \tilde{x}_1 \, \tilde{x}_2}{(\tilde{x}_1+\tilde{x}_2)^2} \right)
+ \ln \left( - \frac{4 \, (\tilde{x}_1 - \tilde{x}_2)^2}{(\tilde{x}_1 + \tilde{x}_2)^2} \right)
\right)
\right]
\notag \\
&\qquad \quad {} +
\left.
\mbox{Li}_2 \left( \frac{1-\tilde{x}_2}{\tilde{x}_1 - \tilde{x}_2} \right)
- \mbox{Li}_2 \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1 - \tilde{x}_2} \right)
+ \mbox{Li}_2 \left( \frac{\tilde{x}_1}{\tilde{x}_1 - \tilde{x}_2} \right)
- \mbox{Li}_2 \left( - \, \frac{\tilde{x}_2}{\tilde{x}_1 - \tilde{x}_2} \right)
\right\}
\label{eq_verif_ir3}
\end{align}
As $\Im(\tilde{x}_1)$ and $\Im(\tilde{x}_2)$ have
opposite signs, eq. (\ref{eq_verif_ir3}) can be rearranged as:
\begin{align}
I_3^n
&=
- \,
\frac{2^{\varepsilon} \, \Gamma(1 + \varepsilon)}{(\tilde{x}_1 - \tilde{x}_2) \, \detgj{1}}
\notag\\
& \quad {} \times
\left\{
- \, \frac{1}{\varepsilon} \,
\left[
\ln \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1} \right)
-
\ln \left( \frac{\tilde{x}_2 - 1}{\tilde{x}_2} \right)
\right]
\right.
\notag \\
&\qquad \quad {}
+
\left[
\ln \left( \detgj{1} + i \, \lambda \, \sigma_s \right)
+
\frac{1}{2} \, \ln \left( - \, (\tilde{x}_1 - \tilde{x}_2)^2 \right)
\right] \,
\left[
\ln \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1} \right)
-
\ln \left( \frac{\tilde{x}_2 - 1}{\tilde{x}_2} \right)
\right]
\notag \\
&\qquad \quad {}
+ \frac{1}{2} \, \ln \left( - \, \frac{1 - \tilde{x}_1}{1 - \tilde{x}_2} \right) \,
\ln \left( (1-\tilde{x}_1) \, (1-\tilde{x}_2) \right)
+ \frac{1}{2} \, \ln \left( - \, \frac{\tilde{x}_2}{\tilde{x}_1} \right) \,
\ln ( \tilde{x}_1 \, \tilde{x}_2 )
\notag \\
&\qquad \quad {}
- \frac{1}{2} \, \ln^2 \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1 - \tilde{x}_2} \right)
+ \frac{1}{2} \, \ln^2 \left( \frac{1 - \tilde{x}_2}{\tilde{x}_1 - \tilde{x}_2} \right)
- \frac{1}{2} \, \ln^2 \left( \frac{- \, \tilde{x}_2}{\tilde{x}_1 - \tilde{x}_2} \right)
+ \frac{1}{2} \, \ln^2 \left( \frac{\tilde{x}_1}{\tilde{x}_1 - \tilde{x}_2} \right)
\notag \\
&\qquad \quad {} +
\left.
2 \, \mbox{Li}_2 \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_2 - 1} \right)
+ \frac{1}{2} \, \ln^2 \left( \frac{1 - \tilde{x}_2}{\tilde{x}_1 - 1} \right)
- 2 \, \mbox{Li}_2 \left( \frac{\tilde{x}_1}{\tilde{x}_2} \right)
- \frac{1}{2} \, \ln^2 \left( - \,\frac{\tilde{x}_2}{\tilde{x}_1} \right) \right\}
\label{eq_verif_ir4}
\end{align}
Using the relation
between $\ln(z)$ and $\ln(-z)$, after some algebra the quantity
\begin{align*}
E &= \;\;\;
\frac{1}{2} \, \ln \left( - \, \frac{1 - \tilde{x}_1}{1 - \tilde{x}_2} \right) \,
\ln \left( (1-\tilde{x}_1) \, (1-\tilde{x}_2) \right)
+ \frac{1}{2} \, \ln \left( - \, \frac{\tilde{x}_2}{\tilde{x}_1} \right) \,
\ln ( \tilde{x}_1 \, \tilde{x}_2 )
\notag \\
&\quad
- \frac{1}{2} \, \ln^2 \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1 - \tilde{x}_2} \right)
+ \frac{1}{2} \, \ln^2 \left( \frac{1 - \tilde{x}_2}{\tilde{x}_1 - \tilde{x}_2} \right)
- \frac{1}{2} \, \ln^2 \left( \frac{- \, \tilde{x}_2}{\tilde{x}_1 - \tilde{x}_2} \right)
+ \frac{1}{2} \, \ln^2 \left( \frac{\tilde{x}_1}{\tilde{x}_1 - \tilde{x}_2} \right)
\notag
\end{align*}
can be rewritten:
\begin{align}
E
&=
\frac{1}{2} \,
\left[ 2 \, \ln (\tilde{x}_1 - \tilde{x}_2) - i \, \pi \, S(\tilde{x}_1) \right] \,
\left[
\ln \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1} \right)
-
\ln \left( \frac{\tilde{x}_2 - 1}{\tilde{x}_2} \right)
\right]
\notag \\
&=
\frac{1}{2} \, \ln \left( - \, (\tilde{x}_1 - \tilde{x}_2)^2 \right) \,
\left[
\ln \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1} \right)
-
\ln \left( \frac{\tilde{x}_2 - 1}{\tilde{x}_2} \right)
\right]
\label{eqaux_vere1}
\end{align}
with $S(\tilde{x}_1) = \mbox{sign}(\Im(\tilde{x}_1))$. Substituting eq. (\ref{eqaux_vere1}) into
eq. (\ref{eq_verif_ir4}), we end up with:
\begin{align}
I_3^n
&=
-
\frac{2^{\varepsilon} \, \Gamma(1 + \varepsilon)}
{(\tilde{x}_1 - \tilde{x}_2) \, \detgj{1}} \,
\notag \\
&\quad {} \times
\left\{
- \frac{1}{\varepsilon} \,
\left[
\ln \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1} \right)
-
\ln \left( \frac{\tilde{x}_2 - 1}{\tilde{x}_2} \right)
\right]
\right.
\notag \\
&\qquad \quad {}
+
\left[
\ln \left( \detgj{1} + i \, \lambda \, \sigma_s \right) +
\ln \left( - (\tilde{x}_1 - \tilde{x}_2)^2 \right)
\right] \,
\left[
\ln \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_1} \right)
-
\ln \left( \frac{\tilde{x}_2 - 1}{\tilde{x}_2} \right)
\right]
\notag \\
&\qquad \quad {}
+ 2 \, \mbox{Li}_2 \left( \frac{\tilde{x}_1 - 1}{\tilde{x}_2 - 1} \right)
+ \frac{1}{2} \, \ln^2 \left( \frac{1 - \tilde{x}_2}{\tilde{x}_1 - 1} \right)
-
\left.
2 \, \mbox{Li}_2 \left( \frac{\tilde{x}_1}{\tilde{x}_2} \right) -
\frac{1}{2} \, \ln^2 \left( -\frac{\tilde{x}_2}{\tilde{x}_1} \right)
\right\}
\notag \\
&=
- \frac{2^{\varepsilon} \, \Gamma(1 + \varepsilon)}{\varepsilon \, \detgj{1}}
\,
\left\{
\left[ 1 - \varepsilon \, \ln \left( \detgj{1} + i \, \lambda \, \sigma_s \right) \right]
\, K(\tilde{x}_1,\tilde{x}_2) - \varepsilon \, J(\tilde{x}_1,\tilde{x}_2)
\right\}
\label{eq_verif_ir5}
\end{align}
where $K(\tilde{x}_1,\tilde{x}_2)$ is given by eq.~(\ref{eqcompk11}) and $J(\tilde{x}_1,\tilde{x}_2)$
by eq.~(\ref{eqcompj5}).
Last we note that the prescription ``$+ i \, \lambda \, \sigma_s$" in
eq.(\ref{eq_verif_ir5}) can be replaced by ``$- i \, \lambda$" as it matters
only when $\detgj{1} < 0$.
Thus eq. (\ref{eq_verif_ir5}) is nothing but eq. (\ref{eqcompintlog1}):
the indirect and direct ways lead to the same result indeed.
\vspace{0.3cm}
\noindent
{\bf 2. Occurrence of a collinear divergence }\\
\noindent
We recap the texture of the $\text{$\cal S$}$ matrix in this case (cf.\ eq.~(\ref{eqcalscoll})):
\begin{equation}
\text{$\cal S$} =
\left(
\begin{array}{ccc}
0 & 0 & s_1 - m_3^2 \\
0 & 0 & s_3 - m_3^2 \\
s_1 - m_3^2 & s_3 - m_3^2 & - 2 \, m_3^2
\end{array}
\right)
\label{eqcalscoll_ver}
\end{equation}
Obviously, we have $\det(\text{$\cal S$}) = 0$. As explained in section
\ref{3point_ir}, (eqs. (\ref{eqdefg3coll}) and (\ref{eqdefv3coll})), the
coefficient $\overline{b}_3$ vanishes whereas $\overline{b}_2$ and $\overline{b}_1$ are different
from zero. So the three-point function is given by:
\begin{align}
I_3^n
&= \;\;\;
\frac{\overline{b}_1}{\det{(G)}} \,
\left[
\frac{\bbj{2}{1}}{\detgj{1}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{1\}},\widetilde{D}_{12} \right) +
\frac{\bbj{3}{1}}{\detgj{1}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{1\}},\widetilde{D}_{13} \right)
\right]
\nonumber \\
&\quad
+ \frac{\overline{b}_2}{\det{(G)}} \,
\left[
\frac{\bbj{1}{2}}{\detgj{2}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{2\}},\widetilde{D}_{21} \right) +
\frac{\bbj{3}{2}}{\detgj{2}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{2\}},\widetilde{D}_{23} \right)
\right]
\label{eq_verif_irc1}
\end{align}
As $\widetilde{D}_{12} = \widetilde{D}_{21} = 2 \, m_3^2$ and $\widetilde{D}_{13} = \widetilde{D}_{23} = 0$,
$L_{3}^{n} ( 0,\Delta_{1}^{\{1\}},\widetilde{D}_{12} )$ and
$L_{3}^{n} ( 0,\Delta_{1}^{\{2\}},\widetilde{D}_{12} )$
are given by eq. (\ref{eqlijsoft41}) whereas
$L_{3}^{n} ( 0,\Delta_{1}^{\{1\}},\widetilde{D}_{13} )$ and
$L_{3}^{n} ( 0,\Delta_{1}^{\{2\}},\widetilde{D}_{23} )$
are given by (\ref{eqlijsoft8}).
\vspace{0.3cm}
\noindent
Let us first focus on the first term of the r.h.s.
of eq. (\ref{eq_verif_irc1}). The relevant reduced $\text{$\cal S$}$ matrix is:
\begin{equation}
\text{$\cal S$}^{\{1\}} =
\left(
\begin{array}{cc}
0 & s_3 - m_3^2 \\
s_3 - m_3^2 & - 2 \, m_3^2
\end{array}
\right)
\label{eqcalscol_ver1}
\end{equation}
whose determinant is: $\detsj{1} = - (s_3- m_3^2)^2$. The $1 \times 1$
associated Gram matrix is: $G^{\{1\}(2)} = ( 2 \, s_3)$ and the reduced
$\bar{b}$ coefficients are given by:
\begin{align}
\frac{\bbj{2}{1}}{\detgj{1}}
= - \, \frac{s_3 + m_3^2}{2 \, s_3}
& \quad {} , \quad {}
\frac{\bbj{3}{1}}{\detgj{1}}
= - \, \frac{s_3 - m_3^2}{2 \, s_3} \label{eqbbar_vercol2}
\end{align}
whereas:
\begin{align}
\Delta_1^{\{1\}}
= \frac{(s_3 - m_3^2)^2}{2 \, s_3}
& \quad {} , \quad {}
\widetilde{D}_{12} + \Delta_1^{\{1\}} = \frac{(s_3 + m_3^2)^2}{2 \, s_3}
\label{eqdtildepdelta1_vercol}
\end{align}
The square of the root of the polynomial
$(\widetilde{D}_{12} + \Delta_1^{\{1\}}) \, z^2 - \Delta_1^{\{1\}} - i \, \lambda$ is:
\begin{equation}
\bar{z}^2 = \frac{(s_3 - m_3^2)^2}{(s_3 + m_3^2)^2} + i \, \lambda \, \sigma_s
\label{eqsqroot_vercol}
\end{equation}
with $\sigma_s = \mbox{sign}(s_3)$. Whereas
\begin{equation}
\sqrt{\bar{z}^2}
= \left| \frac{s_3 - m_3^2}{s_3 + m_3^2} \right| + i \, \lambda \, \sigma_s
\nonumber
\end{equation}
a more handy choice for further manipulations is instead:
\begin{equation}
\bar{z} =
\frac{s_3 - m_3^2}{s_3 + m_3^2} + i \, \lambda \, \sigma_s \, \sigma_r
\label{eqroot_vercol2}
\end{equation}
where $\sigma_r = \mbox{sign}( (s_3 - m_3^2)/(s_3 + m_3^2) )$. Let us note:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{\bbj{2}{1}}{\detgj{1}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{1\}},\widetilde{D}_{12} \right)
+ \frac{\bbj{3}{1}}{\detgj{1}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{1\}},\widetilde{D}_{13} \right)
\label{eq_verif_irc20}
\end{align}
we get:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{2^{\varepsilon} \, \Gamma(1 + \varepsilon)}{2 \, (m_3^2 - s_3)}
\notag\\
&\quad {} \times
\left[
- \, \frac{1}{\varepsilon} \,
\ln \left( -\frac{m_3^2}{s_3} + i \, \lambda \, \sigma_s \, \sigma_r \right)
+ \bar{H}_{0,1}
\left( \widetilde{D}_{12} + \Delta_1^{\{1\}},-\Delta_1^{\{1\}} - i \, \lambda \right)
\right.
\notag \\
&\qquad \quad {} -
\left.
\frac{1}{\varepsilon^2} \,
\frac{\Gamma(1 - \varepsilon)^2}{\Gamma(1 - 2 \, \varepsilon)} \,
\left( - \frac{2 \, (s_3 - m_3^2)^2}{s_3} - i \, \lambda \right)^{-\varepsilon}
\right]
\label{eq_verif_irc2}
\end{align}
Using the definition of $\bar{H}_{0,1}(X,Y)$ (cf.\ eq.~(\ref{eqdefhbarij})) and eq. (\ref{eqdefh05}), we have:
\begin{align}
& \bar{H}_{0,1}(\widetilde{D}_{12} + \Delta_1^{\{1\}},-\Delta_1^{\{1\}} - i \, \lambda)
\notag\\
&=
\mbox{Li}_2
\left( \frac{s_3}{s_3 - m_3^2} - i \, \lambda \sigma_s \, \sigma_r \right)
- \mbox{Li}_2
\left(
- \, \frac{m_3^2}{s_3 - m_3^2} + i \, \lambda \, \sigma_s \, \sigma_r
\right)
\notag\\
&\quad {}
+
\ln
\left( - \frac{m_3^2}{s_3} + i \, \lambda \, \sigma_s \, \sigma_r \right)
\left[
\ln
\left( \frac{(s_3 + m_3^2)^2}{2 \, s_3} + i \, \lambda \, \sigma_s \right)
+ \frac{1}{2} \,
\ln
\left(
\frac{4 \, s_3 \, m_3^2}{(s_3 + m_3^2)^2} - i \, \lambda \sigma_s
\right)
\right.
\notag \\
&\qquad \qquad \qquad \qquad \quad \quad \quad \quad {} +
\left.
\frac{1}{2} \,
\ln
\left(
-\, \frac{4 \, (s_3 - m_3^2)^2}{(s_3 + m_3^2)^2} - i \, \lambda \, \sigma_s
\right)
\right]
\label{eqresubarh_vercol1}
\end{align}
which can be rewritten as:
\begin{align}
&\bar{H}_{0,1}
\left( \widetilde{D}_{12} + \Delta_1^{\{1\}},-\Delta_1^{\{1\}} - i \, \lambda \right)
\notag\\
&=
\ln \left( - \frac{m_3^2}{s_3} + i \, \lambda \, \sigma_s \, \sigma_r \right)
\,
\left[
-\ln \left( 2 \, s_3 - i \, \lambda \, \sigma_s \right)
+ \frac{1}{2} \,
\ln \left( 4 \, s_3 \, m_3^2 - i \, \lambda \sigma_s \right)
\right.
\notag \\
&\quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad \quad \quad {} +
\left.
\frac{1}{2} \,
\ln \left( - \, 4 \, (s_3 - m_3^2)^2 - i \, \lambda \, \sigma_s \right)
\right]
\notag \\
&\quad {}
+ 2 \,
\mbox{Li}_2
\left( \frac{s_3}{s_3 - m_3^2} - i \, \lambda \sigma_s \, \sigma_r \right)
- \frac{\pi^2}{6}
\notag \\
&\quad {}
+
\ln
\left(
- \, \frac{m_3^2}{s_3 - m_3^2} + i \, \lambda \, \sigma_s \, \sigma_r
\right) \,
\ln \left( \frac{s_3}{s_3 - m_3^2} - i \, \lambda \sigma_s \, \sigma_r \right)
\label{eqresubarh_vercol2}
\end{align}
Putting eq. (\ref{eqresubarh_vercol2}) into eq. (\ref{eq_verif_irc2}), the
$\ln(2)$ drop out and we end with:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{\Gamma(1 + \varepsilon)}{2 \, (m_3^2 - s_3)} \,
\notag\\
&\quad {} \times
\left\{
- \, \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
\ln \left( - \frac{(s_3 - m_3^2)^2}{s_3} - i \, \lambda \right)
-
\ln\left( -\frac{m_3^2}{s_3} + i \, \lambda \, \sigma_s \, \sigma_r \right)
\right]
\right.
\notag \\
&\qquad \quad {}
- \frac{1}{2} \,
\ln^2 \left( - \frac{(s_3 - m_3^2)^2}{s_3} - i \, \lambda \right)
+ 2 \,
\mbox{Li}_2
\left( \frac{s_3}{s_3 - m_3^2} - i \, \lambda \sigma_s \, \sigma_r \right)
\notag \\
&\qquad \quad {}
+ \ln \left( - \frac{m_3^2}{s_3} + i \, \lambda \, \sigma_s \, \sigma_r \right)
\,
\left[
\vphantom{\frac{1}{2}}-\ln \left( s_3 - i \, \lambda \, \sigma_s \right)
\right.
\notag \\
&\qquad \quad {}
\left.
+ \frac{1}{2} \,
\ln \left( s_3 \, m_3^2 - i \, \lambda \sigma_s \right)
+ \frac{1}{2} \,
\ln \left( - (s_3 - m_3^2)^2 - i \, \lambda \, \sigma_s \right)
\right]
\notag \\
&\qquad \quad {} +
\left.
\ln
\left(
- \, \frac{m_3^2}{s_3 - m_3^2} + i \, \lambda \, \sigma_s \, \sigma_r
\right) \,
\ln \left( \frac{s_3}{s_3 - m_3^2} - i \, \lambda \sigma_s \, \sigma_r \right)
\right\}
\label{eq_verif_irc3}
\end{align}
$\Sigma_3^n(s_3)$ can be shown to be equal to the following quantity:
\begin{align}
\Upsilon_3^n(s_3)
&= \frac{\Gamma(1 + \varepsilon)}{2 \, (m_3^2 - s_3)} \,
\left\{
- \, \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
2 \, \ln \left( - s_3 + m_3^2 - i \, \lambda \right)
-
\ln \left( m_3^2 - i \, \lambda \right)
\right]
\right.
\notag \\
&\quad
\left.
- \ln^2 \left( - s_3 + m_3^2 - i \, \lambda \right)
+ \frac{1}{2} \, \ln^2 \left( m_3^2 - i \, \lambda \right)
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2 + i \, \lambda} \right)
\right\}
\label{eq_verif_irc4}
\end{align}
To show that,
one has to distinguish the following cases: 1) $s_3 < - \, m_3^2$,
2) $- \, m_3^2 < s_3 < 0$, 3) $0 < s_3 < m_3^2$ and 4) $m_3^2 < s_3$.
For each of them, some tedious algebra performed on the r.h.s. of eqs.
(\ref{eq_verif_irc3}) shows that the two results are
equal.
Let us discuss in detail only the case $s_3 > m_3^2$, for which
$\sigma_s = \sigma_r = +$. The argument of the dilogarithm in eq.
(\ref{eq_verif_irc3}) has a real part greater than $1$, hence:
\begin{equation}
\mbox{Li}_2
\left( \frac{s_3}{s_3 - m_3^2} - i \, \lambda \sigma_s \, \sigma_r \right)
=
\mbox{Li}_2
\left( \frac{s_3}{s_3 - m_3^2} - i \, \lambda \right)
=
\mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2 + i \, \lambda} \right)
\nonumber
\end{equation}
The logarithms in eq. (\ref{eq_verif_irc3}) can be modified in such a way
that their arguments are ratios of positive quantities, $\Sigma_3^n(s_3)$
thus reads:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{\Gamma(1 + \varepsilon)}{2 \, (m_3^2 - s_3)} \,
\notag\\
& \quad {}
\left\{
- \, \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
\ln \left( \frac{(s_3 - m_3^2)^2}{s_3} \right)
-
\ln \left( \frac{m_3^2}{s_3} \right) - 2 \, i \, \pi
\right]
\right.
\notag \\
&\quad
- \frac{1}{2} \,
\left( \ln \left( \frac{(s_3 - m_3^2)^2}{s_3} \right) - i \, \pi \right)^2
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2 + i \, \lambda} \right)
\notag \\
&\quad
+ \left( \ln \left( \frac{m_3^2}{s_3} \right) + i \, \pi \right) \,
\notag \\
&\quad \quad
\times
\left[
\vphantom{\frac{1}{2}} -\ln \left( s_3 \right)
+ \frac{1}{2} \,
\left(
\ln \left( s_3 \, m_3^2 \right) + \ln \left( (s_3 - m_3^2)^2 \right)
- i \, \pi
\right)
\right]
\notag \\
&\quad
\left.
+
\left( \ln \left( \frac{m_3^2}{s_3 - m_3^2} \right) + i \, \pi \right) \,
\ln \left( \frac{s_3}{s_3 - m_3^2} \right)
\right\}
\label{eq_verif_irc5}
\end{align}
Splitting logarithms of ratios, we get:
\begin{align}
\Sigma_3^n(s_3)
&=
\frac{\Gamma(1 + \varepsilon)}{2 \, (m_3^2 - s_3)} \,
\left\{
- \, \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
2 \, \left( \ln \left( s_3 - m_3^2 \right) - i \, \pi \right)
- \ln\left( m_3^2 \right)
\right]
\right.
\notag \\
&\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
-
\left(
\ln^2 \left( s_3 - m_3^2 \right)
- 2 \, i \, \pi \, \ln \left( s_3 - m_3^2 \right) - \pi^2
\right)
\notag \\
&\quad{} \quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
\left.
+ \frac{1}{2} \, \ln^2(m_3^2)
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2 + i \, \lambda} \right)
\right\}
\label{eq_verif_irc6}
\end{align}
In eq.~(\ref{eq_verif_irc6}), for the case at hand, we recognise eq. (\ref{eq_verif_irc4}).
Similar handling can be performed in the other three cases so as to reach the
same conclusion. We thus conclude:
\begin{align}
\Sigma_3^n(s_3)
& = \Upsilon_3(s_3)
\notag \\
&=
\frac{\Gamma(1 + \varepsilon)}{(m_3^2 - s_3)} \,
\left\{
- \, \frac{1}{\varepsilon^2} \,
\left( -s_3 + m_3^2 - i \, \lambda \right)^{-\varepsilon}
+ \frac{1}{2 \, \varepsilon^2} \,
\left( m_3^2 - i \, \lambda \right)^{-\varepsilon}
\right.
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
\left.
+ \mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2 + i \, \lambda} \right)
\right\}
\label{eq_verif_irc7}
\end{align}
So long for the first term of eq. (\ref{eq_verif_irc1}).
The second term can be obtained from the first one by replacing $s_3$ by $s_1$.
The coefficients $\overline{b}_1$ and $\overline{b}_2$ as well as $\det{(G)}$ are easily extracted
from the $\text{$\cal S$}$ matrix cf. eq. (\ref{eqcalscoll_ver}):
\begin{align}
\overline{b}_1 \; = \; (s_3 - m_3^2) \, (s_1 - s_3) &, \quad {}
\overline{b}_2 \; = \; (s_1 - m_3^2) \, (s_3 - s_1)
\notag\\
\det{(G)} &= - (s_1 - s_3)^2
\label{eqdefbb1bb2detg}
\end{align}
Thus we finally get:
\begin{align}
I_3^n
&= \frac{\overline{b}_1}{\det{(G)}} \, \Sigma_3^n(s_3) + \frac{\overline{b}_2}{\det{(G)}} \,
\Sigma_3^n(s_1)
\notag \\
&= \frac{\Gamma(1 + \varepsilon)}{ (s_1 - s_3)} \,
\left\{
- \frac{1}{\varepsilon^2} \,
\left[
\left( -s_3 + m_3^2 - i \, \lambda \right)^{-\varepsilon}
- \left( -s_1 + m_3^2 - i \, \lambda \right)^{-\varepsilon}
\right]
\right.
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
+ \left.
\mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2 + i \, \lambda} \right)
- \mbox{Li}_2 \left( \frac{s_1}{s_1 - m_3^2 + i \, \lambda} \right)
\right\}
\label{eq_verif_irc8}
\notag
\end{align}
which coincides with eq. (\ref{eqdirei3n8}):
the direct and indirect ways lead to the same result.
\vspace{0.3cm}
\noindent
{\bf 3. Concomitant occurrence of a soft and a collinear divergences} \\
\noindent
Here again, we start to recap the texture of the $\text{$\cal S$}$ matrix\footnote{Contrary to
the example $3$ in subsec.~\ref{exp_exemp_ir}, we choose to set $m_3^2 = 0$ because,
for the purpose of this appendix, a non vanishing mass does not bring anything new with
respect to the previous case.}:
\begin{equation}
\text{$\cal S$} =
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & s_3 \\
0 & s_3 & 0
\end{array}
\right)
\label{eqcalscollsof_ver}
\end{equation}
Obviously, $\det(\text{$\cal S$}) = 0$. As in example ``1. Occurrence of a soft divergence'', the coefficients $\overline{b}_2$
and $\overline{b}_3$ vanish whereas $\overline{b}_1 = \det{(G)}$, and, as
$\widetilde{D}_{12} = \widetilde{D}_{13} = 0$, $L(0,\Delta_1^{\{1\}},0)$ is given by
eq. (\ref{eqlijsoft8}), thus:
\begin{align}
I_3^n &=
\frac{\bbj{2}{1}}{\detgj{1}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{1\}},0 \right)
+
\frac{\bbj{3}{1}}{\detgj{1}} \,
L_{3}^{n} \left( 0,\Delta_{1}^{\{1\}},0 \right)
\label{eq_verif_ircs1}
\end{align}
The relevant reduced $\text{$\cal S$}$ matrix is:
\begin{equation}
\text{$\cal S$}^{\{1\}} =
\left(
\begin{array}{cc}
0 & s_3 \\
s_3 & 0
\end{array}
\right)
\label{eqcalscols_ver1}
\end{equation}
whose determinant is $\detsj{1} = - s_3^2$. The $1 \times 1$ associated Gram
``matrix" is $G^{\{1\}(2)} = ( 2 \, s_3)$ and the reduced $\bar{b}$
coefficients are given by:
\begin{align}
\frac{\bbj{2}{1}}{\detgj{1}}
&=
\frac{\bbj{3}{1}}{\detgj{1}}
\; = \;
- \, \frac{1}{2}
\label{eqbbar_vercols2}
\end{align}
whereas:
\begin{align}
\Delta_1^{\{1\}} &= \frac{s_3}{2} \label{eqdelat1_vercols}
\end{align}
We thus obtain:
\begin{align}
I_3^n
&= - L_3^n\left( 0,\Delta_1^{\{1\}},0 \right) \notag \\
&= -\frac{1}{\varepsilon^2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1-2 \, \varepsilon)} \,
\left(
- \,s_3 - i \, \lambda
\right)^{-1-\varepsilon}
\notag \\
&= W\left( \detgj{1}, 0, 0 \right)
\notag
\end{align}
(cf.\ eqs.~(\ref{eqdirei3n1}), (\ref{eqdefwi0}) and (\ref{eqdirei3n4})), so the direct and
indirect ways lead to the same results.
\subsubsection*{Complex mass case}
We now treat the complex mass case. As discussed in section
(\ref{3point_ir}), the only relevant case is the collinear case where the
non vanishing internal mass, say $m_3^2$, is complex\footnote{We follow the convention of appendix~\ref{P2-ImofdetS} of ref.\ \cite{paper2}}:
$m_3^2 = m_R^2 + i \, m_I^2$
with $m_R^2$ and $m_I^2$ real and $m_R^2 > 0$, $m_I^2 < 0$\footnote{
One could be tempted, in this subsec. to recover the real mass case results by setting
$m^2_I = - \lambda$. Doing that could lead to wrong formulae because, when deriving the
complex mass case, we have already assumed that $|m^2_I| \gg \lambda$ and so dropped some $i \, \lambda$ terms.
}. The $\text{$\cal S$}$ matrix is given
by eq. (\ref{eqcalscoll_ver}) and $I_3^n$ by eq. (\ref{eq_verif_irc1}).
As in the real mass case, let us focus on the first line of eq.~(\ref{eq_verif_irc1}), the second line can be obtained by changing $s_3$ in $s_1$.
To
compute $L(0,\Delta_1^{\{1\}},\widetilde{D}_{12})$ and $L(0,\Delta_1^{\{1\}},\widetilde{D}_{13})$,
we have to determine which formulae to use, depending on the sign of
$\Im(\Delta_1^{\{1\}})$. The quantity $\Delta_1^{\{1\}}$ is given by eq. (\ref{eqdtildepdelta1_vercol}) which reads:
\begin{align}
\Delta_1^{\{1\}}
&= \frac{1}{2 \, s_3} \,
\left( (s_3 - m_R^2)^2 - m_I^4 - 2 \, i \, m_I^2 \, (s_3 - m_R^2) \right)
\label{eqdelta1_vercolc}
\end{align}
When $(s_3 - m_R^2)/s_3 > 0$ i.e. either $s_3 > m_R^2$ or $s_3 < 0$,
$L(0,\Delta_1^{\{1\}},\widetilde{D}_{12})$
is given by eq. (\ref{eqlijsoft41}) as in the real mass case,
and when $(s_3 - m_R^2)/s_3 < 0$ i.e. $0 < s_3 < m_R^2$,
$L(0,\Delta_1^{\{1\}},\widetilde{D}_{12})$
is given by eq. (\ref{eqlijsoft61}).
As discussed previously, there is no such dichotomy for
$L(0,\Delta_1^{\{1\}},\widetilde{D}_{13})$
which is given by eq. (\ref{eqlijsoft8}).
\vspace{0.3cm}
\noindent
Let us first consider $s_3 > m_R^2$ or $s_3 < 0$. We define
$\bar{z} = (s_3 - m_3^2)/(s_3 + m_3^2)$ reminiscent of eq.
(\ref{eqroot_vercol2}) and $\Sigma_3^n(s_3)$ is given by eq.
(\ref{eq_verif_irc3}), in which the infinitesimal imaginary parts
$\propto \lambda$ are dropped out except for arguments of logarithms
which do not depend on $m_3^2$, namely:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{\Gamma(1 + \varepsilon)}{2 \, (m_3^2 - s_3)}
\left\{
- \, \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
\ln \left( - \, \frac{(s_3 - m_3^2)^2}{s_3} \right)
-
\ln \left( - \, \frac{m_3^2}{s_3} \right)
\right]
\right.
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {}\quad {}
- \frac{1}{2} \, \ln^2 \left( - \, \frac{(s_3 - m_3^2)^2}{s_3} \right)
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2} \right)
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {}\quad {}
+ \ln \left( - \, \frac{m_3^2}{s_3} \right)
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
\times \left[
\vphantom{\frac{1}{2}}- \ln \left( s_3 - i \, \lambda \, \sigma_s \right)
+ \frac{1}{2} \,
\left[
\ln \left( s_3 \, m_3^2 \right) + \ln \left( - \, (s_3 - m_3^2)^2 \right)
\right]
\right]
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {}\quad {}
\left.
+ \ln \left( - \, \frac{m_3^2}{s_3 - m_3^2} \right) \,
\ln \left( \frac{s_3}{s_3 - m_3^2} \right)
\right\}
\label{eq_verif_ircc3}
\end{align}
Logarithms of products and ratios in eq. (\ref{eq_verif_ircc3}) are further
split. For this purpose we use eq. (\ref{eqdeflnzlnmz}) as well as:
\[
\ln \left( (s_3 - m_3^2)^2 \right)
=
2 \, \ln(s_3 - m_3^2)
- 2 \, i \, \pi \, \theta \left( - s_3 + m_R^2 \right)
\]
and, for any real $a$ and complex $b$:
\begin{align}
\ln \left( \frac{b}{a} \right)
&= - \ln \left( a + i \, \lambda \, S(b) \right) + \ln(b)
\notag\\
\ln(a \, b)
&= \;\;\;\, \ln \left( a - i \, \lambda \, S(b) \right) + \ln(b)
\notag
\end{align}
We also use the notation $S(z) = \mbox{sign}\left( \Im(z) \right)$. Let us note that
$S\left( (s_3 - m_3^2)^2 \right) = \mbox{sign}(s_3 - m_R^2) \equiv \sigma_p$ and that
$\ln( s_3 - i \, \lambda \, \sigma_s)$ is equivalent to
$\ln( s_3 + i \, \lambda)$.
Then, to compactify eq.~(\ref{eq_verif_ircc3}), we take advantage of the following relations:
\begin{align}
\ln\left( - s_3 \pm i \, \lambda \right) &= \ln\left( |s_3| \right) \pm i \, \pi \, \theta(s_3) \notag \\
\ln\left( s_3 \pm i \, \lambda \right) &= \ln\left( |s_3| \right) \pm i \, \pi \, \theta(- s_3) \notag \\
\theta(\pm s_3) &= \frac{(1 \pm \sigma_s)}{2} \notag \\
\theta(- s_3 + m_R^2) &= \frac{1- \sigma_p}{2}
\label{eqnewrelovers3}
\end{align}
Eq. (\ref{eq_verif_ircc3}) can be rewritten:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{\Gamma(1 + \varepsilon)}{2 \, (m_3^2 - s_3)} \,
\left\{
- \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
2 \, \left( \ln \left( s_3 - m_3^2 \right) - i \, \pi \right)
- \ln \left( m_3^2 \right)
\right]
\right.
\notag \\
&\qquad \qquad \qquad \quad {}
- \frac{1}{2} \,
\left[
2 \, \ln \left( s_3 - m_3^2 \right)
- i \, \pi \, \left( 1 - \frac{\sigma_p}{2} + \frac{\sigma_p \, \sigma_s}{2} \right)
\right]^2
\notag \\
&\qquad \qquad \qquad \quad {}
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2} \right)
+ \left( \ln \left( m_3^2 \right) + i \, \pi \, \frac{1+\sigma_s}{2}
\right)
\notag \\
&\qquad \qquad \qquad \quad {}
\times
\left[
\frac{1}{2} \, \ln \left( m_3^2 \right)
+ \ln \left( s_3 - m_3^2 \right) - i \, \pi \, (3 - \sigma_s)
\right]
\notag \\
&\qquad \qquad \qquad \quad {}
+
\left(
\ln \left( m_3^2 \right) - \ln \left( s_3 - m_3^2 \right) + i \, \pi
\right) \,
\left(
i \, \pi \, \frac{1-\sigma_s}{2} - \ln \left( s_3 - m_3^2 \right)
\right) \notag \\
&\qquad \qquad \qquad \quad {}
+ \left. i \, \frac{\pi}{2} \, \ln\left( |s_3| \right) \, (1-\sigma_s) \, (1+\sigma_p)
\vphantom{\frac{1}{\varepsilon^2}}
\right\}
\label{eq_verif_ircc4}
\end{align}
Let us treat the case where $s_3 > m_R^2$. We have $\sigma_p = +$ and
$\sigma_s=+$. By expanding eq. (\ref{eq_verif_ircc4}), we get:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{\Gamma(1 + \varepsilon)}{2 \, (m_3^2 - s_3)} \,
\left\{
- \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
2 \, \left( \ln \left( s_3 - m_3^2 \right) - i \, \pi \right)
- \ln\left( m_3^2\right)
\right]
\right.
\notag \\
&\quad {} \quad {}\quad {} \quad {} \quad {}\quad {} \quad {}
-
\left(
\ln^2 \left( s_3 - m_3^2 \right)
- 2 \, i \, \pi \, \ln \left( s_3 - m^2 \right) - \pi^2
\right)
\notag \\
&\quad {} \quad {}\quad {} \quad {} \quad {}\quad {} \quad {}
\left.
+ \frac{1}{2} \, \ln \left( m_3^2 \right)
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3 - m_3^2} \right)
\right\}
\label{eq_verif_ircc5}
\end{align}
As $\ln(s_3 - m_3^2) - i \, \pi = \ln(-s_3 + m_3^2)$, we recover
$\Upsilon_3^n(s_3)$ given by eq. (\ref{eq_verif_irc4}).
The same exercise can be easily done for the case $s_3 < 0$ leading to
the same conclusion.
\vspace{0.3cm}
\noindent
In the case where $0 < s_3 < m_R^2$,
$L(0,\Delta_1^{\{1\}},\widetilde{D}_{12})$
is given by eq. (\ref{eqlijsoft61}) i.e.:
\begin{align}
L(0,\Delta_1^{\{1\}},\widetilde{D}_{12})
&=
\frac{2^{\varepsilon} \, \Gamma(1+\varepsilon)}
{2 \, (\widetilde{D}_{12} + \Delta_1^{\{1\}}) \, \bar{z}} \,
\left[
\frac{1}{\varepsilon} \,
\left(
i \, \pi \, S(i \, \bar{z} )
+ \ln \left( \frac{1+\bar{z}}{1-\bar{z}} \right)
\right)
\right.
\notag \\
&\qquad \qquad \qquad \qquad \quad {}
- \bar{H}_{0,\infty}(- \widetilde{D}_{12} - \Delta_1^{\{1\}}, - \Delta_1^{\{1\}})
\notag \\
&\qquad \qquad \qquad \qquad \quad {}
- \left.
\bar{H}_{1,\infty}(\widetilde{D}_{12} + \Delta_1^{\{1\}}, - \Delta_1^{\{1\}})
\vphantom{\frac{1}{\varepsilon}} \right]
\label{eqdefnewL12}
\end{align}
where $\bar{z} = \sqrt{\Delta_1^{\{1\}}/(\widetilde{D}_{12}+\Delta_1^{\{1\}})}$. We have chosen for the root of the equation
$(\widetilde{D}_{12} + \Delta_1^{\{1\}})\, z^2 + \Delta_1^{\{1\}} = 0$,
$\tilde{z} = i \, \bar{z}$. The functions $\bar{H}_{1,\infty}(x,y)$
is given by the expression in curly brackets of eq.
(\ref{eqdefh27}) and $\bar{H}_{0,\infty}(x,y)$ by the expression in square
brackets in eq. (\ref{eqdefh23}) of appendix \ref{appF} i.e.:
\begin{align}
\bar{H}_{0,\infty}(- \widetilde{D}_{12} - \Delta_1^{\{1\}}, - \Delta_1^{\{1\}})
&= i \, \pi \, S(i \, \bar{z}) \,
\left[
2 \, \ln \left( 2 \, i \, \bar{z} \right)
+ \ln \left( - \widetilde{D}_{12} - \Delta_1^{\{1\}} \right)
\right.
\notag \\
&\qquad \qquad \qquad {}
+ \left.
\eta \left( -\widetilde{D}_{12} - \Delta_1^{\{1\}}, \bar{z}^2 \right)
\right]
+ \pi^2
\label{eqbarh0inf1} \\
\bar{H}_{1,\infty}(\widetilde{D}_{12} + \Delta_1^{\{1\}}, - \Delta_1^{\{1\}})
&=
\ln \left( \frac{1 + \bar{z}}{1 - \bar{z}} \right) \,
\left[ \vphantom{\frac{\widetilde{D}_{12}}{\widetilde{D}_{12} + \Delta_1^{\{1\}}}}
\ln \left( \widetilde{D}_{12} + \Delta_1^{\{1\}} \right)
+ \frac{1}{2} \, \ln \left( 1 - \bar{z}^2 \right)
+ \ln \left( 2 \, \bar{z} \right)
\right.
\notag \\
&\qquad \qquad \qquad \quad {}
+ \left.
\eta
\left(
\widetilde{D}_{12} + \Delta_1^{\{1\}}, \frac{\widetilde{D}_{12}}{\widetilde{D}_{12} + \Delta_1^{\{1\}}}
\right)
\right]
+ \frac{\pi^2}{2}
\notag \\
&\quad {}
- i \, \pi \, S(\bar{z}) \, \ln \left( \frac{\bar{z} + 1}{ 2 \, \bar{z}} \right)
- \mbox{Li}_2 \left( \frac{\bar{z} + 1}{ 2 \, \bar{z}} \right)
+ \mbox{Li}_2 \left( \frac{\bar{z} - 1}{ 2 \, \bar{z}} \right)
\label{eqbarh1inf1}
\end{align}
The quantities $\widetilde{D}_{12}+\Delta_1^{\{1\}}$ and $\Delta_1^{\{1\}}$
can be expressed in terms of $s_3$ and $m_3^2$ with the group of eqs.~(\ref{eqdtildepdelta1_vercol}). The expression obtained contains
the ratio $(s_3 - m_3^2)/(s_3 + m_3^2)$ given by:
\begin{equation}
\frac{s_3 - m_3^2}{s_3 + m_3^2}
=
\frac{s_3^2 - m_R^4 - m_I^4 - 2 \, i \, m_I^2 \, s_3}{(s_3 + m_R^2)^2 + m_I^4}
\label{eqratios3pmm2}
\end{equation}
which has a negative real part and a positive imaginary part for
$0 < s_3 < m_R^2$. This implies that:
\begin{align}
S \left( i \, \bar{z} \right) &= -1, \quad
S \left( \bar{z} \right) = +1
\label{eqvariouss1}
\end{align}
In addition, since $\Im( (s_3 - m_3^2)^2 ) = - 2 \, m_I^2 \, (s_3 - m_R^2) < 0$
and $\Im( (s_3 + m_3^2)^2 ) = 2 \, m_I^2 \, (s_3 + m_R^2) < 0$, one can show
that:
\[
\eta
\left(
\frac{(s_3+m_3^2)^2}{2 \, s_3}, \frac{4 \, m_3^2 \, s_3}{(s_3+m_3^2)^2}
\right)
= \eta
\left(
- \frac{(s_3+m_3^2)^2}{2 \, s_3}, \frac{(s_3-m_3^2)^2}{(s_3+m_3^2)^2}
\right)
= 0
\]
Putting all things together, $L(0,\Delta_1^{\{1\}},\widetilde{D}_{12})$ becomes:
\begin{align}
\hspace{2em}&\hspace{-2em}L(0,\Delta_1^{\{1\}},\widetilde{D}_{12}) \notag \\
&=
\frac{2^{\varepsilon} \, \Gamma(1+\varepsilon) \, s_3}
{(s_3 + m_3^2) \, (s_3 - m_3^2)} \,
\left\{
\frac{1}{\varepsilon} \,
\left[ \ln \left( \frac{s_3}{m_3^2} \right) - i \, \pi \right] - \frac{10 \, \pi^2}{6}
+ i \, \pi \, \ln \left( \frac{s_3}{s_3 - m_3^2} \right)
\right.
\notag \\
&\qquad \qquad \qquad \qquad \qquad \quad
+ i \, \pi \,
\left[
2 \, \ln \left( 2 \, i \, \frac{s_3-m_3^2}{s_3+m_3^2} \right)
+ \ln \left( - \frac{(s_2-m_3^2)^2}{2 \, s_3} \right)
\right]
\notag \\
&\qquad \qquad \qquad \qquad \qquad \quad
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3-m_3^2} \right)
+ \ln \left( -\frac{m_3^2}{s_3-m_3^2} \right) \,
\ln \left( \frac{s_3}{s_3 - m_3^2} \right)
\notag \\
&\qquad \qquad \qquad \qquad \qquad \quad
- \ln \left( \frac{s_3}{m_3^2} \right) \,
\left[
\ln \left( \frac{(s_3+m_3^2)^2}{2 \, s_3} \right)
+ \frac{1}{2} \, \ln \left( \frac{4 \, m_3^2 \, s_3}{(s_3+m_3^2)^2} \right)
\right.
\notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad
+ \left.
\left.
\ln \left( 2 \, \frac{s_3 - m_3^2}{s_3 + m_3^2} \right)
\right]
\right\}
\label{eqdefnewL1210}
\end{align}
Substituting eq. (\ref{eqdefnewL1210}) into eq. (\ref{eq_verif_irc20})
with the explicit values for $\bbj{2}{1}$, $\bbj{3}{1}$ and $\detgj{1}$
and using eq.(\ref{eqlijsoft8}) for $L(0,\Delta_1^{\{1\}},\widetilde{D}_{13})$, we get:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{\Gamma(1+\varepsilon)}{2 \, (m_3^2 - s_3)} \,
\left\{
- \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
\ln \left( \frac{s_3}{m_3^2} \right)
+
\ln \left( - \frac{(s_3 - m_3^2)^2}{s_3} \right) - i \, \pi
\right]
\right.
\notag \\
&\qquad \qquad \qquad \quad
- \frac{1}{2} \, \ln^2 \left( - \frac{(s_3 - m_3^2)^2}{s_3} \right) - \frac{3 \, \pi^2}{2}
+ i \, \pi \, \ln \left( \frac{s_3}{s_3 - m_3^2} \right)
\notag \\
&\qquad \qquad \qquad \quad
+ i \, \pi \,
\left[
2 \, \ln \left( i \, \frac{s_3-m_3^2}{s_3+m_3^2} \right)
+ \ln \left( - \frac{(s_2-m_3^2)^2}{s_3} \right)
\right]
- \ln \left( \frac{s_3}{m_3^2} \right)
\notag \\
&\qquad \qquad \qquad \quad
\times
\left[
\ln \left( \frac{(s_3+m_3^2)^2}{s_3} \right)
+ \frac{1}{2} \, \ln \left( \frac{m_3^2 \, s_3}{(s_3+m_3^2)^2} \right)
+ \ln \left( \frac{s_3 - m_3^2}{s_3 + m_3^2} \right)
\right]
\notag \\
&\qquad \qquad \qquad \quad
\left.
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3-m_3^2} \right)
+ \ln \left( -\frac{m_3^2}{s_3-m_3^2} \right) \,
\ln \left( \frac{s_3}{s_3 - m_3^2} \right)
\right\}
\label{eq_verif_ircc2}
\end{align}
Keeping in mind that $0 < s_3 < m_R^2$, we split
the logarithms and expand the terms to end with:
\begin{align}
\Sigma_3^n(s_3)
&= \frac{\Gamma(1+\varepsilon)}{2 \, (m_3^2 - s_3)} \,
\left\{
- \frac{1}{\varepsilon^2} + \frac{1}{\varepsilon} \,
\left[
2 \, \ln \left( s_3 - m_3^2 \right) - 2 \, i \, \pi - \ln \left( m_3^2 \right)
\right]
\right.
\notag \\
&\qquad \qquad \qquad \quad
- \left( \ln^2 \left( s_3 - m_3^2 \right) - \pi^2
- 2 \, i \, \pi \, \ln \left( s_3-m_3^2 \right) \right)
\notag \\
&\qquad \qquad \qquad \quad
+ \left.
\frac{1}{2} \, \ln^2 \left( m_3^2 \right)
+ 2 \, \mbox{Li}_2 \left( \frac{s_3}{s_3-m_3^2} \right)
\right\}
\label{eq_verif_ircc3final}
\end{align}
and using $\ln(s_3 - m_3^2) = \ln(-s_3 + m_3^2) + i \, \pi$, we again
recover
eq. (\ref{eq_verif_irc4}).
We note that the same formula holds both for
$\Im(\Delta_{1}^{\{1\}})>0$ i.e. either $s_3<0$ or $s_3> m_R^2$, and for
$\Im(\Delta_{1}^{\{1\}})<0$ i.e. $0< s_3< m_R^2$. This is because in the last
case, the integration contour $\int_{0}^{+ i \infty} + \int_{+\infty}^{1}$ can
actually be deformed into $\int_{0}^{1}$, i.e. eq. (\ref{eqlijsoft61}) can be
deformed into eq. (\ref{eqlijsoft41}) by means of the Cauchy theorem. Indeed,
when $0< s_3< m_R^2$,
$\Im(z^2 \, (\widetilde{D}_{12} + \Delta_{1}^{\{1\}}) - \Delta_{1}^{\{1\}})$ never
vanishes as $z$ spans the real interval $[0,1]$ hence the cut of
$\ln(z^2 \, (\widetilde{D}_{12} + \Delta_{1}^{\{1\}}) - \Delta_{1}^{\{1\}})$ in the
half plane $\{\Re(z) >0\}$ entirely lies inside the ``south-east'' quadrant
$\{\Re(z) >0, \Im(z) <0\}$.
\vspace{0.3cm}
\noindent
$\Sigma_3^n(s_1)$ is read from eq.~(\ref{eq_verif_ircc3final})
by replacing $s_3$ by $s_1$ and the coefficients $\overline{b}_1$ and $\overline{b}_2$ as well as $\det{(G)}$ are
still given by eq.~(\ref{eqdefbb1bb2detg}). So, in the complex mass case the same result
is obtained as in the real mass case for $I_3^n$ and this leads to the conclusion that
for the case of complex masses, the ``direct way'' and the ``indirect way'' also coincide.
\section{Change of contour prescription for the pole in the IR four-point
integral}\label{ir-lambda}
The appendix legitimates the replacement
\begin{equation}\label{subst1}
\int_{0}^{1} du \,
\frac{\left[ u^2 \, P_{ijk} + R_{ij} - i \, \lambda \right]^{-\varepsilon}}
{u^2 \, P_{ijk} + R_{ij} + i \, \lambda}
\to
\int_{0}^{1} du \,
\frac{\left[ u^2 \, P_{ijk} + R_{ij} - i \, \lambda \right]^{-\varepsilon}}
{u^2 \, P_{ijk} + R_{ij} - i \, \lambda}
\end{equation}
when $\lambda \to 0^{+}$ for fixed $0 < - \varepsilon \ll 1$, whenever
$P_{ijk}$ and $R_{ij}$ are both real with $0 < - \, R_{ij}/P_{ijk} < 1$.
Intuitively this replacement is based on the fact that
for any function $f(u)$ analytic along $[0,1]$ and $0 < u_{0} < 1$,
\[
\int_{0}^{1} du \,
\frac{f(u)}{u - u_{0} - i \, \lambda}
-
\int_{0}^{1} du \,
\frac{f(u)}{u - u_{0} + i \, \lambda}
\to
2 i \, \pi \, f(u_{0})
\;\;
\mbox{when} \;\; \lambda \to 0^{+}
\]
which vanishes if $u_{0}$ is a zero of $f(u)$. However the situation is
made trickier when $f(u) = (u-u_{0})^{-\varepsilon}$
thus has a branch point at $u = u_{0}$ and a cut running along part of the
interval of integration. To make the above argument apply one could think of
shifting the branch point and cut by a contour prescription
$- i \, a \lambda$ with $a>1$ so as to pass whether above or below
the pole while remaining on the same side of the cut.
We would then get a residue
value $\propto \lambda^{\; -\varepsilon}$ vanishing in
the limit $\lambda \to 0^{+}$ keeping $\varepsilon < 0$ fixed.
The ``hierarchised lambdalogy" underpinning this disentanglement of pole from
branch point may look awkward to the rigorous reader, let us thus back up
this hand waving argument more rigorously as follows.
\vspace{0.3cm}
\noindent
We first perform a partial fraction decomposition of the pole term:
\begin{eqnarray}
\lefteqn{\frac{1}{u^2 \, P_{ijk} + R_{ij} + i \, s \, \lambda}}
\nonumber\\
& = &
\frac{1}{P_{ijk}} \, \frac{1}{2 \, \sqrt{- R_{ij}/P_{ijk}}} \,
\left[
\frac{1}{u - (u_{0} - i \, s \lambda^{\prime})}
-
\frac{1}{u + (u_{0} - i \, s \lambda^{\prime})}
\right]
\label{subst2}
\end{eqnarray}
where $s = \pm$, $u_{0} = \sqrt{- R_{ij}/P_{ijk}}$ is assumed in $]0,1[$ and
$\lambda^{\prime} = \lambda/(2 u_{0} P_{ijk})$\footnote{We keep track of this
multiplicative change to control various normalisations in the reasoning so as
to check the independence of the conclusion w.r.t. any assumption of
``hierarchised lambdalogy".}. We focus on the pole at $+ u_{0}$ in
decomposition (\ref{subst2}): since the pole at $-u_{0}$ involved in the second
term lies outside the integration region the contour prescription for this pole
is irrelevant and so is the corresponding pole term in the discussion.
We then study the legitimacy of the replacement
\begin{equation}\label{subst3}
\int_{0}^{1} du \,
\frac{\left[ u^2 \, P_{ijk} + R_{ij} - i \, a \lambda \right]^{-\varepsilon}}
{u - u_{0} + i \, \lambda^{\prime}
}
\to
\int_{0}^{1} du \,
\frac{\left[ u^2 \, P_{ijk} + R_{ij} - i \, a \lambda\right]^{-\varepsilon}}
{u - u_{0} - i \, \lambda^{\prime}}
\end{equation}
($a$ being positive yet kept arbitrary).
We consider ``$\delta \equiv$ l.h.s. minus r.h.s." of eq. (\ref{subst3}),
which can be written:
\[
\delta =
\int_{0}^{1} \frac{du \, ( 2 \, i \, \lambda^{\prime})}
{(u - u_{0})^{2} + \lambda^{\prime \, 2}}
\left[
P_{ijk} \,
\left( (u - u_{0})(u + u_{0}) - i \, (2 u_{0} a) \, \lambda^{\prime} \right)
\right]^{-\varepsilon}
\]
we make the change of variable $(u - u_{0}) = |\lambda^{\prime}| \, v$ and
$\delta$ reads:
\begin{equation}\label{subst4}
\delta
=
2 \, i \, \sigma \,
\left( \frac{\lambda}{2 u_{0}} \right) ^{- \varepsilon}
\int_{-u_{0}/|\lambda^{\prime}| }^{(1- u_{0})/|\lambda^{\prime}|}
\frac{dv}{v^2+1}
\left[
\sigma v \left( \sigma \lambda^{\prime} v \, + 2 u_{0} \right)
- i (2 u_{0} a)
\right]^{-\varepsilon}
\end{equation}
(where $\sigma = \mbox{sign}(\lambda^{\prime}) = \mbox{sign}(P_{ijk})$).
For any $b > 1$ and $|\lambda^{\prime}|$ small enough we have, for all real
$v$:
\[
\left|
\sigma v ( \sigma \lambda^{\prime} v \, + 2 u_{0}) - i (2 u_{0} a)
\right|
< \left[ v^4 + (2u_{0})^2 b \, v^2 + (2 u_{0} a)^2 \right] ^{1/2}
\]
and the integral
\[
\int_{- \infty}^{+\infty} dv \,
\frac{[ v^4 + (2u_{0})^2 b \, v^2 + (2 u_{0} a)^2]^{- \varepsilon/2}}{v^2+1}
\]
is convergent when $-\varepsilon>0$ is small enough.
It provides an ``integrable
hat" for the application of Lebesgue's theorem of dominated convergence.
When $\lambda \to 0^{+}$ keeping $-\varepsilon>0$
fixed and small enough, the integral in eq. (\ref{subst4}) has the limit
\[
{\cal L} = (2 u_{0})^{- \varepsilon}
\int_{-\infty}^{+\infty}
\frac{dv}{v^2+1} \left( \sigma v - i a \right)^{-\varepsilon}
\]
which is finite regardless of $a > 0$ and $\varepsilon$ small enough
(its actual value, readily computable using the residue theorem,
is irrelevant for the conclusion).
We thus see that $\delta \sim {\cal O}(\lambda^{- \varepsilon})$ as
anticipated. q.e.d.
\section{Two basic integrals \label{appendJ}}
In what follows $A$ and $B$ are assumed dimensionless and complex valued, the
signs of their real parts is unknown, and the signs of their imaginary parts
may or may not be the same either.
\subsection{First kind}
The computation of the three- and four-point functions in a space-time of arbitrary dimensions,
involves the following extension of the case treated in appendix \ref{P1-appendJ} of ref.\ \cite{paper1}:
\begin{equation}
K(\nu) = \int^{\infty}_0 \frac{d \xi}{(\xi^{\nu}+A) \, (\xi^{\nu}+B)}
\label{eqdefk1ext}
\end{equation}
After partial fraction decomposition the r.h.s. of eq. (\ref{eqdefk1ext}) becomes:
\begin{equation}
K(\nu) = \frac{1}{B-A} \,
\int^{\infty}_0 d \xi \,
\left[ \frac{1}{\xi^{\nu}+A} - \frac{1}{\xi^{\nu}+B} \right]
\label{eqdefk2ext}
\end{equation}
Let us assume that $\nu > 1$ such that
the r.h.s.\ of eq.\ (\ref{eqdefk1ext}) can be split into a difference of two
convergent integrals at infinity, which can be separately computed using
appendix \ref{P1-ap2} of ref. \cite{paper1} with $\mu =1$. $K(\nu)$ thus reads:
\begin{equation}
K(\nu)
=
\frac{1}{B-A} \,
\frac{1}{\nu} \, B \left(1 - \frac{1}{\nu}, \frac{1}{\nu} \right) \,
\left[ A^{\frac{1}{\nu} - 1} - B^{\frac{1}{\nu} - 1} \right]
\label{eqdefk3ext}
\end{equation}
regardless of the signs of $\Im(A)$ vs. $\Im(B)$.
In the case of the three-point function, $\nu = 1/(1-\varepsilon)$ which is slightly less
that $1$ for $\varepsilon < 0$, the r.h.s.\ of eq.~(\ref{eqdefk3ext}) can be analytically
continued in $\nu$ as long as $\nu \neq -1/n$ or $\nu \neq 1/(n+1)$ with $n$ an arbitrary
positive integer. So the result of eq.~(\ref{eqdefk3ext}) can be used in the case where
the dimension of the space-time is shifted by a small positive amount from $n=4$ to $n=4 - 2 \, \varepsilon$.
We also note that the limit $\nu \to 1$ of $K(\nu)$ leads to
eq. (\ref{P1-eqdefk4}) of ref. \cite{paper1}.
\vspace{0.3cm}
\noindent
A practical case met in sec.\ \ref{3point_ir} is $A = - i \, \lambda$ and $B$ remaining an arbitrary complex number with an imaginary part different from $0$.
In the limit $\lambda \to 0^{+}$, eq.~(\ref{eqdefk3ext}) becomes:
\begin{align}
\int^{+\infty}_0
\frac{d \xi}{(\xi^{\nu} - i \, \lambda)\, (\xi^{\nu} + B)}
&= - \frac{1}{\varepsilon} \, (1-\varepsilon) \, \Gamma(1+\varepsilon) \, \Gamma(1-\varepsilon)
\, B^{-1-\varepsilon}
\label{eqmodifk}
\end{align}
This is a well-known fact that the two limits $\lambda \to 0^{+}$ and $\varepsilon \to 0$ do not commute.
\subsection{Second kind}
The most general case for the integral
\begin{equation}
J(\nu)
=
\int^{+\infty}_0
\frac{d \xi}{\left(\xi^{\nu}+A \right) \, \sqrt{\xi^{\nu}+B}}
\label{eqdefj1}
\end{equation}
has been treated in appendix \ref{P2-appendJ} of ref.\ \cite{paper2}, let us just recap the results.
Two cases can be distinguished according to the signs of the imaginary parts of $A$ and $B$.
\vspace{0.3cm}
\noindent
{\bf 1)} $\Im(A)$ and $\Im(B)$ of the same sign
\begin{equation}
J(\nu)
=
\frac{1}{\nu} \,
B \left( \frac{3}{2}- \frac{1}{\nu},\frac{1}{\nu} \right) \,
\int^1_0 dz \, \left( (1-z^2) \, A + z^2 \, B \right)^{-3/2 + 1/\nu}
\label{eqdeffuncj2}
\end{equation}
\vspace{0.3cm}
\noindent
{\bf 2)} $\Im(A)$ and $\Im(B)$ of opposite signs
\begin{align}
J(\nu)
&= - \, \frac{1}{\nu} \,
B \left( \frac{3}{2}-\frac{1}{\nu},\frac{1}{\nu} \right) \notag \\
&\quad {} \times
\left[
e^{- i \, S_B \, \pi/\nu} \,
\int^{+\infty}_0 dz \,
\left( B \, z^2 - (1+z^2) \, A \right)^{-3/2+1/\nu}
\right.
\notag \\
&
\;\;\;\;\;\;\;\;\;\;\;
+
\left.
\int^{+\infty}_{1} dz \,
\left( B \, z^2 + (1-z^2) \, A \right)^{-3/2+1/\nu}
\right]
\label{eqdeffuncj7}
\end{align}
with $S_B = \mbox{sign}\left( \Im\left( B \right) \right)$.
\vspace{0.3cm}
\noindent
Let us remind that the two cases 1) vs. 2) can be reunified by seeing
eq. (\ref{eqdeffuncj7}) as an analytic continuation in $A$ of eq.
(\ref{eqdeffuncj2}) which possibly requires a deformation of the contour
$[0,1]$ originally drawn along the real axis in eq. (\ref{eqdeffuncj2}), cf.\ appendix \ref{P2-appendJ} of \cite{paper2} for more details.
\section{The function $J(x_1,x_2)$}\label{calculJx1x2}
This appendix computes the function
\[
J(x_1,x_2)
= \int^1_0 dx \,
\frac{\ln\left( (x-x_1) \, (x-x_2) \right)}{(x - x_1) \, (x - x_2)}
\]
Using partial fraction decomposition, $J(x_1,x_2)$ can be written as:
\begin{align}
&J(x_1,x_2)
\notag \\
&= \frac{1}{x_1-x_2}
\left[
\quad {} \quad {}
\int^1_0 dx \, \frac{\ln(x-x_1)}{x-x_1}
\quad {} \quad {} \quad {} - \quad {} \quad {} \quad {}
\int^1_0 dx \, \frac{\ln(x-x_2)}{x-x_2}
\right.
\notag \\
&\quad {} \quad {} \quad {}
+ \int^1_0 dx \, \frac{\ln(x-x_2) - \ln(x_1-x_2)}{x-x_1}
\; - \;
\int^1_0 dx \, \frac{\ln(x-x_1) - \ln(x_2-x_1)}{x-x_2}
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {}
+
\left.
\ln(x_1-x_2) \, \int^1_0 dx \, \frac{d x}{x-x_1} \;\; - \;\;
\ln(x_2-x_1) \, \int^1_0 dx \, \frac{d x}{x-x_2}
\quad {}
\right]
\label{eqcompj2}
\end{align}
With the help of appendix~\ref{P1-appF} of \cite{paper1} (cf.\ also appendix B of ref. \cite{tHooft:1978jhc}), we immediately get:
\begin{align}
\int^1_0 dx \, \frac{\ln(x-x_1) - \ln(x_2-x_1)}{x-x_2}
&= R^{\prime}(x_1,x_2) \notag \\
& = \mbox{Li}_2 \left( \frac{x_2}{x_2-x_1} \right)
- \mbox{Li}_2 \left( \frac{x_2-1}{x_2-x_1} \right)
\notag \\
&\quad {}
+ \eta \left(-x_1, \frac{1}{x_2-x_1} \right) \,
\ln \left( \frac{x_2}{x_2-x_1} \right)
\notag \\
&\quad {}
- \eta \left( 1-x_1, \frac{1}{x_2-x_1} \right) \,
\ln \left( \frac{x_2-1}{x_2-x_1} \right)
\label{eqappbth1}
\end{align}
where $R^{\prime}(x_1,x_2)$ is
given by eq.~(\ref{P1-eqdefrprime4}) of \cite{paper1}\footnote{The subtlety discussed in appendix~\ref{P1-appF} of \cite{paper1}
does not show up here because $x_1$ and $x_2$ have imaginary parts of opposite signs.}.
Since $x_1$ and $x_2$ have imaginary parts of opposite signs
all the $\eta$ functions vanish in eq.~(\ref{eqappbth1}).
Then using the Landen identity:
\begin{equation}
\mbox{Li}_2(z) + \mbox{Li}_2 \left( \frac{z}{z-1} \right)
= - \, \frac{1}{2} \, \ln^2(1-z)
\label{eqlanden}
\end{equation}
we can rewrite eq. (\ref{eqappbth1}) as:
\begin{align}
&\int^1_0 dx \,
\frac{\ln(x-x_1) - \ln(x_2-x_1)}{x-x_2}
\notag\\
& \quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
=
\mbox{Li}_2 \left( \frac{x_2-1}{x_1-1} \right)
- \mbox{Li}_2 \left( \frac{x_2}{x_1} \right)
\notag\\
& \quad {} \quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
+ \frac{1}{2}
\left[
\ln^2 \left( x_1-1 \right) \vphantom{\frac{x_1-1}{x_1}}
- \ln^2 \left( x_1 \right) - 2 \, \ln(x_1-x_2) \,
\ln \left( \frac{x_1-1}{x_1} \right)
\right]
\label{eqappbth2}
\end{align}
Substituting into eq. (\ref{eqcompj2}) and easily computing the remaining
$x$ integrals, we get:
\begin{align}
&J(x_1,x_2)
\notag\\
&= \frac{1}{x_1-x_2}
\left\{
\;\;
\frac{1}{2} \, \left[ \ln^2(1-x_1) - \ln^2(-x_1) \right]
-
\frac{1}{2} \, \left[ \ln^2(1-x_2) - \ln^2(-x_2) \right]
\right.
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
+ \mbox{Li}_2 \left( \frac{x_1-1}{x_2-1} \right)
- \mbox{Li}_2 \left( \frac{x_1}{x_2} \right)
- \mbox{Li}_2 \left( \frac{x_2-1}{x_1-1} \right)
+ \mbox{Li}_2 \left( \frac{x_2}{x_1} \right)
\notag\\
&
\quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
+
\frac{1}{2}
\left[
\ln^2 \left( x_2-1 \right) \vphantom{\frac{x_2-1}{x_2}}
-
\ln^2 \left( x_2 \right)
-
2\, \ln(x_2-x_1) \, \ln \left( \frac{x_2-1}{x_2} \right)
\right]
\notag \\
& \quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
-
\frac{1}{2}
\left[
\ln^2 \left( x_1-1 \right) \vphantom{\frac{x_1-1}{x_1}}
-
\ln^2 \left( x_1 \right)
-
2\, \ln(x_1-x_2) \, \ln \left( \frac{x_1-1}{x_1} \right)
\right]
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
+
\left.
\ln(x_1-x_2) \, \ln \left( \frac{x_1-1}{x_1} \right)
-
\ln(x_2-x_1) \, \ln \left( \frac{x_2-1}{x_2} \right)
\;\;
\right\}
\label{eqcompj3}
\end{align}
The following identity:
\[
\ln^2(1-x_1) - \ln^2(-x_1) - \ln^2(x_1-1)+\ln^2(x_1)
= - 2 \, i \, \pi \, S(x_1) \, \ln \left( \frac{x_1-1}{x_1} \right)
\]
where $S(x_1)$ is the sign of the imaginary part of $x_1$,
allows to simplify
eq. (\ref{eqcompj3}) into:
\begin{align}
&J(x_1,x_2)
= \frac{1}{x_1-x_2}
\left\{
\mbox{Li}_2 \left( \frac{x_1-1}{x_2-1} \right)
-
\mbox{Li}_2 \left( \frac{x_1}{x_2} \right)
-
\mbox{Li}_2 \left( \frac{x_2-1}{x_1-1} \right)
+
\mbox{Li}_2 \left( \frac{x_2}{x_1} \right)
\right.
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
\quad {} \quad {} \quad {} \quad {}
+ \left[ 2\, \ln(x_1-x_2) - i \, \pi \, S(x_1) \right] \,
\ln \left( \frac{x_1-1}{x_1} \right)
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
\quad {} \quad {} \quad {} \quad {}\left.
- \left[ 2\, \ln(x_2-x_1) - i \, \pi \, S(x_2) \right] \,
\ln \left( \frac{x_2-1}{x_2} \right)
\right\}
\label{eqcompj4}
\end{align}
Then, we use the identity relating $\mbox{Li}_2 (z)$ and $\mbox{Li}_2(1/z)$ \cite{abramowitz},
and also the following relation:
\begin{equation}
2 \, \ln \left( x_1 - x_2 \right) - i \, \pi \, S(x_1) = 2 \, \ln \left( x_2 - x_1 \right) - i \, \pi \, S(x_2) = \ln \left( - (x_1 - x_2)^2 \right)
\label{eqrelsimp1}
\end{equation}
whose validity relies on the fact that $x_1$ and $x_2$ have
imaginary parts of opposites signs. This yields:
\begin{align}
&J(x_1,x_2)
\notag\\
&= \frac{1}{x_1-x_2}
\left\{
\;\;
2 \, \mbox{Li}_2 \left( \frac{x_1-1}{x_2-1} \right) -
2 \, \mbox{Li}_2 \left( \frac{x_1}{x_2} \right)
+ \frac{1}{2} \, \ln^2 \left( - \frac{x_1-1}{x_2-1} \right)
- \frac{1}{2} \, \ln^2 \left( - \frac{x_1}{x_2} \right)
\right.
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
+
\left.
\ln \left( - (x_1-x_2)^2 \right) \,
\left[
\ln \left( \frac{x_1-1}{x_1} \right) - \ln \left( \frac{x_2-1}{x_2} \right)
\right]
\;\;
\right\}
\label{eqcompj5}
\end{align}
\subsection{Consistency checks and explicit examples}\label{examplefourpoint}
In refs. \cite{Binoth:2005ff,Binoth:1999sp} the infrared structure
of any IR divergent $N$-point one-loop integral
was shown to be carried by IR divergent three-point one-loop functions
resulting from appropriate iterated pinchings. In the following this feature is
explicitly verified for the formulae obtained in this article for the
four-point functions, when compared with those for the three-point functions,
when the latter are formulated most conveniently according to the so-called
``indirect way'' for this purpose.
\vspace{0.3cm}
\noindent
In the case of purely soft divergence i.e. whenever some $\Delta_2^{\{i\}} = 0$
whereas $\widetilde{D}_{ijk} \ne 0$ the various cases cf. eqs.
(\ref{eqlijkir15}), (\ref{eqlijkir16reframed}), (\ref{eqlijkir17reframed}) and
(\ref{eqlijkir18reframed}) can be encompassed in one single formula.
For this purpose let us introduce the following notation,
where for any complex $Q$ we denote
$Q_R \equiv \Re(Q)$ and $Q_I \equiv \Im(Q)$:
\begin{align}
\int_{_{\widetilde{(0,1)}}} du \, F(u)
&=
\left\{
\begin{array}{lcl}
\int_0^{^{- i \, S_{\!_{A}} \, \infty}} du \, F(u) +
\int_{_{+\infty}}^{^1} du \, F(u)
& \mbox{if} & 0 < - \, B_I/A_I <1 \\
& \mbox{and} & A_I \, [ A_R \, B_I - A_I \, B_R ] > 0 \\
& & \\
\int_{_{0}}^{^{1}} du \, F(u) & & \text{otherwise}
\end{array}
\right.
\label{eqdefwtilde01}
\end{align}
with $S_{A} = \mbox{sign}(A_I)$, whether
\[
F(u) \; = \; \frac{\ln(A \, u^2 + B) - \ln(A \, u_0^2 + B)}{u^2 - u_0^2}
\quad {} \mbox{with} \quad {} u_0^2 \neq - \, \frac{B}{A}
\]
or\[
F(u) \; = \; (A \, u^2 + B)^{-1-\varepsilon}
\]
The condition
``$ 0 < - \, B_I/A_I <1$ and $A_I \, [ A_R \, B_I - A_I \, B_R ] > 0$"
is the condition for the discontinuity cuts of $\ln (A \, u^2 + B)$ to
cross the real axis between $0$ and $1$ (cf. appendix~\ref{P2-cut} of \cite{paper2}).
This enables us to rewrite $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$ in a generic way in the various cases as:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) = \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon) \notag \\
& {} \times
\left\{
- \frac{1}{\varepsilon} \,
\int_{\widetilde{(0,1)}} d u \,
\left( u^2 \, P_{ijk} + R_{ij} \right)^{-1-\varepsilon} - U \left( \Delta_3,\Delta_1^{\{ij\}},\widetilde{D}_{ijk} \right)
\right.
\notag \\
&\qquad {}
\;\; + \int_{\widetilde{(0,1)}} \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\left[
\vphantom{\frac{d u }{u^2 \, P_{ijk} + R_{ij}}}
\ln \left( T \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} \right) \right.
\notag \\
&\qquad \qquad{}
- \left.
\ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T \right) + \ln \left( \frac{(P_{ijk} + R_{ij}) \, (T - R_{ij})}{P_{ijk}} \right)
\right]
\notag \\
&\qquad {}
\left.
- \int_0^1 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \, \left[ \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right) + \ln \left( \frac{T - R_{ij}}{T} \right) \right]
\vphantom{\frac{d u }{u^2 \, P_{ijk} + R_{ij}}}
\right\}
\label{eqnewverLijk0}
\end{align}
with:
\begin{align}
\hspace{2em}&\hspace{-2em}U(\Delta_3,\Delta_1^{\{ij\}},\widetilde{D}_{ijk}) \notag \\
&= \left\{
\begin{array}{lcl}
0 & \mbox{if} & \Im(\Delta_3) > 0 \; \Im(\Delta_1^{\{ij\}}) > 0 \\
\int_{_{\Gamma^{+}}} \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\ln \left( \frac{T - R_{ij}}{T} \right) &
\mbox{if} & \Im(\Delta_3) > 0 \; \Im(\Delta_1^{\{ij\}}) < 0 \\
\int_{_{\Gamma^{+}}} \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right) &
\mbox{if} & \Im(\Delta_3) < 0 \; \Im(\Delta_1^{\{ij\}}) > 0 \\
\int_{_{\Gamma^{+}}} \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \, \left[
\ln \left( \frac{T - R_{ij}}{T} \right) \right. &
\mbox{if} & \Im(\Delta_3) < 0 \; \Im(\Delta_1^{\{ij\}}) < 0 \\
\qquad {} + \left. \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right) \right] & &
\end{array}
\right.
\label{eqdeffuncu}
\end{align}
where the contour $\Gamma^{+}$ is the closed contour encircling the ``north-east"
quadrant {\em clockwise} (cf. subsec.~\ref{P2-sectfourpointcomp} of \cite{paper2}).
Depending on the location of the cuts of $ \left( u^2 \, P_{ijk} + R_{ij} \right)^{-1-\varepsilon}$, the first term of eq. (\ref{eqnewverLijk0}) is the same as those
which appear in eqs. (\ref{eqlijsoft40}) or (\ref{eqlijsoft60}), $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$ can
thus be written as:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) &= \frac{1}{T} \, L_3^n(0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) + \tilde{L}_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})
\label{decomp-ir}
\end{align}
with
\begin{align}
\hspace{2em}&\hspace{-2em}\tilde{L}_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) \notag \\
&=
\frac{1}{T} \,
\left\{
\vphantom{\frac{d u }{u^2 \, P_{ijk} + R_{ij}}}
\int_{\widetilde{(0,1)}} \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\Bigg[
\ln \left( T \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} \right) \right.
\notag \\
&\qquad \qquad{}
-
\ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T \right) + \ln \left( \frac{(P_{ijk} + R_{ij}) \, (T - R_{ij})}{P_{ijk}} \right)
\Biggr]
\notag \\
&\qquad \qquad {}
- \int_0^1 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \, \left[ \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right) + \ln \left( \frac{T - R_{ij}}{T} \right) \right]
\notag \\
&\qquad \qquad {}
- \left. U \left( \Delta_3,\Delta_1^{\{ij\}}, \widetilde{D}_{ijk} \right)
\vphantom{\frac{d u }{u^2 \, P_{ijk} + R_{ij}}}
\right\}
\label{eqnewvertLijk1}
\end{align}
and $L_3^n(0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$ is given by eq.~(\ref{eqlijsoft40})
or eq.~(\ref{eqlijsoft62}) depending on the sign of the imaginary part of $\Delta_1^{\{i,j\}}$.
\vspace{0.3cm}
\noindent
For the cases where $\Delta_2^{\{i\}} = 0$ and $\widetilde{D}_{ijk} = 0$, a unique
formula for $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)$ was found above whatever the signs of $\Im(\Delta_3)$ and
$\Im(\Delta_1^{\{ij\}})$. The decomposition of the form (\ref{decomp-ir})
stills holds, the IR divergent part of $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)$ is the same as in the
three-point case (cf.\ eqs.~(\ref{eqlijsoft8}) and~(\ref{eqlijsoft10bis})), whereas now:
\begin{align}
\tilde{L}_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)
&= - \frac{1}{2 \, R_{ij} \, T} \,
\left[
\mbox{Li}_2\left( \frac{R_{ij}}{T} \right) + \left[ \ln(R_{ij}) - \ln(T) \right] \,
\ln \left( \frac{T - R_{ij}}{T} \right)
\right]
\label{eqnewvertLijk2}
\end{align}
Coming back to $I_4^n$ and using the results of subsec.~\ref{indirway}, we have:
\begin{align}
I_4^n
&= \sum_{i \in S_4} \frac{\overline{b}_i}{\det{(G)}} \, \sum_{j \in S_4 \setminus \{i\}}
\frac{\bbj{j}{i}}{\detgj{i}} \, \frac{W\left(\detgj{i,j},\widetilde{D}_{ijk}, \widetilde{D}_{ijl}\right)}{T}
\notag \\
&\quad {}
+ \sum_{i \in S_4} \frac{\overline{b}_i}{\det{(G)}} \, \sum_{j \in S_4 \setminus \{i\}}
\frac{\bbj{j}{i}}{\detgj{i}} \, \sum_{k \in S_4 \setminus \{i,j\}}
\frac{\bbj{k}{i,j}}{\detgj{i,j}} \, \tilde{L}_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})
\label{eqI4nb3}
\end{align}
where $l \in S_4 \setminus \{i,j,k\}$.
The first term in the r.h.s. of eq. (\ref{eqI4nb3}) is nothing but the
combination of the three-point functions such as in refs.
\cite{Binoth:2005ff,Binoth:1999sp} decomposed according to the so called
``direct way" made explicit in subsec.~\ref{dirway}. Note that when
$\widetilde{D}_{ijk} = 0$, the quantity $\tilde{L}_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)$ which does actually
not depend on $k$ factors out from the sum over $k$ which then
yields trivially - 1.
The functions $W(\detgj{i,j},\widetilde{D}_{ijk}, \widetilde{D}_{ijl})$ has been defined by eq. (\ref{eqdefwi0}).
Its arguments have one extra subscript, tracing back the extra pinching which was involved
compared with the three-point case. We recap here the different results concerning this function.
\begin{equation}
W\left(\detgj{i,j},\widetilde{D}_{ijk}, \widetilde{D}_{ijl}\right) = \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \, \int^1_0 dx \,
\left( D^{\{i,j\}(k)}(x) - i \, \lambda \right)^{-1-\varepsilon}
\label{eqdefwi01}
\end{equation}
where
\begin{equation}
D^{\{i,j\}(k)}(x)
=
G^{\{i,j\}(k)} \, x^2 - 2 \, V^{\{i,j\}(k)} \, x - C^{\{i,j\}(k)}
\label{eqremd11}
\end{equation}
with:
\begin{align}
G^{\{i,j\}(k)}
&= - \text{$\cal S$}_{ll} + 2 \, \text{$\cal S$}_{kl} - \text{$\cal S$}_{kk} = \detgj{i,j}
\notag \\
V^{\{i,j\}(k)}
&= \text{$\cal S$}_{kl} - \text{$\cal S$}_{kk} = \frac{1}{2} \, \left[ \detgj{i,j} - \widetilde{D}_{ijk} + \widetilde{D}_{ijl} \right]
\label{eqremd21}\\
C^{\{i,j\}(k)}
&= \text{$\cal S$}_{kk} = - \widetilde{D}_{ijl}
\notag
\end{align}
Note that
$W(\detgj{i,j},\widetilde{D}_{ijk}, \widetilde{D}_{ijl})$ is symmetric under the exchange of $i$ and $j$.
\vspace{0.3cm}
\noindent
* If $\widetilde{D}_{ijk}$ and $\widetilde{D}_{ijl}$ both differ from zero, $W\left(\detgj{i,j},\widetilde{D}_{ijk}, \widetilde{D}_{ijl}\right)$ is given by eq.~(\ref{eqcompintlog1}).
\vspace{0.3cm}
\noindent
* If only $\widetilde{D}_{ijl}$ vanishes, $W\left(\detgj{i,j},\widetilde{D}_{ijk}, 0\right)$ is read from eq.~(\ref{eqdirei3n3}).
\vspace{0.3cm}
\noindent
* If $\widetilde{D}_{ijk}$ and $\widetilde{D}_{ijl}$ both vanish, $W\left(\detgj{i,j},0,0\right)$ is given by eq.~(\ref{eqdirei3n4}).
\vspace{0.3cm}
\noindent
For practical purpose let us stress that we do not have to compute a dedicated
formula for each IR four-point case as in ref. \cite{Ellis:2007qk} where
16 cases were distinguished. Indeed, for each contribution labelled by the
index $i$,
we merely distinguish two cases: either $\Delta_2^{\{i\}} \ne 0$ for which we
use the generic formula suited to the massive case, or $\Delta_2^{\{i\}} = 0$
for which we use the appropriate formula suited to the IR case at hand.
The massive case is split depending on the vanishing of $\Im(\Delta_3)$ (namely if one internal mass squared has an imaginary part different from zero).
The IR case is also divided in two cases: $\widetilde{D}_{ijk} = 0$ and $\widetilde{D}_{ijk} \ne 0$. The latter is furthermore separated
according to the fact that one or several internal masses squared have a non vanishing imaginary part.
All these cases are depicted on the decision tree presented in fig. \ref{dectree}, for each case the appropriate formula is given.
Notice that when $\Im(\Delta_3) \ne 0$, it may appear that $\Delta_2^{\{i\}}$ and/or $\Delta_1^{\{i,j\}}$ are real,
in this case they must be understood as having a ``$+ i \, \lambda$'' prescription.
The expense paid by the present method is a possible proliferation of
dilogarithms, a counteraction against which would require some extra work.
This point will be commented in some more details in the examples studied below.
\tikzstyle{post}=[->,shorten >=1pt,>=stealth,semithick]
\begin{figure}[h!]
\begin{tikzpicture}
\node at (-4,0) [level 3] (c311) {Eq.~(\ref{eqlijkir15r})};
\node at (-8,0) [level 2] (c312) {cases};
\node at (-6.5,2) [level 2] (c31) {$\Im(\Delta_3) = 0$}
edge [post] node[auto] {Yes} (c311)
edge [post] node[auto,swap] {No} (c312);
\node at (-2.6,2) [level 3,text width=3cm] (c32) {Eq.~(\ref{eqdefnl6bis}) for all cases};
\node at (-4,5) [level 2] (c3) {$\widetilde{D}_{ijk} = 0$}
edge [post] node[auto,swap] {No} (c31)
edge [post] node[auto] {Yes} (c32);
\node at (6,2) [level 3] (c21) {Eq.~(\ref{P1-eqlijk14alternaivebbar}) of \cite{paper1}};
\node at (0,2) [level 2] (c22) {cases};
\node at (3,5) [level 2] (c2) {$\Im(\Delta_3) = 0$}
edge [post] node[auto,swap] {No} (c22)
edge [post] node[auto] {Yes} (c21);
\node at (0,8) [level 2] (c1) {$\Delta_2^{\{i\}} = 0$}
edge [post] node[auto] {No} (c2)
edge [post] node[auto,swap] {Yes} (c3);
\node at (0,10) [root] (c0) {$L_4^n(\Delta_3,\Delta_2^{\{i\}},\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$}
edge [post] (c1);
\begin{scope}[every node/.style={level 3}]
\node [below of = c312,xshift=3.5cm] (c3121) {{\footnotesize $\Im(\Delta_3) > 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$: eq.~(\ref{eqlijkir15c})}};
\node [below of = c3121] (c3122) {{\footnotesize $\Im(\Delta_3) > 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$: eq.~(\ref{eqlijkir16reframed})}};
\node [below of = c3122] (c3123) {{\footnotesize $\Im(\Delta_3) < 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$: eq.~(\ref{eqlijkir17reframed})}};
\node [below of = c3123] (c3124) {{\footnotesize $\Im(\Delta_3) < 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$: eq.~(\ref{eqlijkir18reframed})}};
\end{scope}
\begin{scope}[every node/.style={level 3}]
\node [below of = c22,xshift=4.5cm] (c221) {{\scriptsize $\Im(\Delta_3) > 0$, $\Im(\Delta_2^{\{i\}}) > 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$: eq.~(\ref{P2-eqcas1soustraite}) of \cite{paper2}}};
\node [below of = c221] (c222) {{\scriptsize $\Im(\Delta_3) > 0$, $\Im(\Delta_2^{\{i\}}) < 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$: eq.~(\ref{P2-eqcas3eclatesoustrait}) of \cite{paper2}}};
\node [below of = c222] (c223) {{\scriptsize $\Im(\Delta_3) < 0$, $\Im(\Delta_2^{\{i\}}) > 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$: eq.~(\ref{P2-eqcas4eclatesoustrait}) of \cite{paper2}}};
\node [below of = c223] (c224) {{\scriptsize $\Im(\Delta_3) < 0$, $\Im(\Delta_2^{\{i\}}) < 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$: eq.~(\ref{P2-eqcas5eclatesoustrait}) of \cite{paper2}}};
\node [below of = c224] (c225) {{\scriptsize $\Im(\Delta_3) > 0$, $\Im(\Delta_2^{\{i\}}) > 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$: eq.~(\ref{P2-eqcas2aa}) of \cite{paper2}}};
\node [below of = c225] (c226) {{\scriptsize $\Im(\Delta_3) > 0$, $\Im(\Delta_2^{\{i\}}) < 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$: eq.~(\ref{P2-eqcas2bb}) of \cite{paper2}}};
\node [below of = c226] (c227) {{\scriptsize $\Im(\Delta_3) < 0$, $\Im(\Delta_2^{\{i\}}) > 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$: eq.~(\ref{P2-eqcas2cc}) of \cite{paper2}}};
\node [below of = c227] (c228) {{\scriptsize $\Im(\Delta_3) < 0$, $\Im(\Delta_2^{\{i\}}) < 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$: eq.~(\ref{P2-eqcas2dd}) of \cite{paper2}}};
\end{scope}
\foreach \value in {1,...,4}
\draw[->] (c312) |- (c312\value.west);
\foreach \value in {1,...,8}
\draw[post] (c22) |- (c22\value.west);
\end{tikzpicture}
\caption{\footnotesize Decision tree to compute $L_4^n(\Delta_3,\Delta_2^{\{i\}},\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$ for a given sector labelled by $i$, $j$ and $k$. }\label{dectree}
\end{figure}
\vspace{0.3cm}
\noindent
{\bf 1) Two opposite external masses}
\vspace{0.3cm}
\noindent
In this case, all internal masses are zero and two opposite external legs
(say 1 and 3) have non lightlike four momenta (our convention is depicted in
fig. \ref{fig2}). The texture of the $\text{$\cal S$}$ matrix is:
\begin{align}
\text{$\cal S$} &= \left(
\begin{array}{cccc}
0 & 0 & s_{23} & s_1 \\
0 & 0 & s_3 & s_{12} \\
s_{23} & s_3 & 0 & 0 \\
s_1 & s_{12} & 0 & 0
\end{array}
\right)
\label{eqsmatexempl1}
\end{align}
with $s_i = p_i^2$ and $s_{ij} = (p_i + p_j)^2$ (all the momenta are taken ingoing). All the contributions $i$
are such that $\Delta_2^{\{i\}} = 0$ and $\widetilde{D}_{ijk} = 0$. In this case,
$R_{ij} = - \Delta_1^{\{i,j\}}$ is symmetric under the exchange of $i$ and $j$.
The four-point function is given by, cf. eq. (\ref{eqdefnl6}):
\begin{align}
I_4^n
&= \frac{1}{2 \, \det{(\cals)}} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1 - 2 \, \varepsilon)} \,
\sum_{i \in S_4} \, \sum_{j > i} \frac{1}{\Delta_1^{\{i,j\}}} \,
\left(
\frac{\overline{b}_i \, \bbj{j}{i}}{\detgj{i}}
+
\frac{\overline{b}_j \, \bbj{i}{j}}{\detgj{j}}
\right)
\notag \\
&\quad {}
\times \,
\left[
\frac{1}{\varepsilon^2} \,
\left( - 2 \, \Delta_1^{\{i,j\}} \right)^{- \varepsilon}
+ \mbox{Li}_2 \left( \frac{T - R_{ij}}{T - i \, \lambda} \right)
- \frac{\pi^2}{6}
\right]
\label{eqI4nexemple1}
\end{align}
Due to the hollow texture of the reduced $\text{$\cal S$}$ matrices, a bunch of
$\bbj{j}{i}$
coefficients vanish. Eq. (\ref{eqI4nexemple1}) thus simplifies into:
\begin{align}
I_4^n
&= \frac{1}{2 \, \det{(\cals)}} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1 - 2 \, \varepsilon)} \,
\sum_{i=1}^{2} \, \sum_{j=3}^{4} \frac{1}{\Delta_1^{\{i,j\}}} \,
\left(
\frac{\overline{b}_i \, \bbj{j}{i}}{\detgj{i}}
+
\frac{\overline{b}_j \, \bbj{i}{j}}{\detgj{j}}
\right)
\notag \\
&\quad {}
\times \,
\left[ \frac{1}{\varepsilon^2} \,
\left( - 2 \, \Delta_1^{\{i,j\}} \right)^{- \varepsilon}
+ \mbox{Li}_2\left( 1 - \frac{R_{ij} - i \, \lambda}{T - i \, \lambda} \right)
- \frac{\pi^2}{6}
\right]
\label{eqI4nexemple2}
\end{align}
Expressing the $\overline{b}_{i}$ and $\bbj{j}{i}$ coefficient as well as the
various determinants as functions of the $\text{$\cal S$}$ matrix elements, we get:
\begin{align}
I_4^n
&= \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1 - 2 \, \varepsilon)} \, \frac{2}{d} \,
\notag\\
& \quad {}
\left\{
\vphantom{\frac{s_1}{s_1}}
(- s_1 - i \, \lambda)^{-\varepsilon} +
(- s_3 - i \, \lambda)^{-\varepsilon} -
(- s_{12} - i \, \lambda)^{-\varepsilon} -
(- s_{23} - i \, \lambda)^{-\varepsilon}
\right.
\notag \\
&\quad {} \quad {}
- \mbox{Li}_2
\left( 1 - \frac{s_{12} + i \, \lambda}{d/\Sigma + i \, \lambda} \right)
- \mbox{Li}_2
\left( 1 - \frac{s_{23} + i \, \lambda}{d/\Sigma + i \, \lambda} \right)
\notag \\
&\quad {}\quad {}
\left.
+ \mbox{Li}_2
\left( 1 - \frac{s_{1} + i \, \lambda}{d/\Sigma + i \, \lambda} \right)
+ \mbox{Li}_2
\left( 1 - \frac{s_{3} + i \, \lambda}{d/\Sigma + i \, \lambda} \right)
\right\}
\label{eqI4nexemple3}
\end{align}
with:
\begin{align}
d &= s_1 \, s_3 - s_{12} \, s_{23} \label{eqdefdI4n} \\
\Sigma &= s_1 + s_3 - s_{12} - s_{23}
\label{eqdefSigmaI4n}
\end{align}
Eq. (\ref{eqdefSigmaI4n}) involves four dilogarithms as in the
refs. \cite{Duplancic:2000sk,Brandhuber:2004yw}.
The arguments of the dilogarithms in eq. (\ref{eqI4nexemple3}) vs.
in ref. \cite{Duplancic:2000sk} are seemingly different, namely ref.
\cite{Duplancic:2000sk} involves
$\mbox{Li}_2 ( 1 - (w+i \, \lambda)/(d/\Sigma) )$ whereas
$\mbox{Li}_2 ( 1 - (w+i \, \lambda)/(d/\Sigma + i \, \lambda) )$ appears in eq.
(\ref{eqI4nexemple3}), where $w$ stands for $s_1$, $s_3$, $s_{12}$ or
$s_{23}$. However the $i \, \lambda$ prescriptions matter only
when the real parts of the arguments of the dilogarithms are greater than 1 i.e.
whenever $w \, \Sigma/d < 0$, in which case the
signs of $(d/\Sigma) - w$ and of $\Sigma/d$, which respectively control the
signs of the $i \lambda$ prescriptions in either case, are the same: the two
results in eq. (\ref{eqI4nexemple3}) and in ref. \cite{Duplancic:2000sk} are
actually identical.
\vspace{0.3cm}
\noindent
This is to be compared with the formula given in ref. \cite{Ellis:2007qk}. The
latter was taken from ref. \cite{Brandhuber:2004yw} which involves five
dilogarithms instead of four.
The authors of ref. \cite{Brandhuber:2004yw} used the so-called Mantel identity
which entails nine dilogarithms to prove that the four-dilogarithm and
five-dilogarithm results are actually equivalent. The Mantel identity happens
to be a corollary of the Hill identity, the former is derived by applying the
latter three
times to some suitable combinations of variables\footnote{See \cite{lewin},
chap. 1, p. 2-3 and chap. 2, p. 17-18.}.
The continuation of the Hill identity to any arbitrary two complex variables
however requires additional combinations of $\eta$ functions handling the
mismatch between the various discontinuities of the dilogarithms involved,
and these $\eta$ functions are often skipped in the literature\footnote{See
however \cite{vanOldenborgh:1989wn}.}. An even busier modification is then
required for the Mantel identity. This drove
of $\eta$ functions makes the analytical check of the equivalence between the
four-dilogarithm and five-dilogarithm expressions extremely awkward in general,
and to our understanding this drove of $\eta$ functions was not accounted in
ref. \cite{Brandhuber:2004yw}. We did perform numerical tests accounting for
these $\eta$ functions, which verified the equivalence for the configurations
probed.
\vspace{0.3cm}
\noindent
{\bf 2) A simple case with one internal masses}
\vspace{0.3cm}
\noindent
With the same notations of the preceding example, the texture of the $\text{$\cal S$}$ matrix is:
\begin{align}
\text{$\cal S$} &= \left(
\begin{array}{cccc}
0 & 0 & s_{23}-m_3^2 & 0 \\
0 & 0 & 0 & s_{12} \\
s_{23} - m_3^2 & 0 & - 2 \, m_3^2 & s_4 - m_3^2 \\
0 & s_{12} & s_4 - m_3^2 & 0
\end{array}
\right)
\label{eqsmatexempl2}
\end{align}
In this case, the sector $i=1$
has no soft or collinear divergence and is computed using the massive formula. The
other sectors correspond to the cases: $\Delta_2^{\{i\}} = 0$, $\widetilde{D}_{ijk} \ne 0$
and $\Delta_2^{\{i\}} = 0$, $\widetilde{D}_{ijk} = 0$. The four-point amplitude can be
cast in a divergent part and a finite one. The divergent part is given by:
\begin{align}
\left( I^n_4 \right)_{div} &= \sum_{i \in S_4 \setminus \{1\}} \, \frac{\overline{b}_i}{\det{(G)}} \, \sum_{j \in S_4 \setminus \{i\}} \, \frac{\bbj{j}{i}}{\detgj{i}} \, \frac{W\left(\detgj{i,j},\widetilde{D}_{ijk},\widetilde{D}_{ijl}\right)}{T}
\label{eqdefdivpartI44}
\end{align}
As several $\bbj{j}{i}$ vanish due to the hollow texture of reduced $\text{$\cal S$}$
matrices, we actually have to compute\footnote{We will use the following properties: $\detgj{i,j}$ is symmetric under the permutation $i \leftrightarrow j$ and $\widetilde{D}_{ijk}$ is symmetric under any permutation of the set $\{i,j,k\}$}:
\begin{align}
\left( I^n_4 \right)_{div} &= \frac{\overline{b}_2}{\det{(\cals)}} \, \left[ \frac{\bbj{1}{2}}{\detgj{2}} \, W\left( \detgj{1,2},\widetilde{D}_{124},0 \right) + \frac{\bbj{4}{2}}{\detgj{2}} \, W\left( \detgj{2,4},\widetilde{D}_{124},0 \right) \right] \notag \\
&\quad {} + \frac{\overline{b}_3}{\det{(\cals)}} \, \frac{\bbj{1}{3}}{\detgj{3}} \, W\left( \detgj{1,3},0,0 \right) \notag \\
&\quad {} + \frac{\overline{b}_4}{\det{(\cals)}} \, \frac{\bbj{2}{4}}{\detgj{4}} \, W\left( \detgj{2,4},\widetilde{D}_{124},0 \right)
\label{eqdefdivpartI441}
\end{align}
where $W\left( \detgj{1,2},\widetilde{D}_{124},0 \right)$, $W\left( \detgj{2,4},\widetilde{D}_{124},0 \right)$ are given by eq. (\ref{eqdirei3n3}) and
\linebreak $W\left( \detgj{1,3},0,0 \right)$ by eq. (\ref{eqdirei3n4}). We get:
\begin{align}
\left( I^n_4 \right)_{div}
&=
\frac{1}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1 - 2 \, \varepsilon)} \,
\frac{1}{s_{12} \, (s_{23} - m_3^2)} \,
\notag\\
& \quad {}
\left[
\;
\frac{2}{\varepsilon} \,
\left( - s_{23} + m_3^2 - i \, \lambda \right)^{-\varepsilon}
+
\frac{1}{\varepsilon} \,
\left( - s_{12} - i \, \lambda \right)^{-\varepsilon}
\right.
\notag \\
&\quad {}
- \frac{1}{\varepsilon} \, \left( - s_4 + m_3^2 - i \, \lambda \right)^{-\varepsilon} - \frac{1}{2 \, \varepsilon} \left( m_3^2 - i \, \lambda \right)^{-\varepsilon} \notag \\
&\quad {}
\left.
+ \varepsilon \,
\mbox{Li}_2\left( \frac{s_4}{s_4 - m_3^2 + i \, \lambda} \right)
- 2 \, \varepsilon \,
\mbox{Li}_2 \left( \frac{s_{23}}{s_{23} - m_3^2 + i \, \lambda} \right)
+ \varepsilon \, \frac{\pi^2}{12} \;
\right]
\label{eqdefdivpartI442}
\end{align}
Expanding eq. (\ref{eqdefdivpartI442}) in $\varepsilon$, we recover the results
of ref. \cite{Ellis:2007qk} (eq. (4.27)) taken from ref.
\cite{Beenakker:2002nc} for the terms proportional to $1/\varepsilon^2$ and
$1/\varepsilon$. Concerning the finite part, we obtain a host of terms for
which it is cumbersome to verify analytically that they do reduce to the finite
part of eq. (4.27) of ref. \cite{Ellis:2007qk}. We verified numerically that
they do indeed.
\section{Four-point function with Infrared divergences}\label{sectfourpointir}
\begin{figure}[h]
\centering
\parbox[c][43mm][t]{40mm}{\begin{fmfgraph*}(60,40)
\fmfleftn{i}{2} \fmfrightn{o}{2}
\fmf{fermion,label=$p_1$}{i2,v1}
\fmf{fermion,label=$p_2$}{i1,v2}
\fmf{fermion,label=$p_3$}{o1,v3}
\fmf{fermion,label=$p_4$}{o2,v4}
\fmf{fermion,tension=0.5,label=$q_1$}{v1,v2}
\fmf{fermion,tension=0.5,label=$q_2$}{v2,v3}
\fmf{fermion,tension=0.5,label=$q_3$}{v3,v4}
\fmf{fermion,tension=0.5,label=$q_4$}{v4,v1}
\end{fmfgraph*}}
\caption{The box picturing the one-loop four-point function.}
\label{fig2}
\end{figure}
\noindent
In the case where infrared divergences appear, the latter can be regularised by
dimensional regularisation shifting the space-time dimension
$n = 4 - 2 \varepsilon$ slightly above 4 ($\varepsilon < 0$). The Feynman
parametrisation of $I_4^n$ reads:
\begin{eqnarray}
I_4^n
& = &
\Gamma\left(2 + \varepsilon \right)
\int_0^1 \, \prod_{i=1}^4 \, \delta(1- \sum_{i=1}^4 z_i)
\left( -\frac{1}{2} \, Z^{\;T} \cdot
\text{$\cal S$} \cdot Z - i \, \lambda \right)^{-2 - \varepsilon}
\label{eqstartingpointir}
\end{eqnarray} where $Z$ is a column 4-vector whose components are the $z_{i}$.
The power $- 2 - \varepsilon$ in
eq. (\ref{eqstartingpointir}) is not an integer. Notwithstanding, the
tricks and techniques
elaborated in section \ref{P1-sectfourpoint} of ref. \cite{paper1} can be used with a slight adaptation.
Let us sketch the different steps for this special case.
\subsection{Computation of $I_4^{n}$}\label{compI4n}
We make use of identity (\ref{P1-eqFOND1biss}) of ref. \cite{paper1} to shift the power of the
denominator in the integrand, choosing $\mu = 5/2$ and
$\nu = 2/(1-2 \varepsilon)$ so that
$I_4^{n}$ is recast as:
\begin{eqnarray}
I_4^{n}
& = &
\frac{2^{3+\varepsilon}}{B(2 + \varepsilon,1/2-\varepsilon)} \,
\frac{\Gamma(2 + \varepsilon)}{(1-2 \, \varepsilon)} \,
\nonumber \\
&& \mbox{} \times
\int_0^{+\infty} d \xi \,
\int_{\Sigma_{bcd}}
\frac{dx_b \, d x_c \, d x_d}
{(D^{(a)}(x_b,x_c,x_d) + \xi^{\nu} - i \, \lambda)^{5/2}}
\label{eqI4b1}
\end{eqnarray}
Step 1 is very similar to the case $n=4$ and
we get:
\begin{eqnarray}
I_4^{n}
& = &
\frac{2^{3+\varepsilon}}{3 \, B(2 +\varepsilon,1/2 - \varepsilon)} \,
\frac{\Gamma(2 +\varepsilon)}{(1-2 \, \varepsilon)} \,
\nonumber \\
&& \mbox{} \times
\sum_{i=1}^{4} \, \frac{\overline{b}_i}{\det{(G)}} \,
\int_0^{+\infty}
d \xi \, \frac{1}{\Delta_3-\xi^{\nu} + i \, \lambda}
\nonumber\\
& &
\quad{} \quad{} \quad{} \quad{}
\times \int_{\Sigma_{kl}}
\frac{dx_k \, d x_l}
{(D^{\{i\} \, (i^{\prime})}(x_k,x_l)
+ \xi^{\nu} - i \, \lambda)^{3/2}}
\label{eqI4b2}
\end{eqnarray}
with the same notational conventions as in eqs. (\ref{P1-eqI442}) and
(\ref{P1-eqI448}) of sec. \ref{P1-sectfourpoint} of ref. \cite{paper1}.
Likewise, steps 2 and 3 are identical to section \ref{P1-sectfourpoint} of ref. \cite{paper1} but
the power of the variable $\xi$: $\nu$ instead of 2, and will not be repeated
here.
Regarding step 4, the integration shall be performed over the variables
$\xi$, $\rho$ then $\sigma$ in the corresponding $L_4^n(\Delta_3,\Delta_2^{\{i\}},\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$ now given by:
\begin{align}
L_4^n(\Delta_3,\Delta_2^{\{i\}},\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})
&=
\kappa \,\int^{+\infty}_0 d \xi \, \int^{+\infty}_0 d \rho \,
\int^{+\infty}_0 d \sigma
\frac{1}{\xi^{\nu} - \Delta_3 - i \, \lambda} \,
\nonumber \\
&\quad
\mbox{} \times \frac{1}{\xi^{\nu} + \rho^2 - \Delta_2^{\{i\}} - i \, \lambda} \,
\frac{1}{\xi^{\nu} + \rho^2 + \sigma^2 - \Delta_1^{\{i,j\}} - i \, \lambda} \notag \\
&\quad {} \times \frac{1}{(\widetilde{D}_{ijk} + \xi^{\nu} + \rho^2 + \sigma^2 - i \, \lambda)^{1/2}}
\label{eqdefnl1}
\end{align}
with
\[
\kappa =
\frac{2^{4+\varepsilon}}
{3 \, B(2+\varepsilon,1/2-\varepsilon) \, B(3/2,1/2) \, B(1,1/2)}
\frac{\Gamma(2+\varepsilon)}{(1-2 \, \varepsilon)}
\]
reminiscent of eq. (\ref{P1-eqDEFK2}) of sec.
\ref{P1-sectfourpoint} of ref. \cite{paper1}.
Infrared divergences in the four-point function are not dominant Landau-type
singularities but subleading ones. Such an infrared divergence corresponds to
the vanishing determinant $\detsj{i}$ of some reduced $\text{$\cal S$}$ matrix (or
some $\Delta_2^{\{i\}}$), associated
with some three-point functions which are obtained from the four-point function
considered by one pinching \cite{Binoth:2005ff}.
The various cases of vanishing kinematic matrices associated
with three-point functions plagued with infrared soft or collinear
singularities have been evoked in sec. \ref{3point_ir}.
Let us note that the method developed in section \ref{P1-sectfourpoint} of ref. \cite{paper1} is still
valid because we never divide by $\Delta_2^{\{i\}}$ {\em per se} but by
$\Delta_2^{\{i\}}- \xi^{\nu}- \rho^2$.
The divergences will show up when performing the integrations over the
parameters of eq. (\ref{eqdefnl1}). Note also that if we face a case when
IR divergences arise whereas there are some non vanishing internal masses,
only some of the contributions (let us call them sector)``$i,j,k$", not all, are plagued with IR
divergences; the other ones, to which corresponds $\Delta_2^{\{i\}} \ne 0$,
will be treated as in the $n=4$ case.
Besides, in any IR-divergent sector ``$i$'' for which
$\Delta_2^{\{i\}}=0$, we do not have to sum over all three sub-sectors
$j \in S_{4} \setminus \{i\}$ because, as was seen in the
IR-divergent three-point function case, some of the $\bbj{i}{j}$ vanish.
\vspace{0.3cm}
\noindent
Let us consider a sector $i$ which diverges in the IR region.
We have to distinguish two cases : 1) when $\Delta_2^{\{i\}} = 0$ whereas
$\widetilde{D}_{ijk} \ne 0$ in which case there are only soft divergences,
2) when both $\Delta_2^{\{i\}} = 0$ and $\widetilde{D}_{ijk} = 0$ in which case there
are collinear or both soft and collinear divergences.
In this section, we will treat at once
the real and complex mass case. Some of the
cases correspond to real or complex masses, others to
complex masses only. For those corresponding real or complex
masses, we keep the imaginary part $- i \, \lambda$ explicitly having in mind
that with complex masses this $- i \, \lambda$ is ineffective.
Let us stress in passing that we also keep an infinitesimal prescription
$- i \, \lambda$ in the pole term where we put $\Delta_{2}^{\{i\}} = 0$.
\subsection{$\Delta_2^{\{i\}} = 0$ and $\widetilde{D}_{ijk} \ne 0$}\label{subsect52}
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) &=
\kappa \,\int^{+\infty}_0 d \xi \, \int^{+\infty}_0 d \rho \,
\int^{+\infty}_0 d \sigma
\frac{1}
{(\xi^{\nu} - \Delta_3 - i \, \lambda) \, (\xi^{\nu} + \rho^2 - i \, \lambda)}
\nonumber \\
&\quad {} \quad {}
\times
\frac{1}
{(\xi^{\nu} + \rho^2 + \sigma^2 - \Delta_1^{\{i,j\}} - i \, \lambda) \,
(\widetilde{D}_{ijk} + \xi^{\nu} + \rho^2 + \sigma^2 - i \, \lambda)^{1/2}}
\label{eqdefnlir2}
\end{align}
In the complex mass case, $\Im(\Delta_3)$ and
$\Im(\Delta_1^{\{i,j\}})$ have arbitrary signs and $\Im(\widetilde{D}_{ijk})$ is
negative, so we have to distinguish between different cases.
\vspace{0.3cm}
\noindent
Let us define the function $M_2(\xi^{\nu})$ as:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) &= \kappa \,\int^{+\infty}_0 d \xi \, \frac{1}{\xi^{\nu} - \Delta_3 - i \, \lambda} \, M_2(\xi^{\nu})
\label{eqdefM2}
\end{align}
The integrations over $\sigma$ and $\rho$ are identical with those appearing in the massive cases,
we thus borrow the results derived in subsec.\ \ref{P1-fourpointstep4} of ref.\ \cite{paper1} and in subsec.~\ref{P2-sectfourpointcomp} of ref.~\cite{paper2} with
$\Delta_2^{\{i\}} = 0$, explicitly
\footnote{
The integration over $\sigma$ is of the ``second kind'' (cf.\ eqs.~(\ref{eqdeffuncj2}) and (\ref{eqdeffuncj7}))
while the integration over $\rho$ is of the ``first kind'' (cf.\ eq.~(\ref{eqdefk1ext})).
For both integrations, the power $\nu$ appearing in the eqs.~(\ref{eqdefk1ext}), (\ref{eqdeffuncj2}) and (\ref{eqdeffuncj7}) is taken equal to $2$.}:
\begin{itemize}
\item[-] for $\Im(\Delta_1^{\{i,j\}}) \geq 0$,
\begin{align}
M_2(\xi^{\nu})
&=
\frac{1}{2} \, B(1/2,1/2) \,
\int^1_0 \frac{d u}{u^2 \, (\widetilde{D}_{ijk}+\Delta_1^{\{i,j\}}) -\Delta_1^{\{i,j\}}}
\notag \\
&\quad {} \times
\left[
\frac{1}{\left(\xi^{\nu}-i \, \lambda \right)^{1/2}}
-
\frac{1}
{\left(\xi^{\nu}+ u^2 \, (\widetilde{D}_{ijk}+\Delta_1^{\{i,j\}})
-
\Delta_1^{\{i,j\}} - i \, \lambda \right)^{1/2}}
\right]
\label{eqisigir3}
\end{align}
\item[-] for $\Im(\Delta_1^{\{i,j\}}) < 0$,
\begin{align}
M_2(\xi^{\nu})
&=
- \frac{1}{2} \, B(1/2,1/2) \,
\left\{
i \, \int^{+\infty}_0
\frac{d u}{u^2 \, (\widetilde{D}_{ijk}+\Delta_1^{\{i,j\}}) +\Delta_1^{\{i,j\}}}
\right.
\notag \\
&\quad {} \times
\left[
\frac{1}{\left(\xi^{\nu}-i \, \lambda \right)^{1/2}} -
\frac{1}{\left(\xi^{\nu}- u^2 \, (\widetilde{D}_{ijk}+\Delta_1^{\{i,j\}})
-\Delta_1^{\{i,j\}} \right)^{1/2}}
\right]
\notag \\
&\quad {}
+ \int^{+\infty}_1
\frac{d u}{u^2 \, (\widetilde{D}_{ijk}+\Delta_1^{\{i,j\}}) -\Delta_1^{\{i,j\}}}
\notag \\
&\quad {} \times
\left.
\left[
\frac{1}{\left(\xi^{\nu}-i \, \lambda \right)^{1/2}}
-
\frac{1}
{\left(\xi^{\nu}+ u^2 \, (\widetilde{D}_{ijk}+\Delta_1^{\{i,j\}})-\Delta_1^{\{i,j\}}
\right)^{1/2}}
\right]
\right\}
\label{eqisigir4}
\end{align}
\end{itemize}
The integration over $\xi$ to obtain $L_4^n$ is of the ``second kind'' (cf.\ eqs.~(\ref{eqdeffuncj2}) and (\ref{eqdeffuncj7}) with $\nu = 2/(1 - 2 \, \varepsilon)$).
Let us go through the different cases with respect to the sign
of the imaginary part of $\Delta_3$ and $\Delta_1^{\{i,j\}}$. Sticking with the
notations of section \ref{P1-sectfourpoint} of ref. \cite{paper1}, we introduce:
\begin{align*}
P_{ijk} &= \widetilde{D}_{ijk} + \Delta_1^{\{i,j\}} \\
R_{ij} &= - \, \Delta_1^{\{i,j\}} \\
T &= - \, \Delta_3
\end{align*}
The strategy for computing the different integrals is the same for all cases
and very similar to section \ref{P2-sectfourpoint} of ref. \cite{paper2}.
We display the successive steps for the first case tackled only and
we give the final result for the three others.
\subsubsection{$\Im(\Delta_3) > 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$\label{subsubcase1}}
In this case, the three complex numbers $\Delta_3$, $\Delta_1^{\{i,j\}}$ and $\widetilde{D}_{ijk}$ have an imaginary part of the same sign,
We start with eq.~(\ref{eqisigir3}) for $M_2(\xi^{\nu})$ and use eq.~(\ref{eqdeffuncj2}) for the $\xi$ integration to get:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})
&
= F(\varepsilon) \, \int^1_0 \frac{d u}{u^2 \, P_{ijk} + R_{ij}} \notag \\
&\qquad \qquad \quad {} \times \left[
\int^1_0 \frac{d z}
{\left( (1-z^2) \, T - i \, \lambda \right)^{1+\varepsilon}}
\right.
\label{eqlijkir10} \\
&\qquad \qquad \qquad \quad {}
-
\left.
\int^1_0 \frac{d z}
{\left(
z^2 \, (u^2 \, P_{ijk} + R_{ij})+ (1-z^2) \, T - i \, \lambda
\right)^{1+\varepsilon}
}
\right]
\notag
\end{align}
where
\begin{align}
F(\varepsilon)
&= 2^{1+\varepsilon} \, \Gamma(1+\varepsilon)
\label{eqdeffepsilon}
\end{align}
To facilitate the reading, we will define some steps in a similar way as in the subsec. \ref{P1-fourpointstep4} of ref.\ \cite{paper1}. Let us proceed along them. \\
\noindent
{\bf 1)} We change $u=\sqrt{y/x}$ and $z = \sqrt{x}$ and exchange the $y$ and the $x$ integration so that:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})
&=
- \frac{F(\varepsilon)}{4} \,
\int^1_0 \frac{d y}{\sqrt{y}} \, \int^1_y d x \,
\frac{1}{y \, P_{ijk} + x \, R_{ij}} \notag \\
&\qquad \qquad {} \times \left[
\frac{1}{\left( y \, P_{ijk} + x \, (R_{ij} - T) + T
- i \, \lambda \right)^{1+\varepsilon}}
\right.
\notag \\
&\qquad \qquad \qquad {}
-
\left.
\frac{1}{\left( (1-x) \, T - i \, \lambda \right)^{1+\varepsilon}}
\right]
\label{eqlijkir12}
\end{align}
\noindent
{\bf 2)} We set $y = u^2$ and perform a partial fraction decomposition on the variable $x$ to write
$L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$ in the following form:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) \notag \\
&=
- \frac{F(\varepsilon)}{2} \,
\int^1_0 \frac{du}{u^2 \, P_{ijk} T + R_{ij} (T - i \, \lambda)}
\int^1_{u^2} d x \,
\notag \\
&\qquad \qquad {}
\times
\Biggl\{
(T - R_{ij}) \,
\left[
u^2 \, P_{ijk} + x \, (R_{ij} - T) + T - i \, \lambda
\right]^{-1-\varepsilon} \notag \\
&\qquad \qquad \qquad {}
-
T \,
\left[ (1-x) \, T - i \, \lambda \right]^{-1-\varepsilon}
+ \frac{R_{ij}}{u^2 \, P_{ijk} + x \, R_{ij}}
\notag \\
&\qquad {} \qquad \qquad {}
\times
\Biggl[
\left[
u^2 \, P_{ijk} + x \, (R_{ij} - T) + T - i \, \lambda
\right]^{-\varepsilon}
-
\left[ (1-x) \, T - i \, \lambda \right]^{-\varepsilon} \,
\Biggr]
\Biggr\}
\label{eqlijkir131}
\end{align}
In eq. (\ref{eqlijkir131}), the last line provides a contribution of order
$\varepsilon$ only: for the computation of the one-loop four-point function it
can thus be dropped\footnote{Similar truncations of $\varepsilon$ expansions
will be performed everywhere throughout this section. In the perspective of
computing generalised one-loop building blocks to be used in computations beyond
one-loop, one might be led to keep further terms evanescent with $\varepsilon$
to the appropriate order, whenever such terms would hit $1/\varepsilon$ poles
generated by the extra integrations, cf. introduction of \cite{paper1}.}.\\
\noindent
{\bf 3)} Thus, ignoring these terms, the integration in $x$ is readily done providing:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})
&=
\frac{2^{\varepsilon}}{T} \, \frac{\Gamma(1+\varepsilon)}{\varepsilon}
\int^1_0
\frac{d u}{u^2 \, P_{ijk} + R_{ij} - i \, \lambda \, \sigma_0} \,
\label{eqlijkir14} \\
&
\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
\times
\Biggl\{
\left[
u^2 \, (P_{ijk} + R_{ij} - T) + T - i \, \lambda
\right]^{-\varepsilon}
\notag\\
&
\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
-
\left[
u^2 \, P_{ijk} + R_{ij} - i \, \lambda
\right]^{-\varepsilon}
- (T - i \, \lambda)^{-\varepsilon} \, (1-u^2)^{-\varepsilon}
\Biggr\}
\notag
\end{align}
where we have introduced $\sigma_0 = \mbox{sign}(R_{ij}/T)$ in the real mass case.
In the complex mass case all ``$-i \lambda$" and``$- i \sigma_0 \, \lambda$"
contour prescriptions are
ineffective and irrelevant, and all three terms in eq. (\ref{eqlijkir14})
can be straightforwardly expanded in powers of $\varepsilon$. In contrast the
real mass case requires a more cautious treatment which we elaborate below.
We note that, when $u^2 \to - R_{ij}/P_{ijk}$,
\begin{align}
u^2 \, (P_{ijk} + R_{ij} - T) + T & \to (T - R_{ij}) \frac{\widetilde{D}_{ijk}}{P_{ijk}}
\neq 0
\notag\\
1-u^2 & \to \frac{\widetilde{D}_{ijk}}{P_{ijk}} \neq 0
\notag
\end{align}
i.e. the pole at $u^2 = - R_{ij}/P_{ijk}$ is distinct from the branch points
of the first and third functions in the numerator. Therefore we can readily perform an
expansion around $\varepsilon=0$ for the first and third term as:
\begin{align}
\frac{1}{\varepsilon} \,
\left[
u^2 \, (P_{ijk} + R_{ij} - T) + T - i \, \lambda
\right]^{-\varepsilon}
& =
\frac{1}{\varepsilon} -
\ln
\left[
u^2 \, (P_{ijk} + R_{ij} - T) + T - i \, \lambda
\right] + {\cal O}(\varepsilon)
\notag\\
\frac{1}{\varepsilon} \,
(T - i \, \lambda)^{-\varepsilon} \, (1-u^2)^{-\varepsilon}
& =
- \frac{1}{\varepsilon} -
\ln(T - i \, \lambda) - \ln(1-u^2) + {\cal O}(\varepsilon)
\notag
\end{align}
The $1/\varepsilon$ poles cancel between these two contributions leaving only
logarithms. On the other hand the contribution coming from the second term requires
some care since pole and branch point coincide. If this singularity lies outside
the integration region the contour prescription for the pole is irrelevant and
can be dropped. If the singularity lies inside $[0,1]$ and $\sigma_0=+$ the contour
prescription for the pole and cut are the same, there is no pinching. The
integral over $u$ can be performed after an expansion in $\varepsilon$ using
appendix \ref{appF} (cf.\ eq.~(\ref{eqdefh05})). If the singularity lies inside $[0,1]$ and
$\sigma_0 = -$ however, a pinching occurs at the singular point in the limit
$\lambda \to 0^{+}$. A too early expansion in powers
of $\varepsilon$ before performing the integration over $u$ would lead to a
divergence order by order in $\varepsilon$. On the other hand, we note that
for $u^2 = - R_{ij}/P_{ijk}$ the numerator of the second
term - i.e. the pole residue - is $(- \, i \, \lambda)^{-\varepsilon} \to 0$
with $\lambda \to 0^{+}$ for any fixed $\varepsilon <0$. Therefore, up to terms
vanishing $\propto \lambda^{- \varepsilon}$ we can make the replacement
\[
\int_{0}^{1} du \,
\frac{\left[ u^2 \, P_{ijk} + R_{ij} - i \, \lambda \right]^{-\varepsilon}}
{u^2 \, P_{ijk} + R_{ij} + i \, \lambda}
\to
\int_{0}^{1} du \,
\frac{\left[ u^2 \, P_{ijk} + R_{ij} - i \, \lambda \right]^{-\varepsilon}}
{u^2 \, P_{ijk} + R_{ij} - i \, \lambda}
\]
Details are provided in appendix \ref{ir-lambda}.
Finally, we perform a partial expansion around $\varepsilon=0$ and we get:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) = \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon) \notag \\
& {} \times
\left\{
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} - i \, \lambda \, \sigma_0}
\left[
\ln
\left(
(T - i \lambda) \, (1-u^2)
\right) -
\ln
\left(
u^2 \, (P_{ijk} + R_{ij} - T) + T - i \, \lambda
\right)
\right]
\right.
\notag\\
&\qquad{}
\left.
-
\frac{1}{\varepsilon} \,
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} - i \, \lambda}
+
\int^1_0 d u \,
\frac{\ln(u^2 \, P_{ijk} + R_{ij} - i \, \lambda)}
{u^2 \, P_{ijk} + R_{ij} - i \, \lambda}
\right\}
\label{eqlijkir15}
\end{align}
Let us notice that the imaginary part of the argument of each logarithm in eq.~(\ref{eqlijkir15}) keeps a constant sign when $u$ spans $[0,1]$.
This is obviously true for the real mass case and in the case of complex masses, this is easily verified keeping in mind the assumptions: $\Im(T) < 0$ and $\Im(R_{ij}) < 0$.
For the sake of coherence with respect to ref. \cite{paper2}, the pole residue contributions will
be added and subtracted for the two logarithms making up the first term inside the curly
brackets of eq.~(\ref{eqlijkir15}). This latter equation recast in the form:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) = \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon) \notag \\
& {} \times
\left\{
-
\frac{1}{\varepsilon} \,
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} - i \, \lambda}
+
\int^1_0 d u \,
\frac{\ln(u^2 \, P_{ijk} + R_{ij} - i \, \lambda)}
{u^2 \, P_{ijk} + R_{ij} - i \, \lambda}
\right.
\notag \\
&\qquad {}
+ \int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} - i \, \lambda \, \sigma_0} \notag\\
&\qquad \quad{}
\times
\left[ \ln \left( (T - i \lambda) \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} - i \, \lambda\right) \right.
\notag\\
&\qquad \quad{}
-
\left. \ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T - i \, \lambda \right) + \ln \left( \frac{(P_{ijk}+R_{ij}) \, (T - R_{ij})}{P_{ijk}} - i \, \lambda \right) \right]
\notag\\
&\qquad {}
\left.
-
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} - i \, \lambda \, \sigma_0} \,
\ln \left( \frac{T - R_{ij} - i \, \lambda \, \sigma_1}{T - i \, \lambda \, \sigma_1} \right)
\right\}
\label{eqlijkir15r}
\end{align}
with $\sigma_1 = \mbox{sign}( (P_{ijk} + R_{ij})/P_{ijk} )$ for the real mass case and:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) = \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon) \notag \\
& {} \times
\left\{
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} } \,
\left[ \ln \left( T \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} \right) \right]
\right.
\notag \\
&\qquad {} \;
-
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} } \,
\left[ \ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T \right) - \ln \left( \frac{(P_{ijk}+R_{ij}) \, (T - R_{ij})}{P_{ijk}} \right) \right]
\notag\\
&\qquad {} \;
-
\frac{1}{\varepsilon} \,
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} }
+
\int^1_0 d u \,
\frac{\ln(u^2 \, P_{ijk} + R_{ij} )}
{u^2 \, P_{ijk} + R_{ij} }
\notag\\
&\qquad {} \;
\left.
-
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} } \,
\left[ \ln \left( \frac{T - R_{ij}}{T} \right) + \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right) \right]
\right\}
\label{eqlijkir15c}
\end{align}
for the complex mass case. The relevant integrals are given in appendices \ref{appF} of this paper, \ref{P1-appF} of ref. \cite{paper1} and \ref{P2-appF} of ref. \cite{paper2}.
\vspace{0.3cm}
\noindent
Eq. (\ref{eqlijkir15c}) matches eq. (\ref{P2-eqcas1eclate}) of ref. \cite{paper2}
obtained in the corresponding general complex mass case 1.(a), considering the
latter in the limit $\Re(\Delta_{2}^{\{i\}}) = \Re(Q_{i}+T) \to 0$ while keeping
an infinitesimal positive imaginary part $\Im(\Delta_{2}^{\{i\}}) = \lambda$.
One then formally gets:
\begin{align}
\text{r.h.s.} \, (\ref{P2-eqcas1eclate}) \, \text{of \cite{paper2}}
&\to \frac{1}{T} \,
\left\{
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} } \,
\left[ \ln \left( T \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} \right) \right.
\right.
\notag \\
&\qquad {} \;
-
\left. \ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T \right) + \ln \left( \frac{(P_{ijk}+R_{ij}) \, (T - R_{ij})}{P_{ijk}} \right) \right]
\notag\\
&\qquad {} \;
-
\ln\left( Q_i + T \right) \,
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} }
+
\int^1_0 d u \,
\frac{\ln(u^2 \, P_{ijk} + R_{ij} )}
{u^2 \, P_{ijk} + R_{ij} }
\notag\\
&\qquad {} \;
\left.
-
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij} } \,
\left[ \ln \left( \frac{T - R_{ij}}{T} \right) + \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right) \right]
\right\}
\notag
\end{align}
where the divergent term $\ln ( Q_{i}+T )$ corresponds to the ``dressed pole"
$(2 \, e^{- \gamma_{E}})^{\varepsilon}/\varepsilon$.
\subsubsection{$\Im(\Delta_3) > 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$}\label{subsubseccas2}
Compared to the previous case, one uses eq.~(\ref{eqisigir4}) for $M_2(\xi^{\nu})$ and the $\xi$
integration is carried out with the help of eqs.~(\ref{eqdeffuncj2}) and (\ref{eqdeffuncj7}) depending on the different terms.
Then, performing the first step described in subsubsec.\ \ref{subsubcase1}, we obtain:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) \notag \\
&= - \frac{F(\varepsilon)}{4} \left\{ i \, \int_0^{\infty} \frac{dy}{\sqrt{y}} \, \int_0^1 \, \frac{dx}{y \, P_{ijk} - x \, R_{ij}} \, \left( T (1-x) \right)^{-1-\varepsilon} \right. \notag \\
&\qquad \qquad \quad {} + e^{i \, \pi \, \varepsilon} \, \int_0^{\infty} \frac{dy}{\sqrt{y}} \, \int_0^{\infty} \, \frac{dx}{y \, P_{ijk} - x \, R_{ij}} \, \left( - y \, P_{ijk} + x \, R_{ij} - T \, (1+x) \right)^{-1-\varepsilon} \notag \\
&\qquad \qquad \quad {} + i \, \int_0^{\infty} \frac{dy}{\sqrt{y}} \, \int_1^{\infty} \, \frac{dx}{y \, P_{ijk} - x \, R_{ij}} \, \left( - y \, P_{ijk} + x \, R_{ij} + T \, (1-x) \right)^{-1-\varepsilon} \notag \\
&\qquad \qquad \quad {} + \int_1^{\infty} \frac{dy}{\sqrt{y}} \, \int_0^1 \, \frac{dx}{y \, P_{ijk} + x \, R_{ij}} \, \left[ \left( T \, (1-x) \right)^{-1-\varepsilon} \right. \notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad {} - \left. \left( y \, P_{ijk} + x \, R_{ij} + T \, (1-x) \right)^{-1-\varepsilon} \right] \notag \\
&\qquad \qquad \quad {} + \int_0^{1} \frac{dy}{\sqrt{y}} \, \int_0^y \, \frac{dx}{y \, P_{ijk} + x \, R_{ij}} \, \left[ \left( T \, (1-x) \right)^{-1-\varepsilon} \right. \notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {} - \left. \left. \left( y \, P_{ijk} + x \, R_{ij} + T \, (1-x) \right)^{-1-\varepsilon} \right] \vphantom{\frac{dx}{y \, P_{ijk} - x \, R_{ij}}} \right\}
\label{eqlijkir160}
\end{align}
We set $y = u^2$, perform a partial fraction decomposition on the variable $x$ and expand around $\varepsilon=0$. The $x$ integration is readily done and $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$ is written as:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) \notag \\
&= - \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon)
\left\{
- \frac{T^{-\varepsilon}}{\varepsilon} \,
\left[
i \, \int^{+\infty}_0 \, \frac{d u }{u^2 \, P_{ijk} - R_{ij}}
+ \int^{+\infty}_1 \,
\frac{d u }{u^2 \, P_{ijk} + R_{ij}}
\right]
\right.
\notag \\
&\qquad \qquad \qquad \qquad {}
+ i \, \int^{+\infty}_0 \,
\frac{d u}{u^2 \, P_{ijk} - R_{ij}} \,
\left[
\ln \left(\frac{u^2 \, P_{ijk} - R_{ij}}{u^2 \, P_{ijk}} \right)
-
\ln \left( \frac{R_{ij}-T}{R_{ij}} \right)
\right]
\notag \\
&\qquad \qquad \qquad \qquad {}
- \int^{+\infty}_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\left[
\ln \left(\frac{- u^2 \, P_{ijk}}{ R_{ij}} \right)
-
\ln \left( \frac{P_{ijk} \, u^2 + T}{T-R_{ij}} \right)
\right]
\notag \\
&\qquad \qquad \qquad \qquad {}
+ \int^{+\infty}_1 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\ln \left(\frac{u^2 \, P_{ijk} + R_{ij}}{u^2 \, P_{ijk} + T} \right)
\notag \\
&\qquad \qquad \qquad \qquad {}
+
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\left[
\ln \left(\frac{u^2 \, (P_{ijk} + R_{ij} - T) + T}{u^2 \, P_{ijk} + T} \right) \right. \notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {} - \left. \left. \ln(1-u^2)
\vphantom{\ln \left(\frac{u^2 \, (P_{ijk} + R_{ij} - T) + T}{u^2 \, P_{ijk} + T} \right)}
\right]
\right\}
\label{eqlijkir16}
\end{align}
In this case, we have $\Im(R_{ij}) > 0$, $\Im(P_{ijk}) < 0$ and
$\Im(R_{ij}-T) > 0$. Furthermore,
\begin{itemize}
\item
$u^2 \, P_{ijk} - R_{ij} = (1+u^2) \, \Delta_1^{\{i,j\}} + u^2 \, \widetilde{D}_{ijk}$ \\
thus $\Im(u^2 \, P_{ijk} - R_{ij}) < 0$ when $u \in [0,\infty[$,
\item
$u^2 \, P_{ijk} + R_{ij}
= u^2 \, \widetilde{D}_{ijk} + (u^2 - 1) \, \Delta_1^{\{i,j\}}$ \\
thus $\Im(u^2 \, P_{ijk} + R_{ij}) < 0$ when $u \in [1,\infty[$,
\item
$u^2 \, P_{ijk} + T
= u^2 \, (\widetilde{D}_{ijk} + \Delta_1^{\{i,j\}}) - \Delta_3$ \\
thus $\Im(u^2 \, P_{ijk} + T) < 0$ when $u \in [0,\infty[$,
\item
$u^2 \, (P_{ijk}+R_{ij}-T) + T = u^2 \, \widetilde{D}_{ijk} - (1-u^2) \, \Delta_3$ \\
thus $\Im(u^2 \, (P_{ijk}+R_{ij}-T) + T) < 0$ when $u \in [0,1]$.
\end{itemize}
Rearrangements and simplifications similar to those done in the massive case
2.(a) of ref.~\cite{paper2} can be performed, which lead to the following alternative expression:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) = \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon) \notag \\
& {} \times \Bigg\{
-
\frac{1}{\varepsilon} \,
\int_{\usebox{\Gammap}} \, \frac{d u}{u^2 \, P_{ijk} + R_{ij}}
\quad + \quad
\int_{\usebox{\Gammap}} d u \,
\frac{\ln(u^2 \, P_{ijk} + R_{ij})}{u^2 \, P_{ijk} + R_{ij}}
\notag\\
&
\notag\\
&\qquad {} +
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\Bigg[
\ln \left( T \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} \right)
\notag\\
&\qquad {}
-
\ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T \right) + \ln \left( \frac{(P_{ijk} + R_{ij}) \, (T - R_{ij})}{P_{ijk}} \right)
\notag\\
&\qquad {}
- \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right)\Biggr]
-
\int_{\usebox{\Gammap}} \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\ln \left( \frac{T - R_{ij}}{T} \right)
\Biggl\}
\label{eqlijkir16reframed}
\end{align}
\vspace{0.3cm}
\noindent
The contour \raisebox{0.8ex}{$\usebox{\Gammap}$} can be deformed into a contour
$\widehat{(0,1)}^{+}$ stretched from 0 to 1 and which eventually wraps
from above the cut of $\ln(u^2 \, P_{ijk} + R_{ij})$ emerging from
the branch point $u_{0} = \sqrt{- R_{ij}/P_{ijk}}$, whenever the latter lies
in the ``north-east" quadrant $\{\Re(u) > 0, \Im(u) > 0\}$.
\vspace{0.3cm}
\noindent
Eq. (\ref{eqlijkir16reframed}) can be compared with eq. (\ref{P2-eqcas2aa}) of \cite{paper2}
obtained in the corresponding general complex mass case 2.(a), considering the
latter in the limit $\Re(\Delta_{2}^{\{i\}}) = \Re(Q_{i}+T) \to 0$ while keeping
an infinitesimal positive imaginary part $\Im(\Delta_{2}^{\{i\}}) = \lambda$.
Whereas it appeared
convenient to formulate eq. (\ref{P2-eqcas2aa}) of ref. \cite{paper2} in terms of manifestly vanishing
pole residues, this is no longer the case for eq. (\ref{eqlijkir16reframed})
since the pole has become also the branch point of
$\ln(u^2 \, P_{ijk} + (R_{ij} +Q_{i} +T))$
in the limit $(Q_{i}+T) \to 0$. For the purpose of the comparison the $\eta$
functions introduced in eq. (\ref{P2-eqcas2aa}) of ref. \cite{paper2} containing $Q_{i} + T$ shall thus be made explicit in
terms of constant logarithms, part of which then cancel against the constant
logarithms which were subtracted so as to build the explicitly vanishing pole
residues. One then formally gets:
\begin{align}
\hspace{2em}&\hspace{-2em}\text{r.h.s.\ (\ref{P2-eqcas2aa}) of \cite{paper2}} \to \frac{1}{T} \notag \\
& {} \times \Bigg\{
-
\ln \left( Q_i + T \right) \,
\int_{\widehat{(0,1)}^{+}} \, \frac{d u}{u^2 \, P_{ijk} + R_{ij}}
\quad + \quad
\int_{\widehat{(0,1)}^{+}} d u \,
\frac{\ln(u^2 \, P_{ijk} + R_{ij})}{u^2 \, P_{ijk} + R_{ij}}
\notag\\
&\qquad {} +
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\Bigg[
\ln \left( T \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} \right)
\notag\\
&\qquad {}
-
\ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T \right) + \ln \left( \frac{(P_{ijk} + R_{ij}) \, (T - R_{ij})}{P_{ijk}} \right)
\notag\\
&\qquad {}
- \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right)\Biggr]
-
\int_{\widehat{(0,1)}^{+}} \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\ln \left( \frac{T - R_{ij}}{T} \right)
\Biggl\}
\notag
\end{align}
where the divergent term $\ln ( Q_{i}+T )$ corresponds to the ``dressed pole"
$(2 \, e^{- \gamma_{E}})^{\varepsilon}/\varepsilon$.
\subsubsection{$\Im(\Delta_3) < 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$}
In this case, we start with eq.~(\ref{eqisigir3}) for $M_2(\xi^{\nu})$ and use eq.~(\ref{eqdeffuncj7}) for all the $\xi$ integrations.
We give the result for $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})$ after the intermediate step 1 of subsubsec. \ref{subsubcase1}.
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) \notag \\
&= \frac{F(\varepsilon)}{4} \left\{ i \, e^{- i \, \pi \, \varepsilon} \, \int_0^{\infty} \frac{dy}{\sqrt{y}} \, \int_y^{\infty} \, \frac{dx}{y \, P_{ijk} + x \, R_{ij}} \, \left[ \left( y \, P_{ijk} + x \, R_{ij} - T \, (1+x) \right)^{-1-\varepsilon} \right. \right. \notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad {} - \left. \left( - T \, (1+x) \right)^{-1-\varepsilon} \right] \notag \\
&\qquad \qquad \quad {} + \int_1^{\infty} \frac{dy}{\sqrt{y}} \, \int_y^{\infty} \, \frac{dx}{y \, P_{ijk} + x \, R_{ij}} \, \left[ \left( y \, P_{ijk} + x \, R_{ij} + T \, (1-x) \right)^{-1-\varepsilon} \right. \notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad {} - \left. \left( T \, (1-x) \right)^{-1-\varepsilon} \right] \notag \\
&\qquad \qquad \quad {} + \int_0^{1} \frac{dy}{\sqrt{y}} \, \int_1^{\infty} \, \frac{dx}{y \, P_{ijk} + x \, R_{ij}} \, \left[ \left( y \, P_{ijk} + x \, R_{ij} + T \, (1-x) \right)^{-1-\varepsilon} \right. \notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad {} - \left. \left. \left( T \, (1-x) \right)^{-1-\varepsilon} \right] \vphantom{\frac{dx}{y \, P_{ijk} + x \, R_{ij}}} \right\}
\label{eqeqlijkir170}
\end{align}
Then, performing the steps 2 and 3 of this subsubsec.\ yields:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) \notag \\
&= \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon)
\left\{
- \frac{(-T)^{-\varepsilon}}{\varepsilon} \,
\int^{1}_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}}
\right.
\notag \\
&\qquad \qquad \qquad \quad {}
+ i \, \int^{+\infty}_0 \, \frac{d u }{u^2 \, P_{ijk} - R_{ij}} \,
\left[
\ln \left( \frac{u^2 \, (P_{ijk} + R_{ij}-T) - T}{R_{ij}-T} \right)
-
\ln \left( u^2+1 \right) \right]
\notag \\
&\qquad \qquad \qquad \quad {}
+ \int^{+\infty}_1 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\left[
\ln \left(\frac{u^2 \, (P_{ijk} + R_{ij}-T) + T}{R_{ij}-T} \right)
-
\ln \left( u^2 -1 \right)
\right]
\notag \\
&\qquad \qquad \qquad \quad {}
+
\left.
\int^1_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\ln \left( \frac{u^2 \, P_{ijk} + R_{ij}}{R_{ij} - T} \right)
\right\}
\label{eqlijkir17}
\end{align}
In this case, we have $\Im(R_{ij}-T) < 0$.
Furthermore,
\begin{itemize}
\item
$u^2 \, P_{ijk} + R_{ij}
= u^2 \, \widetilde{D}_{ijk} - (1 - u^2) \, \Delta_1^{\{i,j\}}$ \\
thus $\Im(u^2 \, P_{ijk} + R_{ij}) < 0$ when $u \in [0,1]$,
\item
$u^2 \, (P_{ijk}+R_{ij}-T) - T = u^2 \, \widetilde{D}_{ijk} + (1+u^2) \, \Delta_3$ \\
thus $\Im(u^2 \, (P_{ijk}+R_{ij}-T) - T) < 0$ when $u \in [0,\infty[$.
\item
$u^2 \, (P_{ijk}+R_{ij}-T) + T = u^2 \, \widetilde{D}_{ijk} + (u^2-1) \, \Delta_3$ \\
thus $\Im(u^2 \, (P_{ijk}+R_{ij}-T) + T) < 0$ when $u \in [1,\infty[$.
\end{itemize}
Rearrangements and simplifications similar to those done in the massive case
1.(c) of \cite{paper2} can be performed, which lead to the following alternative expression:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) = \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon) \notag \\
& {} \times \Biggl\{
-
\frac{1}{\varepsilon} \,
\int_{0}^{1} \, \frac{d u}{u^2 \, P_{ijk} + R_{ij}}
\quad {} + \quad {}
\int_{0}^{1} d u \,
\frac{\ln(u^2 \, P_{ijk} + R_{ij})}{u^2 \, P_{ijk} + R_{ij}}
\notag\\
&\qquad {} +
\int_{\usebox{\Gammap}} \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\Biggl[
\ln \left( T \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} \right)
\notag \\
&\qquad \qquad {}
-
\ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T \right) + \ln \left( \frac{(P_{ijk} + R_{ij}) \, (T - R_{ij})}{P_{ijk}} \right)
\notag\\
&\qquad \qquad{}
- \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right)
\Biggr]
- \int_{0}^1 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\ln \left( \frac{T - R_{ij}}{T} \right)
\Biggr\}
\label{eqlijkir17reframed}
\end{align}
The deformation of the contour \raisebox{0.8ex}{$\usebox{\Gammap}$} into a contour
$\widehat{(0,1)}^{+}$ stretched from 0 to 1 may eventually wrap
from above the cut of $\ln(u^2 \, (P_{ijk} + R_{ij} -T) +T)$ emerging from
the branch point $u_{0} = \sqrt{- T/(P_{ijk} + R_{ij} -T)}$, whenever the
latter lies in the ``north-east" quadrant.
\vspace{0.3cm}
\noindent
In a way similar to the previous case, eq. (\ref{eqlijkir17reframed})
matches eq. (\ref{P2-eqcas4eclatesoustrait}) of \cite{paper2} obtained in the corresponding
general complex mass case 1.(c), considering the latter in the limit
$\Re(\Delta_{2}^{\{i\}}) = \Re(Q_{i}+T) \to 0$ while keeping
an infinitesimal positive imaginary part $\Im(\Delta_{2}^{\{i\}}) = \lambda$.
\subsubsection{$\Im(\Delta_3) < 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$}
One uses eq.~(\ref{eqisigir4}) for $M_2(\xi^{\nu})$ and the $\xi$
integration is carried out with the help of eqs.~(\ref{eqdeffuncj2}) and (\ref{eqdeffuncj7}) depending on the terms. Then, the step 1
(cf.\ subsubsec.\ \ref{subsubcase1}) leads to:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) \notag \\
&= \frac{F(\varepsilon)}{4} \, \left\{ i \, \left(-T\right)^{-1-\varepsilon} \, \int_0^{\infty} \frac{dy}{\sqrt{y}} \, \int_1^{\infty} \, \frac{dx}{y \, P_{ijk} - x \, R_{ij}} \, (x-1)^{-1-\varepsilon} \right. \notag \\
&\qquad \qquad \quad {} - e^{-i \, \pi \, \varepsilon} \, \left( - T \right)^{-1-\varepsilon} \, \int_0^{\infty} \frac{dy}{\sqrt{y}} \, \int_0^{\infty} \, \frac{dx}{y \, P_{ijk} - x \, R_{ij}} \, (1+x)^{-1-\varepsilon} \notag \\
&\qquad \qquad \quad {} + i \, \int_0^{\infty} \frac{dy}{\sqrt{y}} \, \int_0^{1} \, \frac{dx}{y \, P_{ijk} - x \, R_{ij}} \, \left( - y \, P_{ijk} + x \, R_{ij} + T \, (1-x) \right)^{-1-\varepsilon} \notag \\
&\qquad \qquad \quad {} - i \, e^{- i \, \pi \, \varepsilon} \, \int_0^{\infty} \frac{dy}{\sqrt{y}} \, \int_0^{y} \, \frac{dx}{y \, P_{ijk} + x \, R_{ij}} \, \left[ \left( y \, P_{ijk} + x \, R_{ij} - T \, (1+x) \right)^{-1-\varepsilon} \right. \notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {} - \left. \left( -T \, (1+x) \right)^{-1-\varepsilon} \right] \notag \\
&\qquad \qquad \quad {} - \int_1^{\infty} \frac{dy}{\sqrt{y}} \, \int_1^{y} \, \frac{dx}{y \, P_{ijk} + x \, R_{ij}} \, \left[ \left( y \, P_{ijk} + x \, R_{ij} + T \, (1-x) \right)^{-1-\varepsilon} \right. \notag \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad {} - \left. \left. \left( T \, (1-x) \right)^{-1-\varepsilon} \right] \vphantom{\frac{dx}{y \, P_{ijk} - x \, R_{ij}}} \right\}
\label{eqlijkir180}
\end{align}
At the end of the step 5, we get;
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk}) \notag \\
&=
- \frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon)
\left\{
- \frac{(-T)^{-\varepsilon}}{\varepsilon} \,
\left[
i \, \int^{+\infty}_0 \, \frac{d u }{u^2 \, P_{ijk} - R_{ij}}
+ \int^{+\infty}_1 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}}
\right]
\right.
\notag \\
&\qquad \qquad \qquad \quad {}
+ i \, \int^{+\infty}_0 \, \frac{d u }{u^2 \, P_{ijk} - R_{ij}} \,
\left[
\ln \left(\frac{R_{ij} - u^2 \, P_{ijk}}{R_{ij}} \right)
-
\ln \left( \frac{P_{ijk} \, u^2 - T}{u^2 \, P_{ijk}} \right)
\right]
\notag \\
&\qquad \qquad \qquad \quad {}
- i \, \int^{+\infty}_0 \, \frac{d u }{u^2 \, P_{ijk} - R_{ij}} \,
\left[
\ln \left( \frac{u^2 \, (P_{ijk} + R_{ij} - T) - T}{u^2 \, P_{ijk} - T} \right)
-
\ln \left( u^2+1 \right)
\right]
\notag \\
&\qquad \qquad \qquad \quad {}
-
\int^{+\infty}_1 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\left[
\ln
\left(
\frac{u^2 \, (P_{ijk} + R_{ij} - T) + T}{u^2 \, P_{ijk} + R_{ij}}
\right)
- \ln(u^2-1)
\right]
\notag \\
&\qquad \qquad \qquad \quad {}
-
\left.
\int^{+\infty}_0 \, \frac{d u }{u^2 \, P_{ijk} + R_{ij}} \,
\ln \left( \frac{- u^2 \, P_{ijk}}{ R_{ij}} \right)
\right\}
\label{eqlijkir18}
\end{align}
In this case, we have $\Im(R_{ij}) > 0$, $\Im(P_{ijk}) < 0$.
Furthermore,
\begin{itemize}
\item
$u^2 \, P_{ijk} - R_{ij} = u^2 \, \widetilde{D}_{ijk} + (u^2+1) \, \Delta_1^{\{i,j\}}$ \\
thus $\Im(u^2 \, P_{ijk} - R_{ij}) < 0$ when $u \in [0,\infty[$,
\item
$u^2 \, P_{ijk} + R_{ij}
= u^2 \, \widetilde{D}_{ijk} + (u^2- 1) \, \Delta_1^{\{i,j\}}$ \\
thus $\Im(u^2 \, P_{ijk} + R_{ij}) < 0$ when $u \in [1,\infty[$,
\item
$u^2 \, P_{ijk} - T
= u^2 \, (\widetilde{D}_{ijk} + \Delta_1^{\{i,j\}}) + \Delta_3$ \\
thus $\Im(u^2 \, P_{ijk} - T) < 0$ when $u \in [0,\infty[$,
\item
$u^2 \, (P_{ijk}+R_{ij}-T) - T = u^2 \, \widetilde{D}_{ijk} + (u^2+1) \, \Delta_3$ \\
thus $\Im(u^2 \, (P_{ijk}+R_{ij}-T) - T) < 0$ when $u \in [0,\infty[$.
\item
$u^2 \, (P_{ijk}+R_{ij}-T) + T = u^2 \, \widetilde{D}_{ijk} + (u^2-1) \, \Delta_3$ \\
thus $\Im(u^2 \, (P_{ijk}+R_{ij}-T) + T) < 0$ when $u \in [1,\infty[$.
\end{itemize}
Rearrangements and simplifications similar to those done in the massive case
2.(c) of \cite{paper2} can be performed, which lead to the following alternative expression:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},\widetilde{D}_{ijk})
&=
\frac{2^{\varepsilon}}{T} \, \Gamma(1+\varepsilon) \, \int_{\usebox{\Gammap}} \, \frac{d u}{u^2 \, P_{ijk} + R_{ij}}
\notag\\
&\quad {} \times
\Biggl\{
-
\frac{1}{\varepsilon} \,
+
\ln(u^2 \, P_{ijk} + R_{ij})
+
\ln \left( T \, (1-u^2) \right) - \ln \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}} \right)
\notag\\
& \quad {} \quad
-
\ln \left( u^2 \, (P_{ijk} + R_{ij} - T) + T \right) + \ln \left( \frac{(P_{ijk} + R_{ij}) \, (T - R_{ij})}{P_{ijk}} \right)
\notag \\
& \quad {} \quad
- \eta \left( \frac{T \, (P_{ijk} + R_{ij})}{P_{ijk}}, \frac{T - R_{ij}}{T} \right) - \ln \left( \frac{T - R_{ij}}{T} \right)
\Biggr\}
\label{eqlijkir18reframed}
\end{align}
As in the previous cases, the contours \raisebox{0.8ex}{$\usebox{\Gammap}$} can be deformed into contours
$\widehat{(0,1)}^{+}_{1,2}$ stretched from 0 to 1. They eventually wrap
from above the cuts of $\ln(u^2 \, P_{ijk} + R_{ij})$ emerging from
the branch point $\sqrt{- R_{ij}/P_{ijk}}$ and of
$\ln(u^2 \, (P_{ijk} + R_{ij} -T) +T)$ emerging from the branch point \linebreak
$\sqrt{- T/(P_{ijk} + R_{ij} -T)}$, respectively, whenever either of these
branch points or both lie in the ``north-east" quadrant.
\vspace{0.3cm}
\noindent
In a way similar to the previous case, eq. (\ref{eqlijkir18reframed})
matches eq. (\ref{P2-eqcas2cc}) of \cite{paper2} obtained in the corresponding
general complex mass case 2.(c), considering the latter in the limit
$\Re(\Delta_{2}^{\{i\}}) = \Re(Q_{i}+T) \to 0$ while keeping
an infinitesimal positive imaginary part $\Im(\Delta_{2}^{\{i\}}) = \lambda$.
\subsection{$\Delta_2^{\{i\}} = 0$ and $\widetilde{D}_{ijk} = 0$}\label{casDelta20Dt0}
In this case, we have $P_{ijk} = - R_{ij}$ and $Q_i = -T$, so that
eqs. (\ref{eqisigir3}) and (\ref{eqisigir4}) become:
\begin{itemize}
\item[-] for $\Im(\Delta_1^{\{i,j\}} > 0)$,
\begin{align}
M_2(\xi^{\nu})
&=
\frac{1}{2} \, B(1/2,1/2) \,
\int^1_0 \frac{d z}{\Delta_1^{\{i,j\}} \, (z^2 - 1)}
\notag \\
&\quad {} \times
\left[
\frac{1}{\left(\xi^{\nu}-i \, \lambda \right)^{1/2}}
-
\frac{1}{\left(\xi^{\nu}+ \Delta_1^{\{i,j\}} \, (z^2-1)
- i \, \lambda \right)^{1/2}}
\right]
\label{eqisigir3p}
\end{align}
\item[-] for $\Im(\Delta_1^{\{i,j\}} < 0)$,
\begin{align}
M_2(\xi^{\nu})
&=
- \frac{1}{2} \, B(1/2,1/2) \,
\left\{
i \, \int^{+\infty}_0 \frac{d z}{\Delta_1^{\{i,j\}} \, (z^2+1)}
\right.
\notag \\
&\quad {}
\times
\left[
\frac{1}{\left(\xi^{\nu}-i \, \lambda \right)^{1/2}}
-
\frac{1}{\left(\xi^{\nu}- \Delta_1^{\{i,j\}} \, (1+z^2) \right)^{1/2}}
\right]
\notag \\
&\quad {}
+ \int^{+\infty}_1 \frac{d z}{\Delta_1^{\{i,j\}} \, (z^2-1)}
\notag \\
&\quad {}
\times
\left.
\left[
\frac{1}{\left(\xi^{\nu}-i \, \lambda \right)^{1/2}}
-
\frac{1}{\left(\xi^{\nu}+ \Delta_1^{\{i,j\}} \, (z^2-1) \right)^{1/2}}
\right]
\right\}
\label{eqisigir4p}
\end{align}
\end{itemize}
Here again, the $\xi$ integration will be of the type (\ref{eqdeffuncj2}) or (\ref{eqdeffuncj7})
(cf.\ appendix (\ref{appendJ})).
Let us go through the different cases according the signs of
$\Im(\Delta_3)$ and $\Im(\Delta_1^{\{i,j\}})$.
\subsubsection{$\Im(\Delta_3) > 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$}\label{531}
We proceed along the two first steps described in subsubsec. \ref{subsubcase1}. We borrow the result obtained in eq.~(\ref{eqlijkir12}), set $P_{ijk} = -R_{ij}$
and make the following change of variables $x = y + (1-y) \, v$.
The quantity $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)$ reads now:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0) &= - \frac{F(\varepsilon)}{4 \, R_{ij}} \,
\int^1_0 \frac{d v}{v} \,
\left[
\frac{1}{[ v \, R_{ij} +(1-v) \, T \, - i \, \lambda]^{1+\varepsilon}}
-
\frac{1}{[(1-v) \, T \, - i \, \lambda]^{1+\varepsilon}}
\right]
\notag\\
& \quad {}\quad {}\quad {} \quad {}
\times
\int^1_0 d y \, y^{-1/2} \, (1-y)^{-1-\varepsilon}
\label{eqisigir8p}
\end{align}
The integrals over $y$ and $v$ are unnested and are computed easily using eqs.~(\ref{thirdinty0}), (\ref{thirdinty}) and (\ref{eqsecondontv3}).
Notice that $\Im(v \, R_{ij} +(1-v) \, T \, - i \, \lambda)$ never changes its sign when $v$ spans $[0,1]$, this is obviously true in the real mass case and due to the fact that $\Im(T)$ and $\Im(R_{ij})$ have a same sign imaginary part in the complex mass case.
Inserting the explicit results for the $y$ and $v$ integrals,
we get:
\begin{eqnarray}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)
& = &
\frac{1}{2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1- 2 \, \varepsilon)} \,
\frac{1}{R_{ij} \, T} \,
\nonumber \\
& & {} \times
\left[
\frac{1}{\varepsilon^2} \, (2 \, R_{ij} - i \, \lambda)^{-\varepsilon}
+
\mbox{Li}_2 \left( \frac{T-R_{ij}}{T - i \, \lambda} \right)
- \frac{\pi^2}{6}
\right]
\label{eqdefnl6}
\end{eqnarray}
Equation (\ref{eqdefnl6}) displays explicitly the singularity of $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)$
when $\varepsilon \to 0$. In eq. (\ref{eqdefnl6}) we use the property $\mbox{Li}_2(z) + \mbox{Li}_2(1-z) = \pi^2/6 - \ln(z) \, \ln(1-z)$
to obtain the following alternative form
suitable for further comparisons:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)
&=
\frac{1}{2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1- 2 \, \varepsilon)} \,
\frac{1}{R_{ij} \, T} \,
\nonumber \\
&\quad {} \times
\left\{
\frac{1}{\varepsilon^2} \, (2 \, R_{ij} - i \lambda)^{-\varepsilon}
- \mbox{Li}_2 \left( \frac{R_{ij}- i \lambda}{T- i \lambda} \right)
\right.
\notag\\
& \quad {} \quad {}
\left.
-
\left[
\ln \left(R_{ij}- i \lambda \right) - \ln \left( T- i \lambda \right)
\right] \, \ln \left( \frac{T - R_{ij}}{T- i \lambda} \right)
\right\}
\label{eqdefnl6bis}
\end{align}
\subsubsection{$\Im(\Delta_3) > 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$}
The starting point is eq.~(\ref{eqlijkir160}) with $P_{ijk} = - R_{ij}$.
In the first line of eq. (\ref{eqlijkir160}) we rescale $y = x \, u^2$ so
that the double integral factorises into a product of two unnested integrals
over $u$ and over $x$.
The integrals of the second and third lines of eq. (\ref{eqlijkir160}) yield no
divergences when $\varepsilon \rightarrow 0$, we thus take $\varepsilon=0$ in
them. We then make the following change of variables:
$x = y - (y-1) \, v$ in the penultimate line, and $x = y -(1-y) \, v$ in the
last line. We obtain:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0) \notag \\
&= \frac{F(\varepsilon)}{4 \, R_{ij}} \,
\left\{
i \, T^{-1-\varepsilon} \, \int_0^{+\infty} \frac{2 \, du}{1+u^2} \,
\int_{0}^1 \, \frac{dx}{\sqrt{x}} \, (1-x)^{-1-\varepsilon}
\right.
\notag \\
&\quad {}
+
\quad {}
\int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \int_0^{+\infty}
\frac{dx}{(y+x) \; ( (x+y) \, R_{ij} - T \, (1+x) )}
\notag \\
&\quad {}
+ i \, \int_0^{+\infty} \frac{dy}{\sqrt{y}} \,
\int_1^{+\infty} \frac{dx}{(y+x) \; ( (x+y) \, R_{ij} + T \, (1-x) )}
\label{eqdefnl92} \\
&\quad {}
+ \int_1^{+\infty} \frac{dy}{\sqrt{y}} \, (y-1)^{-1-\varepsilon} \,
\int_1^{\frac{y}{y-1}} \, \frac{dv}{v} \,
\left[
T^{-1-\varepsilon} \, (v-1)^{-1-\varepsilon}
-
\left( - v \, R_{ij} - T \, (1-v) \right)^{-1-\varepsilon}
\right]
\notag \\
&\quad {}
\left.
+ \int_0^{1} \frac{dy}{\sqrt{y}} \, (1-y)^{-1-\varepsilon} \,
\int_0^{\, \frac{y}{1-y}} \frac{dv}{v} \,
\left[
T^{-1-\varepsilon} \, (1+v)^{-1-\varepsilon}
-
\left( - v \, R_{ij} + T \, (1+v) \right)^{-1-\varepsilon}
\right]
\right\} \notag
\end{align}
Let us compute the different terms. Using eq.~(\ref{thirdinty0}), the first integral is readily given by:
\begin{align}
\int_0^{+\infty} \frac{2 \, du}{1+u^2} \,
\int_{0}^1 \, \frac{dx}{\sqrt{x}} \, (1-x)^{-1-\varepsilon}
&=
\pi \, B \left( \frac{1}{2}, - \, \varepsilon \right)
\label{eqfirstintyv}
\end{align}
In the second and third lines of eq. (\ref{eqdefnl92}) the $x$ integration is
easily performed after a partial fraction decomposition on the $x$ variable.
We get:
\begin{align}
\hspace{2em}&\hspace{-2em}
\int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \int_0^{+\infty}
\frac{dx}{(y+x) \; ( (x+y) \, R_{ij} - T \, (1+x) )}
\notag \\
& \quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
= \frac{1}{T} \, \int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \frac{1}{y-1}
\left[ \ln \left( \frac{y \, R_{ij} -T}{R_{ij} - T} \right) - \ln(y) \right]
\label{eqsecondintyv} \\
\hspace{2em}&\hspace{-2em}
\int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \int_1^{+\infty}
\frac{dx}{(y+x) \; ( (x+y) \, R_{ij} + T \, (1-x) )}
\notag \\
& \quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
= \frac{1}{T} \, \int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \frac{1}{y+1} \,
\ln \left( \frac{R_{ij}}{R_{ij} - T} \right)
\label{eqthirdintyv}
\end{align}
We write the last two integrals of eq. (\ref{eqdefnl92}) as:
\begin{align}
\hspace{2em}&\hspace{-2em}
\int_1^{+\infty} \frac{dy}{\sqrt{y}} \, (y-1)^{-1-\varepsilon} \,
\int_1^{\frac{y}{y-1}} \, \frac{dv}{v} \,
\left[
T^{-1-\varepsilon} \, (v-1)^{-1-\varepsilon}
-
\left( - v \, R_{ij} - T \, (1-v) \right)^{-1-\varepsilon}
\right]
\notag \\
&= \int_1^{+\infty} \frac{dy}{\sqrt{y}} \, (y-1)^{-1-\varepsilon} \,
\left[ E_1(y)-E_1(1^{+})\right] \,
\; + \; E_1(1^{+}) \,
B \left( \frac{1}{2} + \varepsilon, - \, \varepsilon \right)
\label{eqfourthintyv} \\
\hspace{2em}&\hspace{-2em}
\int_0^{1} \frac{dy}{\sqrt{y}} \, (1-y)^{-1-\varepsilon} \,
\int_0^{\, \frac{y}{1-y}} \frac{dv}{v} \,
\left[
T^{-1-\varepsilon} \, (1+v)^{-1-\varepsilon}
-
\left( - v \, R_{ij} + T \, (1+v) \right)^{-1-\varepsilon}
\right]
\notag \\
&= \int_0^{1} \frac{dy}{\sqrt{y}} \, (1-y)^{-1-\varepsilon} \,
\left[ E_2(y)-E_2(1^{-}) \right]
\; + \; E_2(1^{-}) \, B\left( \frac{1}{2}, - \, \varepsilon \right)
\label{eqfifthintyv}
\end{align}
with:
\begin{align}
E_1(y)
&= \int_1^{\frac{y}{y-1}} \, \frac{dv}{v} \,
\left[
T^{-1-\varepsilon} \, (v-1)^{-1-\varepsilon}
-
\left( - v \, R_{ij} - T \, (1-v) \right)^{-1-\varepsilon}
\right]
\label{eqdeffunce1} \\
E_2(y)
&= \int_0^{\, \frac{y}{1-y}} \frac{dv}{v} \,
\left[
T^{-1-\varepsilon} \, (1+v)^{-1-\varepsilon}
-
\left( - v \, R_{ij} + T \, (1+v) \right)^{-1-\varepsilon}
\right]
\label{eqdeffunce2}
\end{align}
When $y \to 1^{+}$, the upper bound of the integral
defining the function $E_1(y)$ goes to $+ \infty$ and likewise for $E_2(y)$
when $y \to 1^{-}$. The quantities
$E_1(y)-E_1(1^{+})$ and $E_2(y)-E_2(1^{-})$ are given by integrals between $y/(y-1)$
and $+\infty$ and between $y/(1-y)$ and $+\infty$ respectively.
As, in $E_1(y)-E_1(1^{+})$, $y$ is greater than $1$ and so is the
lower bound $y/(y-1)$, the singular support $v=1$ of the distribution
$(v-1)^{-1-\varepsilon}$ lies outside the range of integration thus the first
term of the r.h.s. of eq. (\ref{eqdeffunce1}) can be taken at $\varepsilon=0$.
In $E_2(y)-E_2(1^{-})$, the integrand is non singular either and the first term of
the r.h.s. of eq. (\ref{eqdeffunce2}) can also be taken at $\varepsilon=0$.
These two terms give:
\begin{align}
\int_1^{+\infty} \frac{dy}{\sqrt{y}} \, (y-1)^{-1-\varepsilon} \,
\left[ E_1(y)-E_1(1^{+}) \right]
&= - \, \frac{1}{T} \, \int_1^{+\infty} \frac{dy}{\sqrt{y}} \, \frac{1}{y-1} \,
\ln \left( \frac{y \, R_{ij} - T}{R_{ij} - T} \right)
\label{eqfourthintyv1t} \\
\int_0^{1} \frac{dy}{\sqrt{y}} \, (1-y)^{-1-\varepsilon} \,
\left[ E_2(y)-E_2(1^{-}) \right]
&= - \, \frac{1}{T} \, \int_0^1 \frac{dy}{\sqrt{y}} \, \frac{1}{y-1} \,
\ln \left( \frac{y \, R_{ij} - T}{R_{ij} - T} \right)
\label{eqfifthintyv1t}
\end{align}
The combined r.h.s. of eqs. (\ref{eqfourthintyv1t}) and (\ref{eqfifthintyv1t})
cancel against the first term in eq. (\ref{eqsecondintyv}).
The expressions of $E_1(1^{+})$ and $E_2(1^{-})$ are computed using eqs.
(\ref{eqresulK3}) and ( \ref{eqresulK4}):
\begin{align}
E_1(1^{+})
&= - \, K_4(T-R_{ij},-T)
\notag \\
&= \frac{1}{T} \,
\left\{
- \, \frac{1}{\varepsilon} \, \left( -R_{ij} \right)^{-\varepsilon}
+ \ln (T) - \ln (T-R_{ij})
\right.
\notag \\
&\quad {} \quad {} \quad {}
\left.
+ \, \varepsilon \,
\left[
\mbox{Li}_2 \left( \frac{R_{ij}}{T} \right)
+ \ln \left( -R_{ij} \right)\, \ln \left( \frac{T-R_{ij}}{T} \right)
\right]
\right\}
\label{eqdefe3de1}\\
E_2(1^{-})
&= - \, K_3(T-R_{ij},T)
\notag \\
&= \frac{1}{T} \, \ln \left( \frac{T-R_{ij}}{T} \right) \,
\left[ 1 - \varepsilon \, \ln(T) \right]
\label{eqdefe2de1}
\end{align}
Putting everything together, we get for $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)$:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0) \notag \\
&= \frac{F(\varepsilon)}{4 \, R_{ij} \, T} \,
\left\{
i \, \pi \, B \left( \frac{1}{2}, - \, \varepsilon \right) \,
\left( 1 - \varepsilon \ln(T) \right)
\right.
\notag\\
& \quad {} \quad {} \quad {} \quad {} \quad {}
- \int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \frac{1}{y-1} \, \ln(y)
+ i \, \int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \frac{1}{y+1} \,
\ln \left( \frac{R_{ij}}{R_{ij}-T} \right)
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {}
+ B \left( \frac{1}{2}, - \, \varepsilon \right) \,
\left[
\ln \left( \frac{T - R_{ij}}{T} \right) \,
\left( 1 - \varepsilon \, \ln(T) \right)
\right]
\notag \\
&\quad {} \quad {} \quad {} \quad {} \quad {}
+ B \left( \frac{1}{2} + \varepsilon, - \, \varepsilon \right)
\left[
- \, \frac{1}{\varepsilon} \left( -R_{ij} \right)^{-\varepsilon}
+ \ln (T) - \ln (T-R_{ij})
\right.
\notag \\
&\quad \quad \quad \quad \quad \quad \quad
\quad \quad \quad \quad \qquad {}
+ \left.
\left.
\varepsilon
\left[
\mbox{Li}_2\left( \frac{R_{ij}}{T} \right)
+ \ln \left( -R_{ij} \right)\, \ln \left( \frac{T-R_{ij}}{T} \right)
\right] \right]
\right\}
\label{eqdefnl93}
\end{align}
Extracting the Euler Beta functions from eqs. (\ref{secondinty0}), (\ref{secondinty}) and (\ref{thirdinty0}), (\ref{thirdinty}) and using the fact that
$\ln(- \, R_{ij}) = \ln(R_{ij}) - i \, \pi$,
eq. (\ref{eqdefnl93}) can be cast in the following form:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0) \notag \\
&= \frac{1}{2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1-2 \, \varepsilon)} \,
\frac{1}{R_{ij} \, T} \,
\notag\\
& \quad {} \times
\left\{
\frac{1}{\varepsilon^2} \, \left( 2 \, R_{ij} \right)^{-\varepsilon}
- \mbox{Li}_2\left( \frac{R_{ij}}{T} \right)
- \left[ \ln \left(R_{ij}\right) - \ln \left( T \right) \right] \,
\ln \left( \frac{T - R_{ij}}{T} \right)
\right\}
\label{eqdefnl95}
\end{align}
Eqs.(\ref{eqdefnl95}) and (\ref{eqdefnl6bis}) have the same analytic
expression.
\subsubsection{$\Im(\Delta_3) < 0$, $\Im(\Delta_1^{\{i,j\}}) > 0$}
We start with eq.~(\ref{eqeqlijkir170}) with $P_{ijk} = - R_{ij}$.
We then set $x = y + (1+y) \, v$ in the first integral of eq.
(\ref{eqeqlijkir170}), $x = y + (y-1) \, v$ in the second integral and
$x = y + (1-y) \, v$ in the third one and we get:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)
&= \frac{F(\varepsilon)}{4 \, R_{ij}} \;
\left\{
i \, \text{e}^{-i \, \pi \, \varepsilon} \, \int^{+\infty}_0 \frac{d y}{\sqrt{y}} \,
(1+y)^{-1-\varepsilon} \,
\right.
\notag\\
& \quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
\times
\int^{+\infty}_0 \frac{d v}{v} \,
\left[
\frac{1}{[ v \, R_{ij} - (1+v) \, T]^{1+\varepsilon}}
-
\frac{1}{[- (1+v) \, T]^{1+\varepsilon}}
\right]
\notag \\
&\quad {} \quad {}\quad {}\quad {}\quad {}\quad {}
+ \quad {}
\int^{+\infty}_1 \frac{d y}{\sqrt{y}} \, (y-1)^{-1-\varepsilon} \,
\notag\\
& \quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
\times
\int^{+\infty}_0 \frac{d v}{v} \,
\left[
\frac{1}{[ v \, R_{ij} - (1+v) \, T]^{1+\varepsilon}}
-
\frac{1}{[- (1+v) \, T]^{1+\varepsilon}}
\right]
\notag \\
& \quad {} \quad {}\quad {}\quad {}\quad {}\quad {}
+ \quad {}
\int^{1}_0 \frac{d y}{\sqrt{y}} \, (1-y)^{-1-\varepsilon} \,
\notag\\
& \quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
\left.
\times
\int^{+\infty}_1 \frac{d v}{v} \,
\left[
\frac{1}{[ v \, R_{ij} + (1-v) \, T ]^{1+\varepsilon}}
-
\frac{1}{[(1-v) \, T]^{1+\varepsilon}}
\right]
\right\}
\label{eqisigir8s}
\end{align}
Here again the integrals over $y$ and $v$ are unnested.
Since the signs of
$\Im(R_{ij})$ and $\Im(T)$ are mutually opposite, let us note that
the imaginary parts of each of the terms raised to the power
$1+\varepsilon$ in denominators in the $v$ integrals remain constant
over the corresponding ranges of integration over $v$.
The first two integrals on $v$ of eq.~(\ref{eqisigir8s}) are given by eq.~(\ref{eqresulK3})
with $A = R_{ij} - T$ and $B = - T$ while the last one is given by eq.~(\ref{eqresulK4}) with
$A^{\prime} = R_{ij} - T$ and $B^{\prime} = T$. As for the $y$ integration, they can be read from
eqs.~(\ref{firstinty0}) to (\ref{thirdinty}).
All ingredients combine into:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0) \notag \\
&=
\frac{1}{2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1- 2 \, \varepsilon)} \,
\frac{1}{R_{ij} \, T} \,
\nonumber \\
&\quad {} \times
\left\{
\frac{1}{\varepsilon^2} \, (2 \, R_{ij})^{-\varepsilon}
- \mbox{Li}_2 \left( \frac{R_{ij}}{T} \right) -
\left[
\ln \left( R_{ij} \right) - \ln \left( - \, T \right) \, - i \, \pi
\right] \,
\ln \left( \frac{T-R_{ij}}{T} \right)
\right\}
\label{eqdefnl8}
\end{align}
In the present case $\Im(- \, R_{ij})$ and $\Im(T)> 0$ so that
$\ln \left( - \,T \right) = \ln \left( T \right) - i \, \pi$
thus eq. (\ref{eqdefnl8}) can be rewritten:
\begin{align}
L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)
&=
\frac{1}{2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}{\Gamma(1- 2 \, \varepsilon)} \,
\frac{1}{R_{ij} \, T} \,
\label{eqdefnl8bis} \\
&\quad {} \times
\left\{
\frac{1}{\varepsilon^2} \, (2 \, R_{ij})^{-\varepsilon}
- \mbox{Li}_2 \left( \frac{R_{ij}}{T} \right) -
\left[ \ln \left( R_{ij} \right) - \ln \left( T \right) \right] \,
\ln \left( \frac{T - R_{ij}}{T} \right)
\right\}
\nonumber
\end{align}
i.e. again the same analytic form as the previous two cases, cf.
eqs. (\ref{eqdefnl6bis}) and (\ref{eqdefnl95}).
\subsubsection{$\Im(\Delta_3) < 0$, $\Im(\Delta_1^{\{i,j\}}) < 0$}
The starting point is now eq.~(\ref{eqlijkir180}) with $P_{ijk} = - R_{ij}$.
In the first two integrals we rescale $y = x \, u^2$ to factorise the double integral into a product of two unnested integrals over $u$ and over $x$.
Using results given in eqs.~(\ref{firstinty0}) and (\ref{secondinty0}), these unnested integrals are readily performed yielding:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0) \notag \\
&= \frac{F(\varepsilon)}{4 \, R_{ij}} \,
\left\{
i \, \pi \, \left( -T \right)^{-1-\varepsilon}
\left[
- i \, \text{e}^{- i \, \pi \, \varepsilon} \,
B \left( \frac{1}{2}, \frac{1}{2}+ \varepsilon \right)
- \,
B \left( \frac{1}{2}+ \varepsilon, - \, \varepsilon \right)
\right]
\right.
\notag \\
&\quad {} \quad {}
- i \, \int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \int_0^1 \frac{dx}{y+x}
\left( R_{ij} \, (x+y) + T \, (1-x) \right)^{-1-\varepsilon}
\notag \\
&\quad {}\quad {}
+ i \, \text{e}^{- i \, \pi \, \varepsilon} \,
\int_0^{+\infty} \frac{dy}{\sqrt{y}} \int_0^y \frac{dx}{y-x}
\notag \\
&\qquad \qquad \qquad {} \times
\left[
\left( - R_{ij} \, (y-x) - T \, (1+x) \right)^{-1-\varepsilon}
- \left( -T \right)^{-1-\varepsilon} \, (1+x)^{-1-\varepsilon} \right]
\notag \\
&\quad {}\quad {}
+ \int_1^{+\infty} \frac{dy}{\sqrt{y}} \int_1^y \frac{dx}{y-x}
\notag \\
&\qquad \qquad \qquad {} \times
\left.
\left[
\left( - R_{ij} \, (y-x) + T \, (1-x) \right)^{-1-\varepsilon}
-
\left( -T \right)^{-1-\varepsilon} \, (x-1)^{-1-\varepsilon}
\right]
\vphantom{\frac{1}{2}} \right\}
\label{eqdefnl101}
\end{align}
In the three remaining integrals in eq. (\ref{eqdefnl101}), the first two ones
remain finite when $\varepsilon \to 0$ and are thus computed in this limit,
which yields:
\begin{align}
&\int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \int_0^1 \frac{dx}{y+x}
\left( R_{ij} \, (x+y) + T \, (1-x) \right)^{-1}
\notag\\
& \quad {} \quad {} \quad {} \quad {}
=
\frac{1}{T} \, \int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \frac{1}{1+y} \,
\left[ \ln\left( \frac{y \, R_{ij} + T}{R_{ij}} \right) - \ln(y) \right]
&
\label{i1}\\
& \int_0^{+\infty} \frac{dy}{\sqrt{y}} \int_0^y \frac{dx}{y-x}
\left[
\left(
- R_{ij} \, (y-x) - T \, (1+x) \right)^{-1-\varepsilon}
-
\left( -T \right)^{-1} \, (1+x)^{-1}
\right]
\notag\\
& \quad {} \quad {} \quad {} \quad {}
=
\frac{1}{T} \, \int_0^{+\infty} \frac{dy}{\sqrt{y}} \, \frac{1}{1+y} \,
\ln \left( \frac{y \, R_{ij} + T}{T} \right)
\label{i2}
\end{align}
Making the change of variable $u=1/y$ we readily see that
\[
\int_{0}^{+\infty} \frac{dy}{\sqrt{y}} \, \frac{\ln(y)}{1+y}
\, = \,
- \int_{0}^{+\infty} \frac{du}{\sqrt{u}} \, \frac{\ln(u)}{1+u}
\, = \, 0
\]
the combination $\{- \, i \times (\ref{i1}) + \, i \times (\ref{i2})\}$ thus
gives $i \, \pi \, \ln(R_{ij}/T)$.
In the
third integral, we make the change of variable
$x = y - (y-1) \, v$, the third integral thus becomes
\[
\int_1^{+\infty} \frac{dy}{\sqrt{y}} \, (y-1)^{-1-\varepsilon} \,
\int_0^{1} \frac{dv}{v}
\left[
\left( -v \, R_{ij} + T \, (v-1) \right)^{-1-\varepsilon}
-
\left( -T \right)^{-1-\varepsilon} \, (1-v)^{-1-\varepsilon}
\right]
\]
The $v$ integration is performed using the result of $K_2(- R_{ij}, - T)$ (cf.\ eq.\
(\ref{eqsecondontv3})). After some algebra, eq. (\ref{eqdefnl101}) thus reads:
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0) \notag \\
&= \frac{1}{2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1 - \varepsilon)}{\Gamma(1 - 2 \, \varepsilon)} \,
\frac{1}{R_{ij} \, T} \,
\left[
\frac{1}{\varepsilon^2} \, \left( 2 \, R_{ij} \right)^{-\varepsilon}
+ \mbox{Li}_2\left( \frac{T - R_{ij}}{T} \right) - \frac{\pi^2}{6}
\right]
\label{eqdefnl103}
\end{align}
Eq. (\ref{eqdefnl103}) is identical to eq. (\ref{eqdefnl6}); since
$\Im(R_{ij})$ and $\Im(T)$ have the same sign as in subsubsec. \ref{531},
eq. (\ref{eqdefnl103}) can be recast into a form identical to eq.
(\ref{eqdefnl6bis}):
\begin{align}
\hspace{2em}&\hspace{-2em}L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0) \notag \\
&=
\frac{1}{2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1 - \varepsilon)}{\Gamma(1 - 2 \, \varepsilon)} \,
\frac{1}{R_{ij} \, T} \,
\notag\\
& \quad {} \times
\left\{
\frac{1}{\varepsilon^2} \, \left( 2 \, R_{ij} \right)^{-\varepsilon}
- \mbox{Li}_2 \left( \frac{R_{ij}}{T} \right)
- \left[
\ln \left( R_{ij} \right) - \ln \left( T \right)
\right] \,
\ln \left( \frac{T-R_{ij}}{T} \right)
\right\}
\label{eqdefnl103bis}
\end{align}
In summary in all four cases $L_4^n(\Delta_3,0,\Delta_1^{\{i,j\}},0)$ takes the same analytical form.
Compared with the multiplicity of forms met in the general complex mass case,
and still with the diverse cases met in subsec. \ref{subsect52}, this
simplification come from the coalescence of the pole and branch points all at
the value 1 which is the end-point singularity causing the appearance of the
soft and collinear singularity in all four cases.
\section{Introduction}\label{intro}
This article is the third of a triptych. The first one \cite{paper1} presented
a method exploiting a Stokes-type identity to compute ``generalised'' (in the sense of the underlying kinematics) one-loop three- and
four-point scalar integrals for the real mass case. The second article \cite{paper2}
extended the results of the first paper to the case of general complex masses.
The present article widens the results of \cite{paper1} and \cite{paper2} to
the case where some internal masses are vanishing leading to infrared divergences.
We refer the reader to ref. \cite{letter} for more details on the motivation of this work.
\vspace{0.3cm}
\noindent
The scalar Feynman integrals for one-loop three- and four-point functions are
all known and have been compiled in a useful article \cite{Ellis:2007qk}.
This article relies mainly on the results of other publications, especially the important
work of Beenakker and Denner \cite{Beenakker:1988jr}.
Let us mention also ref. \cite{Denner:2010tr}
which provides a complete set of results for soft and/or collinear divergent four-point functions using different kind of IR regulators.
The purpose of the present
article is to extend these results for more general kinematics beyond those relevant
for collider processes at the one-loop order. Note that despite the fact that some
internal masses may vanish, the others can be real or complex and we treat both
cases in this article. The soft and collinear divergences are dealt with using
dimensional regularisation, $n = 4 - 2 \, \varepsilon$, and doing an $\varepsilon$ expansion.
\vspace{0.3cm}
\noindent
The outline of this article follows closely the one of our preceding articles \cite{paper1} and \cite{paper2}.
We start by considering the three-point function $I_{3}^{n}$ in a space-time dimension
shifted by a small amount from $4$ to $n$. The kinematics leading to infrared divergences is discussed.
It is considered as a warm-up for sec. \ref{sectfourpointir}.
We successively present two variants of the method. The simplest variant,
labelled ``direct way'', is presented in subsec. \ref{dirway}. It is
well suited for the three-point function, but cannot be extended to the case of the four-point function.
Then, in subsec.~\ref{exp_exemp_ir}, practical implementation of the
results of the preceding subsection is discussed and some explicit examples are computed and compared to \cite{Ellis:2007qk}.
In subsec. \ref{indirway} we present an alternative
coined ``indirect way'' easily applicable to the four-point case which is the subject of sec.~\ref{sectfourpointir}.
We first explain, in subsec.~\ref{compI4n} how to extend the calculation of $I_4^4$ developed
in \cite{paper1} to the case where the infrared divergences are regulated in $n$-dimension.
The net result is that the four-point scalar integral can be decomposed on sectors labelled
by three indices and a three dimensional integral over the first octant of $\mathds{R}^3$ is
associated to each sector. Then, two cases are distinguished depending on the sectors.
In the first case, presented in~\ref{subsect52},
the determinant of the one-pinched kinematical matrix $\text{$\cal S$}$ vanishes but not the internal
mass associated to this sector: this case is met when a soft divergence appears. In the
second case, presented in subsec. \ref{casDelta20Dt0}, both the determinant of the
one-pinched $\text{$\cal S$}$ matrix and the internal mass associated to this sector vanish: this case
is met when a collinear or a soft and collinear divergence shows up. In sec.~\ref{examplefourpoint},
the infrared divergent part of the scalar four-point integral is shown to be proportional
to a three-point scalar integral as it should be. Some explicit examples are given and
compared to the results found in the literature. We then conclude.
Various appendices gather a number of utilities removed from the main text to facilitate its reading.
Accordingly, in appendix~\ref{appendJ}, we complete appendix~\ref{P1-appendJ} of \cite{paper1} and
appendix~\ref{P2-appendJ} of \cite{paper2} by giving a missing case where the power of
the integration variable is not an integer as required by
dimensional regularisation. Appendix~\ref{calculJx1x2} shows how to compute an integral
appearing in the three-point case in closed form.
Appendix \ref{herba} collects a bunch of integrals required to compute the three- and
four-point functions having soft and/or collinear divergences in the case of general complex masses.
Then, appendix~\ref{direcway3pIR}
goes through the examples given in subsec. \ref{exp_exemp_ir} and explains in detail how
the results obtained in the latter subsection can be found again from those derived in
the ``indirect way'' case. Appendix~\ref{appF} provides the way to compute the last integration
in closed form in terms of dilogarithms for the case of infrared divergent integrals.
It complements the appendix~\ref{P1-appF} of \cite{paper1} and appendix~\ref{P2-appF} of \cite{paper2}.
Lastly, appendix~\ref{ir-lambda} proves a tricky point used in sec.~\ref{sectfourpointir}: for
the real mass case, the sign of the vanishing imaginary part of the denominator in the last integral can be safely changed.
\section{Summary and outlook}
In this article we presented an extension of a novel approach developed in companion
articles \cite{paper1} and \cite{paper2} to the case of vanishing internal masses
involving soft and/or collinear divergences.
For this latter case, the method remains very similar to the massive cases: the three- and
four-point functions are split into ``sectors'' whose coefficients are expressed
in terms of algebraic kinematical invariants involved in reduction algorithms.
Each ``sector'' may diverge or not when the IR regulator is sent to zero yielding
to a simple decision tree to compute the relevant integrals. This avoids the computation of
the numerous different integrals over Feynman parameters as it is usually done
in the literature. This extension also applies to general kinematics beyond the
one relevant for one-loop collider processes, offering a potential application to
the calculation of two-loop processes using one-loop (generalised) $N$-point
functions as building blocks as discussed in the introduction of \cite{paper1}.
\vspace{0.3cm}
\noindent
One drawback of the present method is the proliferation of dilogarithms in the expression of the four-point function
computed in closed form. This requires some extra work to be
better apprehended, in order to counteract it.
But as the method used hereby is the same as in the real mass case, up to
slight modifications, any solution found for the latter case can be applied in the infrared divergent case.
This issue will be addressed in a
future article.
\vspace{0.3cm}
\noindent
The last goal is to provide the generalised one-loop building blocks
entering as integrands in the computation of two-loop three- and four-point
functions by means of an extra numerical double integration.
In this respect, let us mention that the expansion around $\varepsilon = 0$ of the results
given in this article has been truncated in order to keep only the divergent and the constant terms.
This is sufficient for any one-loop computation but may be not enough for two-loop applications of the method.
This article contains already a lot of results, so the expansion around $\varepsilon = 0$ at the necessary orders is postponed to a future work.
\section*{In memoriam}
Various ideas and techniques used in this work were initiated by Prof. Shimizu
after a visit to LAPTh. He explained us his ideas about the numerical
computation of scalar two-loop three- and four-point functions, he shared his
notes partly in English, partly in Japanese with us and he encouraged us to
push this project forward. J.Ph. G. would like to thank Shimizu-sensei for
giving him a taste of the Japanese culture and for his kindness.
\section*{Acknowledgements}
We would like to thank P. Aurenche for his support along this project and for a careful reading of the manuscript.
\section{Three-point function with infrared divergences}\label{3point_ir}
\begin{figure}[h]
\centering
\parbox[c][43mm][t]{80mm}{\begin{fmfgraph*}(60,80)
\fmfleftn{i}{1} \fmfrightn{o}{1} \fmftopn{t}{1}
\fmf{fermion,label=$p_1$}{t1,v1}
\fmf{fermion,label=$p_2$}{i1,v2}
\fmf{fermion,label=$p_3$}{o1,v3}
\fmf{fermion,tension=0.5,label=$q_1$}{v1,v2}
\fmf{fermion,tension=0.5,label=$q_2$}{v2,v3}
\fmf{fermion,tension=0.5,label=$q_3$,label.side=right}{v3,v1}
\end{fmfgraph*}}
\caption{\footnotesize
The triangle picturing the one-loop three-point function.}
\label{fig1}
\end{figure}
\noindent
When some internal masses vanish,
divergences of collinear or soft origin appear and the approach shall be
revisited. We regularise these divergences
using dimensional re\-gularisation, shifting the dimension of the space time by a
small positive amount from 4 to $n = 4 - 2 \, \varepsilon$ with
$\varepsilon < 0$. After performing the loop momentum integral,
instead of eq. (\ref{P1-eqSTARTINGPOINT3}) of ref. \cite{paper1} we get\footnote{As in refs. \cite{paper1,paper2}, we assume that the elements of
the kinematic matrix $\text{$\cal S$}$ have been made dimensionless by an appropriate
rescaling.}:
\begin{equation}
I_3^n
=
- \Gamma(1+\varepsilon) \,
\int \prod_{i=1}^3 \, dz_i \,
\delta(1-\sum_{i=1}^3 z_i)\,
\left(
- \, \frac{1}{2} \, z^T \cdot \text{$\cal S$} \cdot z - i \, \lambda
\right) ^{-1-\varepsilon}
\label{eqdefi3n}
\end{equation}
To appropriately shift the power of the denominator in eq. (\ref{eqdefi3n})
so as to apply the Stokes identity (\ref{P1-eqDEFREL1}) of ref. \cite{paper1} as we did in the massive
case, we use the following modified integral representation instead of
identity (\ref{P1-eqFOND1-simple}) of the previous reference (cf. appendix~\ref{P1-ap2} of \cite{paper1}):
\[
\frac{1}{D^{1+\varepsilon}}
=
\frac{\nu}{B(2-1/\nu,1/\nu)} \;
\int^{+\infty}_{0} \, \frac{d \xi}{(D+\xi^{\nu})^2}
\]
with $\nu = 1/(1-\varepsilon)$. Instead of eq. (\ref{P1-eqI341}) of ref. \cite{paper1} we now get:
\begin{align}
I_3^n
&= - \, 2^{1+\varepsilon} \, \frac{\Gamma(1+\varepsilon)}{1-\varepsilon} \,
\frac{1}{B(1+\varepsilon,1-\varepsilon)} \,
\int^{+\infty}_0 d \xi \, \int_{\Sigma_{bc}}
\frac{dx_b \, dx_c}{(D^{(a)}(x_b,x_c) + \xi^{\nu} - i \, \lambda)^2}
\label{eqdefi3n1}
\end{align}
We otherwise proceed as in subsec. \ref{P1-sect3pstep1} of ref. \cite{paper1}.
The counterpart of eq. (\ref{P1-eqI345}) of the same reference now reads:
\begin{align}
I_3^n
&= 2^{\varepsilon} \, \frac{\Gamma(1+\varepsilon)}{1-\varepsilon} \,
\frac{1}{B(1+\varepsilon,1-\varepsilon)} \,
\notag\\
&\quad {} \times
\sum_{i \in S_3} \, \frac{\overline{b}_i}{\det(G)} \,
\int^{+\infty}_0 \frac{d \xi}{\Delta_2 - \xi^{\nu}+ i \, \lambda} \,
\int^1_0 \, \frac{dx}{D^{\{i\}(j)}(x)+ \xi^{\nu} - i \, \lambda}
\label{eqdefi3n2}
\end{align}
where $j \in S_3 \setminus \{i\}$ ($S_3 = \{1,2,3\}$). More precisely, we assume that $j$ is chosen to be $1 + (i \; \mbox{modulo} \; 3)$.
Similarly to what we did for the three-point function in the massive case,
one can also consider both a ``direct way" and an ``indirect way" in the IR
case. We first focus on the ``direct way" which provides a more
straightforward and synthetic
discussion of the various cases at hand. We then illustrate how these
cases are involved in a few examples. The ``indirect way" instead
leads to a cumbersome split-up discussion. Notwithstanding the latter has
its own interest.
The calculation of the four-point one-loop
integral relying on the approach described in this article proceeds
along the ``indirect way'' as we found no extension of the ``direct way'' approach in
this case.
In refs. \cite{Binoth:2005ff,Binoth:1999sp} it was
shown on general grounds using the decomposition\footnote{This decomposition has been discovered before and used for different purposes, see \cite{vanNeerven:1983vr,Kotikov:1991pm,Bern:1993kr,Tarasov:1996br}.}
\begin{equation}\label{decomp-golem}
\det{(\cals)} \, I_4^n(\text{$\cal S$})
=
\sum_{i=1}^{4} \overline{b}_{i} \, I_3^n(\text{$\cal S$}^{\{i\}}) - \det{(G)} \, (1 - 2 \, \varepsilon) \, I_4^{n+2}(\text{$\cal S$})
\end{equation}
that the infrared structure of any IR divergent four-point one-loop
integral is carried by IR divergent three-point one-loop functions resulting
from appropriate iterated pinchings.
Therefore the comparison of the IR structures
in both sides of eq. (\ref{decomp-golem}) proceeds most conveniently via a term
by term comparison using the three-point one-loop functions decomposed according
to the ``indirect way'' as well. In anticipation, we hereby give the key
ingredients to perform this comparison, as well as the general recombination
of these ``indirect way'' ingredients into the more compact expression obtained
from the ``direct way'', thereby checking their equivalence.
The extensive collection of expressions computed in closed form which enable
to perform detailed case-by-case comparisons is gathered in appendix
\ref{direcway3pIR} to lighten the presentation.
\subsection{Direct way}\label{dirway}
\noindent
Soft and/or collinear divergences are caused by some vanishing masses which
make $\det{(\cals)}$ vanish so that $\Delta_2 = 0$, whereas the other internal masses
may or may not vanish as well - and may even be complex.
We will keep the $- \, i \, \lambda$ prescription having in mind that
it is ineffective in the case of complex masses.
\vspace{0.3cm}
\noindent
Starting from eq. (\ref{eqdefi3n2}) and performing the $\xi$ integration using
eq. (\ref{eqmodifk}), we end up with:
\begin{align}
I_3^n
&= \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\sum_{i \in S_3} \, \frac{\overline{b}_i}{\det(G)} \,
\int^1_0 dx \,
\left( D^{\{i\}(j)}(x) - i \, \lambda \right)^{-1-\varepsilon}
\label{eqdirei3n1}
\end{align}
In the general case $D^{\{i\}(j)}(x)$ depends on two internal
masses squared $m_j^2$ and $m_k^2$ such that $m_j^2 = D^{\{i\}(j)}(0)/2 = \widetilde{D}_{ik}/2$ and $m_k^2 = D^{\{i\}(j)}(1)/2 = \widetilde{D}_{ij}/2$,
cf.\ sec.\ \ref{P1-sectthreepoint} of \cite{paper1}.
We introduced the label $k$ which is the only element of the complement of $\{i,j\}$ in $S_3$.
With our assumption on $j$, this implies that $k \equiv 1 + ((i+1) \; \mbox {modulo} \;3)$.
Let us focus on the function $W$ given by:
\begin{equation}
W\left(\detgj{i},\widetilde{D}_{ij}, \widetilde{D}_{ik}\right) = \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \, \int^1_0 dx \,
\left( D^{\{i\}(j)}(x) - i \, \lambda \right)^{-1-\varepsilon}
\label{eqdefwi0}
\end{equation}
We remind (cf.\ eqs.~(\ref{P1-Db}), (\ref{P1-Dc}) and (\ref{P1-Da}) of ref.\ \cite{paper1}):
\begin{equation}
D^{\{i\}(j)}(x)
=
G^{\{i\}(j)} \, x^2 - 2 \, V^{\{i\}(j)} \, x - C^{\{i\}(j)}
\label{eqremd1}
\end{equation}
with
\begin{align}
G^{\{i\}(j)} &= - \text{$\cal S$}_{kk} + 2 \, \text{$\cal S$}_{kj} - \text{$\cal S$}_{jj} = \detgj{i} \notag \\
V^{\{i\}(j)} &= \text{$\cal S$}_{kj} - \text{$\cal S$}_{jj} = \frac{1}{2} \left[ \detgj{i} - \widetilde{D}_{ij} + \widetilde{D}_{ik} \right] \label{eqremd2}\\
C^{\{i\}(j)} &= \text{$\cal S$}_{jj} = - \widetilde{D}_{ik} \notag
\end{align}
The Gram matrix $G^{\{i\}(j)}$ is built from the one-pinched $\text{$\cal S$}$ matrix $\text{$\cal S$}^{\{i\}}$, it is a real
matrix which depends only on a squared external momentum in the three-point case. Notice that, in this
case, $G^{\{i\}(j)}$ is a $1 \times 1$ matrix and $V^{\{i\}(j)}$ a one-dimensional vector, this explains
the notations\footnote{Let us remind that $\text{$\cal S$}_{jk} = \text{$\cal S$}^{\{i\}}_{jk}$ for $j,k \ne i$.} used in
eqs.~(\ref{eqremd1}) and (\ref{eqremd2}).
The knowledge of $\detgj{i}$, $\widetilde{D}_{ij}$ and $\widetilde{D}_{ik}$ fully determines the polynomial $D^{\{i\}(j)}(x)$.
These two internal masses may or may not vanish, hence three
cases to be considered.
\vspace{0.3cm}
\noindent
{\bf a) Neither $m_j^2$ nor $m_k^2$ vanishes}\\
We perform a Taylor expansion\footnote{Here and below, only the
terms in the $\varepsilon$-expansion providing the divergent and finite terms
in the limit $\varepsilon \to 0$ are kept.} of $W\left(\detgj{i},\widetilde{D}_{ij}, \widetilde{D}_{ik}\right)$ in $\varepsilon$:
\begin{align}
\hspace{2em}&\hspace{-2em}W\left(\detgj{i},\widetilde{D}_{ij}, \widetilde{D}_{ik}\right) \notag \\
&= \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \, \left[ \int^1_0 \frac{dx}{D^{\{i\}(j)}(x) - i \, \lambda}
-
\varepsilon
\int^1_0 dx
\frac{\ln \left( D^{\{i\}(j)}(x) - i \, \lambda \right)}
{D^{\{i\}(j)}(x) - i \, \lambda} \right]
\label{eqdirei3n2}
\end{align}
Let us note $x_1$ and $x_2$ the two roots of $D^{\{i\}(j)}(x) - i \lambda$,
given by, cf. eqs. (\ref{eqremd1}), (\ref{eqremd2}):
\begin{align}
x_{\underset{2}{1}}
&=
\frac{
\detgj{i} - \widetilde{D}_{ij} + \widetilde{D}_{ik}
\pm
\sqrt{ {\cal K}\left( \detgj{i},\widetilde{D}_{ij},\widetilde{D}_{ik} \right) + i \, \lambda \, S_G}
}{2 \, \detgj{i}}
\label{eqroot12}
\end{align}
where ${\cal K}$ is the K\"all\'en function:
\begin{equation}
{\cal K}(x,y,z) = x^2 + y^2 + z^2 - 2 \, x \, y - 2 \, x \, z - 2 \, y \, z
\label{eqkallenfunc}
\end{equation}
and $S_G = \mbox{sign}(\detgj{i})$. Then, we introduce $J(x_1,x_2)$ and
$K(x_1,x_2)$ defined by:
\begin{align}
K(x_1,x_2)
&= \int^1_0 dx \, \frac{1}{(x - x_1) \, (x - x_2)}
\label{eqcompk1} \\
J(x_1,x_2)
&= \int^1_0 dx \,
\frac{\ln\left( (x-x_1) \, (x-x_2) \right)}{(x - x_1) \, (x - x_2)}
\label{eqcompj1}
\end{align}
As it will become clear in the forthcoming paragraph on the origin of infrared
singularities, only
the case with $m_j^2$ and $m_k^2$ both real matters in practice, which makes
the explicit calculation of $J(x_1,x_2)$ somewhat simpler\footnote{With real
masses, $x_1$ and $x_2$ in
eq. (\ref{eqroot12}) have imaginary parts of opposite signs. This namely
simplifies splittings and recombinations of logarithms of ratios in the
explicit calculation of the function $J(x_1,x_2)$ computed in appendix
\ref{calculJx1x2}.}. The latter is provided in appendix \ref{calculJx1x2}.
The $x$ integration in the function $K(x_1,x_2)$ straightforwardly gives:
\begin{align}
K(x_1,x_2)
&= \frac{1}{x_1-x_2} \,
\left[
\ln \left( \frac{x_1-1}{x_1} \right)
-
\ln \left( \frac{x_2-1}{x_2} \right)
\right]
\label{eqcompk11}
\end{align}
Thus $W\left(\detgj{i},\widetilde{D}_{ij}, \widetilde{D}_{ik}\right)$ reads:
\begin{align}
\hspace{2em}&\hspace{-2em}W\left(\detgj{i},\widetilde{D}_{ij}, \widetilde{D}_{ik}\right) \notag \\
&=
\frac{1}{\varepsilon} \, \frac{\Gamma(1+\varepsilon)}{\detgj{i}}
\left\{
\left[
1 - \varepsilon \, \ln \left(\frac{\detgj{i}}{2} - i \, \lambda \right)
\right] \, K(x_1,x_2)
- \varepsilon \, J(x_1,x_2)
\right\}
\label{eqcompintlog1}
\end{align}
\vspace{0.3cm}
\noindent
{\bf b) One and only one of $m_j^2$ and $m_k^2$ vanishes}\\
Let us assume that the vanishing internal mass is $m_j^2$. $D^{\{i\}(j)}(x)$
becomes:
\begin{equation}
D^{\{i\}(j)}(x)
= x \, \left( G^{\{i\}(j)} \, x - 2 \, V^{\{i\}(j)} \right)
\label{eqnewquad1}
\end{equation}
From eqs.~(\ref{eqdefwi0}) and (\ref{eqremd2}), $W\left(\detgj{i},\widetilde{D}_{ij}, 0\right)$ is thus of the form
\begin{align}
W\left(\detgj{i},\widetilde{D}_{ij}, 0\right)
&= \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \, \int^1_0 dx \, x^{-1-\varepsilon} \, (a \, x + z)^{-1-\varepsilon}
\label{eqdefyint1}
\end{align}
where $a = \detgj{i}$ is real and
$z = - \detgj{i} + \widetilde{D}_{ij} - i \, \lambda$ is complex.
As $z$ and $a \, x + z$ have imaginary parts of the same sign
$(a \, x + z)^{-1-\varepsilon}$ can be split as follows:
\[
(a \, x + z)^{-1-\varepsilon}
=
z^{-1-\varepsilon} \, \left( 1 + \frac{a}{z} \, x \right)^{-1-\varepsilon}
\]
The r.h.s. of eq. (\ref{eqdefyint1}) involves the Gauss hypergeometric function
$_{2}F_{1}$:
\begin{equation}
W\left(\detgj{i},\widetilde{D}_{ij}, 0\right) =
- \, \frac{2^{\varepsilon}}{\varepsilon^2} \, \Gamma(1+\varepsilon) \, z^{-1-\varepsilon} \,
_{2}F_{1}
\left( 1+\varepsilon,-\varepsilon;1-\varepsilon;- \, \frac{a}{z} \right)
\label{eqdefyint2}
\end{equation}
We use the identity \cite{abramowitz}
\begin{align*}
_{2}F_{1}(a,b;c;w) &= \frac{\Gamma(c) \, \Gamma(c-a-b)}{\Gamma(c-a) \, \Gamma(c-b)} \; _{2}F_{1}(a,b;a+b-c+1;1-w) \\
&\quad {} + (1-w)^{c-a-b} \, \frac{\Gamma(c) \, \Gamma(a+b-c)}{\Gamma(a) \, \Gamma(b)} \; _{2}F_{1}(c-a,c-b;c-a-b+1;1-w)
\end{align*}
and the Pfaff identity
\[
_{2}F_{1}(a,b;c;w)
=
(1-w)^{-b} \, _{2}F_{1} \left( c-a,b;c;\frac{w}{w-1} \right)
\]
to rewrite:
\begin{align}
&_{2}F_{1}
\left( 1+\varepsilon,-\varepsilon;1-\varepsilon;- \, \frac{a}{z} \right)
\notag\\
&=
2 \, \frac{\Gamma^2(1-\varepsilon)}{\Gamma(1- 2 \, \varepsilon)} \,
\left( - \frac{a}{z} \right)^{\varepsilon}
- \left( \frac{a+z}{z} \right)^{-\varepsilon} \,
\left( - \frac{a}{z} \right)^{2 \, \varepsilon} \,
_{2}F_{1}
\left( - 2 \, \varepsilon, - \varepsilon; 1-\varepsilon; \frac{a+z}{a} \right)
\label{eqsplit2F1}
\end{align}
Performing a Taylor expansion in $\varepsilon$ we get:
\[
_{2}F_{1}
\left( - 2 \, \varepsilon, - \, \varepsilon ;1 - \varepsilon; \tau \right)
=
1 + 2 \, \varepsilon^2 \, \mbox{Li}_2(\tau)
\]
and splitting $\ln( (a+z)/z) = \ln(a+z) - \ln(z)$, we rewrite $W\left(\detgj{i},\widetilde{D}_{ij}, 0\right)$as:
\begin{align}
W\left(\detgj{i},\widetilde{D}_{ij}, 0\right)
&= \frac{2^{\varepsilon}}{\varepsilon^2} \, \Gamma(1+\varepsilon) \, \frac{1}{z} \,
\left\{
\left( a+z \right)^{-\varepsilon} \,
\left( - \frac{a}{z} \right)^{2 \, \varepsilon} \,
\left[ 1 + 2 \, \varepsilon^2 \, \mbox{Li}_2 \left( \frac{a+z}{a} \right) \right]
\right.
\notag \\
&\qquad \qquad \qquad \qquad {}
-
\left.
2 \; \frac{\Gamma^2(1-\varepsilon)}{\Gamma(1- 2 \, \varepsilon)} \,
\left( z \right)^{-\varepsilon} \,
\left( - \frac{a}{z} \right)^{\varepsilon}
\right\}
\label{eqdefyint3}
\end{align}
Making explicit $z = - \detgj{i} + \widetilde{D}_{ij} - i \, \lambda$,
$a+z = \widetilde{D}_{ij} - i \, \lambda$ we get:
\begin{align}
\hspace{2em}&\hspace{-2em}W\left(\detgj{i},\widetilde{D}_{ij}, 0 \right) \notag \\
&=
- \frac{2^{\varepsilon}}{\varepsilon^2} \, \Gamma(1+\varepsilon) \; \frac{1}{\detgj{i} - \widetilde{D}_{ij}} \notag \\
&\quad {} \times
\left\{
\left( \frac{\detgj{i}}{\detgj{i} - \widetilde{D}_{ij} + i \, \lambda } \right)^{2 \, \varepsilon} \, \, \left(\widetilde{D}_{ij} - i \, \lambda \right)^{-\varepsilon} \, \left[ 1 + 2 \, \varepsilon^2 \, \mbox{Li}_2 \left( \frac{\widetilde{D}_{ij} - i \, \lambda}{\detgj{i}} \right) \right]
\right. \notag \\
&\qquad \qquad {} - \left.
2 \, \frac{\Gamma^2(1-\varepsilon)}{\Gamma(1 - 2 \, \varepsilon)} \, \left( \frac{\detgj{i}}{\detgj{i} - \widetilde{D}_{ij} + i \, \lambda } \right)^{\varepsilon} \, \left( \widetilde{D}_{ij} - \detgj{i} - i \, \lambda \right)^{- \varepsilon}
\vphantom{\mbox{Li}_2 \left( \frac{- \text{$\cal S$}_{kk} - i \, \lambda}{G^{\{i\}(j)}_{kk}} \right)} \right\}
\label{eqcompwcasb4}
\end{align}
This formula is manifestly well-behaved as $\widetilde{D}_{ij} \rightarrow 0$ ($m_k^2 \rightarrow 0$) yet it is not
handy to expand around $\varepsilon=0$. A more practical alternative
may be obtained as follows. Firstly, we use the identities relating $\mbox{Li}_2(1-w)$,
$\mbox{Li}_2(w)$ and $\mbox{Li}_2(1/w)$ to change the argument of the $\mbox{Li}_2$
function, and the following relations:
\begin{align}
\ln \left( \frac{a}{z} \right) &= \ln \left( - \frac{a}{z} \right) - i \, \pi \, S(a \, z) \\
\ln \left( \frac{a+z}{a} \right) &= \ln \left( -\frac{a+z}{a} \right) + i \, \pi \, S(a \, z) \\
\ln \left( -\frac{a+z}{a} \right) &= \ln \left( \frac{a+z}{z} \right) - \ln \left( - \frac{a}{z} \right)
\label{eqrelalamormoilenoeud}
\end{align}
with
\begin{equation}
S(z) = \mbox{sign}\left( \Im(z) \right)
\label{eqdeffuncS0}
\end{equation}
so that the $\mbox{Li}_2$ function can be rewritten as:
\begin{align}
\mbox{Li}_2 \left( \frac{a+z}{a} \right) &= \mbox{Li}_2 \left( - \frac{a}{z} \right) - \frac{\pi^2}{6} - \frac{1}{2} \, \ln^2 \left( - \frac{a}{z} \right) + \ln \left( \frac{a+z}{z} \right) \, \ln \left( - \frac{a}{z} \right)
\label{eqchangdilog1}
\end{align}
Secondly, we Taylor expand around $\varepsilon=0$ the $(-a/z)$ terms in eq.
(\ref{eqdefyint3}). We thus get:
\begin{align}
W\left(\detgj{i},\widetilde{D}_{ij}, 0\right)
&= -\,\frac{2^{\varepsilon}}{\varepsilon} \, \frac{\Gamma(1+\varepsilon)}{z} \,
\left[
\frac{2}{\varepsilon} \, (z)^{-\varepsilon}
-
\frac{1}{\varepsilon} \, (a+z)^{-\varepsilon}
-
2 \, \varepsilon \, \mbox{Li}_2 \left( - \frac{a}{z} \right)
\right]
\label{eqdefyint6}
\end{align}
i.e. making explicit $z$ and $a$ in terms of $\detgj{i}$ and $\widetilde{D}_{ij}$:
\begin{align}
W\left(\detgj{i},\widetilde{D}_{ij}, 0\right)
&= \frac{1}{\varepsilon} \, \frac{\Gamma(1+\varepsilon)}{\detgj{i} - \widetilde{D}_{ij}} \notag \\
&\quad {} \times
\left\{
\frac{2}{\varepsilon} \,
\left[ \frac{1}{2} \, \left( \widetilde{D}_{ij} - \detgj{i} \right) - i \, \lambda \right]^{-\varepsilon} -
\frac{1}{\varepsilon} \, \left( \frac{\widetilde{D}_{ij}}{2} - i \, \lambda \right)^{-\varepsilon}
\right.
\notag \\
&\qquad \quad {} -
\left.
2 \, \varepsilon \,
\mbox{Li}_2
\left(
\frac{\detgj{i}}{\detgj{i} - \widetilde{D}_{ij} + i \, \lambda }
\right)
\right\}
\label{eqdirei3n3}
\end{align}
which is both well behaved when $\widetilde{D}_{ij} \rightarrow 0$ ($m_k^2 \rightarrow 0$) and more compact.
\vspace{0.3cm}
\noindent
{\bf c) Both $m_j^2$ and $m_k^2$ vanish}
\noindent
The function $D^{\{i\}(j)}(x)$ becomes:
\begin{equation}
D^{\{i\}(j)}(x) = - \, G^{\{i\}(j)} \, x \, (1-x)
\label{eqnewquad2}
\end{equation}
and we immediately get:
\begin{align}
W\left(\detgj{i},0,0\right)
&= -\frac{1}{\varepsilon^2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^2(1-\varepsilon)}
{\Gamma(1 - 2 \, \varepsilon)} \,
\left( - \, \frac{\detgj{i}}{2} - i \, \lambda \right)^{-1-\varepsilon}
\label{eqdirei3n4}
\end{align}
In the limit $\widetilde{D}_{ij} \rightarrow 0$ ($m_k^2 \rightarrow 0$),
eq.~(\ref{eqdirei3n3}) or eq.~(\ref{eqcompwcasb4}) smoothly becomes eq. (\ref{eqdirei3n4}) as expected.
\subsection{Practical implementation of the preceding cases and
explicit examples}\label{exp_exemp_ir}
The various cases reviewed above may or may not be involved
in a specific computation
because some coefficients weighing the $W\left(\detgj{i},\widetilde{D}_{ij},\widetilde{D}_{ik}\right)$ may vanish.
In particular, as seen on eq. (\ref{eqdirei3n1}),
when $\Delta_2=0$, the three-point function in
dimension $4 - 2 \, \varepsilon$ is
the sum of three two-point functions in
dimension $2 - 2 \, \varepsilon$
These
two-point functions correspond to the three distinct pinchings
of the
internal propagators of the three-point function. At first sight,
one should worry that some of these two-point functions in low dimensions
may badly diverge due to a threshold singularity which is however not present
in the three-point function!
For example, one of the pinchings of a three-point function having IR/collinear
singularities would lead to a two-point function with the external legs on the
mass shell of one of the propagators whereas the other propagator is massless.
This would lead to a polynomial $D^{\{i\}(j)}(x) \propto x^2$ or $(1-x)^2$.
Fortunately the corresponding $\overline{b}$ coefficients weighting
such pathological terms identically vanish, and the discussion which follows,
illustrated with examples, elucidates why it happens so.
Let us note $p_i^2 = s_i$ with $i=1,2,3$.
\vspace{0.3cm}
\noindent
{\bf 1.} A soft divergence occurs
when the kinematic matrix $\text{$\cal S$}$ has a vanishing line (and corresponding
column). This happens whenever a massless propagator connects
two vertices in which enter external momenta on the
mass shells of the two other propagators.
As the external momenta are real this case can occur
only when the non vanishing internal masses are real.
Let us assume, cf. fig. 1, that the internal mass squared $m_1^2$ vanishes
whereas the external four-momenta $p_1$ and $p_2$ satisfy the mass shell
conditions $s_1=m_3^2$, $s_2=m_2^2$.
The $\text{$\cal S$}$ matrix has the following texture:
\begin{equation}
\text{$\cal S$}^{soft} =
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & -2 \, m_2^2 & s_3 - m_2^2 - m_3^2 \\
0 & s_3 - m_2^2 - m_3^2 & - 2 \, m_3^2
\end{array}
\right)
\label{eqcalssoft}
\end{equation}
If one singles out row and column $1$ in $\text{$\cal S$}^{soft}$,
the two-component vector $V^{(1)}$ is readily seen to vanish and so do the
coefficients $\overline{b}_2$ and $\overline{b}_3$ which are proportional to the two
components of $(G^{(1)})^{-1} \cdot V^{(1)}$:
thus only $\overline{b}_1$ differs from zero, cf. eq. (\ref{P1-eqdefdelta2}) of ref. \cite{paper1}.
\vspace{0.3cm}
\noindent
To illustrate this point, let us consider the case where $m_1=0$, $m_2=m_3=m$,
$s_1=s_2=m^2$ and $s_3$ arbitrary. In this case, since $\overline{b}_1$ is the
only non vanishing coefficient, the polynomial $D^{\{1\}(2)}(x)$ involved in
$I_3^n$
corresponds to the one appearing in the two-point function obtained by pinching
the internal line with four-momentum $q_1$ (cf.\ fig.~\ref{fig1}). This polynomial involves two
masses (equal here) and this example corresponds to case {\bf a} of the
preceding section. In this simple case, the two roots of the polynomial $D^{\{1\}(2)}(x)$ is given by:
\begin{equation}
x_{1,2} = \frac{1}{2} \pm \frac{1}{2} \, \sqrt{ 1 - \frac{4 \, ( m^2 - i \,\lambda)}{s_3}}
\label{eqroot12ex}
\end{equation}
with the property:
\[
1 - x_1 = x_2
\]
Injecting this property into eqs.\ (\ref{eqcompk11}) and (\ref{eqcompj5}), we get for the functions $J(x_1,x_2)$ and $K(x_1,x_2)$ in eq.~(\ref{eqcompintlog1}):
\begin{align}
J(x_1,x_2) &= \frac{2}{x_1-x_2} \, \left\{ \mbox{Li}_2 \left( \frac{x_2}{x_1} \right) - \mbox{Li}_2 \left( \frac{x_1}{x_2} \right) \right. \notag \\
&\quad + \left. \ln \left( -\frac{x_2}{x_1} \right) \, \ln \left( - (x_1-x_2)^2 \right) \right\} \label{eqcompj6} \\
K(x_1,x_2) &= \frac{2}{x_1-x_2} \, \ln \left( -\frac{x_2}{x_1} \right) \label{eqcompk2}
\end{align}
We check numerically that we recover the result of \cite{Ellis:2007qk}.
\vspace{0.3cm}
\noindent
{\bf 2.} A collinear divergence occurs
when two internal masses vanish whereas the external four-momentum which
enters into the vertex connecting the two adjacent massless propagators is
lightlike (massless collinear splitting at this vertex).
Note that the non vanishing internal mass can be real or complex.
Let us assume that the labels of the two massless propagators are
$1$ and $2$, with $s_2=0$: the $\text{$\cal S$}$ matrix has the following texture:
\begin{equation}
\text{$\cal S$}^{coll} =
\left(
\begin{array}{ccc}
0 & 0 & s_1 - m_3^2 \\
0 & 0 & s_3 - m_3^2 \\
s_1 - m_3^2 & s_3 - m_3^2 & - 2 \, m_3^2
\end{array}
\right)
\label{eqcalscoll}
\end{equation}
If one singles out row and column $3$ in $\text{$\cal S$}^{coll}$, the Gram matrix
$G^{(3)}$ and the vector $V^{(3)}$ read:
\begin{align}
G^{(3)} &= \left(
\begin{array}{cc}
2 \, s_1 & s_1+s_3\\
s_1+s_3 & 2 \, s_3
\end{array}
\right)
\label{eqdefg3coll} \\
V^{(3)} &= \left(
\begin{array}{c}
s_1 + m_3^2 \\
s_3 + m_3^2
\end{array}
\right)
\label{eqdefv3coll}
\end{align}
A simple calculation yields:
\begin{align}
\sum_{i \in S_3 \setminus \{3\}} \,
\left[ \left( G^{(3)} \right) ^{-1} \cdot V^{(3)} \right]_i
&= 1
\label{eqsumgm1v}
\end{align}
so that $\overline{b}_3$ given by eq. (\ref{P1-eqdefdelta2}) of ref. \cite{paper1}
vanishes, whereas
$\overline{b}_{1}$ and $\overline{b}_{2}$ generically differ from zero.
\vspace{0.3cm}
\noindent
Let us compute $I_3^n$ for this specific case. As $\overline{b}_3 = 0$, the
relevant polynomials $D^{\{i\}(j)}(x)$ are those of the two-point
functions obtained in the two pinching configurations
(cf.\ figure (\ref{fig1})) where
either the internal line with four-momentum $q_1$
or the one with the four-momentum $q_2$ shall be pinched.
As these two lines are associated to vanishing
masses and the third propagator is associated to a non vanishing mass,
the two polynomials both have one vanishing mass, this corresponds
to case {\bf b} of the previous section.
Starting with eq. (\ref{eqdirei3n1}) $I_3^n$ reads:
\begin{align}
I_3^n
&= \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\left[
\frac{\overline{b}_1}{\det(G)} \,
\int^1_0 dx \,
\left( D^{\{1\}(2)}(x) - i \, \lambda \right)^{-1-\varepsilon}
\right.
\notag \\
&
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\left.
\, + \frac{\overline{b}_2}{\det(G)} \,
\int^1_0 dx \,
\left( D^{\{2\}(3)}(x) - i \, \lambda \right)^{-1-\varepsilon} \right]
\label{eqdirei3n5}
\end{align}
It is better to change $x \leftrightarrow 1-x$ in the second integral in order to have
a polynomial of the type (\ref{eqnewquad1}) and write $I_3^n$ as:
\begin{align}
I_3^n
&= \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\left[
\,
\frac{\overline{b}_1}{\det(G)} \,
\int_0^1 dx \,
\left( D^{\{1\}(2)}(x) - i \, \lambda \right)^{-1-\varepsilon}
\right.
\notag \\
&
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\left.
+ \,
\frac{\overline{b}_2}{\det(G)} \,
\int_0^1 dx \,
\left( D^{\{2\}(1)}(x) - i \, \lambda \right)^{-1-\varepsilon}
\right] \notag \\
&= \frac{\overline{b}_1}{\det(G)} \, W\left( \detgj{1},\widetilde{D}_{12},0 \right) + \frac{\overline{b}_2}{\det(G)} \, W\left( \detgj{2},\widetilde{D}_{21},0 \right)
\label{eqdirei3n6}
\end{align}
Noting that $\overline{b}_1 = (s_3-m_3^2) \, (s_1-s_3)$ and
$\overline{b}_2 = (s_1-m_3^2) \, (s_3-s_1)$, determining $\detgj{1}$, $\detgj{2}$ and $\widetilde{D}_{12}$ from the $\text{$\cal S$}$ matrix
elements (cf.\ eq.~(\ref{eqremd2})) and applying directly the result of
eq. (\ref{eqdirei3n3}), we get:
\begin{align}
I_3^n
&= \frac{\Gamma(1+\varepsilon)}{s_1-s_3} \,
\left\{ - \, \frac{1}{\varepsilon^2}
\left[
\left( {} - s_3+m_3^2 - i \, \lambda \right)^{-\varepsilon}
\;\; - \;\;
\left( {} -s_1+m_3^2 -i \, \lambda \right)^{-\varepsilon}
\right]
\right.
\notag \\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
+
\left.
\mbox{Li}_2 \left( \frac{s_3}{s_3-m_3^2+i \, \lambda} \right)
\; - \;\;
\mbox{Li}_2 \left( \frac{s_1}{s_1-m_3^2+i \, \lambda} \right) \;
\right\}
\label{eqdirei3n8}
\end{align}
Using the Landen identity (\ref{eqlanden}) we recover the formula (4.8) of ref. \cite{Ellis:2007qk}
after some algebra.
\vspace{0.3cm}
\noindent
{\bf 3.} Both a soft and a collinear divergence may occur at the same time,
thereby proceeding from both cases {\bf 1.} and {\bf 2.} above.
\vspace{0.3cm}
\noindent
Let us take the example of case {\bf 2} and specify $s_1 = m_3^2$.
The texture of the $\text{$\cal S$}$ matrix becomes:
\begin{equation}
\text{$\cal S$}^{cs} =
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & s_3 - m_3^2 \\
0 & s_3 - m_3^2 & - 2 \, m_3^2
\end{array}
\right)
\label{eqcalscs}
\end{equation}
Only $\overline{b}_1$ does not vanish, so $I_3^n$ reads simply:
\begin{align}
I_3^n
&= \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\frac{\overline{b}_1}{\det(G)} \,
\int^1_0 dx \,
\left( D^{\{1\}(2)}(x) - i \, \lambda \right)^{-1-\varepsilon} \notag \\
&= \frac{\overline{b}_1}{\det(G)} \, W\left( \detgj{1},\widetilde{D}_{12},0 \right)
\label{eqdirei3n9}
\end{align}
Using eq. (\ref{eqdirei3n3}) and expressing $\detgj{1}$ and $\widetilde{D}_{12}$ in terms of
the $\text{$\cal S$}$ matrix elements (cf.\ eq.~(\ref{eqremd2})), we get:
\begin{align}
I_3^n &=
\frac{\Gamma(1+\varepsilon)}{m_3^2-s_3} \,
\left\{
- \, \frac{1}{\varepsilon^2}
\left( {} - s_3+m_3^2 - i \, \lambda \right)^{-\varepsilon}
+ \frac{1}{2 \, \varepsilon^2} \,
\left( m_3^2 - i \, \lambda \right)^{-\varepsilon}
\right.
\notag \\
&\quad \qquad \qquad \qquad \qquad \qquad \quad {}
\left.
+ \mbox{Li}_2 \left( \frac{s_3}{s_3-m_3^2+i \, \lambda} \right)
\right\}
\label{eqdirei3n10}
\end{align}
After some algebra, we recover\footnote{In contrast to the present eq.
(\ref{eqdirei3n10}), eq. (4.11) of ref. \cite{Ellis:2007qk} contains
a factor $\Gamma^2(1-\varepsilon)/\Gamma(1-2\varepsilon)$ and an extra term
$+\pi^2/12$ inside the brackets. Yet they cancel against each other when
performing the $\varepsilon$-expansion at the appropriate order.}
formula (4.11) of ref. \cite{Ellis:2007qk}.
\vspace{0.3cm}
\subsection{Indirect way}\label{indirway}
The present subsection provides the calculation according to the
``indirect way". The results presented are valid for both real
and complex masses; unless explicitly specified the $i \, \lambda$ prescription
is kept having in mind that it is ineffective in the case of complex masses.
Let us start from eq. (\ref{eqdefi3n2}).
The $x$ integration is traded for a $\rho$ integration in a way very similar to
the four-dimensional case (see subsubsec.~\ref{P1-subsubsectindway} of ref.\ \cite{paper1}) and we get:
\begin{align}
I_3^n
&= \sum_{i \in S_3} \, \frac{\overline{b}_i}{\det(G)} \,
\sum_{j \in S_3 \setminus \{i\}} \, \frac{\bbj{j}{i}}{\detgj{i}} \,
L_3^n \left( 0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} \right)
\label{eqdef3n3}
\end{align}
with:
\begin{align}
L_3^n \left( 0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} \right)
&= \kappa_{_{IR}} \,
\int^{+\infty}_0 \frac{d \xi}{\xi^{\nu} - i \, \lambda}
\notag\\
&\quad {} \quad {}
\times
\int^{+\infty}_0
\frac{d \rho}{
(\xi^{\nu} + \rho^2 - \Delta_1^{\{i\}} - i \, \lambda)
(\xi^{\nu} + \rho^2+ \widetilde{D}_{ij} - i \, \lambda)^{1/2}
}
\label{eqlijsoft1}
\end{align}
and:
\[
\kappa_{_{IR}} =2^{\varepsilon} \, \frac{\Gamma(1+\varepsilon)}{(1-\varepsilon)} \,
\frac{1}{B(1+\varepsilon,1-\varepsilon)}, \quad \nu = \frac{1}{1-\varepsilon}
\]
\noindent
To handle the cases with soft and/or collinear divergences, the two relevant configurations are
1) $\widetilde{D}_{ij} \neq 0$ and 2) $\widetilde{D}_{ij} = 0$.
Note that when both $\Delta_2$ and $\Delta_1^{\{i\}}$ vanish
$L_3^n ( 0, 0, \widetilde{D}_{ij} )$ is
weighting a vanishing $\overline{b}$, therefore it shall not be considered (cf.\ subsec.~\ref{exp_exemp_ir}).
\vspace{0.3cm}
\noindent
The $\rho$ integration can be done using appendix \ref{P2-appendJ} as in the four
dimensional case and as the outcome of this integration, we shall distinguish two cases depending on the sign of
$\Im(\Delta_1^{\{i\}})$.
\vspace{0.3cm}
\noindent
{\bf 1) $\widetilde{D}_{ij} \neq 0$}
\vspace{0.3cm}
\noindent
{\bf 1.a) $\Im(\Delta_1^{\{i\}}) > 0$}\\
This case covers in particular real masses. After the $\rho$ integration,
$L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )$ becomes:
\begin{align}
&L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )
\notag\\
&= \kappa_{_{IR}} \,\int^{+\infty}_0 d \xi \,
\int^{1}_0 d z \,
\frac{1}
{
(\xi^{\nu} - i \, \lambda)
(\xi^{\nu} - (1-z^2) \, \Delta_1^{\{i\}} + z^2 \, \widetilde{D}_{ij} - i \, \lambda)
}
\label{eqlijsoft3}
\end{align}
The $\xi$ integration is performed first, using
eq.~(\ref{eqmodifk}) of the appendix \ref{appendJ}
and $L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )$ becomes:
\begin{align}
L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )
&= - \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\int^1_0 dz
\left(
z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}} - i \, \lambda
\right)^{-1-\varepsilon}
\label{eqlijsoft40}
\end{align}
Expanding\footnote{Let us remind that only the terms in the
$\varepsilon$-expansion which provide the divergent and finite terms in the
limit $\varepsilon \to 0$ are kept.}
the r.h.s. of eq. (\ref{eqlijsoft40}) around $\varepsilon=0$ then gives:
\begin{align}
L_{3}^{n} \left( 0,\Delta_{1}^{\{i\}},\widetilde{D}_{ij} \right)
&= 2^{\varepsilon} \, \Gamma(1+\varepsilon) \,
\left[
- \, \frac{1}{\varepsilon} \,
\int^1_0
\frac{dz}
{z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}} - i \, \lambda}
\right.
\notag\\
& \quad {} \quad {} \quad {} \quad {} \quad {} \quad {} \quad {}
\left.
+ \int^1_0 dz \,
\frac{\ln(z^2 \,(\widetilde{D}_{ij}+\Delta_1^{\{i\}})-\Delta_1^{\{i\}} - i \,\lambda)}
{z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}} - i \, \lambda}
\right]
\label{eqlijsoft41}
\end{align}
\vspace{0.3cm}
\noindent
{\bf 1.b) $\Im(\Delta_1^{\{i\}}) < 0$}\\
This case occurs only with complex masses,
in a way such that the $i \, \lambda$ prescriptions are overshadowed and
ineffective: the latter are therefore dropped in this part.
After $\rho$ integration performed as in the four-dimensional case,
we get (see eq.~(\ref{eqdeffuncj7}) of appendix~\ref{appendJ}):
\begin{align}
&L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )
\notag \\
&=
- \kappa_{_{IR}} \,\int^{+\infty}_0
\frac{d \xi}{\xi^{\nu} - i \, \lambda} \,
\left[
i \int^{+\infty}_0
\frac{dz}{- \xi^{\nu} + \widetilde{D}_{ij} \, z^2 + (1+z^2) \, \Delta_1^{\{i\}}}
\right.
\notag \\
& \quad {}\quad {}\quad {}\quad {}\quad {}\quad {} \quad {}\quad {}
\quad {}\quad {}\quad {}
\left.
+ \int^{+\infty}_1 \frac{dz}
{\xi^{\nu} + \widetilde{D}_{ij} \, z^2 - (1-z^2) \, \Delta_1^{\{i\}}}
\right]
\label{eqlijsoft5}
\end{align}
Using identity (\ref{eqmodifk}),
$L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )$ reads:
\begin{align}
L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )
&= \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\left[
- \, i \int^{+\infty}_0 dz
\left(
-z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}}
\right)^{-1-\varepsilon}
\right.
\notag \\
&\quad {}\quad {}\quad {}\quad {} \quad {} \quad {} \quad {}
+
\left.
\int^{+\infty}_1 dz \;
\left(\,
z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}}
\right)^{-1-\varepsilon}
\;
\right]
\label{eqlijsoft60}
\end{align}
Expanding the terms in the square bracket in eq.
(\ref{eqlijsoft60}) around $\varepsilon=0$ then gives:
\begin{align}
&L_{3}^{n} \left( 0,\Delta_{1}^{\{i\}},\widetilde{D}_{ij} \right)
\notag \\
&= 2^{\varepsilon} \, \Gamma(1+\varepsilon) \,
\left\{
\;
\frac{1}{\varepsilon} \,
\left[
\; i \, \int^{+\infty}_0
\frac{dz}{z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) + \Delta_1^{\{i\}}}
+
\int^{+\infty}_1
\frac{dz}{z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}}}
\;
\right]
\right.
\notag \\
&\quad \quad \quad \quad \quad \quad \quad {}
-i \int^{+\infty}_0 dz \,
\frac{\ln(-z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}})}
{z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) + \Delta_1^{\{i\}}}
\notag\\
&\quad \quad \quad \quad \quad \quad \quad {}
\left.
- \;\; \int^{+\infty}_1 dz \,
\frac{\ln(z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}})}
{z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}}}
\right\}
\label{eqlijsoft61}
\end{align}
For eqs. (\ref{eqlijsoft41}) and (\ref{eqlijsoft61}), the $z$ integration
can be then performed using appendix \ref{appF}.\\
\vspace{0.3cm}
\noindent
Since the cuts of
$( z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) -\Delta_1^{\{i\}})^{-1-\varepsilon}$ and
$\ln ( z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}} )$ are the same,
the discussion carried to extend the ``indirect way'' to the general complex mass
case holds likewise here (cf.\ sec.~\ref{P2-sectthreepoint} of ref.\ \cite{paper2}).
Eq. (\ref{eqlijsoft60}) can be rewritten as:
\begin{align}
&L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )
\notag\\
&= - \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\left\{ \int_{0}^{i \, \infty} + \int_{+\infty}^{1} \right\} dz
\left(
\, z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}}
\right)^{-1-\varepsilon}
\label{eqlijsoft62}
\end{align}
In eq. (\ref{eqlijsoft62}) let us add a vanishing contribution along
the ``contour at $\infty$" in the north-east quadrant
$\{\Re(z) > 0$, $\Im(z) > 0\}$ so as to concatenate the two
contributions. The connected contour thus obtained can in turn be deformed into
a finite contour $\widehat{(0,1)}_{i,j}$ stretched from 0 to 1 as pictured on
fig. \ref{contour} thereby unifying eqs. (\ref{eqlijsoft40}) and
(\ref{eqlijsoft62}):
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.8]{contour.pdf}
\end{center}
\caption{\footnotesize Location of the relevant discontinuity cut
${\cal C}_{i,j}$ with respect to the two half straight lines $[0, + i \infty[$
and $[1,+\infty[$ and deformation of the contour $\widehat{(0,1)}$ partly
wrapping the extremity of the cut.}\label{contour}
\end{figure}
\begin{align}
L_3^n (0, \Delta_{1}^{\{i\}}, \widetilde{D}_{ij} )
&= - \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\int_{\widehat{(0,1)}_{i,j}} dz
\left(
\, z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}}
\right)^{-1-\varepsilon}
\label{eqlijsoft63}
\end{align}
\vspace{0.3cm}
\noindent
{\bf 2) $\widetilde{D}_{ij} = 0$}\\
Here again we shall distinguish two cases depending on the sign of
$\Im(\Delta_1^{\{i\}})$.
\vspace{0.3cm}
\noindent
{\bf 2.a) $\Im(\Delta_1^{\{i\}}) > 0$}\\
This case covers in particular real masses.
The calculation initially amounts to setting $\widetilde{D}_{ij} = 0$ in eq.
(\ref{eqlijsoft40}), which becomes:
\begin{align}
L_{3}^{n} \left( 0,\Delta_{1}^{\{i\}},0 \right)
&= - \, \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\left( - \Delta_1^{\{i\}} - i \, \lambda \right)^{-1-\varepsilon} \,
\int^1_0 dz \left(1-z^2 \right)^{-1-\varepsilon}
\label{eqlijsoft7}
\end{align}
The integration over $z$ is performed using eq.~(\ref{thirdinty}) and
$L_{3}^{n}(0,\Delta_{1}^{\{i\}},0) $ becomes:
\begin{align}
L_{3}^{n} \left( 0,\Delta_{1}^{\{i\}},0 \right)
&= \frac{1}{\varepsilon^2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma(1-\varepsilon)^2}{\Gamma(1-2 \, \varepsilon)} \,
\left( - 2 \, \Delta_1^{\{i\}} - i \, \lambda \right)^{-1-\varepsilon}
\label{eqlijsoft8}
\end{align}
\noindent
{\bf 2.b) $\Im(\Delta_1^{\{i\}}) < 0$} \\
Here again, this case occurs only with complex masses,
with the $i \, \lambda$ prescriptions overshadowed and therefore dropped.
The calculation initially amounts anew to setting $\widetilde{D}_{ij} = 0$ in eq.
(\ref{eqlijsoft60}) which becomes:
\begin{align}
L_{3}^{n} \left( 0,\Delta_{1}^{\{i\}},0 \right)
&= \frac{2^{\varepsilon} \, \Gamma(1+\varepsilon)}{\varepsilon} \,
\left[
- i \, \left( - \Delta_1^{\{i\}} \right)^{-1-\varepsilon} \,
\int^{+\infty}_0 dz \left(1+z^2 \right)^{-1-\varepsilon}
\right.
\notag\\
& \quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}\quad {}
\left.
- \left( \Delta_1^{\{i\}} \right)^{-1-\varepsilon} \,
\int^{+\infty}_1 dz \left(z^2-1 \right)^{-1-\varepsilon}
\right]
\label{eqlijsoft9}
\end{align}
The $z$ integrals are computed in appendix \ref{herba} (eqs.~(\ref{firstinty}) and (\ref{secondinty})).
Hence for $L_{3}^{n}(0,\Delta_{1}^{\{i\}},0)$:
\begin{align}
L_{3}^{n} \left( 0,\Delta_{1}^{\{i\}},0 \right)
&=
- \, \frac{2^{-\varepsilon}}{2 \, \varepsilon^2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^{2}(1- \varepsilon)}{\Gamma(1 - 2 \,\varepsilon)} \,
\frac{1}{\cos(\pi \, \varepsilon)}
\notag \\
&
\;\;\;\;\;\;\;\; {} \times
\left[
i \, \sin(\pi \, \varepsilon) \,
\left( - \Delta_1^{\{i\}} \right)^{-1-\varepsilon} \,
+
\left( \Delta_1^{\{i\}} \right)^{-1-\varepsilon} \,
\right]
\label{eqlijsoft10}
\end{align}
Since $\Im(\Delta_1^{\{i\}}) < 0$, $( \Delta_1^{\{i\}} )^{-1-\varepsilon}$ may
be rewritten as $- \, \text{e}^{\, i \, \pi \, \varepsilon}\,
( - \, \Delta_1^{\{i\}} )^{-1-\varepsilon}$, so that
$L_{3}^{n}(0,\Delta_{1}^{\{i\}},\widetilde{D}_{ij})$ simplifies into:
\begin{align}
L_{3}^{n} \left( 0,\Delta_{1}^{\{i\}},0 \right)
&=
\frac{1}{\varepsilon^2} \, \Gamma(1+\varepsilon) \,
\frac{\Gamma^{2}(1- \varepsilon)}{\Gamma(1 - 2 \,\varepsilon)} \,
\left( - 2 \, \Delta_1^{\{i\}} \right)^{-1-\varepsilon} \,
\label{eqlijsoft10bis}
\end{align}
which coincides with eq. (\ref{eqlijsoft8}).
\vspace{0.3cm}
\noindent
Let us show now the equivalence between the ``indirect way'' and the ``direct way'' starting
from the integral representation of $L_3^n(0,\Delta_1^{\{i\}},\widetilde{D}_{ij})$ and disregarding
the fact that some $\widetilde{D}_{ij}$ may or may not vanish.
Thus, $I_3^n$ now reads:
\begin{align}
I_3^n
&= - \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\sum_{i \in S_3} \, \frac{\overline{b}_i}{\det(G)} \,
\sum_{j \in S_3 \setminus \{i\}} \, \frac{\bbj{j}{i}}{\detgj{i}}
\notag \\
&\quad \quad \quad \quad
\times
\int_{\widehat{(0,1)}_{i,j}} dz \,
\left(
\, z^2 \, (\widetilde{D}_{ij}+\Delta_1^{\{i\}}) - \Delta_1^{\{i\}}
\right)^{-1-\varepsilon}
\label{eqlijsoft64}
\end{align}
Sticking to the general complex mass case, we perform the following change of
variable: $s = \bbj{j}{i} \, z$ in such a way that the two integrands
corresponding to the sum over $j$ (at $i$ fixed) in eq. (\ref{eqlijsoft64}) are
the same (use eqs.~(\ref{P1-eqtruc1}) and (\ref{P1-eqtruc3}) of \cite{paper1}).
Specifying the two elements of $S_3 \setminus \{i\}$ to be
$k \equiv 1 + ((i+1)$ modulo $3)$ and $l \equiv 1 + (i$ modulo $3)$, the two
integrals are concatenated into a single one integrated along the contour
${\cal I}^{(i)}_{k,l} \equiv -\bbj{k}{i}\widehat{(0,1)}_{i,k} \cup
\bbj{l}{i}\widehat{(0,1)}_{i,l}$ in the complex $s$-plane:
\begin{align}
I_3^n
&= - \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\sum_{i \in S_3} \, \frac{\overline{b}_i}{\det(G) \detgj{i}} \,
\notag \\
&\quad \quad \quad
\times
\int_{{\cal I}^{(i)}_{k,l}} ds
\left(
\frac{s^2 + \detsj{i}}{\detgj{i}}
\right)^{-1-\varepsilon}
\label{eqlijsoft65}
\end{align}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.4]{triangle1.pdf}
\end{center}
\caption{\footnotesize Example of a contour deformation involving a
triangle with one distorted side, for which no cut crosses the straight base
$[ -\bbj{k}{i},\bbj{l}{i}]$.}\label{triangle1}
\end{figure}
\vspace{0.3cm}
\noindent
As in the case $\Delta_2 \ne 0$ treated in \cite{paper2}, the contour ${\cal I}^{(i)}_{k,l}$ can be
deformed into the straight line $\left[ -\bbj{k}{i},\bbj{l}{i} \right]$ as depicted on figure \ref{triangle1} :
\begin{align}
I_3^n &= - \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\sum_{i \in S_3} \, \frac{\overline{b}_i}{\det(G) \detgj{i}} \,
\notag \\
&\quad
\times
\int_{- \bbj{k}{i}}^{\bbj{l}{i}} ds
\left(
\frac{s^2 + \detsj{i}}{\detgj{i}}
\right)^{-1-\varepsilon}
\label{eqlijsoft66}
\end{align}
Performing the change of variable:
$s = -\bbj{k}{i} - \detgj{i} \,u$ and following the procedure given by eqs.~(\ref{P1-cvar1}) to (\ref{P1-eqdefgi}) of ref.\ \cite{paper1} leads to:
\begin{align}
I_3^n &= \frac{2^{\varepsilon}}{\varepsilon} \, \Gamma(1+\varepsilon) \,
\sum_{i \in S_3} \, \frac{\overline{b}_i}{\det(G)} \,
\int_{0}^{1} du \left( D^{\{i\} \, (l)}(u) \right)^{-1-\varepsilon}
\nonumber
\end{align}
which is namely eq. (\ref{eqdirei3n1}).
\vspace{0.3cm}
\noindent
It is instructive to recover the results of the ``direct way'' from the ones of
the ``indirect way'' using for the latter the closed form formulae. These formulae
can be obtained for the case $\widetilde{D}_{ij} \ne 0$ by using results of appendix \ref{appF}
and for $\widetilde{D}_{ij} = 0$ using eq.~(\ref{eqlijsoft8}). This exercise is performed
with great details in appendix~\ref{direcway3pIR}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,187
|
<?xml version="1.0" encoding="UTF-8"?>
<!--
Licensed to the Apache Software Foundation (ASF) under one
or more contributor license agreements. See the NOTICE file
distributed with this work for additional information
regarding copyright ownership. The ASF licenses this file
to you under the Apache License, Version 2.0 (the
"License"); you may not use this file except in compliance
with the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing,
software distributed under the License is distributed on an
"AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
KIND, either express or implied. See the License for the
specific language governing permissions and limitations
under the License.
-->
<simple-methods xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="http://ofbiz.apache.org/dtds/simple-methods-v2.xsd">
<simple-method method-name="createBlogEntry" short-description="Create a new Blog Entry">
<set field="contentAssocTypeId" value="PUBLISH_LINK"/>
<set field="ownerContentId" from-field="parameters.blogContentId"/>
<set field="contentIdFrom" from-field="parameters.blogContentId"/>
<if-empty field="parameters.statusId">
<set field="parameters.statusId" value="CTNT_INITIAL_DRAFT"/>
</if-empty>
<if-empty field="parameters.templateDataResourceId">
<set field="parameters.templateDataResourceId" value="BLOG_TPL_TOPLEFT"/>
</if-empty>
<!-- determine of we need to create complex template structure or simple content structure -->
<if-empty field="parameters.contentName">
<add-error>
<fail-property resource="ContentUiLabels" property="ContentArticleNameIsMissing"/>
</add-error>
</if-empty>
<check-errors/>
<!-- complex template structure (image & text) -->
<set field="createMain.dataResourceId" from-field="parameters.templateDataResourceId"/>
<set field="createMain.contentAssocTypeId" from-field="contentAssocTypeId"/>
<set field="createMain.contentName" from-field="parameters.contentName"/>
<set field="createMain.description" from-field="parameters.description"/>
<set field="createMain.statusId" from-field="parameters.statusId"/>
<set field="createMain.contentIdFrom" from-field="contentIdFrom"/>
<set field="createMain.partyId" from-field="userLogin.partyId"/>
<set field="createMain.ownerContentId" from-field="ownerContentId"/>
<set field="createMain.dataTemplateTypeId" value="SCREEN_COMBINED"/>
<set field="createMain.mapKey" value="MAIN"/>
<call-service service-name="createContent" in-map-name="createMain">
<result-to-field result-name="contentId" field="contentId"/>
</call-service>
<!-- reset contentIdFrom to new contentId -->
<set field="contentAssocTypeId" value="SUB_CONTENT"/>
<set field="contentIdFrom" from-field="contentId"/>
<if-not-empty field="parameters._uploadedFile_fileName">
<!-- upload a picture -->
<set field="createImage.dataResourceTypeId" value="LOCAL_FILE"/>
<set field="createImage.dataTemplateTypeId" value="NONE"/>
<set field="createImage.mapKey" value="IMAGE"/>
<set field="createImage.ownerContentId" from-field="ownerContentId"/>
<set field="createImage.contentName" from-field="parameters.contentName"/>
<set field="createImage.description" from-field="parameters.description"/>
<set field="createImage.statusId" from-field="parameters.statusId"/>
<set field="createImage.contentAssocTypeId" from-field="contentAssocTypeId"/>
<set field="createImage.contentIdFrom" from-field="contentIdFrom"/>
<set field="createImage.partyId" from-field="userLogin.partyId"/>
<set field="createImage.isPublic" value="Y"/>
<set field="createImage.uploadedFile" from-field="parameters.uploadedFile"/>
<set field="createImage._uploadedFile_fileName" from-field="parameters._uploadedFile_fileName"/>
<set field="createImage._uploadedFile_contentType" from-field="parameters._uploadedFile_contentType"/>
<call-service service-name="createContentFromUploadedFile" in-map-name="createImage">
<result-to-field result-name="contentId" field="imageContentId"/>
</call-service>
</if-not-empty>
<if-not-empty field="parameters.articleData">
<!-- create text data -->
<set field="createText.dataResourceTypeId" value="ELECTRONIC_TEXT"/>
<set field="createText.contentPurposeTypeId" value="ARTICLE"/>
<set field="createText.dataTemplateTypeId" value="NONE"/>
<set field="createText.mapKey" value="MAIN"/>
<set field="createText.ownerContentId" from-field="ownerContentId"/>
<set field="createText.contentName" from-field="parameters.contentName"/>
<set field="createText.description" from-field="parameters.description"/>
<set field="createText.statusId" from-field="parameters.statusId"/>
<set field="createText.contentAssocTypeId" from-field="contentAssocTypeId"/>
<set field="createText.textData" from-field="parameters.articleData"/>
<set field="createText.contentIdFrom" from-field="contentIdFrom"/>
<set field="createText.partyId" from-field="userLogin.partyId"/>
<set field="createText.mapKey" value="ARTICLE"/>
<log level="info" message="calling createTextContent with map: ${createText}"/>
<call-service service-name="createTextContent" in-map-name="createText">
<result-to-field result-name="contentId" field="textContentId"/>
</call-service>
</if-not-empty>
<if-not-empty field="contentId">
<if-not-empty field="parameters.summaryData">
<!-- create the summary data -->
<set field="createSummary.dataResourceTypeId" value="ELECTRONIC_TEXT"/>
<set field="createSummary.contentPurposeTypeId" value="ARTICLE"/>
<set field="createSummary.dataTemplateTypeId" value="NONE"/>
<set field="createSummary.mapKey" value="SUMMARY"/>
<set field="createSummary.ownerContentId" from-field="ownerContentId"/>
<set field="createSummary.contentName" from-field="parameters.contentName"/>
<set field="createSummary.description" from-field="parameters.description"/>
<set field="createSummary.statusId" from-field="parameters.statusId"/>
<set field="createSummary.contentAssocTypeId" from-field="contentAssocTypeId"/>
<set field="createSummary.textData" from-field="parameters.summaryData"/>
<set field="createSummary.contentIdFrom" from-field="contentIdFrom"/>
<set field="createSummary.partyId" from-field="userLogin.partyId"/>
<call-service service-name="createTextContent" in-map-name="createSummary"/>
</if-not-empty>
</if-not-empty>
<field-to-result field="contentIdFrom" result-name="contentId"/>
<field-to-result field="parameters.blogContentId" result-name="blogContentId"/>
</simple-method>
<simple-method method-name="updateBlogEntry" short-description="Update a existing Blog Entry">
<set field="showNoResult" value="Y"/>
<call-simple-method method-name="getBlogEntry"/>
<if>
<condition>
<or>
<if-compare-field field="parameters.contentName" operator="not-equals" to-field="contentName"/>
<if-compare-field field="parameters.description" operator="not-equals" to-field="description"/>
<if-compare-field field="parameters.summaryData" operator="not-equals" to-field="summaryData"/>
<if-compare-field field="parameters.templateDataResourceId" operator="not-equals" to-field="templateDataResourceId"/>
<if-compare-field field="parameters.statusId" operator="not-equals" to-field="statusId"/>
</or>
</condition>
<then>
<set-service-fields service-name="updateContent" map="parameters" to-map="updContent"/>
<set field="updContent.dataResourceId" from-field="parameters.templateDataResourceId"/>
<call-service service-name="updateContent" in-map-name="updContent"/>
<if-compare-field field="parameters.statusId" operator="not-equals" to-field="statusId">
<if-not-empty field="imageContent">
<set field="imageContent.status.Id" from-field="parameters.statusId"/>
<store-value value-field="imageContent"/>
</if-not-empty>
</if-compare-field>
</then>
</if>
<!-- new article text -->
<if-empty field="articleText">
<if-not-empty field="parameters.articleData">
<set field="ownerContentId" from-field="parameters.blogContentId"/>
<set field="contentAssocTypeId" value="SUB_CONTENT"/>
<set field="contentIdFrom" from-field="contentId"/>
<set field="createText.dataResourceTypeId" value="ELECTRONIC_TEXT"/>
<set field="createText.contentPurposeTypeId" value="ARTICLE"/>
<set field="createText.dataTemplateTypeId" value="NONE"/>
<set field="createText.mapKey" value="ARTICLE"/>
<set field="createText.ownerContentId" from-field="ownerContentId"/>
<set field="createText.contentName" from-field="parameters.contentName"/>
<set field="createText.description" from-field="parameters.description"/>
<set field="createText.statusId" from-field="parameters.statusId"/>
<set field="createText.contentAssocTypeId" from-field="contentAssocTypeId"/>
<set field="createText.textData" from-field="parameters.articleData"/>
<set field="createText.contentIdFrom" from-field="contentIdFrom"/>
<set field="createText.partyId" from-field="userLogin.partyId"/>
<call-service service-name="createTextContent" in-map-name="createText"/>
</if-not-empty>
</if-empty>
<!-- update article text -->
<if-not-empty field="articleText">
<if-compare-field field="parameters.articleData" operator="not-equals" to-field="articleData">
<set field="articleText.textData" from-field="parameters.articleData"/>
<store-value value-field="articleText"/>
</if-compare-field>
</if-not-empty>
<!-- create summary text -->
<if-empty field="summaryData">
<if-not-empty field="parameters.summaryData">
<!-- create the summary data -->
<set field="ownerContentId" from-field="parameters.blogContentId"/>
<set field="contentAssocTypeId" value="SUB_CONTENT"/>
<set field="contentIdFrom" from-field="contentId"/>
<set field="createSummary.dataResourceTypeId" value="ELECTRONIC_TEXT"/>
<set field="createSummary.contentPurposeTypeId" value="ARTICLE"/>
<set field="createSummary.dataTemplateTypeId" value="NONE"/>
<set field="createSummary.mapKey" value="SUMMARY"/>
<set field="createSummary.ownerContentId" from-field="ownerContentId"/>
<set field="createSummary.contentName" from-field="parameters.contentName"/>
<set field="createSummary.description" from-field="parameters.description"/>
<set field="createSummary.statusId" from-field="parameters.statusId"/>
<set field="createSummary.contentAssocTypeId" from-field="contentAssocTypeId"/>
<set field="createSummary.textData" from-field="parameters.summaryData"/>
<set field="createSummary.contentIdFrom" from-field="contentIdFrom"/>
<set field="createSummary.partyId" from-field="userLogin.partyId"/>
<call-service service-name="createTextContent" in-map-name="createSummary"/>
</if-not-empty>
</if-empty>
<!-- update summary text -->
<if-not-empty field="summaryData">
<if-compare-field field="parameters.summaryData" operator="not-equals" to-field="summaryData">
<set field="summaryText.textData" from-field="parameters.summaryData"/>
<store-value value-field="summaryText"/>
</if-compare-field>
</if-not-empty>
<if-not-empty field="parameters._uploadedFile_fileName">
<if-not-empty field="imageContent">
<entity-and entity-name="ContentAssoc" list="oldAssocs" filter-by-date="true">
<field-map field-name="contentId" from-field="contentId"/>
<field-map field-name="contentIdTo" from-field="imageContent.contentId"/>
<field-map field-name="mapKey" value="IMAGE"/>
</entity-and>
<first-from-list list="oldAssocs" entry="oldAssoc"/>
<now-timestamp field="oldAssoc.thruDate"/>
<store-value value-field="oldAssoc"/>
</if-not-empty>
<!-- upload a picture -->
<set field="createImage.dataResourceTypeId" value="LOCAL_FILE"/>
<set field="createImage.dataTemplateTypeId" value="NONE"/>
<set field="createImage.mapKey" value="IMAGE"/>
<set field="createImage.ownerContentId" from-field="parameters.contentId"/>
<set field="createImage.contentName" from-field="parameters.contentName" default-value="${contentName}"/>
<set field="createImage.description" from-field="parameters.description" default-value="${description}"/>
<set field="createImage.statusId" from-field="parameters.statusId" default-value="${statusId}"/>
<set field="createImage.contentAssocTypeId" value="SUB_CONTENT"/>
<set field="createImage.contentIdFrom" from-field="parameters.contentId"/>
<set field="createImage.partyId" from-field="userLogin.partyId"/>
<set field="createImage.isPublic" value="Y"/>
<set field="createImage.uploadedFile" from-field="parameters.uploadedFile"/>
<set field="createImage._uploadedFile_fileName" from-field="parameters._uploadedFile_fileName"/>
<set field="createImage._uploadedFile_contentType" from-field="parameters._uploadedFile_contentType"/>
<call-service service-name="createContentFromUploadedFile" in-map-name="createImage"/>
</if-not-empty>
<field-to-result field="parameters.contentId" result-name="contentId"/>
<field-to-result field="parameters.blogContentId" result-name="blogContentId"/>
</simple-method>
<simple-method method-name="getOwnedOrPublishedBlogEntries" short-description="Get blog entries that the user owns or are published">
<entity-condition entity-name="ContentAssocViewTo" use-cache="false" list="unfilteredList">
<condition-list combine="and">
<condition-expr field-name="contentIdStart" operator="equals" from-field="parameters.contentId"/>
<condition-expr field-name="caContentAssocTypeId" operator="equals" value="PUBLISH_LINK"/>
</condition-list>
<order-by field-name="caFromDate DESC"/>
</entity-condition>
<filter-list-by-date list="unfilteredList" to-list="blogItems"/>
<set field="blogList[]"/>
<iterate list="blogItems" entry="blogItem">
<set-service-fields service-name="genericContentPermission" map="blogItem" to-map="mapIn"/>
<set field="mapIn.ownerContentId" from-field="parameters.contentId"/>
<set field="mapIn.mainAction" value="VIEW"/>
<call-service service-name="genericContentPermission" in-map-name="mapIn">
<result-to-field result-name="hasPermission" field="hasPermission"/>
</call-service>
<if-compare field="hasPermission" operator="equals" value="true" type="Boolean">
<set field="blogList[]" from-field="blogItem"/>
<else>
<set field="mapIn.mainAction" value="UPDATE"/>
<call-service service-name="genericContentPermission" in-map-name="mapIn">
<result-to-field result-name="hasPermission" field="hasPermission"/>
</call-service>
<if-compare field="hasPermission" operator="equals" value="true" type="Boolean">
<set field="blogList[]" from-field="blogItem"/>
</if-compare>
</else>
</if-compare>
</iterate>
<field-to-result field="blogList" result-name="blogList"/>
<field-to-result field="parameters.blogContentId" result-name="blogContentId"/>
</simple-method>
<simple-method method-name="getBlogEntry" short-description="Get all the info for a blog article">
<if-empty field="parameters.contentId">
<field-to-result field="parameters.blogContentId" result-name="blogContentId"/>
<return/>
</if-empty>
<entity-one entity-name="Content" value-field="content"/>
<get-related value-field="content" relation-name="FromContentAssoc" list="rawAssocs"/>
<filter-list-by-date list="rawAssocs" to-list="assocs"/>
<iterate list="assocs" entry="assoc">
<if-compare field="assoc.mapKey" value="ARTICLE" operator="equals">
<get-related-one value-field="assoc" relation-name="ToContent" to-value-field="mainContent"/>
<get-related-one value-field="mainContent" relation-name="DataResource" to-value-field="dataResource"/>
<get-related-one value-field="dataResource" relation-name="ElectronicText" to-value-field="articleText"/>
</if-compare>
<if-compare field="assoc.mapKey" value="SUMMARY" operator="equals">
<get-related-one value-field="assoc" relation-name="ToContent" to-value-field="summaryContent"/>
<get-related-one value-field="summaryContent" relation-name="DataResource" to-value-field="dataResource"/>
<get-related-one value-field="dataResource" relation-name="ElectronicText" to-value-field="summaryText"/>
</if-compare>
<if-compare field="assoc.mapKey" value="IMAGE" operator="equals">
<get-related-one value-field="assoc" relation-name="ToContent" to-value-field="imageContent"/>
</if-compare>
</iterate>
<if-empty field="showNoResult">
<field-to-result field="content.contentId" result-name="contentId"/>
<field-to-result field="content.contentName" result-name="contentName"/>
<field-to-result field="content.description" result-name="description"/>
<field-to-result field="content.statusId" result-name="statusId"/>
<if-not-empty field="imageContent">
<field-to-result field="content.dataResourceId" result-name="templateDataResourceId"/>
</if-not-empty>
<field-to-result field="articleText.textData" result-name="articleData"/>
<field-to-result field="summaryText.textData" result-name="summaryData"/>
<field-to-result field="imageContent.contentId" result-name="imageContentId"/>
<field-to-result field="mainContent.contentId" result-name="articleContentId"/>
<field-to-result field="summaryContent.contentId" result-name="summaryContentId"/>
<field-to-result field="parameters.blogContentId" result-name="blogContentId"/>
<else>
<set from-field="content.contentId" field="contentId"/>
<set from-field="content.contentName" field="contentName"/>
<set from-field="content.description" field="description"/>
<set from-field="content.statusId" field="statusId"/>
<set from-field="content.dataResourceId" field="templateDataResourceId"/>
<set from-field="articleText.textData" field="articleData"/>
<set from-field="summaryText.textData" field="summaryData"/>
<set from-field="imageContent.dataResourceId" field="imageDataResourceId"/>
</else>
</if-empty>
</simple-method>
</simple-methods>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,127
|
#ifndef OPENSSL_HEADER_SHA_H
#define OPENSSL_HEADER_SHA_H
#include <openssl/base.h>
#if defined(__cplusplus)
extern "C" {
#endif
/* SHA-1. */
/* SHA_CBLOCK is the block size of SHA-1. */
#define SHA_CBLOCK 64
/* SHA_DIGEST_LENGTH is the length of a SHA-1 digest. */
#define SHA_DIGEST_LENGTH 20
/* TODO(fork): remove */
#define SHA_LBLOCK 16
#define SHA_LONG uint32_t
/* SHA1_Init initialises |sha| and returns one. */
OPENSSL_EXPORT int SHA1_Init(SHA_CTX *sha);
/* SHA1_Update adds |len| bytes from |data| to |sha| and returns one. */
OPENSSL_EXPORT int SHA1_Update(SHA_CTX *sha, const void *data, size_t len);
/* SHA1_Final adds the final padding to |sha| and writes the resulting digest
* to |md|, which must have at least |SHA_DIGEST_LENGTH| bytes of space. It
* returns one. */
OPENSSL_EXPORT int SHA1_Final(uint8_t *md, SHA_CTX *sha);
/* SHA1 writes the digest of |len| bytes from |data| to |out| and returns
* |out|. There must be at least |SHA_DIGEST_LENGTH| bytes of space in
* |out|. */
OPENSSL_EXPORT uint8_t *SHA1(const uint8_t *data, size_t len, uint8_t *out);
/* SHA1_Transform is a low-level function that performs a single, SHA-1 block
* transformation using the state from |sha| and 64 bytes from |block|. */
OPENSSL_EXPORT void SHA1_Transform(SHA_CTX *sha, const uint8_t *block);
struct sha_state_st {
uint32_t h0, h1, h2, h3, h4;
uint32_t Nl, Nh;
uint32_t data[16];
unsigned int num;
};
/* SHA-224. */
/* SHA224_CBLOCK is the block size of SHA-224. */
#define SHA224_CBLOCK 64
/* SHA224_DIGEST_LENGTH is the length of a SHA-224 digest. */
#define SHA224_DIGEST_LENGTH 28
/* SHA224_Init initialises |sha| and returns 1. */
OPENSSL_EXPORT int SHA224_Init(SHA256_CTX *sha);
/* SHA224_Update adds |len| bytes from |data| to |sha| and returns 1. */
OPENSSL_EXPORT int SHA224_Update(SHA256_CTX *sha, const void *data, size_t len);
/* SHA224_Final adds the final padding to |sha| and writes the resulting digest
* to |md|, which must have at least |SHA224_DIGEST_LENGTH| bytes of space. */
OPENSSL_EXPORT int SHA224_Final(uint8_t *md, SHA256_CTX *sha);
/* SHA224 writes the digest of |len| bytes from |data| to |out| and returns
* |out|. There must be at least |SHA224_DIGEST_LENGTH| bytes of space in
* |out|. */
OPENSSL_EXPORT uint8_t *SHA224(const uint8_t *data, size_t len, uint8_t *out);
/* SHA-256. */
/* SHA256_CBLOCK is the block size of SHA-256. */
#define SHA256_CBLOCK 64
/* SHA256_DIGEST_LENGTH is the length of a SHA-256 digest. */
#define SHA256_DIGEST_LENGTH 32
/* SHA256_Init initialises |sha| and returns 1. */
OPENSSL_EXPORT int SHA256_Init(SHA256_CTX *sha);
/* SHA256_Update adds |len| bytes from |data| to |sha| and returns 1. */
OPENSSL_EXPORT int SHA256_Update(SHA256_CTX *sha, const void *data, size_t len);
/* SHA256_Final adds the final padding to |sha| and writes the resulting digest
* to |md|, which must have at least |SHA256_DIGEST_LENGTH| bytes of space. */
OPENSSL_EXPORT int SHA256_Final(uint8_t *md, SHA256_CTX *sha);
/* SHA256 writes the digest of |len| bytes from |data| to |out| and returns
* |out|. There must be at least |SHA256_DIGEST_LENGTH| bytes of space in
* |out|. */
OPENSSL_EXPORT uint8_t *SHA256(const uint8_t *data, size_t len, uint8_t *out);
/* SHA256_Transform is a low-level function that performs a single, SHA-1 block
* transformation using the state from |sha| and 64 bytes from |block|. */
OPENSSL_EXPORT void SHA256_Transform(SHA256_CTX *sha, const uint8_t *data);
struct sha256_state_st {
uint32_t h[8];
uint32_t Nl, Nh;
uint32_t data[16];
unsigned int num, md_len;
};
/* SHA-384. */
/* SHA384_CBLOCK is the block size of SHA-384. */
#define SHA384_CBLOCK 128
/* SHA384_DIGEST_LENGTH is the length of a SHA-384 digest. */
#define SHA384_DIGEST_LENGTH 48
/* SHA384_Init initialises |sha| and returns 1. */
OPENSSL_EXPORT int SHA384_Init(SHA512_CTX *sha);
/* SHA384_Update adds |len| bytes from |data| to |sha| and returns 1. */
OPENSSL_EXPORT int SHA384_Update(SHA512_CTX *sha, const void *data, size_t len);
/* SHA384_Final adds the final padding to |sha| and writes the resulting digest
* to |md|, which must have at least |SHA384_DIGEST_LENGTH| bytes of space. */
OPENSSL_EXPORT int SHA384_Final(uint8_t *md, SHA512_CTX *sha);
/* SHA384 writes the digest of |len| bytes from |data| to |out| and returns
* |out|. There must be at least |SHA384_DIGEST_LENGTH| bytes of space in
* |out|. */
OPENSSL_EXPORT uint8_t *SHA384(const uint8_t *data, size_t len, uint8_t *out);
/* SHA384_Transform is a low-level function that performs a single, SHA-1 block
* transformation using the state from |sha| and 64 bytes from |block|. */
OPENSSL_EXPORT void SHA384_Transform(SHA512_CTX *sha, const uint8_t *data);
/* SHA-512. */
/* SHA512_CBLOCK is the block size of SHA-512. */
#define SHA512_CBLOCK 128
/* SHA512_DIGEST_LENGTH is the length of a SHA-512 digest. */
#define SHA512_DIGEST_LENGTH 64
/* SHA512_Init initialises |sha| and returns 1. */
OPENSSL_EXPORT int SHA512_Init(SHA512_CTX *sha);
/* SHA512_Update adds |len| bytes from |data| to |sha| and returns 1. */
OPENSSL_EXPORT int SHA512_Update(SHA512_CTX *sha, const void *data, size_t len);
/* SHA512_Final adds the final padding to |sha| and writes the resulting digest
* to |md|, which must have at least |SHA512_DIGEST_LENGTH| bytes of space. */
OPENSSL_EXPORT int SHA512_Final(uint8_t *md, SHA512_CTX *sha);
/* SHA512 writes the digest of |len| bytes from |data| to |out| and returns
* |out|. There must be at least |SHA512_DIGEST_LENGTH| bytes of space in
* |out|. */
OPENSSL_EXPORT uint8_t *SHA512(const uint8_t *data, size_t len, uint8_t *out);
/* SHA512_Transform is a low-level function that performs a single, SHA-1 block
* transformation using the state from |sha| and 64 bytes from |block|. */
OPENSSL_EXPORT void SHA512_Transform(SHA512_CTX *sha, const uint8_t *data);
struct sha512_state_st {
uint64_t h[8];
uint64_t Nl, Nh;
union {
uint64_t d[16];
uint8_t p[64];
} u;
unsigned int num, md_len;
};
#if defined(__cplusplus)
} /* extern C */
#endif
#endif /* OPENSSL_HEADER_SHA_H */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,516
|
package nbd
import (
"crypto/tls"
"crypto/x509"
"encoding/binary"
"flag"
"fmt"
"io/ioutil"
"net"
"os"
"path"
"sync"
"sync/atomic"
"testing"
"text/template"
"time"
"github.com/zero-os/0-Disk/errors"
)
const ConfigTemplate = `
servers:
- protocol: unix
address: {{.TempDir}}/nbd.sock
exports:
- name: foo
driver: {{.Driver}}
path: {{.TempDir}}/nbd.img
{{if .NoFlush}}
flush: false
fua: false
{{end}}
{{if .TLS}}
tls:
keyfile: {{.TempDir}}/server-key.pem
certfile: {{.TempDir}}/server-cert.pem
cacertfile: {{.TempDir}}/client-cert.pem
servername: localhost
clientauth: requireverify
{{end}}
logging:
`
var longtests = flag.Bool("longtests", false, "enable long tests")
var noFlush = flag.Bool("noflush", false, "Disable flush and FUA (for benchmarking - do not use in production")
type TestConfig struct {
TLS bool
TempDir string
Driver string
NoFlush bool
}
type NbdInstance struct {
t *testing.T
quit chan struct{}
closed bool
closedMutex sync.Mutex
plainConn net.Conn
tlsConn net.Conn
conn net.Conn
transmissionFlags uint16
TestConfig
}
var nextHandle uint64
func getHandle() uint64 {
return atomic.AddUint64(&nextHandle, 1)
}
func StartNbd(t *testing.T, tc TestConfig) *NbdInstance {
ni := &NbdInstance{
t: t,
quit: make(chan struct{}),
TestConfig: tc,
}
if TempDir, err := ioutil.TempDir("", "nbdtest"); err != nil {
t.Fatalf("Could not create test directory: %v", err)
} else {
ni.TempDir = TempDir
}
if err := ioutil.WriteFile(path.Join(ni.TempDir, "server-key.pem"), []byte(testServerKey), 0644); err != nil {
t.Fatalf("Could not write server key")
}
if err := ioutil.WriteFile(path.Join(ni.TempDir, "server-cert.pem"), []byte(testServerCert), 0644); err != nil {
t.Fatalf("Could not write server cert")
}
if err := ioutil.WriteFile(path.Join(ni.TempDir, "client-key.pem"), []byte(testClientKey), 0644); err != nil {
t.Fatalf("Could not write client key")
}
if err := ioutil.WriteFile(path.Join(ni.TempDir, "client-cert.pem"), []byte(testClientCert), 0644); err != nil {
t.Fatalf("Could not write client key")
}
confFile := path.Join(ni.TempDir, "gonbdserver.conf")
tpl := template.Must(template.New("config").Parse(ConfigTemplate))
cf, err := os.Create(confFile)
if err != nil {
t.Fatalf("cannot create config file: %v", err)
}
if err := tpl.Execute(cf, ni.TestConfig); err != nil {
t.Fatalf("executing template: %v", err)
}
cf.Close()
oldArgs := os.Args
os.Args = []string{
"gonbdserver",
"-f",
"-c",
confFile,
}
RegisterFlags()
flag.Parse()
control := &Control{
quit: ni.quit,
}
go Run(control)
time.Sleep(100 * time.Millisecond)
os.Args = oldArgs
return ni
}
func (ni *NbdInstance) CloseConnection() {
// fmt.Fprintf(os.Stderr, ">>>> CloseConnection()\n")
ni.closedMutex.Lock()
defer ni.closedMutex.Unlock()
if ni.closed {
return
}
if ni.plainConn != nil {
ni.plainConn.Close()
ni.plainConn = nil
}
if ni.tlsConn != nil {
ni.tlsConn.Close()
ni.tlsConn = nil
}
close(ni.quit)
ni.closed = true
}
func (ni *NbdInstance) Close() {
ni.CloseConnection()
time.Sleep(100 * time.Millisecond)
os.RemoveAll(ni.TempDir)
}
// make an appropriate TLS config
func (ni *NbdInstance) getTLSConfig(t *testing.T) (*tls.Config, error) {
keyFile := path.Join(ni.TempDir, "client-key.pem")
certFile := path.Join(ni.TempDir, "client-cert.pem")
caFile := path.Join(ni.TempDir, "server-cert.pem")
// Load client cert
cert, err := tls.LoadX509KeyPair(certFile, keyFile)
if err != nil {
return nil, err
}
// Load CA cert
caCert, err := ioutil.ReadFile(caFile)
if err != nil {
return nil, err
}
caCertPool := x509.NewCertPool()
caCertPool.AppendCertsFromPEM(caCert)
// Setup HTTPS client
tlsConfig := &tls.Config{
Certificates: []tls.Certificate{cert},
RootCAs: caCertPool,
ServerName: "localhost",
}
tlsConfig.BuildNameToCertificate()
return tlsConfig, nil
}
func (ni *NbdInstance) Connect(t *testing.T) error {
var err error
ni.plainConn, err = net.Dial("unix", path.Join(ni.TempDir, "nbd.sock"))
if err != nil {
return err
}
ni.conn = ni.plainConn
ni.conn.SetDeadline(time.Now().Add(time.Second))
var magic uint64
if err = binary.Read(ni.conn, binary.BigEndian, &magic); err != nil {
return errors.Wrap(err, "Read of magic errored")
}
if magic != NBD_MAGIC {
return errors.New("Bad magic")
}
var optsMagic uint64
if err = binary.Read(ni.conn, binary.BigEndian, &optsMagic); err != nil {
return errors.Wrap(err, "Read of opts magic errored")
}
if optsMagic != NBD_OPTS_MAGIC {
return errors.New("Bad magic")
}
var handshakeFlags uint16
if err = binary.Read(ni.conn, binary.BigEndian, &handshakeFlags); err != nil {
return errors.Wrap(err, "Read of handshake flags errored")
}
if handshakeFlags != NBD_FLAG_FIXED_NEWSTYLE|NBD_FLAG_NO_ZEROES {
return errors.New("Unexpected handshake flags")
}
var clientFlags uint32 = NBD_FLAG_C_FIXED_NEWSTYLE | NBD_FLAG_C_NO_ZEROES
if err = binary.Write(ni.conn, binary.BigEndian, clientFlags); err != nil {
return errors.New("Could not send client flags")
}
t.Logf("Connected")
if ni.TLS {
tlsOpt := nbdClientOpt{
NbdOptMagic: NBD_OPTS_MAGIC,
NbdOptID: NBD_OPT_STARTTLS,
NbdOptLen: 0,
}
if err = binary.Write(ni.conn, binary.BigEndian, tlsOpt); err != nil {
return errors.New("Could not send start tls option")
}
var tlsOptReply nbdOptReply
if err := binary.Read(ni.conn, binary.BigEndian, &tlsOptReply); err != nil {
return errors.New("Could not receive Tls option reply")
}
if tlsOptReply.NbdOptReplyMagic != NBD_REP_MAGIC {
return errors.Newf("Tls option reply had wrong magic (%x)", tlsOptReply.NbdOptReplyMagic)
}
if tlsOptReply.NbdOptID != NBD_OPT_STARTTLS {
return errors.New("Tls option reply had wrong id")
}
if tlsOptReply.NbdOptReplyType != NBD_REP_ACK {
return errors.New("Tls option reply had wrong reply type")
}
if tlsOptReply.NbdOptReplyLength != 0 {
return errors.New("Tls option reply had bogus length")
}
tlsConfig, err := ni.getTLSConfig(t)
if err != nil {
return errors.Wrap(err, "Could not get TLS config")
}
tls := tls.Client(ni.conn, tlsConfig)
ni.tlsConn = tls
ni.conn = tls
ni.plainConn.SetDeadline(time.Time{})
ni.conn.SetDeadline(time.Now().Add(time.Second))
// explicitly handshake so we get an error here if there is an issue
if err := tls.Handshake(); err != nil {
return errors.Wrap(err, "TLS handshake failed: %s")
}
}
listOpt := nbdClientOpt{
NbdOptMagic: NBD_OPTS_MAGIC,
NbdOptID: NBD_OPT_LIST,
NbdOptLen: 0,
}
if err = binary.Write(ni.conn, binary.BigEndian, listOpt); err != nil {
return errors.New("Could not send list option")
}
exports := 0
listloop:
for {
var listOptReply nbdOptReply
if err := binary.Read(ni.conn, binary.BigEndian, &listOptReply); err != nil {
return errors.Wrap(err, "Could not receive list option reply")
}
if listOptReply.NbdOptReplyMagic != NBD_REP_MAGIC {
return errors.Newf("List option reply had wrong magic (%x)", listOptReply.NbdOptReplyMagic)
}
if listOptReply.NbdOptID != NBD_OPT_LIST {
return errors.New("List option reply had wrong id")
}
switch listOptReply.NbdOptReplyType {
case NBD_REP_ACK:
break listloop
case NBD_REP_SERVER:
var namelen uint32
if err := binary.Read(ni.conn, binary.BigEndian, &namelen); err != nil {
return errors.Wrap(err, "Could not receive list option reply name length")
}
name := make([]byte, namelen, namelen)
if err := binary.Read(ni.conn, binary.BigEndian, &name); err != nil {
return errors.Wrap(err, "Could not receive list option reply name")
}
if listOptReply.NbdOptReplyLength > namelen+4 {
junk := make([]byte, listOptReply.NbdOptReplyLength-namelen-4, listOptReply.NbdOptReplyLength-namelen-4)
if err := binary.Read(ni.conn, binary.BigEndian, &junk); err != nil {
return errors.Wrap(err, "Could not receive list option reply name junk")
}
}
t.Logf("Found export '%s'", string(name))
exports++
default:
return errors.New("List option reply type was unexpected")
}
}
if exports != 1 {
return errors.New("Unexpected number of exports")
}
ni.conn.SetDeadline(time.Time{})
return nil
}
func (ni *NbdInstance) Abort(t *testing.T) error {
var err error
opt := nbdClientOpt{
NbdOptMagic: NBD_OPTS_MAGIC,
NbdOptID: NBD_OPT_ABORT,
NbdOptLen: 0,
}
if err = binary.Write(ni.conn, binary.BigEndian, opt); err != nil {
return errors.Wrap(err, "Could not send start abort option")
}
var optReply nbdOptReply
if err := binary.Read(ni.conn, binary.BigEndian, &optReply); err != nil {
return errors.Wrap(err, "Could not receive abort option reply")
}
if optReply.NbdOptReplyMagic != NBD_REP_MAGIC {
return errors.Newf("abort option reply had wrong magic (%x)", optReply.NbdOptReplyMagic)
}
if optReply.NbdOptID != NBD_OPT_ABORT {
return errors.New("abort option reply had wrong id")
}
if optReply.NbdOptReplyType != NBD_REP_ACK {
return errors.New("abort option reply had wrong reply type")
}
if optReply.NbdOptReplyLength != 0 {
return errors.New("abort option reply had bogus length")
}
return nil
}
func (ni *NbdInstance) Go(t *testing.T) error {
var err error
export := "foo"
opt := nbdClientOpt{
NbdOptMagic: NBD_OPTS_MAGIC,
NbdOptID: NBD_OPT_GO,
NbdOptLen: uint32(2 + 2*1 + 4 + len(export)),
}
if err = binary.Write(ni.conn, binary.BigEndian, opt); err != nil {
return errors.Wrap(err, "Could not send go option")
}
var numInfoElements uint16 = 1
if err = binary.Write(ni.conn, binary.BigEndian, numInfoElements); err != nil {
return errors.Wrap(err, "Could not send number of elements for go option")
}
var infoElement uint16 = NBD_INFO_BLOCK_SIZE
if err = binary.Write(ni.conn, binary.BigEndian, infoElement); err != nil {
return errors.Wrap(err, "Could not send go info element")
}
nameLength := uint32(len(export))
if err = binary.Write(ni.conn, binary.BigEndian, nameLength); err != nil {
return errors.Wrap(err, "Could not send go export length")
}
if err = binary.Write(ni.conn, binary.BigEndian, []byte(export)); err != nil {
return errors.Wrap(err, "Could not send go export name")
}
infoloop:
for {
var optReply nbdOptReply
if err := binary.Read(ni.conn, binary.BigEndian, &optReply); err != nil {
return errors.Wrap(err, "Could not receive go option reply")
}
if optReply.NbdOptReplyMagic != NBD_REP_MAGIC {
return errors.Newf("Go option reply had wrong magic (%x)", optReply.NbdOptReplyMagic)
}
if optReply.NbdOptID != NBD_OPT_GO {
return errors.New("Go option reply had wrong id")
}
switch optReply.NbdOptReplyType {
case NBD_REP_ACK:
break infoloop
case NBD_REP_INFO:
var infotype uint16
if err := binary.Read(ni.conn, binary.BigEndian, &infotype); err != nil {
return errors.Wrap(err, "Could not receive go option reply name length")
}
switch infotype {
case NBD_INFO_EXPORT:
if optReply.NbdOptReplyLength != 12 {
return errors.New("Bad length in NBD_INFO_EXPORT")
}
var exportSize uint64
var transmissionFlags uint16
if err := binary.Read(ni.conn, binary.BigEndian, &exportSize); err != nil {
return errors.Wrap(err, "Could not receive NBD_INFO_EXPORT export size")
}
if err := binary.Read(ni.conn, binary.BigEndian, &transmissionFlags); err != nil {
return errors.Wrap(err, "Could not receive NBD_INFO_EXPORT transmission flags")
}
ni.transmissionFlags = transmissionFlags
t.Logf("Transmission flags: FLUSH=%v, FUA=%v",
transmissionFlags&NBD_FLAG_SEND_FLUSH != 0,
transmissionFlags&NBD_FLAG_SEND_FUA != 0)
default:
t.Logf("Ignoring info type %d", infotype)
if optReply.NbdOptReplyLength > 2 {
junk := make([]byte, optReply.NbdOptReplyLength-2, optReply.NbdOptReplyLength-2)
if err := binary.Read(ni.conn, binary.BigEndian, &junk); err != nil {
return errors.Wrap(err, "Could not receive go option reply name junk")
}
}
}
default:
return errors.New("List option reply type was unexpected")
}
}
return nil
}
func (ni *NbdInstance) CreateFile(t *testing.T, size int64) error {
filename := path.Join(ni.TempDir, "nbd.img")
file, err := os.Create(filename)
if err != nil {
return err
}
defer file.Close()
return file.Truncate(size)
}
func (ni *NbdInstance) Disconnect(t *testing.T) error {
var err error
cmd := nbdRequest{
NbdRequestMagic: NBD_REQUEST_MAGIC,
NbdCommandFlags: 0,
NbdCommandType: NBD_CMD_DISC,
NbdHandle: getHandle(),
NbdOffset: 0,
NbdLength: 0,
}
if err = binary.Write(ni.conn, binary.BigEndian, cmd); err != nil {
return errors.Wrap(err, "Could not send disconnect command")
}
time.Sleep(100 * time.Millisecond)
return nil
}
func doTestConnection(t *testing.T, tls bool) {
ni := StartNbd(t, TestConfig{TLS: tls, NoFlush: *noFlush})
defer ni.Close()
if err := ni.Connect(t); err != nil {
t.Logf("Error on connect: %v", err)
t.Fail()
return
}
if err := ni.Abort(t); err != nil {
t.Logf("Error on abort: %v", err)
t.Fail()
return
}
}
func TestConnection(t *testing.T) {
doTestConnection(t, false)
}
func TestConnectionTls(t *testing.T) {
doTestConnection(t, true)
}
func doTestConnectionIntegrity(t *testing.T, transationLog []byte, tls bool, driver string) {
if _, ok := backendMap[driver]; !ok {
t.Skip(fmt.Sprintf("Skipping test as driver %s not built", driver))
return
}
ni := StartNbd(t, TestConfig{TLS: tls, Driver: driver, NoFlush: *noFlush})
defer ni.Close()
if err := ni.CreateFile(t, 50*1024*1024); err != nil {
t.Logf("Error on create file: %v", err)
t.Fail()
return
}
if err := ni.Connect(t); err != nil {
t.Logf("Error on connect: %v", err)
t.Fail()
return
}
if err := ni.Go(t); err != nil {
t.Logf("Error on go: %v", err)
t.Fail()
return
}
it := ni.NewIntegrityTest(t, transationLog)
defer it.Close()
if err := it.Run(); err != nil {
t.Logf("Error on Integrity Test: %v", err)
t.Fail()
return
}
}
func TestConnectionIntegrity(t *testing.T) {
doTestConnectionIntegrity(t, []byte(testTransactionLog), false, "file")
}
func TestConnectionIntegrityTls(t *testing.T) {
doTestConnectionIntegrity(t, []byte(testTransactionLog), true, "file")
}
func TestConnectionIntegrityHuge(t *testing.T) {
if !*longtests {
t.Skip("Skipping this test as long tests not enabled (use -longtests to enable)")
} else {
doTestConnectionIntegrity(t, []byte(testHugeTransactionLog), false, "file")
}
}
func TestConnectionIntegrityHugeTls(t *testing.T) {
if !*longtests {
t.Skip("Skipping this test as long tests not enabled (use -longtests to enable)")
} else {
doTestConnectionIntegrity(t, []byte(testHugeTransactionLog), true, "file")
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,184
|
Q: Marrying Statistical Mechanics and Differential Geometry Recently, I have been trying to get a more thorough understanding of the mathematics at the fundaments of theoretical physics. I liked V.I. Arnol'd's introduction to mechanics and the book by Ratiu and Marsden, but what really blew me away were Marian Fecko's book and Ferederic P. Schuller's lectures on the "Geometric Anatomy of Theoretical Physics". Schuller, especially, has this really clear vision of introducing more an more structured mathematical objects, starting with topological manifolds and peaking at Yang-Mills fields/connections on associated fiber bundles.
With that, the mathematical objects that pop up in physics have quite nice definitions and I personally found this way of trying to rigorously define the spaces we work in and objects we work with very refreshing.
Now I would like to expand on this a bit and I was wondering if there is a nice article that just picks off at a description of classical mechanics in terms of phase space as a symplectic manifold (or something along these lines) and slaps a nice coat of $\sigma$-algebra on top and equips the whole thing with a probability measure. Non-rigorously, that's in most StatMech textbooks, but I was wondering if someone knows a nice reference where this is carried out with some clean-cut mathematical definitions and gory technicalities. I'm also ok with a detour via quantum mechanics or QFT, but I'm especially interested in how one can take the phase space geometry and supplement it with probability theory.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,286
|
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,006
|
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