10, if T > 10 *W*log2*W*, then *T* > *W**l**o**g*2*T*, yielding the further relaxed the condition on *T* as \n$$T \\geq \\frac{d}{\\epsilon^4} \\; polylog(d,\\frac{1}{\\epsilon},\\frac{1}{\\delta}).$$ \n[equivalence] Consider an arbitrary ground truth permutation *π*\\*. For all time steps *t* ∈ [1, *n*], let *y**t* = *π*\\*(*x**t*). Consider the online prediction game of predicting *y**t* + 1 at each time step given previous observation *y*1 : *t* without knowing *π*\\* but knowing $\\Tilde{p}$. Then, $\\Tilde{p}'(y\\_{t+1}| y\\_{1:t})$ is equivalent to the multiplicative weights algorithm’s prediction of *y**t* + 1 with the Hedge strategy of Freund and Schapire, where it\n* Considers *d*! experts corresponding to guessing each permutation *π*ʹ is the ground truth permutation.\n* Maintains a weight *w**π*ʹ(*t*) for each expert at time step *t*, and the weights are initially as $\\mathcal{\\Tilde{P}}(\\pi)$.\n* Picks a distribution across experts $p\\_{\\pi'}^{(t)}=\\frac{w\\_{\\pi'}^{(t)}}{\\Phi^{(t)}}$ where Φ(*t*) = ∑*j**w**j*(*t*).\n* Produces prediction of *y**t* + 1 as $ \\sum\\_{\\pi'}p\\_{\\pi'}^{(t)}\\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t+1} | y\\_{1:t}) $\n* Receives a cost vector of $m^{(t)}\\_{\\pi'}=-\\frac{1}{\\epsilon} \\log{\\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t+1} | y\\_{1:t})}$.\n* Updates the weights *w**i*(*t* + 1) = *w**i*(*t*)exp( − *ε**m**i*(*t*)) and repeat\nWe can first see that $p\\_{\\pi'}^{(t)} = \\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t})$ by induction:\nBase case: $p\\_{\\pi'}^{(0)} = \\mathcal{\\Tilde{P}}(\\pi)$ by assumption.\nInductive Case:\nWith the cost vector as $m^{(t-1)}\\_{\\pi'}=-\\frac{1}{\\epsilon} \\log{\\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t} | y\\_{1:t-1})}$, the update at step *t* is $w\\_{\\pi'}^{(t)}=w\\_{\\pi'}^{(t-1)} \\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t} | y\\_{1:t-1})$. Therefore, the probability over any particular expert *π*ʹ is\n \n$$\\begin{aligned}\n\\label{mul}\n p\\_{\\pi'}^{(t)}&=\\frac{w\\_{\\pi'}^{(t)}}{\\Phi^{(t)}}\\\\\n &=\\frac{w\\_{\\pi'}^{(t-1)} \\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t} | y\\_{1:t-1})}{\\sum\\_j w\\_{j}^{(t-1)} \\mathcal{\\Tilde{P}}\\_{j}(y\\_{t} | y\\_{1:t-1})}\\\\\n &=\\frac{p\\_{\\pi'}^{(t-1)}\\Phi^{(t-1)} \\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t} | y\\_{1:t-1})}{\\sum\\_j p\\_{j}^{(t-1)}\\Phi^{(t-1)} \\mathcal{\\Tilde{P}}\\_{j}(y\\_{t} | y\\_{1:t-1})}\\\\\n &=\\frac{\n p\\_{\\pi'}^{(t-1)} \n \\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t} | y\\_{1:t-1})}{\\sum\\_j p\\_{j}^{(t-1)} \\mathcal{\\Tilde{P}}\\_{j}(y\\_{t} | y\\_{1:t-1})}\\\\\\end{aligned}$$ \nThis is equivalent to the update given by Bayes’ rule when plugging in $p\\_{\\pi'}^{(t)} = \\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t})$ :\n \n$$\\begin{aligned}\n \\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t}) &= \\frac{\n \\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t-1})\n \\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t} | y\\_{1:t-1})}\n {\\mathcal{\\Tilde{P}}(y\\_{t} | y\\_{1:t-1})}\\end{aligned}$$ \nSo we can conclude that $p\\_{\\pi'}^{(t)} = \\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t})$, i.e. the process of updating the probability distribution across experts within the prediction game is equivalent to the process of the language model updating the probabilities $\\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t+1}) $ across permutations *π*ʹ. And this means that the algorithm’s prediction $\\sum\\_{\\pi'}p\\_{\\pi'}^{(t)}\\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t+1} | y\\_{1:t}) = \\sum\\_{\\pi'} \\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t}) \\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t+1} | y\\_{1:t}) = \\mathcal{\\Tilde{P}}(y\\_{t+1}| y\\_{1:t}) = \n\\Tilde{p}'(y\\_{t+1}| y\\_{1:t}) $\n[lemma:mweights] When using the Hedge strategy for the multiplicative weights algorithm, the average difference between the weighted distribution across experts and any particular expert *π* is bounded as \n$$\\frac1T \\sum\\_t^T \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t})}\n {\\Tilde{p}'(y\\_{t+1} | y\\_{1:t})} \n \\leq 2\\epsilon^2$$ \n for *ε* ≤ 1 and for $T\\geq \\left(4\n \\log^2 \\left(\\frac{d}\\sigma \\right)\n\\log (d!)\\right)/\\epsilon^4$.\nConsider an arbitrary expert *π*.\nWe first show that the cost vectors are bounded by $\\rho= -\\frac{1}{\\epsilon} \\log \\frac\\sigma{d}$: Recall we defined $m^{(t)}\\_{\\pi}=-\\frac{1}{\\epsilon} \\log{\\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t})}$. By the definition of our redistributed probability distribution, at time step *t* ∈ [1, …, *T*],\n \n$$\\begin{aligned}\n \\frac\\sigma{d} &\\leq \\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t}) \\leq 1\\\\\n \\log \\frac\\sigma{d} &\\leq \\log\\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t}) \\leq 0 \\\\\n 0 &\\leq m^{(t)}\\_{\\pi} \\leq -\\frac{1}{\\epsilon} \\log \\frac\\sigma{d}\\\\\n 0 &\\leq m^{(t)}\\_{\\pi} \\leq -\\frac{1}{\\epsilon} \\log \\frac\\sigma{d}.\\end{aligned}$$ \nBy corollary 16.3 in, if we have cost vectors *m*(*t*) ∈ [ − *ρ*, *ρ*]*d*!, then for time *T* ≥ (4*ρ*2log(*d*!))/*ε*2 where *ε* ≤ 1,\n \n$$\\begin{aligned}\n \\frac1T\\sum\\_t^T p^{(t)}\\cdot m^{(t)} \\leq \\frac1T \\sum\\_t^T m\\_{\\pi}^{(t)}+2\\epsilon.\\end{aligned}$$ \nNote that we can simplify $T\\geq \\left(4\n \\log^2 \\left(\\frac{d}\\sigma\\right)\n\\log (d!)\\right)/\\epsilon^4$.\nWe can now bound\n \n$$\\begin{aligned}\n \\frac1T\\sum\\_t^T \\left( p^{(t)}\\cdot m^{(t)} - m\\_{\\pi}^{(t)}\\right) &\\leq 2\\epsilon\\\\ \n \\frac1T\\sum\\_t^T \n \\left(\n \\sum\\_{\\pi'}\n p^{(t)}\\_{\\pi'} m^{(t)}\\_{\\pi'} - m\\_{\\pi}^{(t)}\\right) \n &\\leq 2\\epsilon\\\\\n \\frac1T\\sum\\_t^T \n \\left(\n \\sum\\_{\\pi'}\n \\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t}) \n \\left(-\\frac{1}{\\epsilon} \\log{\\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t+1} | y\\_{1:t})}\\right) \n - \\left(-\\frac{1}{\\epsilon} \\log{\\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t})}\\right) \n \\right)\n &\\leq 2\\epsilon\\\\\n \\frac{1}{\\epsilon T}\\sum\\_t^T \n \\sum\\_{\\pi'}\n \\left(\n \\mathcal{\\Tilde{P}}(\\pi' | y\\_{1:t}) \n \\left(\\log{\\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t})} - \\log{\\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t+1} | y\\_{1:t})}\\right) \n \\right)\n &\\leq 2\\epsilon\\\\\n \\frac1T \\sum\\_t^T \\mathbb{E}\\_{\\pi'} \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t})}\n {\\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t+1} | y\\_{1:t})} \n &\\leq 2\\epsilon^2\\end{aligned}$$ \nBy Jensen’s inequality, we also have that\n \n$$\\begin{aligned}\n \\frac1T \\sum\\_t^T \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t})}\n {\\mathbb{E}\\_{\\pi'} \n \\mathcal{\\Tilde{P}}\\_{\\pi'}(y\\_{t+1} | y\\_{1:t})} \n &\\leq 2\\epsilon^2 \\\\\n \\frac1T \\sum\\_t^T \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_{\\pi}(y\\_{t+1} | y\\_{1:t})}\n {\\Tilde{p}'(y\\_{t+1} | y\\_{1:t})} \n &\\leq 2\\epsilon^2 \\end{aligned}$$ \n[martingale] Let\n \n$$\\begin{aligned}\n Z\\_i &= \\sum\\_{t=1}^i\n \\left(\n D\\_{KL}(\n \\mathcal{\\Tilde{P}}\\_I(x\\_{t+1} | x\\_{1:t})\n \\|\n \\mathcal{\\Tilde{P}}(x\\_{t+1} | x\\_{1:t})\n ) - \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_I(x\\_{t+1} | x\\_{1:t})}\n {\\mathcal{\\Tilde{P}}(x\\_{t+1} | x\\_{1:t})} \n \\right)\n \\end{aligned}$$ \n*Z**i* is a martingale.\nConsider\n \n$$\\begin{aligned}\n \\mathbb{E}\\_{x\\_{i+1}\\sim \\mathcal{\\Tilde{P}}\\_I} \\left[Z\\_i\\right]\n &=\n \\mathbb{E}\\_{x\\_{i+1}\\sim \\mathcal{\\Tilde{P}}\\_I} \\left[\n \\sum\\_{t=1}^i\n \\left(\n D\\_{KL}(\n \\mathcal{\\Tilde{P}}\\_I(x\\_{t+1} | x\\_{1:t})\n \\|\n \\mathcal{\\Tilde{P}}(x\\_{t+1} | x\\_{1:t})\n ) - \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_I(x\\_{t+1} | x\\_{1:t})}\n {\\mathcal{\\Tilde{P}}(x\\_{t+1} | x\\_{1:t})} \n \\right)\n \\right] \\\\ \n&= \\mathbb{E}\\_{x\\_{i+1}\\sim \\mathcal{\\Tilde{P}}\\_I} \\left[\n D\\_{KL}(\n \\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})\n \\|\n \\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})\n ) - \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})}\n {\\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})} + Z\\_{i-1}\n \\right]\\end{aligned}$$ \nObserve that *Z**i* − 1 has no dependence on *x**i* + 1.\n \n$$\\begin{aligned}\n\\mathbb{E}\\_{x\\_{i+1}\\sim \\mathcal{\\Tilde{P}}\\_I} \\left[Z\\_i\\right]\n &= \\mathbb{E}\\_{x\\_{i+1}\\sim \\mathcal{\\Tilde{P}}\\_I} \\left[\n \\mathbb{E}\\_{x\\_{i+1}\\sim \\mathcal{\\Tilde{P}}\\_I} \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})}\n {\\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})} \\right]\n - \\mathbb{E}\\_{x\\_{i+1}\\sim \\mathcal{\\Tilde{P}}\\_I} \\left[ \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})}\n {\\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})} \n \\right] + Z\\_{i-1}\\\\\n &= Z\\_{i-1}\\end{aligned}$$ \nTherefore, *Z**i* is a martingale.\n∣*Z**i* − *Z**i* − 1∣ ≤ *c**i* where $c\\_i = 2 |\\log\\frac{d}{\\sigma}|$\nWe have\n \n$$\\begin{aligned}\n |Z\\_i-Z\\_{i-1}| &= \\left|\n D\\_{KL}(\n \\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})\n \\|\n \\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})\n ) - \\log \\frac\n {\\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})}\n {\\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})} \\right|\\end{aligned}$$ \nIn our redistributed probability distribution $\\mathcal{\\Tilde{P}}$, we have $\\frac\\sigma{d} \\leq \\mathcal{\\Tilde{P}}\\_{\\pi}(x\\_i | x\\_{1:i-1}) \\leq 1$ for any *π* at any time *i*. Therefore,\n \n$$\\begin{aligned}\n \\log\\frac\\sigma{d} \n \\leq \n\\log \\frac\n {\\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})}\n {\\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})}\n \\leq \\log \\frac{d}{\\sigma}.\\end{aligned}$$ \nAlso, we can find an upper bound for the KL divergence by maximizing $\\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})$ to 1 and minimizing $\\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})$ to $\\frac\\sigma{d}$ so that\n \n$$\\begin{aligned}\n D\\_{KL}(\n \\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})\n \\|\n \\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})\n ) &= \\sum\\_{x\\_{i+1}} \\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})\n \\log\\frac{\\mathcal{\\Tilde{P}}\\_I(x\\_{i+1} | x\\_{1:i})}{\\mathcal{\\Tilde{P}}(x\\_{i+1} | x\\_{1:i})} \\\\\n &\\leq \\log\\frac{d}{\\sigma} \\end{aligned}$$ \nWe can maximize ∣*Z**i* − *Z**i* − 1∣ by maximizing the first term and minimizing the second term, or vice versa. In the first case, $|Z\\_i - Z\\_{i-1}|\\leq |\\log\\frac{d}\\sigma - \\log\\frac\\sigma{d}|=2|\\log\\frac{d}{\\sigma}|$. In the other case, $|Z\\_i - Z\\_{i-1}| \\leq |0-\\log\\frac{d}\\sigma| = |\\log\\frac{d}\\sigma|$.\nTherefore, ∣*Z**i* − *Z**i* − 1∣ ≤ *c**i* where $c\\_i = 2|\\log\\frac{d}{\\sigma}|$.\n[lemma:azuma] By Azuma’s inequality, with probability 1 − *δ*, we have that ∥*Z**T*∥ ≤ *b* where $b=2 \\log\\frac{d}{\\sigma} \\sqrt{-8T \\log \\frac{1}{\\delta}}$\nBy Azuma’s inequality, for all positive reals *b*,\n \n$$\\begin{aligned}\n P(Z\\_T-Z\\_1\\geq b) &\\leq \\exp\\left(\\frac{-b^2}{2\\sum\\_{k=2}^T c^2\\_k}\\right) \\\\\n P(Z\\_T-Z\\_1\\leq b) &\\geq 1 - \\exp\\left(\\frac{-b^2}{2\\sum\\_{k=2}^T c^2\\_k}\\right)\\\\\n &\\geq 1 - \\exp\\left(\\frac{-b^2}{8\\sum\\_{k=2}^T \\log^2\\frac{d}{\\sigma}}\\right)\\end{aligned}$$ \nWe can rewrite in terms of $\\delta = \\exp\\left(\\frac{-b^2}{8\\sum\\_{k=2}^T \\log^2\\frac{d}{\\sigma}}\\right)$ so\n \n$$\\begin{aligned}\n b &= \\sqrt{-\\left({8\\sum\\_{k=2}^T \\log^2\\frac{d}{\\sigma}}\\right) \\log \\delta} \\\\\n &\\leq \\log\\frac{d}{\\sigma} \\sqrt{-8T \\log \\frac{1}{\\delta}} \\end{aligned}$$ \nTherefore, \n*P*(*Z**T* − *Z*1 ≤ *b*) ≥ 1 − *δ* \nModel Architecture Details\n==========================\nIn addition, we add a learnable scaling and bias parameter to the result of the embedding layer, so that the model can still learn to scale it as needed.\nConvergence on other datasets\n=============================\nFigure [fig:convegence3] shows the perplexity of lexinvariantLMs across the three different datasets. Note that Github converges significantly faster than standard Engish text like Wiki-40B, since code is more structured and easier to decipher the token permutation.\n [fig:convegence3]\nCode Deciphering Full Examples\n==============================\nJava:\n```\nbinary_search()z\n if (high >= low)z\n mid = (high + low) / 2;\n if (arr[mid] == x)\n return mid;\n if (arr[mid] > x)z\n high = mid - 1;\n return binary_search();\n } elsez\n low = mid + 1;\n return binary_search();\n }\n } elsez\n return -1;\n }\n}\nvoid func2()z\n```\nPython:\n```\nbinary_search()z\n if (high >= low)z\n mid = (high + low) // 2\n if (arr[mid] == x)z\n return mid\n if (arr[mid] > x)z\n high = mid - 1 \n return binary_search() \n elsez\n low = mid + 1\n return binary_search()\n elsez\n return -1\ndef func2()z\n```\nSemantic Deciphering Full Example\n=================================\n`’crash!’ ’aaah!’ i looked up from my cup of coffee. ’crash!’ - that was the cafe window. and ’aaah!’ - that was kate. people in the cafe shouted. kate and i ran to the window. there was no one there. then i turned to kate and put my arm around her. ’are you all right?’ i asked. ’yes,’ she said. ’i think so.’ ’what is it?’ some one shouted and a short red-faced man ran into the room. the man took my arm. ’matt! what are you doing to kate?’ he asked. ’nothing, papa,’ kate replied. ’it wasn’t him. it was from out in the street.’ the red-faced man looked at the window and then at me. he turned to his daughter. ’are you ok, kate?’ he asked. kate gave him a little smile. ’yes, i think i am, papa,’ she said. then her father spoke to me. ’sorry, matt. i heard kate and i thought...’ ’that’s ok, paolo,’ i answered. it was ok. you see, this is soho, in the centre of london. in the day it’s famous for music and films. at night people come and eat and drink in the restaurants. expensive restaurants and cheap restaurants; italian restaurants and chinese restaurants. and day and night there are internet cafes like the web cafe. in soho you can buy any thing and any one. there are lots of nice people in soho. but there are also lots of people who are not very nice. i know because i live and work here. i often take a drink to a shop or cafe. i’m not rich and famous. and i don’t know a lot. but i do know soho. what one here is a drink - restaurants - music - coffee - father the one here that drink is`\nExample prediction of the lexinvariantwith 32k vocabulary train on the Pile:\n `- coffee. and i`\nSynthetic Reasoning Task\n========================\nTable [tab:syn-freq] shows a variant of the synthetic reasoning task results in Subsection [tab:syn], where the symbols are instead sampled proportion to the token frequencies. Although the improvement still generally holds, the standard LM with character-based vocabulary becomes significantly better. We believe that this is because the model can get a significant advantage by guessing among the most common letter.\n[htb]\nll cccc Dataset & Vocab & & \n(lr)3-4 (lr)5-6 & & Standard & LI & Standard & LI \n &char & 72.80 & 90.95 & 40.63 & 60.47 \n&32k & 61.20 & 90.95 & 40.55 & 54.55 \n &char & 75.55 & 63.45 & 42.71 & 59.86 \n&32k & 41.05 & 57.95 & 26.81 & 51.86 \n &char & 66.00 & 86.75 & 36.62 & 70.77 \n&32k & 59.25 & 78.45 & 37.46 & 65.04 \n[tab:syn-freq]\n Language Models Regularized with Lexinvariance and BIG-bench Results\n=====================================================================\nAs described in the main paper, we implement a lexinvariance regularized Model in a way similar to embedding dropout. Note that one problem in implementing it naively by using random Gaussian embeddings and learned embedding in a mixture is that the two would become quickly distinguishable from each other during training since learned embeddings often have larger norms, allowing the model simply ignore the randomized tokens. So instead of using random Gaussian embedding matrices in place of a learned embedding matrix, we explored another approach for training a lexinvariantregularized LM: training a standard LM with learnable embedding matrix over sequences partially applied with a random token permutation *B**p*(*x*1, *π*), …, *B**p*(*x*1, *π*), where *B**p*(*x**i*, *π*) = *π*(*x**i*) with probability *p* and *B**p*(*x**i*, *π*) = *x**i* with probability 1 − *p*. Since each token can be remapped to any other token with equal chance, the produced model should ideally also be lexinvariantwhen *p* = 1, though with no strict guarantees. In practice, we found the models trained this way behave very similarly to models with random Gaussian embedding.\nWe evaluate our model over BIG-bench tasks where the language model performance scales well, and we prioritize evaluating generative tasks over multiple-choice tasks. Tasks we evaluated on:\ngre reading comprehension.mul, linguistics puzzles.gen, linguistics puzzles.gen, rhyming.gen, tellmewhy.gen, simple arithmetic multiple targets json.gen, simple arithmetic json subtasks.gen, disfl qa.gen, arithmetic.gen, bridging anaphora resolution barqa.gen, matrixshapes.gen, sufficient information.gen, logical args.mul, novel concepts.mul, code line description.mul, unnatural in context learning.gen, unit interpretation.mul, english proverbs.mul, general knowledge.mul, geometric shapes.gen, human organs senses.mul, contextual parametric knowledge conflicts.gen, crass ai.mul, auto categorization.gen, penguins in a table.gen, hindu knowledge.mul, english russian proverbs.mul, modified arithmetic.gen, cryobiology spanish.mul, evaluating information essentiality.mul, intent recognition.mul, understanding fables.mul, figure of speech detection.mul, empirical judgments.mul, simple ethical questions.mul, swahili english proverbs.mul, language identification.mul, phrase relatedness.mul, nonsense words grammar.mul, undo permutation.mul, object counting.gen, identify odd metaphor.mul, elementary math qa.mul, social iqa.mul, parsinlu qa.mul, metaphor understanding.mul, timedial.mul, causal judgment.mul, list functions.gen, implicatures.mul, date understanding.mul, codenames.gen, fact checker.mul, physics.mul, abstract narrative understanding.mul, emojis emotion prediction.mul, metaphor boolean.mul, strategyqa.gen, ascii word recognition.gen, auto debugging.gen, cause and effect.mul, conlang translation.gen, cryptonite.gen, cs algorithms.mul, dyck languages.mul, gender inclusive sentences german.gen, hindi question answering.gen, international phonetic alphabet transliterate.gen, irony identification.mul, logical fallacy detection.mul, movie dialog same or different.mul, operators.gen, paragraph segmentation.gen, parsinlu reading comprehension.gen, repeat copy logic.gen, rephrase.gen, simple arithmetic json.gen, simple arithmetic multiple targets json.gen, sports understanding.mul, word unscrambling.gen, hyperbaton.mul, linguistic mappings.gen, anachronisms.mul, indic cause and effect.mul, question selection.mul, hinglish toxicity.mul, snarks.mul, vitaminc fact verification.mul, international phonetic alphabet nli.mul, logic grid puzzle.mul, natural instructions.gen, entailed polarity.mul, list functions.gen, conceptual combinations.mul, goal step wikihow.mul, logical deduction.mul, conlang translation.gen, strange stories.mul, odd one out.mul, mult data wrangling.gen, temporal sequences.mul, analytic entailment.mul, disambiguation qa.mul, sentence ambiguity.mul, swedish to german proverbs.mul, logical sequence.mul, chess state tracking.gen, reasoning about colored objects.mul, implicit relations.mul, riddle sense.mul, physical intuition.mul, simple arithmetic json multiple choice.mul, geometric shapes.gen, gem.gen, simp turing concept.gen, common morpheme.mul, qa wikidata.gen, international phonetic alphabet transliterate.gen, similarities abstraction.gen, rephrase.gen, emoji movie.gen, qa wikidata.gen, word sorting.gen, emoji movie.gen, qa wikidata.gen, periodic elements.gen, hindi question answering.gen\nBellow, we plot the net percentage of tasks improved/deproved in each of the BIG-bench categories, out of the tasks that are changed by at least a threshold amount.\n![BIG-bench results with 0,1,2 and 3 shots.](figures/latest.png \"fig:\") [fig:mylabel]\nCompute\n=======\nWe use one TPU v3-8 for all our pretraining runs. It takes approximately 23 hours for each pretraining run.\nBroader Impacts\n===============\nOur work primarily provides a scientific exploration and understanding of the properties of lexinvariantlanguage models. More broadly, these properties could potentially help improve the robustness, generalizability, and reasoning ability of LMs in the future works. In general we don’t foresee more specific negative societal impacts from this work other than general misuse of language models.\n---\n1. We specifically consider lexical symbols as tokens, not necessarily words or other linguistic units.[↩](#fnref1)\n2. The convergence rate could be better depending on the language distribution, such as on math and code, where the symbols should have clear functional meaning in context. We explore this empirically in the experiment section.[↩](#fnref2)\n"},"data_id":{"kind":"string","value":"arxiv_0181639"}}},{"rowIdx":45388,"cells":{"content":{"kind":"string","value":"Stabilization of Age-Structured Chemostat Hyperbolic PDE with Actuator Dynamics\n===============================================================================\nIntroduction\n============\nMotivation: Chemostat as a benchmark for nonlinear control and for control in epidemiology\n------------------------------------------------------------------------------------------\nMany industrial biotechnology processes can be described by a nonlinear PDE (Partial Differential Equation) of population dynamics that are structured around age. Reflecting the maturation of enzymes, microbial organisms, animal or plant cells and tissues, these models of biological and biochemical systems are often exploited to favor the desired functionalities and achieve cost-effective bioreactors’ production rate. The word “chemostat” denotes a biological process which is fed a nutrient at a certain rate and from which a bioproduct-nutrient mix is being extracted/removed at the same rate. This rate is referred to as the “dilution rate.” The relevance of chemostats in control engineering extends beyond their own application in biotechnology (such as the manufacturing of pharmaceuticals). First, a chemostat is a nonlinear control problem—even when the limiting substrate(s) nonlinear dynamics are neglected—due to the fact that its dynamic model contains a product of the population density and the dilution rate. Second, the relevance of the chemostat is also in the fact that it is an example of a “positive system,” namely, a control system whose state is subject to an inequality constraint.\nThird, and perhaps most important at present, chemostat is a special case of more complex population dynamics such as those that arise in epidemiology. Before one takes on control design of SIR-type (susceptible-infected-recovered) models in epidemiology, in which control is being applied to more than one population (for example, to the susceptible and the infected populations), it is necessary to understand how to control single-population chemostats. Just as there is a parallel between the biomass in a chemostat and the infected population in epidemiology, as well as between the substrate in a chemostat and the susceptible population in epidemiology, there is likewise a parallel between the treatment/therapy being administered as an input in epidemiology and the dilution input of a chemostat.\nHence, the study of control of an age-structured chemostat that we undertake here is an important first step towards designing controllers for *age-structured* models in epidemiology.\nAge-structured population dynamics\n----------------------------------\nAge-structured chemostat models are described by a particular first order hyperbolic Partial Differential Equation (PDE) with a non-local boundary condition: the McKendrick-von Foerster equation (see References and the references therein). Age-structured models are extensions of the chemostat models described by Ordinary Differential Equations (ODEs; see Reference ). A particular and deep mathematical tool has been developed for the study of age-structured models: the ergodic theorem, also known as the asymptotic behavior (see References for similar results on asymptotic similarity and Reference for the proof of the ergodic theorem by means of Lyapunov arguments).\nControl studies for age-structured models are rare. Optimal control problems have been studied (see References and the references therein). On the other hand, feedback control of infinite-dimensional population dynamics was introduced in Reference and further considered in References. More specifically, the use of the ergodic theorem and the corresponding study of linear integral delay equations in References led to a special nonlinear infinite-dimensional change of variables, from the age-structured population density (a population density that is a function of the continuum age) into\n* a scalar variable that represents the controllable mode of the bilinear non-local hyperbolic PDE system which is directly actuated by the dilution rate input, and\n* an infinite-dimensional uncontrollable but exponentially stable dynamical system described by linear integral delay equations,\nand has subsequently been the cornerstone of the control design and stability analysis of age-structured population dynamics.\nThe control studies in References and used the dilution rate as the control input as in many other control studies of chemostat models described by ODEs (see References ). Actuation by dilution rate is both physical and plausible and it amounts to harvesting the population. In simplistic terms, a steady level of harvesting is needed to maintain the population at a sustained level (equilibrium) and, when the population exceeds such an equilibrium, it is over-harvested by (over)dilution, whereas when the population drops below such an equilibrium, it is under-harvested by (under)dilution. How this over- and under-harvesting is to be done exactly is highly non-trivial because the population is age-structured, meaning that its state is functional/infinite-dimensional, due to which one cannot really speak of overpopulation and underpopulation in a bulk sense. One has, instead, to take into account that the population can vary from the equilibrium in infinitely many ways (overpopulated old with underpopulated young, vice-versa, and so on). This infinite variety in the state profiles, in addition to the infinite number of states (age-specific densities), along with the nonlinearity of the problem, is one of the theoretical attractions to this control design problem.\nChemostat with dynamic actuation of dilution\n--------------------------------------------\nWhile dilution is a plausible form of actuation, it is not possible to actuate it instantaneously. As with any other actuation, it is governed by its own dynamics. In the case of dilution, it is affected by the dynamics of pipes and valves that are used to run the nutrients into a chemostat and the population-nutrient mix out of the chemostat. In biochemical engineering, the dilution rate is defined as the ratio of the inlet volumetric flow rate to the reactor volume. Since the volume of the reactor is not necessarily constant, it follows that the dynamics of the dilution rate can play an important role.\nIn this paper we design stabilizing feedback laws for the age-structured chemostat with dilution dynamics governed by a single, scalar integrator. In the most advanced versions of our designs, we ensure that not only the population density, but also the dilution rate, remains in a specific interval (and positive). Dilution rate being positive means that we are not feeding population back into the chemostat but are only harvesting/removing population.\nWhile the age-structured chemostat with a constant dilution rate is a PDE system with one eigenvalue at the origin and all other eigenvalues in the left half plane, meaning that it is not open-loop unstable, and meaning that it can, at worst, settle at an undesirable age profile for the population (provided the constant dilution is at a population-sustaining level), the system with dilution dynamics modeled by an integrator has two eigenvalues at the origin, meaning that it is open-loop unstable. This is one more way of noting the need for a more sophisticated control design for this PDE system once the dynamics of the dilution actuation are introduced.\nIn this paper we tackle three design challenges:\n* we overcome the dynamics of the integrator modeling the dilution actuation,\n* we perform the design and the analysis using feedback of only a sensor of bulk population, which has an unknown profile in sensing populations of different ages,\n* we ensure that the dilution rate that drives the population to the desired age-structured equilibrium from all initial positive population density age profiles remains itself in a specific interval (and positive).\nWhile in the initial work on control of the age-structured chemostat the state space of the systems was, so to speak, the “positive orthant” of infinite dimension (the age-structured population density), in this work the state space increases by one positive-valued state when the dilution dynamics are modeled by an integrator and becomes, so to speak, a “positive orthant” of dimension ∞ + 1. From a mathematical point of view, the paper deals with a global feedback stabilization problem for a constrained nonlinear infinite-dimensional control system which is defined on a specific open set. Finite-dimensional control systems defined on open sets were recently studied in Reference.\n***Organization:*** The structure of the paper is as follows. Section [secpopDynamicsActuatorDynamicsBackstepping-model] introduces the mathematical model as well as the stabilization problem. Sections [secpopDynamicsActuatorDynamicsBackstepping] and [secBacksteppingStabilityProof] study the backstepping design of the problem with unrestricted dilution rate assuming that the state is available, while Section [secRelaxedBackstepping] provides globally stabilizing output feedback laws. Sections [secSafety] and [sec-constrained-miroslav] are devoted to the feedback stabilization problem of the more demanding case where the dilution rate is restricted to take values in a specific interval.\n***Notation:***\n* ${\\mathbb R}\\_+ $ denotes the interval [0, + ∞).\n* Let *U* be an open subset of a metric space and $\\Omega \\subseteq {\\mathbb R}^{m} $ be a set. By *C*0(*U*; Ω), we denote the class of continuous mappings on *U*, which take values in Ω. When $U\\subseteq {\\mathbb R}^{n} $, by *C*1(*U*; Ω), we denote the class of continuously differentiable functions on *U*, which take values in Ω. When $U=[a,b)\\subseteq {\\mathbb R}$ (or $U=[a,b]\\subseteq {\\mathbb R}$) with *a* < *b*, *C*0([*a*, *b*); Ω) (or *C*0([*a*, *b*]; Ω)) denotes all functions *f* : [*a*, *b*) → Ω (or *f* : [*a*, *b*] → Ω), which are continuous on (*a*, *b*) and satisfy $\\mathop{\\lim }\\limits\\_{s\\to a^{+} } \\left(f(s)\\right)=f(a)$ (or $\\mathop{\\lim }\\limits\\_{s\\to a^{+} } \\left(f(s)\\right)=f(a)$ and $\\mathop{\\lim }\\limits\\_{s\\to b^{-} } \\left(f(s)\\right)=f(b)$). When $U=[a,b)\\subseteq {\\mathbb R}$, *C*1([*a*, *b*); Ω) denotes all functions *f* : [*a*, *b*) → Ω which are continuously differentiable on (*a*, *b*) and satisfy $\\mathop{\\lim }\\limits\\_{s\\to a^{+} } \\left(f(s)\\right)=f(a)$ and $\\mathop{\\lim }\\limits\\_{h\\to 0^{+} } h^{-1} \\left(f(a+h)-f(a)\\right)=\\mathop{\\lim }\\limits\\_{s\\to a^{+} } f'(s)$.\n* $\\mathcal K\\_{\\infty} $ is the class of all strictly increasing, unbounded functions $a\\in C^{0} ( {\\mathbb R}\\_+ ; {\\mathbb R}\\_+ )$, with *a*(0) = 0 (see Reference ).\n* KL is the class of functions $\\beta:{\\mathbb R}\\_+ \\times {\\mathbb R}\\_+ \\to {\\mathbb R}\\_+$ which satisfy the following: For each *t* ≥ 0, the mapping *β*( ⋅ , *t*) is of class K, and, for each *s* ≥ 0, the mapping *β*(*s*, ⋅ ) is nonincreasing with lim*t* → ∞*β*(*s*, *t*) = 0 (see Reference ).\n* For any subset $S\\subseteq {\\mathbb R}$ and for any *A* > 0, *P**C*1([0, *A*]; *S*) denotes the class of all functions *f* ∈ *C*0([0, *A*]; *S*) for which there exists a finite (or empty) set *B* ⊂ (0, *A*) such that: (i) the derivative *f*ʹ(*a*) exists at every *a* ∈ (0, *A*) ∖ *B* and is a continuous function on (0, *A*) ∖ *B*, (ii) all meaningful right and left limits of *f*ʹ(*a*) when *a* tends to a point in *B*⋃{0, *A*} exist and are finite.\n* Let a function $f\\in C^{0} ({\\mathbb R}\\_+ \\times [0,A])$ be given, where *A* > 0 is a constant. We use the notation *f*[*t*] to denote the profile at certain *t* ≥ 0, i.e., (*f*[*t*])(*a*) = *f*(*a*, *t*) for all *a* ∈ [0, *A*].\n* Let a function $x\\in C^{0} ([-A,+\\infty );{\\mathbb R})$ be given, where *A* > 0 is a constant. We use the notation $x\\_{t} \\in C^{0} ([-A,0];{\\mathbb R})$ to denote the “*A* − history” of *x* at certain *t* ≥ 0, i.e., (*x**t*)( − *a*) = *x*(*t* − *a*) for all *a* ∈ [0, *A*].\n* Let a function $f\\in C^0({\\mathbb R}\\_{+}; {\\mathbb R})$ be given. By $D^+ f:{\\mathbb R}\\_+ \\to {\\mathbb R}$, *t* ↦ (*D*+*f*)(*t*) = limsup*h* → 0+{*h*− 1(*f*(*t* + *h*) − *f*(*t*))}, we denote the upper Dini derivative.\n* Let *A*, *B* be logical statements. Their logical conjunction is denoted by *A* ∧ *B*.\nPopulation Model with Actuator Dynamics\n=======================================\nWe consider the age distribution of the population of a microorganism in a bioreactor governed by the following equations\n[eqsystemOpenLoop]\n \n$$\\begin{aligned}\n{3}\n \\text{PDE:}& \\quad &&\\frac{\\partial {f}}{\\partial t}(a,t) +\\frac{\\partial f }{\\partial a}(a,t)=-\\left(\\mu(a)+D(t)\\right)f(a,t) \\label{eq\\_PDE}\\\\\n \\text{ODE:}& \\quad &&\\dot D (t)=u(t) \\label{eq\\_ODE}\\\\\n \\text{OE:}& \\quad &&y(t) = \\int\\_0^A p(a)f(a,t) {{\\rm d}}a\\label{eq\\_output}\\\\\n \\text{BC:}& \\quad &&f(0,t)=\\int\\_0^A k(a)f(a,t) {{\\rm d}}a \\label{eq\\_OLBC} \\end{aligned}$$ \nwhere *a* ∈ [0, *A*] is the age variable, *A* is the maximum reproductive age of the microorganism (a constant), $t\\in {\\mathbb R}\\_+$ is time, *f*(*a*, *t*) > 0 denotes the distribution of the microbial mass in the reactor at age *a* ∈ [0, *A*] and time *t*, *μ* and *k* are the mortality rate profile and birth rate profile, respectively, and *p* is the sensor kernel. The functions $\\mu :[0,A]\\to {\\mathbb R}\\_+ $ and $k:[0,A]\\to {\\mathbb R}\\_+ $ are assumed to be continuous in age *a*, and the function $p :[0,A]\\to {\\mathbb R}\\_+ $ continuously differentiable in *a* with $\\int\\_0^Ap(a){{\\rm d}}a >0$. The population of microorganisms grows at a rate regulated by the dilution rate *D*(*t*), which is the control input of model. Accounting for the inlet and outlet rate of the bioreactor, the PDE expresses the microbial mass balance as a variation of the *McKendrick-von Foerster* equation. The internal boundary feedback, is the renewal condition, which expresses the reproduction of the microorganism as the current mass of the newborns *f*(0, *t*). From a mathematical point of view, the boundary condition involves non-local terms. The measured output defined in is a weighted average of the mass of the microorganism with an age-specific kernel *p*(*a*). \n In chemostat reactors, the dilution rate is defined as the ratio of the inlet volumetric flow rate to the reactor volume. Although, the reactor’s inlet flow and outlet volumetric flow rates are readily adjustable, the volume of the reactor is not necessarily constant and its variations can be described by the dynamic actuation introduced as. Indeed, the population balance is a realistic extension of the model studied in Reference, which does not account for a dynamic actuation and consequently assumes no possible variations of the reactor volume. Equation defines an appropriate control input $u(t) \\in {\\mathbb R}$ that depends on the reactor volume, the reactor inlet and outlet volumetric flow rates as well as on the time derivative of the reactor inlet volumetric flow rate. \nIt should also be noticed that model is derived by neglecting the dependence of the growth of the microorganism on the concentration of a limiting substrate. However, resource-based models that incorporate limiting substrates or uptake of nutrients are more suitable to describe the dynamics of continuous microbial culture and might exhibit limit cycles that are induced by the behavior of a limiting resource assuming constant dilution rates. Model presupposes that the growth or decay of a living population only depends on irreversible incidences of birth and death and is therefore suitable to predict the evolution of macro-populations in demography, epidemiology and ecology. \nThe state of model is $(f[t],D(t)) \\in \\mathcal{F} \\times {\\mathbb R}$, where F is the function space defined by the following equation:\n \n$$\\begin{aligned}\n \\mathcal{F} = \\left\\{ f \\in PC^1([0,A];(0,\\infty)): f(0)=\\int\\_0^A k(a)f(a) {{\\rm d}}a\\right\\}.\\end{aligned}$$ \nThe results of this contribution encompass two different cases for the state space. For the first case, the state space is defined as $\\mathcal{F} \\times {\\mathbb R}$, allowing for the dilution rate *D*(*t*) to take arbitrary real values. However, as noted above, the dilution rate is the ratio of the inlet volumetric flow rate to the reactor volume and consequently, it is physically meaningful to restrict its set of admissible values to the interval $I \\subseteq {\\mathbb R}\\_+$, which is much more demanding as the state is constrained and the state space defined as F × *I*. \nOur main assumption for model is the existence of a constant *D*\\* > 0 such that\n \n$$\\begin{aligned}\n1=\\int \\_{0}^{A}k(a){ \\ifthenelse{ \\equal{}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\right)}}\n}\\left(-D^{\\*} a-\\int \\_{0}^{a}\\mu (s)ds \\right)da, \\end{aligned}$$ \nwhich is the analogue of the Lotka-Sharpe equation (see Reference ) and equivalent to assuming population viability. When the above assumption holds, model admits a family of equilibrium points (*f*\\*, *D*\\*) (as the model without actuator dynamics ), given by the following equation for arbitrary *M* > 0:\n \n$$\\begin{aligned}\n{f^\\ast}(a) = M { \\ifthenelse{ \\equal{}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\right)}}\n}\\left(-D^{\\*} a-\\int \\_{0}^{a}\\mu (s)ds \\right). \\end{aligned}$$ \n[eqsteadyState] The existence of a continuum of equilibria for model implies that none of the equilibrium points is asymptotically stable. Therefore, if we want the bioreactor to operate on a specific equilibrium point, then we need to stabilize such an equilibrium point by means of feedback control. \nBackstepping Design with Unrestricted Dilution\n==============================================\nOur objective is to stabilize a desired steady-state for a chosen *M* > 0 or equivalently, stabilize an appropriate output setpoint\n \n$$\\begin{aligned}\n\\label{eq\\_ystar\\_def}\n {y^\\ast }= \\int\\_0^A p(a) {f^\\ast}(a) \\ {{\\rm d}}a >0.\\end{aligned}$$ \nOur design is build upon the control law proposed in Reference for the stabilization of system without actuator dynamics. We recall the output-feedback controller \n \n$${D\\_\\mathrm{nom}}(t) = {D^\\ast }+ {k\\_1}\\ln \\frac{y(t)}{{y^\\ast }},~~{k\\_1}>0\n \\label{eq\\_nominalControl}$$ \nthat stabilizes the equilibrium corresponding to *y*\\*. To extend controller to the case with actuator dynamics described by system, a backstepping approach is adopted. Defining the dilution rate error as follows\n \n$$\\begin{aligned}\n \\delta(t) \\coloneqq D(t) - {D\\_\\mathrm{nom}}(t) \\label{eq\\_defintionDelta}\\end{aligned}$$ \nand taking its time-derivative along the output *y*(*t*) given by, with the help of integration by parts, the following ODE is derived\n \n$$\\begin{aligned}\n \\dot \\delta (t) &= u(t) - {k\\_1}\\frac{\\dot y(t)}{y(t)}\\\\\n &= u(t) + {k\\_1}D(t) + \\frac{{k\\_1}}{y(t)}\\left[ p(A)f(A,t) - p(0) f(0,t) - \\int\\_0^A \\Tilde{p}(a) f(a,t) {{\\rm d}}a \\right]\\end{aligned}$$ \nwhere\n \n$$\\Tilde{p}(a) \\coloneqq p'(a) -p(a)\\mu(a).$$ \nAt this point, we aim at designing a control law *u*(*t*) that achieves exponential convergence of the error *δ*(*t*). Selecting *k*2 > 0 such that\n \n$$\\dot\\delta (t) = -{k\\_2}\\delta(t),$$ \nthe following controller is designed\n[eqcontrollerBackstepping]\n \n$$\\begin{aligned}\n u(t) &= {u\\_\\mathrm{c}}(t) + {u\\_\\mathrm{s}}(t)\\\\\n {u\\_\\mathrm{c}}(t) &= - {k\\_1}D(t) - \\frac{{k\\_1}}{y(t)}\\left[ p(A)f(A,t) - p(0) f(0,t) - \\int\\_0^A \\Tilde{p}(a) f(a,t) {{\\rm d}}a \\right]\\label{eq\\_cancelinggeneral}\\\\\n {u\\_\\mathrm{s}}(t) &= -{k\\_2}\\left( D(t) - {D^\\ast }-{k\\_1}\\ln \\frac{y(t)}{{y^\\ast }}\\right) = -{k\\_2}\\delta(t),\n \\end{aligned}$$ \nwhich is a superposition of canceling and stabilizing terms *u*c(*t*) and *u*s(*t*). In the most general case, the controller requires full measurement of the plant and the actuator states *f*(*a*, *t*) and *D*(*t*), respectively. Furthermore, the feasibility of controller depends on the knowledge of the parameters *p*, *p*ʹ, *μ* and *D*\\*. However, assuming constant kernel *p* and mortality rate *μ*, the canceling terms of backstepping controller reduces to the following feedback law\n \n$${u\\_\\mathrm{c}}(t) = - D(t) - \\frac{p}{y(t)}\\left[ f(A,t) - f(0,t) \\right] - \\mu,$$ \nwhich removes the burden of full-state measurement of the plant’s state.\nTo illustrate the stabilizing effect of the control law, we first present simulation results before conducting a stability analysis. Following Galerkin’s approach, the approximate population density, $\\hat f: [0,A]\\times {\\mathbb R}\\_+ \\to {\\mathbb R}$ is defined as follows\n \n$$\\begin{aligned}\n \\hat f(a,t) = \\sum\\_{j=1}^N \\phi\\_j(a)\\xi\\_j(t) \\eqqcolon \\phi(a)^\\top \\xi(t)\\end{aligned}$$ \nwhere $\\phi\\_j: [0,A]\\to {\\mathbb R}$ are the trial functions satisfying the boundary condition, chosen as in Reference with *N* = 6, and the temporal weights $\\xi\\_j:{\\mathbb R}\\_+ \\to {\\mathbb R}$ are solutions to the initial value problem below\n \n$$\\begin{aligned}\n \\dot \\xi(t) &= (A\\_\\phi - D(t) I)\\xi(t), \\quad t\\geq 0\\\\\n \\xi(0) &= \\xi\\_0\n \\end{aligned}$$ \nwhere $A\\_\\phi \\in {\\mathbb R}^{N\\times N}$ is determined by the choice of trial functions *ϕ**j*, and $\\xi\\_0 \\in {\\mathbb R}^{N}$ is computed for a given initial population density *f*0 ∈ F of system. A parameter set of the initial condition *f*0(*a*), birth rate *k*(*a*), death rate *μ*(*a*), steady-state input *D*\\* and maximum age of reproduction *A* used throughout this work is given below\n[eqSimulationParameterAndInitialCondition]\n \n$$\\begin{aligned}\n f\\_0(a) = 8 - 3a + \\phi\\_2(a), \\quad \\mu(a) = \\frac{1}{20-5a}, \\quad k(a) = a, \\quad p(a) = 1 + \\frac{a^2}{10},\\label{eq\\_SimulationParameterSet}\\\\\n {D^\\ast }\\approx 0.48,\\quad A=2, \\quad \\phi\\_2(a) = \\sin(\\omega\\_1 a){ \\ifthenelse{ \\equal{\\sigma\\_1 a}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\sigma\\_1 a \\right)}}\n}\\frac{{f^\\ast}(a)}{{f^\\ast}(0)},\\quad \\omega\\_1 \\approx 3.82, \\quad \\sigma\\_1 \\approx 0.91. \\label{eq\\_SimulationInitialCondition}\n \\end{aligned}$$ \nAs shown in a representative simulation example in Figure [figActuatorDynamicsstaticControllerAllFinal], controller achieves convergence to the desired output *y*\\* = *y*des. However, the controller stabilizes the equilibrium profile without restricting the dilution rate *D*(*t*) to physically meaningful values.\n [figActuatorDynamicsstaticControllerAllFinal]\nClosed-Loop Stability for All Positive Initial Population Densities\n===================================================================\nFor the stability proof, we invoke the transformation $\\varPi: \\mathcal{F}\\to {\\mathbb R}\\times C^0([-A,0]),\\ f[t] \\mapsto (\\eta(t),\\psi\\_t)$ mapping the age-profile *f*[*t*] to its unstable “malthusian” and “asymptotic” mode\n[eqtransformation]\n \n$$\\begin{aligned}\n \\eta (t)&= \\ln\\Pi(f[t])\\label{eta}\\\\\n \\psi(t-a)&=\\frac{f(a,t)}{{f^\\ast}(a)\\Pi(f[t])}-1\\label{eq\\_psi}\n \\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n \\Pi(f) &= \\frac{\\int\\_0^A \\pi(a) f(a){{\\rm d}}a}{\\int\\_0^A ak(a){f^\\ast}(a){{\\rm d}}a},\\quad \\pi(a)= \\int\\_a^A k(s){ \\ifthenelse{ \\equal{\\int\\_s^a\\left(\\mu(\\alpha)+{D^\\ast }\\right){{\\rm d}}\\alpha}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\int\\_s^a\\left(\\mu(\\alpha)+{D^\\ast }\\right){{\\rm d}}\\alpha \\right)}}\n}{{\\rm d}}s, \\quad a\\in[0,A].\\label{eq\\_adjointEFpi}\\end{aligned}$$ \nEquation has been proven to be a valid transformation, i.e., the right hand side is indeed a function of (*t* − *a*). Transforming the open-loop system in light of, one receives\n[eqOpenLoopTransformed]\n \n$$\\begin{aligned}\n \\dot \\eta (t) &= {D^\\ast }- D(t) \\label{eq\\_OLetaDynamics}\\\\ \n \\dot D (t) &= u (t) \\label{eq\\_OLinput}\\\\\n \\psi(t) &= \\int\\_0^A \\tilde k(a) \\psi(t-a)\\, {{\\rm d}}a ~.\\label{eq\\_OLIDEdynamics}\n \\end{aligned}$$ \nAfter transforming the system consisting of and, we arrive at the dynamics\n[eqclosedLoopTransformed]\n \n$$\\begin{aligned}\n{4}\n \\dot \\eta (t) &= -{k\\_1}\\eta (t) -\\delta (t) - {k\\_1}v(\\psi\\_t)\\label{eq\\_etaDynamics}\\\\ \n \\dot \\delta (t) &= - {k\\_2}\\delta (t)\\label{eq\\_deltaDynamics}\\\\\n \\psi(t) &= \\int\\_0^A \\tilde k(a) \\psi(t-a)\\ {{\\rm d}}a \\label{eq\\_IDEdynamics}\n \\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n \\tilde k(a) \\coloneqq k(a) \\frac{{f^\\ast}(a)}{{f^\\ast}(0)}\n, \\qquad\n g(a) \\coloneqq \\frac{p(a){f^\\ast}(a)}{\\int\\_0^A p(a){f^\\ast}(a){{\\rm d}}a}, \\qquad v(\\psi\\_t) \\coloneqq \\ln \\left (1+\\int\\_0^Ag(a)\\psi(t-a){{\\rm d}}a \\right ),\n \\label{ggg}\\end{aligned}$$ \nand the equilibrium (*η*, *δ*, *ψ*) = 0 is to be proven asymptotically stable. This can be achieved using a member of the family of Control Lyapunov Functionals provided in Reference. For this, we have to make a technical assumption:\n[Technical assumption on birth kernel] There exists a constant *λ* > 0 such that $\\int\\_0^A|\\tilde k(a) -r\\lambda \\int \\_a^A\\tilde k(s) {{\\rm d}}s|{{\\rm d}}a<1$ where $r^{-1} = \\int\\_0^A \\tilde k (a) a {{\\rm d}}a$. Let *σ* > 0 be a sufficiently small constant that satisfies the inequality\n \n$$\\begin{aligned}\n \\int\\_0^A\\left|\\tilde k(a) -r\\lambda \\int \\_a^A\\tilde k(s) {{\\rm d}}s\\right| { \\ifthenelse{ \\equal{\\sigma a}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\sigma a \\right)}}\n}{{\\rm d}}a<1.\n \\label{doy}\\end{aligned}$$ \n[asstechnicalbirthKernel]\nAssumption 1 is a mild technical assumption, since it is satisfied by arbitrary mortality rate *μ* for every birth kernel *k* that has a finite number of zeros on [0, *A*]. From a modeling standpoint, this means that individuals of ages zero to *A* are reproducing at *almost every* age (see Reference ).\nThe state *ψ* of the internal dynamics is restricted (see Reference ) to the set\n \n$$\\begin{aligned}\n \\mathcal S = \\left\\{ \\psi \\in C^0([-A,0];(-1,\\infty)): P(\\psi) = 0 \\land \\psi(0) = \\int\\_0^A \\tilde k(a)\\psi(-a){{\\rm d}}a \\right\\}\n \\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n P(\\psi) = \\int\\_0^A\\psi(-a)\\int\\_a^A\\tilde k(s) {{\\rm d}}s{{\\rm d}}a\\left(\\int\\_0^Aa\\tilde k(a){{\\rm d}}a\\right)^{-1}.\n \\end{aligned}$$ \n[Lyapunov stability of backstepping controller with *unrestricted dilution*] [thmBacksteppingLyapunov1]\nLet Assumption [asstechnicalbirthKernel] hold. Then for every *k*1, *k*2 > 0, there exists a function *α*1 ∈ K∞ such that for every $(f\\_{0},D\\_{0}) \\in \\mathcal{F} \\times {\\mathbb R}$ the unique solution $(f[t],D(t)) \\in \\mathcal{F} \\times {\\mathbb R}$ of the closed-loop system with and initial condition (*f*[0], *D*(0)) = (*f*0, *D*0) exists for all *t* ≥ 0 and satisfies the following stability estimate for all *t* ≥ 0\n \n$$\\begin{aligned}\n R\\_{1}(f[t],D(t)) \\leq { \\ifthenelse{ \\equal{-\\frac{\\sigma\\_1}{2} t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -\\frac{\\sigma\\_1}{2} t \\right)}}\n}\\;\\alpha\\_1 \\left(R\\_{1}(f\\_{0},D\\_{0})\\right) \\label{eq\\_expBoundOriginalStateDtilde}\n \\end{aligned}$$ \nwhere $\\sigma\\_1 = \\min\\left(\\frac{{k\\_{1}}}{2},{k\\_{2}},\\sigma\\right) > 0$ and\n \n$$\\begin{aligned}\n R\\_{1}(f,D):= \\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f(a)}{{f^\\ast}(a)} \\right| + \\left| D-{D^\\ast }\\right|\n \\label{eq\\_measureR1(f,F)}\n \\end{aligned}$$ \nfor all $(f,D) \\in \\mathcal{F} \\times {\\mathbb R}$.\n[Proof of Theorem [thmBacksteppingLyapunov1]] Lemma 4.1 of Reference provides existence and uniqueness of the solution $\\psi\\_t \\in \\mathcal S$ and also\n \ninf*t* ≥ − *A**ψ*(*t*) ≥ min*t* ∈ [ − *A*, 0]*ψ*(*t*) > − 1, ∀*t* ≥ 0. \nNotice that the second inequality is given by $\\psi\\_0 \\in \\mathcal S$ and is guaranteed for all physical initial conditions *f*0 ∈ F. Indeed, the initial condition of is given by\n \n$$\\psi\\_0(a) = \\psi(-a) = \\frac{f\\_0(a)}{{f^\\ast}(a) \\Pi (f\\_0)} -1 > -1, \\quad \\forall a \\in [0,A] \\label{eq\\_functionalPsiV}$$ \nwhere *f*0(*a*), *f*\\*(*a*) and Π(*f*0) are positive on the domain [0, *A*]. Since *g*(*a*) ≥ 0, $\\int\\_0^A g(a){{\\rm d}}a = 1$ according to and inequality holds, the map\n \n$$\\label{eq-vdef}\n v(\\psi\\_t) \\coloneqq \\ln \\left (1+\\int\\_0^Ag(a)\\psi(t-a){{\\rm d}}a \\right ), \\quad \\textrm{for} \\quad t \\geq 0$$ \nis well-defined and continuous. Thus, the ODE subsystem of the closed-loop system, namely, equations, locally admits a unique solution. Consider first the Lyapunov function\n \n$$U\\_1(\\eta,\\delta) = \\frac{1}{2}\\left(\\eta^2+{b\\_{1}}\\delta^2\\right), \\quad \\forall(\\eta,\\delta) \\in {\\mathbb R}^2$$ \nwhere *b*1 > 0 is a constant to be chosen. The time derivative of *U*1(*η*(*t*), *δ*(*t*)) along the solutions of, can be upper bounded for all times *t* ≥ 0 for which the solution (*η*(*t*), *δ*(*t*)) exists as follows:\n \n$$\\begin{aligned}\n \\frac{{{\\rm d}}}{{{\\rm d}}t}U\\_1(\\eta(t),\\delta(t)) &\\leq -\\frac{{k\\_1}}{4} \\eta(t)^2 - \\left({b\\_{1}}{k\\_2}-\\frac{1}{{k\\_1}}\\right) \\delta(t)^2 + \\frac{{k\\_1}}{2} v(\\psi\\_t)^2. \\label{eq\\_BoundODELyap}\n \\end{aligned}$$ \nInequality was derived using the inequalities $-\\eta(t) \\delta(t) \\leq \\frac{{k\\_1}}{4} \\eta(t)^2 + \\frac{1}{{k\\_1}} \\delta(t)^2$ and $- \\eta(t) v(\\psi\\_t) \\leq \\frac{1}{2} v(\\psi\\_t)^2 + \\frac{1}{2} \\eta(t)^2$. Next, as in the proof of Reference, we use Corollary 4.6 of the latter with *C*(*ψ**t*) = (1 + min(0, min*a* ∈ [0, *A*]*ψ**t*(*a*)))− 2 and $b(s) = \\frac{1}{2} s^2$ under Assumption [asstechnicalbirthKernel] with sufficiently small parameter *σ* > 0 to obtain that the functional\n \n$$\\begin{aligned}\n\\label{eq-G}\n G(\\psi\\_t) \\coloneqq \\frac{\\max\\_{a\\in [0,A]} |\\psi(t-a)|{ \\ifthenelse{ \\equal{-\\sigma a}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -\\sigma a \\right)}}\n}}{1+\\min(0,\\min\\_{a\\in [0,A]}\\psi(t-a))}\n \\end{aligned}$$ \nsatisfies\n \n*D*+*G*(*ψ**t*)2 ≤ − 2*σ**G*(*ψ**t*)2, ∀*t* ≥ 0 \nalong solutions *ψ**t* of the IDE. Now, Reference also provides that\n \n$$|v(\\psi\\_t)|\\leq G(\\psi\\_t){ \\ifthenelse{ \\equal{\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\sigma A \\right)}}\n}, \\quad \\forall t \\geq 0.\\label{eq-vleqG}$$ \nGiven this fact, the bounds and and defining the Lyapunov functional\n \n$$\\begin{aligned}\n V\\_1(\\eta,\\delta,\\psi) &=& U\\_1(\\eta,\\delta) + \\frac{1}{2}{b\\_{2}} G(\\psi)^2, \n \\nonumber\\\\\n &=& \\frac{1}{2}\\left(\\eta^2 + {b\\_{1}}\\delta^2 + {b\\_{2}} G(\\psi)^2\\right), \\quad \\forall (\\eta,\\delta,\\psi)\\in {\\mathbb R}\\times {\\mathbb R}\\times \\mathcal{S}\\label{eq\\_LyapV1Definition}\n \\end{aligned}$$ \nwith ${b\\_{1}} = \\frac{2}{{k\\_2}{k\\_1}}$, ${b\\_{2}} = \\frac{{k\\_1}}{\\sigma}{ \\ifthenelse{ \\equal{2\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( 2\\sigma A \\right)}}\n}$, we derive the differential inequality\n \n*D*+*V*1(*η*(*t*), *δ*(*t*), *ψ**t*) ≤ − *σ*1*V*1(*η*(*t*), *δ*(*t*), *ψ**t*), ∀*t* ≥ 0 \nwhere $\\sigma\\_1 = \\min\\left(\\frac{{k\\_{1}}}{2},{k\\_{2}},\\sigma\\right) > 0$. Since estimate implies that all solutions stay bounded for all times for which they exist, we can conclude existence for all times. The differential inequality in conjunction with Lemma 2.12 on pages 77-78 in Reference implies the following estimate for all *t* ≥ 0:\n \n$$\\begin{aligned}\n V\\_1(\\eta(t),\\delta(t),\\psi\\_t) \\leq { \\ifthenelse{ \\equal{-\\sigma\\_1 t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -\\sigma\\_1 t \\right)}}\n} V\\_1(\\eta(0),\\delta(0),\\psi\\_{0}).\n \\label{doula}\n \\end{aligned}$$ \nWe next show that the estimate holds. Recall from that\n \n$$\\begin{aligned}\n \\delta(t) &= \\tilde D(t) - {k\\_{1}} \\left( \\eta(t) + v(\\psi\\_t) \\right), \n\\quad t \\geq 0\\label{eq\\_deltaInverseTransform}\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n \\tilde D(t) = D(t) - {D^\\ast }, \\quad t \\geq 0.\\end{aligned}$$ \nDefine the positive definite quadratic function $U\\_2: {\\mathbb R}^3 \\to {\\mathbb R}$\n \n$$\\begin{aligned}\n U\\_2(\\eta,\\tilde D, v) = \\eta^2 + \\frac{{b\\_{2}}}{2} { \\ifthenelse{ \\equal{-2\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -2\\sigma A \\right)}}\n} v^2 + \\left( \\tilde D - {k\\_{1}}\\eta - {k\\_{1}}v \\right)^2\\end{aligned}$$ \nwhere *v* is given by. Clearly, there exist constants $\\underline{c},\\overline{c} >0$ such that for all $ (\\eta,\\tilde D, v)\\in {\\mathbb R}^3$ the following inequalities hold:\n \n$$\\begin{aligned}\n \\underline{c}\\left(\\eta^2 + \\tilde D^2 + v^2 \\right) \\leq U\\_2(\\eta,\\tilde D, v) \\leq \\overline{c}\\left(\\eta^2 + \\tilde D^2 + v^2 \\right). \\label{eq\\_upperAndLowerBoundU2}\\end{aligned}$$ \nConsequently, in conjunction with,,, implies the following estimate:\n \n$$\\begin{aligned}\n V\\_1(\\eta(t),\\delta(t),\\psi\\_t) &= \\frac{1}{2}\\left(U\\_2(\\eta(t),\\tilde D(t), v(\\psi\\_t)) + {b\\_{2}} G(\\psi\\_t)^2 - \\frac{{b\\_{2}}}{2} { \\ifthenelse{ \\equal{-2\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -2\\sigma A \\right)}}\n} v(\\psi\\_t)^2\\right)\\\\\n &\\geq \\frac{1}{2}\\left(\\underline{c}\\left(\\eta(t)^2 + \\tilde D(t)^2 + v(\\psi\\_t)^2 \\right) + \\frac{{b\\_{2}}}{2} G(\\psi\\_t)^2 \\right), \\quad t \\geq 0.\\label{eq\\_lyapDtildebound}\\end{aligned}$$ \nUsing, we conclude that there exists an appropriate constant $\\tilde c>0$ (independent of the solution), for which the following estimate holds:\n \n$$\\begin{aligned}\n |\\eta(t)| + { \\ifthenelse{ \\equal{\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\sigma A \\right)}}\n} G(\\psi\\_t) + \\left|\\tilde D(t)\\right| &\\leq \\tilde c\\sqrt{ V\\_1(\\eta(t),\\delta(t),\\psi\\_t)}, \\quad t \\geq 0.\\label{eq\\_lowerBoundV\\_1}\\end{aligned}$$ \nFrom Reference,\n \n$$\\begin{aligned}\n \\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f[t](a)}{{f^\\ast}(a)} \\right| \\leq |\\eta(t)| + { \\ifthenelse{ \\equal{\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\sigma A \\right)}}\n} G(\\psi\\_t), \\quad t \\geq 0,\\label{eq\\_KK17\\_EtaPsiBoundToF\\_upper}\n \\end{aligned}$$ \ni.e., we arrive at the left-hand side of after combining and. To complete the proof of, we need to show that there exists $ \\tilde\\alpha \\in \\mathcal{K}\\_\\infty$ such that\n \n$$\\begin{aligned}\n V\\_1(\\eta(0),\\delta(0),\\psi\\_0) \\leq \\tilde\\alpha\\left(\\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f\\_0(a)}{{f^\\ast}(a)} \\right| + \\left|\\tilde D(0)\\right|\\right). \\label{eq\\_initialLyapboundClassK}\n \\end{aligned}$$ \nWe first note that Reference provides the existence of $\\tilde\\alpha\\_1,~\\tilde\\alpha\\_2 \\in \\mathcal{K}\\_\\infty$\n \n$$\\begin{aligned}\n |\\eta\\_0| \\leq \\tilde\\alpha\\_1\\left(\\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f\\_0(a)}{{f^\\ast}(a)} \\right| \\right),\\quad G(\\psi\\_0) \\leq \\tilde\\alpha\\_2 \\left(\\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f\\_0(a)}{{f^\\ast}(a)} \\right|\\right), \\label{eq\\_KK17technicalbounds}\n \\end{aligned}$$ \nand, using, we upper bound\n \n$$\\begin{aligned}\n V\\_1(\\eta,\\delta,\\psi) \\leq \\frac{1}{2}\\left(\\overline{c}\\left(\\eta^2 + \\tilde D(\\eta,\\delta,\\psi)^2 \\right) + \\tilde b G(\\psi)^2 \\right),\\quad (\\eta,\\delta,\\psi) \\in {\\mathbb R}\\times {\\mathbb R}\\times \\mathcal{S}\\label{eq\\_boundingforalphatilde}\\end{aligned}$$ \nfor an appropriate constant $\\tilde b > 0$. The inequalities and allow us to conclude that there exists appropriate $\\tilde \\alpha$ (independent of the solution) for which holds. Combining,, and, we obtain estimate.\nNotice that the exponential convergence rate *σ*1 of is the minimum of three constants: the nominal control gain *k*1, the backstepping convergence rate *k*2 and the convergence rate of the internal dynamics *σ*.\nEasier-to-Implement Backstepping Controller: Output Feedback without Sensor Model\n=================================================================================\nRecall the canceling controller\n \n$${u\\_\\mathrm{c}}(t) = {k\\_1}\\left( -D(t)+{D^\\ast }+ v\\_1(\\psi\\_t)\\right)\\label{eq\\_cancelinggeneral\\_recall}\\\\$$ \nwhere the tedious term\n \n$$\\begin{gathered}\n v\\_1(\\psi\\_t) = -{D^\\ast }+ \\frac{1}{\\int\\_0^A p(a) {f^\\ast}(a) (1+\\psi(t-a)){{\\rm d}}a} \\\\ \n \\left [ p(A) {f^\\ast}(A) (1+\\psi(t-A)) - p(0) {f^\\ast}(0) (1+\\psi(t)) - \\int\\_0^A \\tilde p(a) {f^\\ast}(a) (1+\\psi(t-a)){{\\rm d}}a \\right] \\label{eq\\_functionalPsi}\\end{gathered}$$ \nis a functional of the internal state variable *ψ* and thus decays exponentially to *v*1(0) = 0. We relax the canceling control from exact cancellation to now only canceling the steady-state value\n \n*u*c(*t*) = *k*1( − *D*(*t*) + *D*\\*) \ndrastically reducing the measurement requirements. This approach achieves convergence *y*(*t*) → *y*\\* to the desired output, which is illustrated by means of a representative simulation example in Figure [figActuatorDynamicsstaticControllerRelaxedAllFinal].\n [figActuatorDynamicsstaticControllerRelaxedAllFinal]\nFinding the closed-loop dynamics of, and\n[eqclosedLoopSuperlateBotchingTransformed]\n \n$$\\begin{aligned}\n{4}\n \\dot \\eta (t) &= -{k\\_1}\\eta (t) -\\delta (t) - {k\\_1}v(\\psi\\_t)\\label{eq\\_etaDynamicsSuperlateBotching}\\\\ \n \\dot \\delta (t) &= -{k\\_2}\\delta(t) -{k\\_1}v\\_1(\\psi\\_t)\\label{eq\\_deltaDynamicsSuperlateBotching}\\\\\n \\psi(t) &= \\int\\_0^A \\tilde k(a) \\psi(t-a)\\, {{\\rm d}}a, \\label{eq\\_IDEdynamicsSuperlateBotching}\n \\end{aligned}$$ \nwe notice that, as a result of simplifying the controller, the dynamics of *δ* have become more complex, and as a consequence the Lyapunov-like stability proof becomes harder to establish.\nIn our proof of stability, we make use of the following result, whose proof we state in the Appendix.\n[lemmaT1boundMain] There exists a constant *c*1 > 0 such that every solution *ψ**t* of the IDE satisfies the following estimate for all *t* ≥ 0:\n \n$$\\begin{aligned}\n \\label{eq\\_T\\_1\\_boundMain}\n |v\\_1(\\psi\\_t)| \\leq \\sqrt{c\\_1}\\frac{ \\max\\_{a\\in[0,A]}|\\psi(t-a)|}{1+\\min\\_{a\\in [0,A]}\\psi(t-a)}. \\end{aligned}$$ \n[Lyapunov stability of *static output feedback* backstepping controller with unrestricted dilution][thmBacksteppingRelaxed2] Let Assumption [asstechnicalbirthKernel] hold. Then for every *k*1, *k*2 > 0, there exists a function *α*2 ∈ K∞ such that for every $(f\\_{0},D\\_{0}) \\in \\mathcal{F} \\times {\\mathbb R}$ the unique solution $(f[t],D(t)) \\in \\mathcal{F} \\times {\\mathbb R}$ of the closed-loop system with\n \n$$\\label{eq\\_controllerBackstepping+}\nu(t) = - ({k\\_1}+{k\\_2})\\left(D(t) - {D^\\ast }\\right)\n+ {k\\_1}{k\\_2}\\ln \\frac{y(t)}{{y^\\ast }}$$ \nand initial condition (*f*[0], *D*(0)) = (*f*0, *D*0) exists for all *t* ≥ 0 and satisfies the following stability estimate for all *t* ≥ 0\n \n$$\\begin{aligned}\n R\\_{1}(f[t],D(t)) \\leq { \\ifthenelse{ \\equal{-\\frac{\\sigma\\_1}{2} t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -\\frac{\\sigma\\_1}{2} t \\right)}}\n}\\;\\alpha\\_2 \\left(R\\_{1}(f\\_{0},D\\_{0})\\right) \\label{eq\\_expBoundOriginalStateDtildeRelaxedController}\n \\end{aligned}$$ \nwhere $\\sigma\\_1 = \\min \\left(\\frac{{k\\_{1}}}{2},\\frac{{k\\_{2}}}{2},\\sigma\\right) > 0$ and *R*1 is defined in.\n[Proof of Theorem [thmBacksteppingRelaxed2]] Again, like in, we know that the map *t* ↦ *v*(*ψ**t*) is well-defined for all *t* ≥ 0 and is continuous. Thus, the closed-loop system locally admits a solution. We first show by considering the function\n \n$$U\\_1(\\eta,\\delta) = \\frac{1}{2}(\\eta^2+{b\\_{1}}\\delta^2), \\quad (\\eta, \\delta) \\in {\\mathbb R}^2,$$ \nwhere *b*1 > 0 is a constant to be chosen. The time derivative of *U*1(*η*(*t*), *δ*(*t*)) along solutions of, can be upper bounded for all times *t* ≥ 0 for which the solution (*η*(*t*), *δ*(*t*)) exists\n \n$$\\begin{aligned}\n \\frac{{{\\rm d}}}{{{\\rm d}}t}U\\_1(\\eta(t),\\delta(t)) &\\leq -\\frac{{k\\_1}}{4}\\eta(t)^2 - \\left(\\frac{1}{2}{b\\_{1}}{k\\_2}-\\frac{1}{{k\\_1}}\\right)\\delta(t)^2 \\notag\\\\\n &+ \\frac{{k\\_1}}{2}v(\\psi\\_t)^2 + {b\\_{1}}\\frac{{k\\_1}^2}{2{k\\_2}}v\\_1(\\psi\\_t)^2 \\label{eq\\_BoundODELyapRelaxedFinal}.\n \\end{aligned}$$ \nThe bound was achieved using the inequalities $-\\eta(t)\\delta(t)\\leq \\frac{{k\\_{1}}}{4}\\eta(t)^2 + \\frac{1}{{k\\_{1}}}\\delta(t)^2$, $-\\eta(t)v(\\psi\\_t) \\leq \\frac{1}{2}\\eta(t)^2 + \\frac{1}{2}v(\\psi\\_t)^2$ and $-\\delta(t)v\\_1(\\psi\\_t) \\leq \\frac{{k\\_{2}}}{2}\\delta(t)^2 + \\frac{1}{2{k\\_{2}}}v\\_1(\\psi\\_t)^2$.\nNext, consider the Lyapunov functional\n \n$$\\begin{aligned}\n & V\\_2(\\eta(t),\\delta(t),\\psi\\_t) = U\\_1(\\eta(t),\\delta(t)) + \\frac{{b\\_{2}}}{2} G(\\psi\\_t)^2 = \\frac{1}{2} \\left( \\eta(t)^2 + {b\\_{1}}\\delta(t)^2 + {b\\_{2}} G(\\psi\\_t)^2 \\right) \\label{eq\\_lyapunovFunctionalRelaxedControl}\n \\end{aligned}$$ \nwhere *b*2 > 0 is a constant to be chosen and *G* is defined in.\nUsing Lemma [lemmaT1boundMain] and definition, it holds that\n \n$$\\begin{aligned}\n\\label{eq\\_v2ConnectionToG\\_2}\n |v\\_1(\\psi\\_t)|\\leq { \\ifthenelse{ \\equal{\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\sigma A \\right)}}\n}G(\\psi\\_t), \\quad \\forall t\\geq 0.\n \\end{aligned}$$ \nUsing the above estimate,, and defining ${b\\_{1}} = \\frac{4}{{k\\_2}{k\\_1}}$, ${b\\_{2}} = \\left( \\frac{{k\\_1}}{\\sigma} + {b\\_{1}}\\frac{c\\_1{k\\_1}^2}{\\sigma {k\\_2}} \\right) { \\ifthenelse{ \\equal{2\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( 2\\sigma A \\right)}}\n}$, we obtain the differential inequality\n \n*D*+*V*2(*η*(*t*), *δ*(*t*), *ψ**t*) ≤ − *σ*1*V*2(*η*(*t*), *δ*(*t*), *ψ**t*), ∀*t* ≥ 0 \nwhere $\\sigma\\_1 = \\min\\left (\\frac{1}{2}{k\\_1},\\frac{1}{2}{k\\_2},\\sigma\\right ) > 0$. Since estimate implies that all solutions stay bounded for all times for which they exist, we can conclude existence for all times.\nThe rest of the proof is exactly the same with the proof of Theorem [thmBacksteppingLyapunov1].\nGlobal Stabilization with State Constraints\n===========================================\nThe proposed controller unfortunately does not ensure that the dilution rate *D*(*t*) remains positive. This can be observed in an example simulation in Figure [figActuatorDynamicsstaticControllerAllFinal]. Whenever the signed control error *y*(*t*) − *y*\\* takes large negative values, the controller tries to “add population.”\nA classical approach to guarantee positivity of the dilution rate is to introduce a control barrier function (CBF)\n \n*h*(*η*, *D*, *ψ*) = *D*, \nwhich imposes a positivity constraint on the state *D*, and a safety filter of the nominal control law *u*0 = *u*c + *u*s, so that the overall control is given by\n \n$$u = u\\_0 + u\\_\\mathrm p = {u\\_\\mathrm{c}}+ {u\\_\\mathrm{s}}+ u\\_\\mathrm p \n \\label{eq\\_SafetyAugmentedControl}$$ \nwhere $u\\_\\mathrm p$ is the safety filter/override component of the controller that “penalizes” the negativity of the dilution and overrides the nominal feedback *u*0 so that the dilution is kept positive.\nThe safety filter for a system that has actuator dynamics $\\dot D =u$ and CBF *h* = *D* is particularly simple and given in Reference\n \n$$u\\_\\mathrm p = \\max \\{0,-u\\_0 - {k\\_3}D\\}, \\quad {k\\_3}>0, \n \\label{eq\\_safetyFilter}$$ \nwhich ensures that\n \n$$\\begin{aligned}\n \\dot h(t) &\\geq - {k\\_3}h(t),\n \\end{aligned}$$ \nnamely, that $\\dot D(t) \\geq - {k\\_3}D(t)$, i.e., $D(t) \\geq D(0) { \\ifthenelse{ \\equal{-{k\\_3}t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -{k\\_3}t \\right)}}\n} >0$ for *D*(0) > 0.\nThe representative example simulation in Figure [figActuatorDynamicsstaticControllerSafetyAllFinal] shows that the safety filter ensures the positivity of *D*(*t*) without impacting the convergence to the desired equilibrium. However, proving that the region of attraction of an equilibrium contains the entire safe set in the presence of a safety filter is impossible in general and has eluded us with this particular system as well.\n [figActuatorDynamicsstaticControllerSafetyAllFinal]\nFor this reason, we investigate the possibility of ensuring the positivity of *D*(*t*) with a stabilizing controller different than the nominal control. We next study system with state space $\\mathcal{F} \\times ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$, where $0 \\leq {{\\underline{D\\mkern-4mu}\\mkern4mu }}<{D^\\ast }<{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}$ are positive constants. The values of the constants ${{\\underline{D\\mkern-4mu}\\mkern4mu }}, {{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}$ are determined by the technical characteristics of the bioreactor but from a mathematical point of view the values of the constants ${{\\underline{D\\mkern-4mu}\\mkern4mu }}, {{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}$ are considered to be arbitrary given constants.\nConsider the diffeomorphism $\\varPhi:{\\mathbb R}\\to ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$ defined by\n \n$$\\begin{aligned}\n D = \\varPhi(\\zeta) = {{\\underline{D\\mkern-4mu}\\mkern4mu }}+ \\frac{{a\\_1}{ \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}}{{a\\_2}+{ \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}}\n \\label{diffeo1}\\end{aligned}$$ \nwith inverse $\\varPhi^{-1}:({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}) \\to {\\mathbb R}$\n \n$$\\begin{aligned}\n \\zeta = \\varPhi^{-1}(D) = \\ln \\left (\\frac{{a\\_2}(D-{{\\underline{D\\mkern-4mu}\\mkern4mu }})}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D} \\right )\n \\label{diffeo2}\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n {a\\_1}&= {{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}- {{\\underline{D\\mkern-4mu}\\mkern4mu }}> 0, ~~{a\\_2}= \\frac{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-{D^\\ast }}{{D^\\ast }-{{\\underline{D\\mkern-4mu}\\mkern4mu }}} > 0.\n \\label{diffeo3}\\end{aligned}$$ \nNotice that by virtue of, and it holds that $\\varPhi^{-1}({D^\\ast })=0$ and $\\varPhi(0)={D^\\ast }$.\nLet *k*1, *k*2, *k*3 > 0 be given arbitrary positive constants (the controller gains) and consider the static output feedback law\n \n$$\\begin{aligned}\n u(t) = \\frac{(D(t)-{{\\underline{D\\mkern-4mu}\\mkern4mu }})({{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D(t))}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}- {{\\underline{D\\mkern-4mu}\\mkern4mu }}}\\left [({k\\_{1}} + {k\\_{2}}) ({D^\\ast }-D(t)) - {k\\_{3}}\\left ( \\varPhi^{-1}(D(t))- {k\\_{2}} \\ln (y(t)/{y^\\ast }) \\right) \\right]. \\label{eq\\_IAS1\\_controller}\\end{aligned}$$ \nOur main result in this section is stated next.\n [figsimulationActDynKarafyllis]\n[Global asymptotic stabilization with dilution constrained to a finite positive interval] [thmConstrainedDilutionGlobalAsymStability] Let Assumption [asstechnicalbirthKernel] hold. Then for every *k*1, *k*2, *k*3 > 0, there exists a non-increasing function $\\varphi : {\\mathbb R}\\_+ \\to (0,+\\infty) $ and a function *α* ∈ K∞ such that for every $(f\\_{0},D\\_{0}) \\in \\mathcal{F} \\times ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$ the unique solution $(f[t],D(t)) \\in \\mathcal{F} \\times ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$ of the closed-loop system with and initial condition (*f*[0], *D*(0)) = (*f*0, *D*0) exists for all *t* ≥ 0 and satisfies the following stability estimate for all *t* ≥ 0\n \n$$\\begin{aligned}\n R\\_{2}(f[t],D(t)) \\leq { \\ifthenelse{ \\equal{- \\varphi \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( - \\varphi \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) t \\right)}}\n} \\alpha \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) \n \\label{eq\\_IAS2\\_DecayBound}\n \\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n R\\_{2}(f,D):=\\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f(a)}{{f^\\ast}(a)} \\right| + |\\varPhi^{-1}(D)|\n \\label{eq\\_IAS2\\_DecayBound22}\n \\end{aligned}$$ \nfor all $(f,D) \\in \\mathcal{F} \\times ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$.\n*Discussion of Theorem [thmConstrainedDilutionGlobalAsymStability]:* The stability estimate shows Global Asymptotic Stability of the equilibrium point (*f*\\*, *D*\\*) for the closed-loop system with. However, there are important differences with the corresponding stability estimate of Theorem [thmBacksteppingRelaxed2].\n* The equilibrium point (*f*\\*, *D*\\*) for system with is stabilized in a special measure: the measure $R\\_{2}(f,D)=\\max\\_{a\\in [0,A]}\\left| \\ln (f(a)/{f^\\ast}(a)) \\right| + |\\varPhi^{-1}(D)|$ and not the measure *R*1(*f*, *D*) = max*a* ∈ [0, *A*]∣ln(*f*(*a*)/*f*\\*(*a*))∣ + ∣*D* − *D*\\*∣ that was used in Theorem [thmBacksteppingRelaxed2]. This is a consequence of the fact that the measure *R*1(*f*, *D*) does not take into account the state constraint for *D*. On the other hand, the measure *R*2(*f*, *D*) takes into account all state constraints (notice that *R*2(*f*, *D*) “blows up” when the state tends to a point on the boundary of the state space). In other words, the measure *R*2(*f*, *D*) is a size functional for system with (58) and state space $\\mathcal{F} \\times ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$ in the sense of Reference, while the measure *R*1(*f*, *D*) is not a size functional.\n* An interesting special case of this result is the case where the dilution has no upper bound but it must remain positive, namely, ${{\\underline{D\\mkern-4mu}\\mkern4mu }}= 0, {{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}= +\\infty$. Since ${a\\_1}/{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}= 1$ and ${a\\_2}/{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}=1/{D^\\ast }$, the transformation in this case simplifies to $ \\zeta = \\varPhi^{-1}(D) = \\ln \\left ({D}/{{D^\\ast }} \\right )$, and the controller simplifies to\n \n$$\\begin{aligned}\n u(t) =D(t)\\left [({k\\_{1}} + {k\\_{2}}) ({D^\\ast }- D(t)) + {k\\_{3}}\\ln \\left(\\frac{{D^\\ast }}{D(t)} \\left(\\frac{y(t)}{{y^\\ast }}\\right)^{{k\\_{2}}} \\right) \\right], \\label{eq\\_IAS1\\_controller\\*}\\end{aligned}$$ \nwhereas the state space becomes the “positive orthant” F × (0, + ∞) and the measure *R*2 becomes *R*2(*f*, *D*) = max*a* ∈ [0, *A*]∣ln(*f*(*a*)/*f*\\*(*a*))∣ + ∣ln(*D*/*D*\\*)∣.\n* Theorem [thmConstrainedDilutionGlobalAsymStability] guarantees Global Asymptotic Stabilization by means of the output feedback law with exponential convergence rate exactly as Theorem [thmBacksteppingRelaxed2] does. To see this, notice that estimate implies that for every *ε* > 0 and for every $(f\\_{0},D\\_{0}) \\in \\mathcal{F} \\times ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$ there exists *T* ≥ 0 for which $R\\_{2}(f[T],D(T)) \\leq { \\ifthenelse{ \\equal{- \\varphi \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) T}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( - \\varphi \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) T \\right)}}\n} \\alpha \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) \\leq \\epsilon$. Using the semigroup property and we get the estimate\n \n$$\\begin{aligned}\n\\begin{array}{l} {R\\_{2}(f[t],D(t)) \\leq { \\ifthenelse{ \\equal{- \\varphi \\left( \\epsilon \\right) (t-T)}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( - \\varphi \\left( \\epsilon \\right) (t-T) \\right)}}\n} \\alpha \\left( R\\_{2}(f[T],D(T)) \\right)} \\\\ {\\leq { \\ifthenelse{ \\equal{- \\varphi \\left( \\epsilon \\right) (t-T)}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( - \\varphi \\left( \\epsilon \\right) (t-T) \\right)}}\n} \\alpha \\left( \\alpha \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) \\right)}\n\\end{array}\\end{aligned}$$ \nfor all *t* ≥ *T*. Combining, we get\n \n$$\\begin{aligned}\nR\\_{2}(f[t],D(t)) \\leq { \\ifthenelse{ \\equal{- \\varphi \\left( \\epsilon \\right) t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( - \\varphi \\left( \\epsilon \\right) t \\right)}}\n} \\left(1+\\frac{\\alpha \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) }{\\epsilon} \\right) ^ \\frac{\\varphi \\left( \\epsilon \\right)}{\\varphi \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right)} \\alpha \\left( \\alpha \\left( R\\_{2}(f\\_{0},D\\_{0}) \\right) \\right)\\end{aligned}$$ \nfor all *t* ≥ 0. The above estimate shows a uniform exponential convergence rate *φ*(*ε*) > 0, for every *ε* > 0. Consequently, the additional state constraint $D \\in ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$, does not exclude the possibility of a uniform exponential convergence rate.\nThe proof of Theorem [thmConstrainedDilutionGlobalAsymStability] shows that the controller gains *k*1, *k*2, *k*3 > 0 strongly affect the convergence properties of the solutions of the closed-loop system with. Similarly, the parameters ${{\\underline{D\\mkern-4mu}\\mkern4mu }}, {{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}$ strongly affect the convergence properties of the solutions of the closed-loop system with : a tighter state constraint leads to slower convergence and higher overshoots. The behavior of the solutions of the closed-loop with is illustrated by means of representative simulation example in Figure [figsimulationActDynKarafyllis].\nHaving discussed Theorem [thmConstrainedDilutionGlobalAsymStability] in detail, we next provide its proof.\n[Proof of Theorem [thmConstrainedDilutionGlobalAsymStability]] Using the transformations,, we get for the closed-loop system with :\n[eqClosedLoopKaraf]\n \n$$\\begin{aligned}\n \\dot \\eta(t) &= \\frac{{a\\_1}{a\\_2}(1-{ \\ifthenelse{ \\equal{\\zeta(t)}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta(t) \\right)}}\n})}{({a\\_2}+1)({a\\_2}+{ \\ifthenelse{ \\equal{\\zeta(t)}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta(t) \\right)}}\n})} := f\\_1(\\zeta(t))\\label{eq\\_etaDynamicsKarafCL}\\\\\n \\dot \\zeta(t) &= ({k\\_{1}} + {k\\_{2}})f\\_1(\\zeta(t)) - {k\\_{3}}\\left (\\zeta(t) - {k\\_{2}} \\eta(t) -{k\\_{2}}v(\\psi\\_t) \\right) \\label{eq\\_zetaDynamicsKarafCL}\\\\\n \\psi(t) &= \\int\\_0^A \\tilde k(a) \\psi(t-a)\\, {{\\rm d}}a.\\label{eq\\_IDEdynamicsKarafCL}\n \\end{aligned}$$ \nwhere *v*(*ψ**t*) is defined by and. Again, like in, we know that the map *t* ↦ *v*(*ψ**t*) is well-defined and the closed-loop locally admits a unique solution. Consider the functional *G*(*ψ**t*) defined by and define the function\n \n$$U\\_3(\\eta,\\zeta) = \\frac{1}{2}\\eta^2 + \\frac{{b\\_{1}}}{2}(\\zeta-{k\\_{2}}\\eta)^2\n \\label{eq\\_LyapFcnU3}$$ \nwhere ${b\\_{1}}= \\frac{1}{{k\\_{1}}{k\\_{2}}}$. Using definition and equations,, we find that\n \n$$\\begin{aligned}\n \\frac{{{\\rm d}}}{{{\\rm d}}t}U\\_3(\\eta(t),\\zeta(t)) &= \\frac{1}{{k\\_{2}}}\\zeta(t) f\\_1(\\zeta(t)) - {b\\_{1}}{k\\_{3}}(\\zeta(t)-{k\\_{2}}\\eta(t))^2 + {b\\_{1}}{k\\_{2}}{k\\_{3}}(\\zeta(t)-{k\\_{2}}\\eta(t))v(\\psi\\_{t})\n \\label{dada1}\n \\end{aligned}$$ \nfor all *t* ≥ 0 for which the solution of exists. Equation in conjunction with the inequality 2*k*2(*ζ* − *k*2*η*)*v*(*ψ*) ≤ (*ζ* − *k*2*η*)2 + *k*22*v*(*ψ*)2, gives for all *t* ≥ 0 for which the solution of exists:\n \n$$\\begin{aligned}\n \\frac{{{\\rm d}}}{{{\\rm d}}t}U\\_3(\\eta(t),\\zeta(t)) \n &\\leq \\frac{1}{{k\\_{2}}}\\zeta(t) f\\_1(\\zeta(t)) - \\frac{1}{2}{b\\_{1}}{k\\_{3}}(\\zeta(t)-{k\\_{2}}\\eta(t))^2 + \\frac{1}{2}{b\\_{1}}{k\\_{2}}^2{k\\_{3}}v(\\psi\\_{t})^2.\n \\label{dadaP1}\n \\end{aligned}$$ \nUsing we obtain from the following differential inequality\n \n$$\\begin{aligned}\n \\frac{{{\\rm d}}}{{{\\rm d}}t}U\\_3(\\eta(t),\\zeta(t)) \n &\\leq \\frac{1}{{k\\_{2}}}\\zeta(t) f\\_1(\\zeta(t)) - \\frac{1}{2}{b\\_{1}}{k\\_{3}}(\\zeta(t)-{k\\_{2}}\\eta(t))^2 + \\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{2} { \\ifthenelse{ \\equal{2\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( 2\\sigma A \\right)}}\n} G(\\psi\\_{t})^2\n \\label{dadaP2}\n \\end{aligned}$$ \nfor all *t* ≥ 0 for which the solution of exists. On the other hand, the differential inequality in conjunction with Lemma 2.12 on pages 77-78 in Reference implies the following estimate\n \n$$G(\\psi\\_t) \\leq { \\ifthenelse{ \\equal{- \\sigma t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( - \\sigma t \\right)}}\n} G(\\psi\\_{0})\n \\label{dadaP3}$$ \nfor all *t* ≥ 0 and consequently, we obtain from,, the following differential inequality that holds for all *t* ≥ 0 for which the solution of exists:\n \n$$\\begin{aligned}\n \\frac{{{\\rm d}}}{{{\\rm d}}t}U\\_3(\\eta(t),\\zeta(t)) \n &\\leq \\frac{1}{{k\\_{2}}}\\zeta(t) f\\_1(\\zeta(t)) - \\frac{1}{2}{b\\_{1}}{k\\_{3}}(\\zeta(t)-{k\\_{2}}\\eta(t))^2 + \\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{2} { \\ifthenelse{ \\equal{-2\\sigma (t-A)}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -2\\sigma (t-A) \\right)}}\n} G(\\psi\\_{0})^2.\n \\label{dadaP4}\n \\end{aligned}$$ \nNotice that by virtue of the fact that $f\\_1(\\zeta)=\\frac{{a\\_1}{a\\_2}(1-{ \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n})}{({a\\_2}+1)({a\\_2}+{ \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n})}$ (which implies that *ζ**f*1(*ζ*) ≤ 0 for all $\\zeta \\in {\\mathbb R}$), we obtain from the following estimate for all *t* ≥ 0 for which the solution of exists:\n \n$$\\begin{aligned}\n \\frac{{{\\rm d}}}{{{\\rm d}}t}U\\_3(\\eta(t),\\zeta(t)) \n &\\leq \\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{2} { \\ifthenelse{ \\equal{-2\\sigma (t-A)}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -2\\sigma (t-A) \\right)}}\n} G(\\psi\\_{0})^2.\n \\label{dadaP5}\n \\end{aligned}$$ \nThe above estimate in conjunction with Lemma 2.12 on pages 77-78 in Reference implies that\n \n$$\\begin{aligned}\nU\\_3(\\eta(t),\\zeta(t)) \\leq U\\_3(\\eta(0),\\zeta(0)) + \\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{2\\sigma} { \\ifthenelse{ \\equal{2\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( 2\\sigma A \\right)}}\n} G(\\psi\\_{0})^2\n\\label{dadaP6}\n \\end{aligned}$$ \nfor all *t* ≥ 0 for which the solution of exists. For the positive definite quadratic function *U*3 defined by, there exist constants $\\underline{c},\\overline{c}>0$ such that for all $(\\eta,\\zeta)\\in {\\mathbb R}^2$ it holds that\n \n$$\\begin{aligned}\n \\underline c \\left( \\eta^2 + \\zeta ^2\\right) \\leq U\\_3(\\eta,\\zeta) \\leq \\overline{c}\\left( \\eta^2 + \\zeta ^2\\right).\n \\label{dada6}\n \\end{aligned}$$ \nInequalities, show that the component (*η*(*t*), *ζ*(*t*)) of the solution of is bounded for all times *t* ≥ 0 for which the solution exists. Thus, the solution of exists for all *t* ≥ 0 and satisfies\n \n$$\\begin{aligned}\n\\eta^2 (t) + \\zeta^2 (t) \\leq \\frac{\\overline{c}}{\\underline{c}} \\left( \\eta^2 (0) + \\zeta^2 (0) \\right) + \\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{2\\sigma \\underline{c}} { \\ifthenelse{ \\equal{2\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( 2\\sigma A \\right)}}\n} G(\\psi\\_{0})^2\n\\label{dadaP7}\n \\end{aligned}$$ \nfor all *t* ≥ 0. The fact that $f\\_1(\\zeta)=\\frac{{a\\_1}{a\\_2}(1-{ \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n})}{({a\\_2}+1)({a\\_2}+{ \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n})}$ and, imply the existence of a non-increasing, positive mapping *φ*(*s*) > 0 defined for all *s* ≥ 0 with the property that for all *s* ≥ 0 the following estimate holds for all $(\\eta,\\zeta)\\in {\\mathbb R}^2$ with $\\zeta^2 \\leq \\frac{\\overline{c}}{\\underline{c}} s^2 + \\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{2\\sigma \\underline{c}} { \\ifthenelse{ \\equal{2\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( 2\\sigma A \\right)}}\n} s^2$:\n \n$$\\begin{aligned}\n \\frac{1}{{k\\_{2}}}\\zeta f\\_1(\\zeta) - \\frac{1}{2}{b\\_{1}}{k\\_{3}}(\\zeta-{k\\_{2}}\\eta)^2 \n \\leq -2 \\varphi(s) U\\_3(\\eta,\\zeta).\n \\label{dadaP8}\n \\end{aligned}$$ \nIt follows from and, that the following estimate holds for all *t* ≥ 0:\n \n$$\\begin{aligned}\n \\frac{{{\\rm d}}}{{{\\rm d}}t}U\\_3(\\eta(t),\\zeta(t)) \n &\\leq -2 \\varphi \\left( p\\_{0} \\right) U\\_3(\\eta(t),\\zeta(t)) + \\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{2} { \\ifthenelse{ \\equal{-2\\sigma (t-A)}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -2\\sigma (t-A) \\right)}}\n} G(\\psi\\_{0})^2\n \\label{dadaP9}\n \\end{aligned}$$ \nwhere *p*0 : = ∣*η*(0)∣ + ∣*ζ*(0)∣ + *G*(*ψ*0). Without loss of generality, we may assume that *φ*(*s*) ≤ *σ* for all *s* ≥ 0. The differential inequality in conjunction with Lemma 2.12 on pages 77-78 in Reference implies the following estimate for all *t* ≥ 0:\n \n$$\\begin{aligned}\n { \\ifthenelse{ \\equal{2 \\varphi \\left( p\\_{0} \\right) t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( 2 \\varphi \\left( p\\_{0} \\right) t \\right)}}\n} U\\_3(\\eta(t),\\zeta(t)) \n &\\leq U\\_3(\\eta(0),\\zeta(0)) + \\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{4 \\varphi \\left( p\\_{0} \\right)} { \\ifthenelse{ \\equal{2 \\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( 2 \\sigma A \\right)}}\n} G(\\psi\\_{0})^2.\n \\label{dadaP10}\n \\end{aligned}$$ \nCombining,,, we get for all *t* ≥ 0\n \n$$\\begin{aligned}\n \\begin{array}{l} { |\\eta (t)| + |\\zeta (t)| + { \\ifthenelse{ \\equal{\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\sigma A \\right)}}\n} G(\\psi\\_{t}) } \\\\\n {\\leq K \\left( p\\_{0} \\right) { \\ifthenelse{ \\equal{- \\varphi \\left( p\\_{0} \\right) t}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( - \\varphi \\left( p\\_{0} \\right) t \\right)}}\n}\n \\left( |\\eta (0)| + |\\zeta (0)| + { \\ifthenelse{ \\equal{\\sigma A}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\sigma A \\right)}}\n} G(\\psi\\_{0}) \\right) }\n \\end{array}\n \\label{dadaP11}\n \\end{aligned}$$ \nwhere *K*(*s*) is the non-decreasing function defined for *s* ≥ 0 by the equation\n \n$$\\begin{aligned}\n K(s):= 3\\sqrt{\\frac{\\overline{c}}{\\underline{c}}} + \\sqrt {\\frac{{b\\_{1}}{k\\_{2}}^2{k\\_{3}}}{ \\underline{c} \\varphi(s) }}. \n \\label{dadaP12}\n \\end{aligned}$$ \nUsing,,,, the fact that *p*0 : = ∣*η*(0)∣ + ∣*ζ*(0)∣ + *G*(*ψ*0) and definition, we obtain the stability estimate for an appropriate *α* ∈ K∞. The proof is complete.\nFull-State Stabilization under State Constraints\n================================================\nThe design that we just presented is mindful of three practical requirements:\n1. dilution must remain above a given positive lower bound ${{\\underline{D\\mkern-4mu}\\mkern4mu }}>0$;\n2. dilution must obey a given upper bound ${{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}< +\\infty$;\n3. only a bulk concentration *y*(*t*) is available for measurement, rather than an age-structured density *f*(*a*, *t*).\nThe price we pay for meeting all these practical requirements is a design that is quite complicated and that offers little insight and generalizability to other situations. In the remainder of this section we present an alternative design, which fails to obey two of the above requirements but is clear, elegant, and systematic.\nFirst, we dispose of the requirement for a strictly positive lower bound ${{\\underline{D\\mkern-4mu}\\mkern4mu }}>0$ and only require the dilution to remain positive. This is practically reasonable. Dilution/harvesting can be easily completely shut off. Second, we lift the fixed upper bound requirement ${{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}< +\\infty$ and require the dilution to merely be bounded.\nThird, we allow the measurement of the full state of the system, namely, of the age-structured density *f*(*a*, *t*). This is a stronger requirement as the age distribution of the population cannot be measured. However, the size distribution of the population, which is proportional to the age distribution can be measured. We allow the full-state measurement only for pedagogical reasons, and without loss of generality. The analysis is much cleaner, clearer, and exact when the full *f*(*a*, *t*) is measured, as opposed to when only *y*(*t*) is measured.\nSo, we start by setting ${{\\underline{D\\mkern-4mu}\\mkern4mu }}= 0$ and ${{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}= +\\infty$. Then, we recall that\n \n$$\\begin{aligned}\n \\eta &=& \\ln \\Pi(f)\\\\\n \\zeta &=& \\ln\\frac{D}{{D^\\ast }}.\\end{aligned}$$ \nNext, we introduce a backstepping transformation\n \n*z* = *ζ* − *c*1*η*, *c*1 > 0. \nStraightforward calculation then yields\n \n$$\\begin{aligned}\n\\dot\\eta &=& {D^\\ast }\\left(1- { \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}\\right)\\nonumber\\\\\n&=& {D^\\ast }\\left(1- { \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n}\\right)+{D^\\ast }{ \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n} \\left(1- { \\ifthenelse{ \\equal{z}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( z \\right)}}\n}\\right)\\\\\n\\dot \\zeta &=& \\frac{u}{D} \\\\\n\\dot z &=& \\frac{u}{D} - c\\_1 {D^\\ast }\\left(1- { \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}\\right). \\end{aligned}$$ \nBefore we design a controller, we introduce two positive definite radially unbounded functions,\n \n$$\\begin{aligned}\n\\omega(z) &=& { \\ifthenelse{ \\equal{z}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( z \\right)}}\n} -1-z\\\\\n\\mu(z) &=& \\sinh^2\\left(\\frac{z}{2}\\right).\\end{aligned}$$ \nNext, we introduce two Lyapunov functions,\n \n$$\\begin{aligned}\n V\\_1 &=& \\omega(-c\\_1\\eta) \\\\\n V\\_2 &=& \\omega(\\zeta - c\\_1\\eta) = \\omega(z)\\end{aligned}$$ \nand the overall Lyapunov function\n \n*V* = *θ**V*1 + *V*2 = *θ**ω*( − *c*1*η*) + *ω*(*ζ* − *c*1*η*), *θ* > 0. \nNoting that $\\omega'(z) = { \\ifthenelse{ \\equal{z}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( z \\right)}}\n} -1$, as well as that\n \n$$\\left({ \\ifthenelse{ \\equal{z}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( z \\right)}}\n} -1\\right)\\left({ \\ifthenelse{ \\equal{-z}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( -z \\right)}}\n} -1\\right) = - 4 \\sinh^2(z/2),$$ \nafter a lengthy calculation, with intermediate steps that produce\n \n$$\\begin{aligned}\n\\dot V\\_1 &= & - 4{D^\\ast }c\\_1 \\mu(-c\\_1\\eta) - {D^\\ast }c\\_1 \\left( { \\ifthenelse{ \\equal{z}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( z \\right)}}\n} -1\\right)\\left( { \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n} -1\\right)\n\\\\\n\\dot V\\_2 &=& {D^\\ast }\\left( { \\ifthenelse{ \\equal{z}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( z \\right)}}\n} -1\\right)\n\\left(\\frac{u}{{D^\\ast }D} - c\\_1 \\left(1- { \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}\\right) \\right),\\end{aligned}$$ \nwe note that the feedback law\n \n$$\\label{eq-bkst-cont-mk}\nu = {D^\\ast }D \\left\\{\nc\\_1 \\left[\\theta \\left( { \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n} -1\\right) + 1- { \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}\n\\right]\n+ c\\_2 \\left({ \\ifthenelse{ \\equal{c\\_1\\eta -\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta -\\zeta \\right)}}\n} -1\\right)\n\\right\\},$$ \nwith *c*2 > 0, produces a particularly elegant Lyapunov derivative,\n \n$$\\dot V = - 4{D^\\ast }\\left[\n \\theta c\\_1 \\mu(-c\\_1\\eta) + c\\_2 \\mu(\\zeta-c\\_1\\eta)\n \\right],$$ \nwhich is negative definite and radially unbounded. Global asymptotic stability of the equilibrium *η* = *ζ* = 0 then follows. With the global exponential stability of the decoupled *ψ*-system, the global asymptotic stability of the (*η*, *ψ*, *ζ*)-system follows.\nThe controller, rewritten in the original (*f*, *D*)-variables as\n \n$$\\label{eq-bkst-cont-mk+}\nu = {D^\\ast }D \\left\\{\nc\\_1 \\left[\\theta \\left( \\left(\\Pi(f)\\right)^{c\\_1} -1\\right) + 1- \\frac{D}{{D^\\ast }}\n\\right]\n+ c\\_2 \\left(\\frac{{D^\\ast }}{D}\\left(\\Pi(f)\\right)^{c\\_1} -1\\right)\n\\right\\},$$ \nwith gains *c*1, *c*2, *θ* > 0, guarantees that there exists a class ${\\cal KL}$ function *β* such that\n \n$$\\begin{aligned}\n \\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f(a,t)}{{f^\\ast}(a)} \\right| + \\left|\\ln \\frac{D(t)}{{D^\\ast }} \\right| \\\n \\leq \n \\beta\\left(\\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f\\_0(a)}{{f^\\ast}(a)} \\right| + \\left|\\ln \\frac{D\\_0}{{D^\\ast }} \\right|,t\\right), \\qquad \\forall t\\geq 0\\end{aligned}$$ \nfor all *f*0 ∈ F and *D*0 > 0.\nThis design with *D* ∈ (0, + ∞) calls for a reexamination of the more difficult problem where $D\\in ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$ and $0 \\leq {{\\underline{D\\mkern-4mu}\\mkern4mu }}<{D^\\ast }<{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}$. We omit much of the detail and summarize the design and analysis.\nLet us denote\n \n$$\\delta\\_1 = \\frac{{D^\\ast }-{{\\underline{D\\mkern-4mu}\\mkern4mu }}}{{D^\\ast }}, \\qquad n = \\frac{{D^\\ast }- {{\\underline{D\\mkern-4mu}\\mkern4mu }}}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-{D^\\ast }}.$$ \nOne gets\n \n$$\\begin{aligned}\n \\dot\\eta &=& {D^\\ast }\\delta\\_1 \\frac{1-{ \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}}{1+n { \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}}, \\qquad \\zeta = \\ln\\left(\\frac{1}{n} \\frac{D-{{\\underline{D\\mkern-4mu}\\mkern4mu }}}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D}\\right)\\\\\n \\dot \\zeta &=& \\frac{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D}{(D-{{\\underline{D\\mkern-4mu}\\mkern4mu }})({{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D)}u\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n\\dot V\\_1 &= & - 4{D^\\ast }c\\_1 \\frac{\\delta\\_1}{1+n{ \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n}}\\mu(-c\\_1\\eta) \n \\nonumber\\\\\n && - {D^\\ast }c\\_1 \\frac{\\delta\\_1(1+n)}{\\left(1+n { \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n}\\right)^2}\\left({ \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n} -1\\right)\\left( { \\ifthenelse{ \\equal{z}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( z \\right)}}\n} -1\\right).\\end{aligned}$$ \nWith further calculations, one obtains\n \n$$\\label{eq-Vdot-most-general}\n \\dot V = - 4{D^\\ast }\\left[\n \\theta c\\_1 \\frac{\\delta\\_1}{1+n{ \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n}} \\mu(-c\\_1\\eta)+ c\\_2 \\mu(\\zeta-c\\_1\\eta)\n \\right]$$ \nwith the controller\n \n$$\\begin{aligned}\n\\label{eq-bkst-cont-mk++}\nu &=& \\frac{{D^\\ast }}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-{{\\underline{D\\mkern-4mu}\\mkern4mu }}}(D-{{\\underline{D\\mkern-4mu}\\mkern4mu }})({{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D) \n\\left\\{\nc\\_1 \\left[\\theta \\frac{\\delta\\_1(1+n)}{\\left(1+n { \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n}\\right)^2}\\left( { \\ifthenelse{ \\equal{c\\_1\\eta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta \\right)}}\n} -1\\right) \n+ \\frac{1- { \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}}{1+n{ \\ifthenelse{ \\equal{\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( \\zeta \\right)}}\n}}\n\\right]\n\\right. \\nonumber\\\\\n&& \\left. + c\\_2 \\left({ \\ifthenelse{ \\equal{c\\_1\\eta -\\zeta}{} }\n {{\\mathrm{exp}}}\n {{\\mathrm{exp}\\left( c\\_1\\eta -\\zeta \\right)}}\n} -1\\right)\n\\right\\}\n\\nonumber\\\\\n&=&\\frac{{D^\\ast }}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-{{\\underline{D\\mkern-4mu}\\mkern4mu }}} (D-{{\\underline{D\\mkern-4mu}\\mkern4mu }})({{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D)\\left\\{\nc\\_1 \\left[\\theta \\frac{\\delta\\_1(1+n)}{\\left(1+n\\left(\\Pi(f)\\right)^{c\\_1}\\right)^2}\\left( \\left(\\Pi(f)\\right)^{c\\_1} -1\\right) + 1- \\frac{1}{n} \\frac{D-{{\\underline{D\\mkern-4mu}\\mkern4mu }}}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D}\n\\right]\n\\right. \\nonumber\\\\\n&& \\left.+ c\\_2 \\left(\nn\\frac{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D}{D-{{\\underline{D\\mkern-4mu}\\mkern4mu }}}\n\\left(\\Pi(f)\\right)^{c\\_1} -1\\right)\n\\right\\}. \\end{aligned}$$ \nThis is clearly a considerably more complicated controller than but the simplicity of the negative definite, radially unbounded makes this controller worth consideration.\nThe controller with gains *c*1, *c*2, *θ* > 0, guarantees that there exists a class ${\\cal KL}$ function *β* such that\n \n$$\\begin{aligned}\n &\n \\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f(a,t)}{{f^\\ast}(a)} \\right| + \\left|\\ln\\left(\n \\frac{D(t)-{{\\underline{D\\mkern-4mu}\\mkern4mu }}}{{D^\\ast }- {{\\underline{D\\mkern-4mu}\\mkern4mu }}} \n \\frac{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-{D^\\ast }}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D(t)}\n \\right) \\right| \\\n \\nonumber\\\\ &\n \\leq \n \\beta\\left(\\max\\_{a\\in [0,A]}\\left| \\ln \\frac{f\\_0(a)}{{f^\\ast}(a)} \\right| \n + \\left|\\ln\\left(\n \\frac{D\\_0-{{\\underline{D\\mkern-4mu}\\mkern4mu }}}{{D^\\ast }- {{\\underline{D\\mkern-4mu}\\mkern4mu }}} \n \\frac{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-{D^\\ast }}{{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}}-D\\_0}\n \\right)\\right|,t\\right), \\qquad \\forall t\\geq 0\\end{aligned}$$ \nfor all *f*0 ∈ F and $D\\_0\\in ({{\\underline{D\\mkern-4mu}\\mkern4mu }},{{\\mkern 1.5mu\\overline{\\mkern-1.5muD\\mkern-1.5mu}\\mkern 1.5mu}})$.\nConclusion\n==========\nThe interest in extending the foundational design and analysis results in Reference goes in many directions. Among them, the most interesting at this stage are in incorporating additional dynamics, which may arise for a variety of reasons (the presence of actuator dynamics, a substrate, input delay, additional species, etc.). In this paper we made what is probably most natural non-trivial step—incorporating actuator dynamics in the form of a single state. Compensating these actuator dynamics was not exceptionally difficult but ensuring that, in addition to the population density remaining positive, the state of the actuator dynamics remains positive as well, or even remains within a given positive interval, is far from elementary and charts a path towards further generalizations.\nAmong the further generalizations are those that bring into the model additional infinite-dimensional states. One such addition is an input delay, which can be compensated using predictor feedback. Other additions of infinite dimensional states are additional species. Even two species can be interconnected in several ways, of which some require no innovations in the control design, while other types of interconnections do. Such generalizations of the chemostat problem into population systems, of which epidemiology is one possible application, open exciting possibilities for future research.\nAuthor contributions\n--------------------\nAll the authors have contributed to conceiving the problems and writing the article. The simulations were produced by Paul-Erik Haacker.\nFinancial disclosure\n--------------------\nThe work of Miroslav Krstic was funded by the NSF grant ECCS-2151525 and the AFOSR grant FA9550-22-1-0265. The work of Mamadou Diagne was funded by the NSF CAREER Award CMMI-2302030 and the NSF grant CMMI-2222250.\nConflict of interest\n--------------------\nThe authors declare no potential conflict of interests.\nProof of technical lemmas\n=========================\n[Proof of Lemma [lemmaT1boundMain].]We rewrite\n \n$$\\begin{aligned}\n -v\\_1(\\psi\\_t) = \\frac{-1}{\\int\\_0^Ap(a){f^\\ast}(a)(1+\\psi(t-a)){{\\rm d}}a} T\\_3(\\psi\\_t)\n \\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n T\\_3(\\psi\\_t) &= p(A){f^\\ast}(A)(T\\_4(\\psi\\_t)-\\psi(t-A)) - p(0){f^\\ast}(0)(T\\_4(\\psi\\_t)-\\psi(t))\\label{eq\\_def\\_T3}\\\\\n &- \\int\\_0^A \\tilde p(a) {f^\\ast}(a) {{\\rm d}}a T\\_4(\\psi\\_t) + \\int\\_0^A \\tilde p(a) {f^\\ast}(a) \\psi(t-a){{\\rm d}}a\\\\\n T\\_4(\\psi\\_t) &= \\frac{1}{{y^\\ast }}\\int\\_0^A p(a){f^\\ast}(a) \\psi(t-a){{\\rm d}}a.\n \\end{aligned}$$ \nBy means of the definition of *y*\\* and the nonnegativity of *p*, *f*\\* on [0, *A*], it holds that\n \n$$\\begin{aligned}\n |T\\_4(\\psi\\_t)| \\leq \\max\\_{a\\in[0,A]}|\\psi(t-a)|. \\label{eq\\_T4\\_bound}\n \\end{aligned}$$ \nTaking the absolute value of and using we arrive at\n \n$$\\begin{aligned}\n |T\\_3(\\psi\\_t)| \\leq \\sqrt{c\\_1}{y^\\ast }\\max\\_{a\\in[0,A]}|\\psi(t-a)| \\label{eq\\_T3\\_bound}\n \\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n \\frac{1}{2} \\sqrt{c\\_1} {y^\\ast }= p(A){f^\\ast}(A) + p(0){f^\\ast}(0) + \\int\\_0^A|\\tilde p(a)|{f^\\ast}(a){{\\rm d}}a >0.\n \\end{aligned}$$ \nWe note, that *c*1 is independent of the chosen setpoint *y*\\*. With nonnegativity of *p*(*a*) and *f*\\*(*a*) for *a* ∈ [0, *A*] and 1 + *ψ*(*t* − *a*) > 0 for all $(a,t) \\in [0,A] \\times {\\mathbb R}\\_+$, we note that\n \n$$\\begin{aligned}\n |v\\_1(\\psi\\_t)| = \\frac{1}{\\int\\_0^Ap(a){f^\\ast}(a)(1+\\psi(t-a)){{\\rm d}}a} |T\\_3(\\psi\\_t)|\n \\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n \\int\\_0^A p(a) {f^\\ast}(a) \\psi(t-a){{\\rm d}}a \\geq {y^\\ast }\\min\\_{a\\in[0,A]} \\psi(t-a)\n \\end{aligned}$$ \nyields using.\n10 Kuritz K, Stöhr D, Pollak N, Allgöwer F. On the relationship between cell cycle analysis with ergodic principles and age-structured cell population models. *Journal of theoretical biology* 2017; 414: 91–102.\nBilly F, Clairambault J, Delaunay F, Feillet CA, Robert N. Age-structured cell population model to study the influence of growth factors on cell cycle dynamics.. *Mathematical Biosciences and Engineering* 2012: xx.\nRong L, Feng Z, Perelson AS. Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy. *SIAM Journal on Applied Mathematics* 2007; 67(3): 731–756.\nKarafyllis I, Krstic M. Stability of integral delay equations and stabilization of age-structured models. *ESAIM: Control, Optimisation and Calculus of Variations* 2017; 23(4): 1667–1714.\nKurth AC, Schmidt K, Sawodny O. Tracking-control for age-structured population dynamics with self-competition governed by integro-PDEs. *Automatica* 2021; 133: 109850.\nKurth AC, Sawodny O. Control of age-structured population dynamics with intraspecific competition in context of bioreactors. *Automatica* 2023; 152: 110944.\nDe Leenheer P, Aeyels D. Stabilization of positive linear systems. *Systems & control letters* 2001; 44(4): 259–271.\nDe Leenheer P, Aeyels D. Stabilization results for positive systems with first integrals. *IFAC Proceedings Volumes* 2001; 34(6): 881–886.\nInaba H. *Age-structured population dynamics in demography and epidemiology*. Springer. 2017.\nMartcheva M. *An introduction to mathematical epidemiology*. 61. Springer. 2015.\nToth D, Kot M. Limit cycles in a chemostat model for a single species with age structure. *Mathematical biosciences* 2006; 202(1): 194–217.\nMcKendrick AG. Applications of mathematics to medical problems. *Proceedings of the Edinburgh Mathematical Society* 1925; 44: 98–130.\nBrauer F, Castillo-Chavez C. *Mathematical models in population biology and epidemiology*. 2. Springer. 2012.\nKermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. *Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character* 1927; 115(772): 700–721.\nBoucekkine R, Hritonenko N, Yatsenko Y. *Optimal control of age-structured populations in economy, demography, and the environment*. Routledge. 2011.\nCharlesworth B. *Evolution in age-structured populations*. 2. Cambridge University Press Cambridge. 1994.\nRundnicki R, Mackey MC. Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations. *Journal of Mathematical Analysis and Applications* 1994; 187(2): 548–566.\nSmith HL, Waltman P. *The theory of the chemostat: dynamics of microbial competition*. 13. Cambridge university press. 1995.\nInaba H. A semigroup approach to the strong ergodic theorem of the multistate stable population process. *Mathematical Population Studies* 1988; 1(1): 49–77.\nInaba H. Asymptotic properties of the inhomogeneous Lotka-Von Foerster system. *Mathematical Population Studies* 1988; 1(3): 247–264.\nFeichtinger G, Tragler G, Veliov VM. Optimality conditions for age-structured control systems. *Journal of Mathematical Analysis and Applications* 2003; 288(1): 47–68.\nSun B. Optimal control of age-structured population dynamics for spread of universally fatal diseases II. *Applicable Analysis* 2014; 93(8): 1730–1744.\nSchmidt K, Karafyllis I, Krstic M. Yield trajectory tracking for hyperbolic age-structured population systems. *Automatica* 2018; 90: 138–146.\nKarafyllis I, Kravaris C, Syrou L, Lyberatos G. A vector Lyapunov function characterization of input-to-state stability with application to robust global stabilization of the chemostat. *European Journal of Control* 2008; 14(1): 47–61.\nKarafyllis I, Kravaris C, Kalogerakis N. Relaxed Lyapunov criteria for robust global stabilisation of non-linear systems. *International Journal of Control* 2009; 82(11): 2077–2094.\nGouze JL, Robledo G. Robust control for an uncertain chemostat model. *International Journal of Robust and Nonlinear Control* 2006; 16(3): 133–155.\nDe Leenheer P, Smith H. Feedback control for chemostat models. *Journal of Mathematical Biology* 2003; 46(1): 48–70.\nDimitrova N, Krastanov M. Nonlinear adaptive stabilizing control of an anaerobic digestion model with unknown kinetics. *International Journal of Robust and Nonlinear Control* 2012; 22(15): 1743–1752.\nSontag ED. Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning. *Systems & Control Letters* 2022; 161: 105138.\nKarafyllis I, Jiang ZP. *Stability and stabilization of nonlinear systems*. Springer Science & Business Media. 2011.\nToth D. Strong resonance and chaos in a single-species chemostat model with periodic pulsing of resource. *Chaos, Solitons & Fractals* 2008; 38(1): 55–69.\nToth D. Bifurcation structure of a chemostat model for an age-structured predator and its prey. *Journal of Biological Dynamics* 2008; 2(4): 428–448.\nKurth AC, Schmidt K, Sawodny O. Inversion-based and optimal feedforward control for population dynamics with input constraints and self-competition in chemostat reactor applications. *Journal of Dynamic Systems, Measurement, and Control* 2021; 143(5).\nSchmidt K, Sawodny O. Efficient simulation of semilinear populations models for age-structured bio reactors. In: IEEE. ; 2017: 1716–1721.\nKrstic M. Inverse Optimal Safety Filters. *IEEE Transactions on Automatic Control* 2023: 1-16. [doi: 10.1109/TAC.2023.3278788](\\doibase 10.1109/TAC.2023.3278788)\n\nStabilization of Age-Structured Chemostat Hyperbolic PDE with Actuator Dynamics\n===============================================================================\nIntroduction\n============\nMotivation: Chemostat as a benchmark for nonlinear control and for control in epidemiology\n------------------------------------------------------------------------------------------\nMany industrial biotechnology processes can be described by a nonlinear PDE (Partial Differential Equation) of population dynamics that are structured around age. Reflecting the maturation of enzymes, microbial organisms, animal or plant cells and tissues, these models of biological and biochemical systems are often exploited to favor the desired functionalities and achieve cost-effective bioreactors’ production rate. The word “chemostat” denotes a biological process which is fed a nutrient at a certain rate and from which a bioproduct-nutrient mix is being extracted/removed at the same rate. This rate is referred to as the “dilution rate.” The relevance of chemostats in control engineering extends beyond their own application in biotechnology (such as the manufacturing of pharmaceuticals). First, a chemostat is a nonlinear control problem—even when the limiting substrate(s) nonlinear dynamics are neglected—due to the fact that its dynamic model contains a product of the population density and the dilution rate. Second, the relevance of the chemostat is also in the fact that it is an example of a “positive system,” namely, a control system whose state is subject to an inequality constraint.\nThird, and perhaps most important at present, chemostat is a special case of more complex population dynamics such as those that arise in epidemiology. Before one takes on control design of SIR-type (susceptible-infected-recovered) models in epidemiology, in which control is being applied to more than one population (for example, to the susceptible and the infected populations), it is necessary to understand how to control single-population chemostats. Just as there is a parallel between the biomass in a chemostat and the infected population in epidemiology, as well as between the substrate in a chemostat and the susceptible population in epidemiology, there is likewise a parallel between the treatment/therapy being administered as an input in epidemiology and the dilution input of a chemostat.\nHence, the study of control of an age-structured chemostat that we undertake here is an important first step towards designing controllers for *age-structured* models in epidemiology.\nAge-structured population dynamics\n----------------------------------\nAge-structured chemostat models are described by a particular first order hyperbolic Partial Differential Equation (PDE) with a non-local boundary condition: the McKendrick-von Foerster equation (see References and the references therein). Age-structured models are extensions of the chemostat models described by Ordinary Differential Equations (ODEs; see Reference ). A particular and deep mathematical tool has been developed for the study of age-structured models: the ergodic theorem, also known as the asymptotic behavior (see References for similar results on asymptotic similarity and Reference for the proof of the ergodic theorem by means of Lyapunov arguments).\nControl studies for age-structured models are rare. Optimal control problems have been studied (see References and the references therein). On the other hand, feedback control of infinite-dimensional population dynamics was introduced in Reference and further considered in References. More specifically, the use of the ergodic theorem and the corresponding study of linear integral delay equations in References led to a special nonlinear infinite-dimensional change of variables, from the age-structured population density (a population density that is a function of the continuum age) into\n* a scalar variable that represents the controllable mode of the bilinear non-local hyperbolic PDE system which is directly actuated by the dilution rate input, and\n* an infinite-dimensional uncontrollable but exponentially stable dynamical system described by linear integral delay equations,\nand has subsequently been the cornerstone of the control design and stability analysis of age-structured population dynamics.\nThe control studies in References and used the dilution rate as the control input as in many other control studies of chemostat models described by ODEs (see References ). Actuation by dilution rate is both physical and plausible and it amounts to harvesting the population. In simplistic terms, a steady level of harvesting is needed to maintain the population at a sustained level (equilibrium) and, when the population exceeds such an equilibrium, it is over-harvested by (over)dilution, whereas when the population drops below such an equilibrium, it is under-harvested by (under)dilution. How this over- and under-harvesting is to be done exactly is highly non-trivial because the population is age-structured, meaning that its state is functional/infinite-dimensional, due to which one cannot really speak of overpopulation and underpopulation in a bulk sense. One has, instead, to take into account that the population can vary from the equilibrium in infinitely many ways (overpopulated old with underpopulated young, vice-versa, and so on). This infinite variety in the state profiles, in addition to the infinite number of states (age-specific densities), along with the nonlinearity of the problem, is one of the theoretical attractions to this control design problem.\nChemostat with dynamic actuation of dilution\n--------------------------------------------\nWhile dilution is a plausible form of actuation, it is not possible to actuate it instantaneously. As with any other actuation, it is governed by its own dynamics. In the case of dilution, it is affected by the dynamics of pipes and valves that are used to run the nutrients into a chemostat and the population-nutrient mix out of the chemostat. In biochemical engineering, the dilution rate is defined as the ratio of the inlet volumetric flow rate to the reactor volume. Since the volume of the reactor is not necessarily constant, it follows that the dynamics of the dilution rate can play an important role.\nIn this paper we design stabilizing feedback laws for the age-structured chemostat with dilution dynamics governed by a single, scalar integrator. In the most advanced versions of our designs, we ensure that not only the population density, but also the dilution rate, remains in a specific interval (and positive). Dilution rate being positive means that we are not feeding population back into the chemostat but are only harvesting/removing population.\nWhile the age-structured chemostat with a constant dilution rate is a PDE system with one eigenvalue at the origin and all other eigenvalues in the left half plane, meaning that it is not open-loop unstable, and meaning that it can, at worst, settle at an undesirable age profile for the population (provided the constant dilution is at a population-sustaining level), the system with dilution dynamics modeled by an integrator has two eigenvalues at the origin, meaning that it is open-loop unstable. This is one more way of noting the need for a more sophisticated control design for this PDE system once the dynamics of the dilution actuation are introduced.\nIn this paper we tackle three design challenges:\n* we overcome the dynamics of the integrator modeling the dilution actuation,\n* we perform the design and the analysis using feedback of only a sensor of bulk population, which has an unknown profile in sensing populations of different ages,\n* we ensure that the dilution rate that drives the population to the desired age-structured equilibrium from all initial positive population density age profiles remains itself in a specific interval (and positive).\nWhile in the initial work on control of the age-structured chemostat the state space of the systems was, so to speak, the “positive orthant” of infinite dimension (the age-structured population density), in this work the state space increases by one positive-valued state when the dilution dynamics are modeled by an integrator and becomes, so to speak, a “positive orthant” of dimension ∞ + 1. From a mathematical point of view, the paper deals with a global feedback stabilization problem for a constrained nonlinear infinite-dimensional control system which is defined on a specific open set. Finite-dimensional control systems defined on open sets were recently studied in Reference.\n***Organization:*** The structure of the paper is as follows. Section [secpopDynamicsActuatorDynamicsBackstepping-model] introduces the mathematical model as well as the stabilization problem. Sections [secpopDynamicsActuatorDynamicsBackstepping] and [secBacksteppingStabilityProof] study the backstepping design of the problem with unrestricted dilution rate assuming that the state is available, while Section [secRelaxedBackstepping] provides globally stabilizing output feedback laws. Sections [secSafety] and [sec-constrained-miroslav] are devoted to the feedback stabilization problem of the more demanding case where the dilution rate is restricted to take values in a specific interval.\n***Notation:***\n* ${\\mathbb R}\\_+ $ denotes the interval [0, + ∞).\n* Let *U* be an open subset of a metric space and $\\Omega \\subseteq {\\mathbb R}^{m} $ be a set. By *C*0(*U*; Ω), we denote the class of continuous mappings on *U*, which take values in Ω. When $U\\subseteq {\\mathbb R}^{n} $, by *C*1(*U*; Ω), we denote the class of continuously differentiable functions on *U*, which take values in Ω. When $U=[a,b)\\subseteq {\\mathbb R}$ (or $U=[a,b]\\subseteq {\\mathbb R}$) with *a* < *b*, *C*0([*a*, *b*); Ω) (or *C*0([*a*, *b*]; Ω)) denotes all functions *f* : [*a*, *b*) → Ω (or *f* : [*a*, *b*] → Ω), which are continuous on (*a*, *b*) and satisfy $\\mathop{\\lim }\\limits\\_{s\\to a^{+} } \\left(f(s)\\right)=f(a)$ (or $\\mathop{\\lim }\\limits\\_{s\\to a^{+} } \\left(f(s)\\right)=f(a)$ and $\\mathop{\\lim }\\limits\\_{s\\to b^{-} } \\left(f(s)\\right)=f(b)$). When $U=[a,b)\\subseteq {\\mathbb R}$, *C*1([*a*, *b*); Ω) denotes all functions *f* : [*a*, *b*) → Ω which are continuously differentiable on (*a*, *b*) and satisfy $\\mathop{\\lim }\\limits\\_{s\\to a^{+} } \\left(f(s)\\right)=f(a)$ and $\\mathop{\\lim }\\limits\\_{h\\to 0^{+} } h^{-1} \\left(f(a+h)-f(a)\\right)=\\mathop{\\lim }\\limits\\_{s\\to a^{+} } f'(s)$.\n* $\\mathcal K\\_{\\infty} $ is the class of all strictly increasing, unbounded functions $a\\in C^{0} ( {\\mathbb R}\\_+ ; {\\mathbb R}\\_+ )$, with *a*(0) = 0 (see Reference ).\n* KL is the class of functions $\\beta:{\\mathbb R}\\_+ \\times {\\mathbb R}\\_+ \\to {\\mathbb R}\\_+$ which satisfy the following: For each *t* ≥ 0, the mapping *β*( ⋅ , *t*) is of class K, and, for each *s* ≥ 0, the mapping *β*(*s*, ⋅ ) is nonincreasing with lim*t* → ∞*β*(*s*, *t*) = 0 (see Reference ).\n* For any subset $S\\subseteq {\\mathbb R}$ and for any *A* > 0, *P**C*1([0, *A*]; *S*) denotes the class of all functions *f* ∈ *C*0([0, *A*]; *S*) for which there exists a finite (or empty) set *B* ⊂ (0, *A*) such that: (i) the derivative *f*ʹ(*a*) exists at every *a* ∈ (0, *A*) ∖ *B* and is a continuous function on (0, *A*) ∖ *B*, (ii) all meaningful right and left limits of *f*ʹ(*a*) when *a* tends to a point in *B*⋃{0, *A*} exist and are finite.\n* Let a function $f\\in C^{0} ({\\mathbb R}\\_+ \\times [0,A])$ be given, where *A* > 0 is a constant. We use the notation *f*[*t*] to denote the profile at certain *t* ≥ 0, i.e., (*f*[*t*])(*a*) = *f*(*a*, *t*) for all *a* ∈ [0, *A*].\n* Let a function $x\\in C^{0} ([-A,+\\infty );{\\mathbb R})$ be given, where *A* > 0 is a constant. We use the notation $x\\_{t} \\in C^{0} ([-A,0];{\\mathbb R})$ to denote the “*A* − history” of *x* at certain *t* ≥ 0, i.e., (*x**t*)( − *a*) = *x*(*t* − *a*) for all *a* ∈ [0, *A*].\n* Let a function $f\\in C^0({\\mathbb R}\\_{+}; {\\mathbb R})$ be given. By $D^+ f:{\\mathbb R}\\_+ \\to {\\mathbb R}$, *t* ↦ (*D*+*f*)(*t*) = limsup*h* → 0+{*h*− 1(*f*(*t* + *h*) − *f*(*t*))}, we denote the upper Dini derivative.\n* Let *A*, *B* be logical statements. Their logical conjunction is denoted by *A* ∧ *B*.\nPopulation Model with Actuator Dynamics\n=======================================\nWe consider the age distribution of the population of a microorganism in a bioreactor governed by the following equations\n[eqsystemOpenLoop]\n \n$$\\begin{aligned}\n{3}\n \\text{PDE:}& \\quad &&\\frac{\\partial {f}}{\\partial t}(a,t) +\\frac{\\partial f }{\\partial a}(a,t)=-\\left(\\mu(a)+D(t)\\right)f(a,t) \\label{eq\\_PDE}\\\\\n \\text{ODE:}& \\quad &&\\dot D (t)=u(t) \\label{eq\\_ODE}\\\\\n \\text{OE:}& \\quad &&y(t) = \\int\\_0^A p(a)f(a,t) {{\\rm d}}a\\label{eq\\_output}\\\\\n \\text{BC:}& \\quad &&f(0,t)=\\int\\_0^A k(a)f(a,t) {{\\rm d}}a \\label{eq\\_OLBC} \\end{aligned}$$ \nwhere *a* ∈ [0, *A*] is the age variable, *A* is the maximum reproductive age of the microorganism (a constant), $t\\in {\\mathbb R}\\_+$ is time, *f*(*a*, *t*) > 0 denotes the distribution of the microbial mass in the reactor at age *a* ∈ [0, *A*] and time *t*, *μ* and *k* are the mortality rate profile and birth rate profile, respectively, and *p* is the sensor kernel. The functions $\\mu :[0,A]\\to {\\mathbb R}\\_+ $ and $k:[0,A]\\to {\\mathbb R}\\_+ $ are assumed to be continuous in age *a*, and the function $p :[0,A]\\to {\\mathbb R}\\_+ $ continuously differentiable in *a* with $\\int\\_0^Ap(a){{\\rm d}}a >0$. The population of microorganisms grows at a rate regulated by the dilution rate *D*(*t*), which is the control input of model. Accounting for the inlet and outlet rate of the bioreactor, the PDE expresses the microbial mass balance as a variation of the *McKendrick-von Foerster* equation. The internal boundary feedback, is the renewal condition, which expresses the reproduction of the microorganism as the current mass of the newborns *f*(0, *t*). From a mathematical point of view, the boundary condition involves non-local terms. The measured output defined in is a weighted average of the mass of the microorganism with an age-specific kernel *p*(*a*). \n In chemostat reactors, the dilution rate is defined as the ratio of the inlet volumetric flow rate to the reactor volume. Although, the reactor’s inlet flow and outlet volumetric flow rates are readily adjustable, the volume of the reactor is not necessarily constant and its variations can be described by the dynamic actuation introduced as. Indeed, the population balance is a realistic extension of the model studied in Reference, which does not account for a dynamic actuation and consequently assumes no possible variations of the reactor volume. Equation defines an appropriate control input $u(t) \\in {\\mathbb R}$ that depends on the reactor volume, the reactor inlet and outlet volumetric flow rates as well as on the time derivative of the reactor inlet volumetric flow rate. \nIt should also be noticed that model is derived by neglecting the dependence of the growth of the microorganism on the concentration of a limiting substrate. However, resource-based models that incorporate limiting substrates or uptake of nutrients are more suitable to describe the dynamics of continuous microbial culture and might exhibit limit cycles that are induced by the behavior of a limiting resource assuming constant dilution rates. Model presupposes that the growth or decay of a living population only depends on irreversible incidences of birth and death and is therefore suitable to predict the evolution of macro-populations in demography, epidemiology and ecology. \nThe state of model is $(f[t],D(t)) \\in \\mathcal{F} \\times {\\mathbb R}$, where F is the function space defined by the following equation:\n \n$$\\begin{aligned}\n \\mathcal{F} = \\left\\{ f \\in PC^1([0,A];(0,\\infty)): f(0)=\\int\\_0^A k(a)f(a) {{\\rm d}}a\\right\\}.\\end{aligned}$$ \nThe results of this contribution encompass two different cases for the state space. For the first case, the state space is defined as $\\mathcal{F} \\times {\\mathbb R}$, allowing for the dilution rate *D*(*t*) to take arbitrary real values. However, as noted above, the dilution rate is the"},"data_id":{"kind":"string","value":"arxiv_0181644"}}},{"rowIdx":45389,"cells":{"content":{"kind":"string","value":"]\n================================================\nAn important property of Algorithm [alg:uav-positioning] is that the segment of search trajectory in Step [enu:alg-two-phase-update-loop]) does not “turn back”, and so does that in Step [enu:alg-two-phase-upadte-loop-left]).\n[Monotonicity][lem:monotonicity] Step [enu:alg-two-phase-update-loop]) in Algorithm [alg:uav-positioning] strictly and monotonically increases *ρ*(*t*), and it also monotonically increases ∣*θ*(*t*)∣. Similar property holds in Step [enu:alg-two-phase-upadte-loop-left]).\nIn Step [enu:alg-two-phase-update-loop]a), *ρ*(*t*) strictly and monotonically increases, while *θ*(*t*) keeps unchanged. In Step [enu:alg-two-phase-update-loop]b), we have (omitting the higher order term)\n \n$$\\begin{aligned}\n\\rho+\\Delta\\rho & =\\|\\mathbf{x}+\\Delta\\mathbf{x}-\\mathbf{x}\\_{\\text{u}}\\|\\\\\n & =\\sqrt{\\|\\mathbf{x}-\\mathbf{x}\\_{\\text{u }}\\|^{2}+2(\\mathbf{x}-\\mathbf{x}\\_{\\text{u}})^{\\text{T}}\\Delta\\mathbf{x}+\\|\\Delta\\mathbf{x}\\|^{2}}\\\\\n & =\\rho\\Big(1+\\frac{1}{\\rho^{2}}(\\mathbf{x}-\\mathbf{x}\\_{\\text{u}})^{\\text{T}}\\Delta\\mathbf{x}+o(\\|\\mathbf{x}\\|)\\Big)\\end{aligned}$$ \nand since (**x** − **x**u)/*ρ* = **M**(*θ*)**u** from ([eq:polar-representation-xy-plane]), we have\n \n$$\\begin{aligned}\n\\Delta\\rho & =\\mathbf{u}^{\\text{T}}\\mathbf{M}(\\theta)^{\\text{T}}\\Delta\\mathbf{x}\\\\\n & =\\mathbf{u}^{\\text{T}}\\mathbf{M}(\\theta)^{\\text{T}}\\mathbf{M}(\\theta)\\mathbf{u}\\gamma\\\\\n & \\qquad\\qquad+\\gamma\\rho\\mathbf{u}^{\\text{T}}\\mathbf{M}(\\theta)^{\\text{T}}\\frac{d}{d\\theta}\\mathbf{M}(\\theta)\\mathbf{u}\\Big(-\\frac{\\partial F\\_{k}}{\\partial\\theta}\\Big)^{-1}\\frac{\\partial F\\_{k}}{\\partial\\rho}\\end{aligned}$$ \nwhich equals to *γ*, being strictly positive. Therefore, Step [enu:alg-two-phase-update-loop]) strictly increases *ρ*(*t*).\nIn addition, Step [enu:alg-two-phase-update-loop]b) moves on the contour of *F**k*(*ρ*, *θ*) = *C*, whose dynamics is give by \n$$\\frac{\\partial F\\_{k}(\\rho,\\theta)}{\\partial\\rho}d\\rho+\\frac{\\partial F\\_{k}(\\rho,\\theta)}{\\partial\\theta}d\\theta=0$$ \n in which ∂*F**k*(*ρ*(*t*), *θ*(*t*))/∂*ρ* ≤ 0 according to Lemma [lem:property-one-side] (with a straight-forward generalization from *F*1 to *F**k*) and ∂*F**k*(*ρ*, *θ*)/∂∣*θ*∣ > 0 according to Lemma [lem:partial-optimality]. As a result, Step [enu:alg-two-phase-update-loop]b) monotonically increases ∣*θ*(*t*)∣.\n\nWe now show that the boundary of the search region P in ([eq:search-region]) for the *θ* > 0 branch is given by a semi-circle as illustrated in Fig. [fig:geometry-search-region] (the blue semi-circle). To see this, we first note that the boundary of P for *θ* > 0 is given by (*ρ*cos*θ*, *ρ*sin*θ*), where *ρ* = *L*cos*θ* and 0 ≤ *θ* ≤ *π*/2. Therefore, $\\overline{\\text{\\textbf{ad}}}=L\\cos\\theta$, $\\overline{\\text{\\textbf{ac}}}=L$, $\\overline{\\text{\\textbf{dc}}}=\\sqrt{(L\\cos^{2}\\theta-L)^{2}+(L\\cos\\theta\\sin\\theta-0)^{2}}$, which yields $(\\overline{\\text{\\textbf{ad}}})^{2}+(\\overline{\\text{\\textbf{dc}}})^{2}=(\\overline{\\text{\\textbf{ac}}})^{2}$ and hence point **d** is on the semi-circle with the diameter given by line segment $\\overline{\\text{\\textbf{ac}}}$.\nFrom the monotone property in Lemma [lem:monotonicity], the search trajectory is a convex curve starting from point **a** at the user and ending at the semi-circle, *e.g.*, point **d**. Note that the length of any curve $\\breve{\\text{\\textbf{ad}}}$ is less than $\\overline{\\text{\\textbf{ab}}}+\\overline{\\text{\\textbf{bd}}}=L\\cos^{2}\\theta+L\\cos\\theta\\sin\\theta$, where the maximum value over 0 ≤ *θ* < *π*/2 can be numerically evaluated to be roughly 1.2*L*.\nAlgorithm [alg:uav-positioning] consists of the search on the BS-user axis with maximum length *L*, and *K* − 1 loops for the off-BS-user axis searches, where each loop consists of searches on the left and right branches each with maximum length 1.2*L*. As a result, the total length of the search trajectory is upper bounded by (2.4*K* − 1.4)*L*.\n10 [1]#1 url@samestyle [2]#2 [2]l@#1=l@#1#2\nY. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: opportunities and challenges,” *IEEE Commun. Mag.*, vol. 54, no. 5, pp. 36–42, 2016.\nM. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Unmanned aerial vehicle with underlaid device-to-device communications: Performance and tradeoffs,” *IEEE Trans. Wireless Commun.*, vol. 15, no. 6, pp. 3949–3963, 2016.\nZ. Xiao, P. Xia, and X.-G. Xia, “Enabling UAV cellular with millimeter-wave communication: Potentials and approaches,” *IEEE Commun. Mag.*, vol. 54, no. 5, pp. 66–73, 2016.\nB. Van Der Bergh, A. Chiumento, and S. Pollin, “LTE in the sky: trading off propagation benefits with interference costs for aerial nodes,” *IEEE Commun. Mag.*, vol. 54, no. 5, pp. 44–50, 2016.\nA. Merwaday and I. Guvenc, “UAV assisted heterogeneous networks for public safety communications,” in *Proc. Wireless Commun. and Networking Conf. Workshops*, 2015, pp. 329–334.\nD.-T. Ho, P. Sujit, T. A. Johansen, and J. B. De Sousa, “Performance evaluation of cooperative relay and particle swarm optimization path planning for UAV and wireless sensor network,” in *Proc. IEEE Globecom Workshops*, 2013, pp. 1403–1408.\nV. Sharma, M. Bennis, and R. Kumar, “UAV-assisted heterogeneous networks for capacity enhancement,” *IEEE Commun. Lett.*, vol. 20, no. 6, pp. 1207–1210, 2016.\nO. Jian, Z. Yi, L. Min, and L. Jia, “Optimization of beamforming and path planning for UAV-assisted wireless relay networks,” *Chinese Journal of Aeronautics*, vol. 27, no. 2, pp. 313–320, 2014.\nY. Zeng and R. Zhang, “Energy-efficient UAV communication with trajectory optimization,” *IEEE Trans. Wireless Commun.*, vol. 16, no. 6, pp. 3747–3760, 2017.\nY. Jin, Y. D. Zhang, and B. K. Chalise, “Joint optimization of relay position and power allocation in cooperative broadcast wireless networks,” in *Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing*, 2012, pp. 2493–2496.\nD. H. Choi, B. H. Jung, and D. K. Sung, “Low-complexity maneuvering control of a UAV-based relay without location information of mobile ground nodes,” in *Proc. IEEE Symposium on Computers and Commun.*, 2014, pp. 1–6.\nA. Chamseddine, G. Charland-Arcand, O. Akhrif, S. Gagné, F. Gagnon, and D. Couillard, “Optimal position seeking for unmanned aerial vehicle communication relay using only signal strength and angle of arrival,” in *IEEE Conf. on Decision and Control*, Dec 2014, pp. 976–981.\nF. Jiang and A. L. Swindlehurst, “Optimization of UAV heading for the ground-to-air uplink,” *IEEE J. Sel. Areas Commun.*, vol. 30, no. 5, pp. 993–1005, 2012.\nM. M. Azari, F. Rosas, K.-C. Chen, and S. Pollin, “Ultra reliable UAV communication using altitude and cooperation diversity,” *IEEE Trans. Commun.*, vol. 66, no. 1, pp. 330–344, 2018.\nS. ur Rahman and Y.-Z. Cho, “UAV positioning for throughput maximization,” *EURASIP J. on Wireless Commun. and Networking*, vol. 2018, no. 1, p. 31, 2018.\nJ. Lyu, Y. Zeng, R. Zhang, and T. J. Lim, “Placement optimization of UAV-mounted mobile base stations,” *IEEE Commun. Lett.*, vol. 21, no. 3, pp. 604–607, 2017.\nA. Al-Hourani, S. Kandeepan, and A. Jamalipour, “Modeling air-to-ground path loss for low altitude platforms in urban environments,” in *Proc. IEEE Global Telecomm. Conf.*, 2014, pp. 2898–2904.\nM. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Drone small cells in the clouds: Design, deployment and performance analysis,” in *Proc. IEEE Global Telecomm. Conf.*, 2015, pp. 1–6.\nA. Hourani, K. Sithamparanathan, and S. Lardner, “Optimal LAP altitude for maximum coverage,” *IEEE Commun. Lett.*, no. 99, pp. 1–4, 2014.\nM. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Optimal transport theory for power-efficient deployment of unmanned aerial vehicles,” in *Proc. IEEE Int. Conf. Commun.*, Kuala Lumpur, Malaysia, May 2016, pp. 1–6.\n——, “Efficient deployment of multiple unmanned aerial vehicles for optimal wireless coverage,” *IEEE Commun. Lett.*, vol. 20, no. 8, 2016.\nQ. Feng, J. McGeehan, E. K. Tameh, and A. R. Nix, “Path loss models for air-to-ground radio channels in urban environments,” in *Proc. IEEE Semiannual Veh. Technol. Conf.*, vol. 6, 2006, pp. 2901–2905.\n“Study on enhanced LTE support for aerial vehicles (release 15),” 3GPP TR 36.777, Tech. Rep., 2017. [Online]. Available: <http://www.3gpp.org>\nJ. Chen and D. Gesbert, “Optimal positioning of flying relays for wireless networks: A LOS map approach,” in *Proc. IEEE Int. Conf. Commun.*, Paris, France, May 2017.\nT. Bai and R. W. Heath, “Coverage and rate analysis for millimeter-wave cellular networks,” *IEEE Trans. Wireless Commun.*, vol. 14, no. 2, pp. 1100–1114, 2015.\nJ. Chen, U. Yatnalli, and D. Gesbert, “Learning radio maps for UAV-aided wireless networks: A segmented regression approach,” in *Proc. IEEE Int. Conf. Commun.*, Paris, France, May 2017.\nJ. Chen, O. Esrafilian, D. Gesbert, and U. Mitra, “Efficient algorithms for air-to-ground channel reconstruction in UAV-aided communications,” in *Proc. IEEE Global Telecomm. Conf.*, Dec. 2017, Wi-UAV workshop.\nR. U. Nabar, H. Bolcskei, and F. W. Kneubuhler, “Fading relay channels: Performance limits and space-time signal design,” *IEEE J. Sel. Areas Commun.*, vol. 22, no. 6, pp. 1099–1109, 2004.\nJ. N. Laneman, D. N. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” *IEEE Trans. Inf. Theory*, vol. 50, no. 12, pp. 3062–3080, 2004.\nT. Wang, A. Cano, G. B. Giannakis, and J. N. Laneman, “High-performance cooperative demodulation with decode-and-forward relays,” *IEEE Trans. Commun.*, vol. 55, no. 7, pp. 1427–1438, 2007.\n---\n1. This paper has been accepted in IEEE Transactions on Wireless Communications. This research was funded, in part, by one or more of the following grants: No. ZDSYS201707251409055, No. 2017ZT07X152, No. 2018B030338001, No. 2018YFB1800800, and ERC under the European Union Horizon 2020 research and innovation program (Agreement no. 670896). This paper was presented, in part, at the IEEE International Conference on Communications, Paris, France, 2017.[↩](#fnref1)\n2. Junting Chen is with the School of Science and Engineering and the Future Network of Intelligence Institute (FNii) at the Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China (e-mail: juntingc@cuhk.edu.cn).[↩](#fnref2)\n3. David Gesbert is with Department of Communication Systems, EURECOM, 06410 Sophia-Antipolis, France (e-mail: gesbert@eurecom.fr).[↩](#fnref3)\n4. More generally, one can design the cost function *f* in terms of the UAV position **x** to capture other system characteristics, such as antenna pattern and effect of small-scale fading.[↩](#fnref4)\n5. The original problem in considered a diversity scheme that combines the signal from the relay and the signal from the BS. Such a strategy also leads to problem P1 under high snr.[↩](#fnref5)\n6. For mathematical completeness, the partial derivative is defined as $\\frac{\\partial f(x\\_{0},y\\_{0})}{\\partial x}=\\lim\\_{t\\uparrow0}\\frac{1}{t}[f(x\\_{0}+t,y\\_{0})-f(x\\_{0},y\\_{0})]$ throughout this paper.[↩](#fnref6)\n7. We found from our experiment that around 22% users have LOS condition on the BS-user link.[↩](#fnref7)\n\nEfficient Local Map Search Algorithms for the Placement of Flying Relays\n========================================================================\nThis paper studies the optimal uav placement problem for wireless networking. The uav operates as a flying wireless relay to provide coverage extension for a bs and deliver capacity boost to a user shadowed by obstacles. While existing methods rely on statistical models for potential blockage of a direct propagation link, we propose an approach capable of leveraging local terrain information to offer performance guarantees. The proposed method allows to strike the best trade-off between minimizing propagation distances to ground terminals and discovering good propagation conditions. The algorithm only requires several propagation parameters, but it is capable to avoid deep propagation shadowing and is proven to find the globally optimal UAV position. Only a *local* exploration over the target area is required, and the maximum length of search trajectory is linear to the geographical scale. Hence, it lends itself to online search. Significant throughput gains are found when compared to other positioning approaches based on statistical propagation models.\nIntroduction\n============\nOne significant challenge for wireless communication networks is the rapid increase of demand for high data rate and low latency wireless service. As a promising solution to future communication networks, substantial attention has been brought on the exploitation of uav as flying relays to connect bs with the users in communication outage.\nIn UAV relaying over a dense urban environment, a fundamental challenge is the shadowing at the user side, where the degree of obstruction depends on the geographical environment. For example, the link between a UAV and a user may be in deep shadow when the UAV is located at the east side of a building, whereas, the propagation condition may be significantly improved when the UAV moves to the north side. It is difficult to know such fine-grained propagation condition prior to physically flying a UAV to a target position for assessment. Existing techniques that model the possible obstruction include specifying a larger path loss exponent, adding additional power loss, and constructing a random variable that describes the statistics of the shadowing. However, these models still over-simplify the actual propagation, because they implicitly assume that the degree of obstruction is *homogeneous* everywhere, meaning that the path loss is statistically the same everywhere given the same propagation distance (and elevation angle). Consequently, in a BS-UAV-user relay network, these models would predict the best UAV relay on the BS-user axis. In reality, however, a UAV may find significantly better propagation condition *off* the BS-user axis. We will demonstrate that substantial performance gain can be achieved when a more realistic fine-grained propagation model is exploited. The main goal of the paper is to develop an efficient and blockage-adaptive search strategy to explore fine-grained propagation condition for the optimal UAV position.\nRelated Works\n-------------\nUAV optimization strategies are mainly developed based on specific air-to-ground path loss models. In, the path loss was modeled as a deterministic function of the UAV-to-user distance, irrelevant to specific UAV positions. Thanks to the simplicity of these distance-based models, developed solutions to UAV navigation problems, studied optimization strategies for mimo communications with UAVs with multiple antennas, and optimized the UAV position for cooperative communications. The models used in imply that the path loss is the same under the same distance, but more detailed research in suggests that air-to-ground propagation should also depend on the elevation angle of the UAV-user link.\nTo capture the dependency of obstruction on elevation angles, in, the path loss was modeled as a random variable of the propagation distance and the elevation angle. Specifically, for the UAV position and coverage optimization problems studied in, the path loss was modeled as the average of the path loss under the LOS case and that under the NLOS case, where the larger the elevation angle, the higher the LOS probability. The work considered the shadowing statistics as a function of elevation angle. Yet, these models implicitly assume that the degree of obstruction is homogeneous for the same distance and elevation angle. In practice, however, the degree of obstruction may vary from one location to another.\nChallenges and Our Contributions\n--------------------------------\nAlthough it is theoretically possible to associate each UAV and user position pair with a channel quality to capture the variation of the terrain, it is almost infeasible to implement a search strategy for the optimal UAV position. On one hand, it is time and energy prohibitive to maneuver a UAV for channel quality assessment at every possible position. On the other hand, even a 3D city map or a radio map can be constructed, it is computationally expensive to exhaustively search of the optimal UAV position. Moreover, the city map or radio map may not be updated or reliable in practice.\nTherefore, it is essential to develop an air-to-ground path loss model to capture the variation of the terrain, and at the same time, consists of some nice structure that can be exploited to efficiently search for the optimal UAV position. Specifically, two main issues are to be addressed:\n* **How to model the air-to-ground path loss for the communication between ground users and low-altitude UAVs?** We need a model that not only captures the various degrees of obstruction due to the complex terrain, but also facilitates a low complexity search strategy for optimal UAV positioning.\n* **How to plan an efficient search trajectory for the optimal UAV position?** We desire to build a search path to find the globally optimal UAV position, while the maximum search length is only linear to the radius of the search area.\nTo answer these questions, we develop a *nested segmented* air-to-ground propagation model, which, for each user position, partitions the UAV search area into several segments, and associates each segment with a path loss model, such as LOS model, obstructed LOS model and nlos model. This feature matches with the observation obtained from the experimental study in. In addition, we impose that as the UAV moves away from the user, it can only enter propagation segments with a higher degree of obstruction, as shown in Fig. [fig:geometry-property] (b). Such a requirement is consistent with many existing air-to-ground models from both academia and industry, where the lower the elevation angle, the higher the probability for which the UAV-user link is obstructed. Using such a model, we develop a *shaded-contour-exploration* algorithm to search for the *globally* optimal UAV position, and the algorithm only requires several propagation parameters but not the entire radio map. We prove that global optimum can be attained under a linear search trajectory. While this idea was partially exploited in our preliminary work, the work focused only on the simplest case of two propagation segments and the optimality proof was not presented.\nTo summarize, the key contributions are made as follows:\n* We propose a nested segmented propagation model to capture the fine-grained degree of obstruction in air-to-ground propagation.\n* We develop a shaded-contour-exploration strategy to find the optimal UAV position for a single user relay system, with proven global optimality and linear search complexity. Substantial performance gain for the case of clustered multiple users is also numerically demonstrated.\n* We perform numerical experiments to evaluate the performance of the UAV relay system and compare with existing approaches from existing models. Substantial throughput gain is found from a simulated Manhattan-like urban environment.\nThe rest of the paper is organized as follows. Section [sec:system-model-dual] establishes the nested segmented air-to-ground propagation model. Section [sec:framework-property] develops the search algorithm, and Section [sec:global-convergence] establishes theoretical results on the global optimality. Numerical results are demonstrated in Section [sec:numerical], and conclusions are given in Section [sec:conclusion].\nSystem Model\n============\nConsider a cellular network in an urban environment, where the BS is placed on rooftop or on a tower that is higher than all the buildings. Due to the possibly dense distribution of buildings and vegetation, it is likely that signals transmitted from the BS are significantly obstructed from users on street levels. Consider to deploy a UAV as a flying relay that connects a user with the BS. Assume that the UAV moves at a *fixed* height *H*d > *H*b, where *H*b is the height of the BS. Denote the horizontal positions of the UAV, BS, and user, respectively, as **x**, **x**b, **x**u ∈ R2, and hence, (**x**, *H*d), (**x**b, *H*b), (**x**u, 0) ∈ R3 are, respectively, the positions of the UAV, BS, and user in 3D.\nNote that the signal from the UAV can still be obstructed from the user due to local obstacles surrounding the user. On the other hand, the UAV relay cannot move too close to the user because it needs to balance the relay link with the BS. To address this dilemma, the goal of this paper is to optimize the horizontal position **x** ∈ R2 for the UAV.\nChannel Model for the UAV-BS Link\n---------------------------------\nDefine the *channel* as the deterministic power gain averaged over small scale fading. The BS-UAV channel is modeled as\n \n*g*b(**x**) = *β*0*d*b(**x**)− *α*0 \nwhere $d\\_{\\text{b}}(\\mathbf{x})=\\sqrt{\\|\\mathbf{x}-\\mathbf{x}\\_{\\text{b}}\\|^{2}+(H\\_{\\text{d}}-H\\_{\\text{b}})^{2}}$ is the distance from the UAV at (**x**, *H*d) to the BS at (**x**b, *H*b), and the constants *α*0 > 1 and *β*0 > 0 are the classical path loss exponent and offset parameters. Such a model corresponds to the common scenario where the altitudes *H*b of the BS and *H*d of the UAV are large enough such that there is *always* a LOS condition for the BS-UAV link. We thus focus on modeling the UAV-user link in Section [subsec:nested-segmented-model].\nChannel Model for the UAV-user Link: A Nested Segmented Model\n-------------------------------------------------------------\nIn the classical large-scale fading channel model, the channel gain is modeled as *G*dB = *b* − *a*log10*d* + *ξ*, where *ξ* is a random variable to capture the shadowing effect. Inspired by the geometry in the ray-tracing propagation modeling as illustrated in Fig. [fig:geometry-property], we propose to split *G*dB into several components each associated with a set of parameters *a*, *b* and a random component *ξ**k* for a specific degree of link obstruction.\nSpecifically, let D ⊆ R2 be the domain of all possible UAV positions **x** at constant altitude *H*d. Consider a partition of D into *K* disjoint segments $\\mathbb{D}=\\mathcal{D}\\_{1}(\\tilde{\\mathbf{x}}\\_{\\text{u}})\\cup\\mathcal{D}\\_{2}(\\tilde{\\mathbf{x}}\\_{\\text{u}})\\cup\\dots\\cup\\mathcal{D}\\_{K}(\\tilde{\\mathbf{x}}\\_{\\text{u}})$, where $\\mathcal{D}\\_{k}\\cap\\mathcal{D}\\_{j}=\\varnothing$, for *k* ≠ *j*, and $\\mathcal{D}\\_{k}(\\tilde{\\mathbf{x}}\\_{\\text{u}})$ denotes the region of UAV locations for which the UAV maintains a degree-*k* of LOS obstruction from the user. The proposed *segmented* propagation model for *G*dB is specified as:\n \n$$G\\_{\\text{dB}}(\\mathbf{x})=\\sum\\_{k=1}^{K}\\big(b\\_{k}-a\\_{k}\\log\\_{10}d\\_{\\text{u}}(\\mathbf{x})+\\xi\\_{k}\\big)\\mathbb{I}\\{\\mathbf{x}\\in\\mathcal{D}\\_{k}\\}\\label{eq:uav-user-channle-model-with-noise}$$ \nwhere $d\\_{\\text{u}}(\\mathbf{x})=\\sqrt{\\|\\mathbf{x}-\\mathbf{x}\\_{\\text{u}}\\|^{2}+(H\\_{\\text{d}}-H\\_{\\text{u}})^{2}}$ is the distance from the UAV located at (**x**, *H*d) to the user at $\\tilde{\\mathbf{x}}\\_{\\text{u}}=(\\mathbf{x}\\_{\\text{u}},H\\_{\\text{u}})$, *a**k* and *b**k* are some parameters, and I{*A*} is an indicator function taking value 1 if condition *A* is satisfied, and 0 otherwise. The random variable *ξ**k* captures the residual shadowing effect. The segment parameters {*α**k*, *β**k*, D*k*} are assumed to satisfy the following conditions:\n1. The propagation segment D*k* exhibits a higher degree of LOS obstruction than D*k* − 1, *i.e.*, for *k* = 2, 3, …, *K* and any UAV position **x**, \n*b**k* − *a**k*log10*d*u(**x**) < *b**k* − 1 − *a**k* − 1log10*d*u(**x**) \n or, in the linear scale representation,\n \n*β**k**d*u(**x**)− *α**k* < *β**k* − 1*d*u(**x**)− *α**k* − 1 \nwhere *b**k* = 10log10*β**k* and *a**k* = 10*α**k*.\n2. The propagation segments D*k* are *nested* along any directions from the user, *i.e.*, for any **x** ∈ D*k* and 0 ≤ *ρ* ≤ 1\n \n**x**u + *ρ*(**x** − **x**u) ∈ D*j*, for some *j* ≤ *k*. \nIn other words, when the UAV moves towards the user, the UAV-user channel tends to become *always* less obstructed as there are fewer obstacles in between the UAV and the user, as illustrated in Fig. [fig:geometry-property].\nIt is intuitive that if the parameters {*α**k*, *β**k*, D*k*} are obtained correctly, the variance of the residual term *ξ**k* can be substantially reduced compared to that of the random component *ξ* in the classical channel model. For example, the variance of the shadowing in the LOS case is believed to be much smaller than that in the combined LOS and NLOS case.\nWith such an insight, we focus on optimizing the UAV network based on the average channel gain $\\bar{G}\\_{\\text{dB}}(\\mathbf{x})\\triangleq\\mathbb{E}\\left\\{ G\\_{\\text{dB}}(\\mathbf{x})\\big|\\{\\mathcal{D}\\_{k}\\}\\right\\} $ given the propagation segments, where the expectation is taken over *ξ**k*. Therefore, assuming *ξ**k* with zero mean, the deterministic channel in linear scale $10\\log\\_{10}g\\_{\\text{u}}(\\mathbf{x})\\triangleq\\bar{G}\\_{\\text{dB}}(\\mathbf{x})$ can be written as:\n \n*g*u(**x**) = ∑*k* = 1*K**β**k**d*u(**x**)− *α**k*I{**x** ∈ D*k*}. \nThe problem of learning *α**k*, *β**k*, and the statistics of *ξ**k* from measurement data has been partially attempted in, and it is not the focus of this paper. Here, we assume that *α**k*, *β**k*, and the statistics of *ξ**k* are perfectly known, but we still need to (partially) determine D*k*, *i.e.*, the boundaries shown in Fig. [fig:geometry-property] (b), to help search for the optimal UAV position. Note that learning the entire D*k* is much more difficult and time consuming (or even prohibitive) than learning *α**k* and *β**k*. The goal of this paper is to optimize the UAV position by only *partially* exploring D*k*.\nProblem Formulation and Application Examples\n--------------------------------------------\nConsider an objective cost function *f*(*g*u, *g*b) of the UAV-user channel gain *g*u and the BS-UAV channel gain *g*b. Assume that *f*(*x*, *y*) is a continuous and decreasing function in *x* and *y*, respectively. A generic UAV positioning problem can be formulated as follows[4](#fn4) \n$$\\mathscr{P}:\\qquad\\underset{\\mathbf{x}\\in\\mathbb{R}^{2}}{\\text{minimize}}\\quad f(g\\_{\\text{u}}(\\mathbf{x}),g\\_{\\text{b}}(\\mathbf{x})).$$ \nThe problem formulation P can capture many applications for a variety of relay transmission strategies. Three examples are illustrated as follows, where we choose some simple formulations for easy demonstration of the proposed algorithm.\nConsider that the transmission from the BS to the UAV is modeled as $y\\_{\\text{r}}=\\sqrt{P\\_{\\text{b}}g\\_{\\text{b}}}a\\_{\\text{b}}s+n\\_{\\text{r}}$, and that from the UAV to the user is modeled as $y\\_{\\text{u}}=\\sqrt{P\\_{\\text{u}}g\\_{\\text{u}}}a\\_{\\text{u}}s\\_{\\text{r}}+n\\_{\\text{u}}$, where *n*r, *n*u ∼ CN(0, 1) are the receive noise at the UAV relay and the user, respectively, *s*, *s*r ∼ N(0, 1) are the transmit signals from the BS and the UAV relay, respectively, and *P*u and *P*b are transmission powers at the UAV and the BS, respectively. The variables *a*b and *a*u model the small scale fading on the BS-UAV link and the UAV-user link, respectively. For Rayleigh fading channels, ∣*a*b∣2 and ∣*a*u∣2 are assumed to follow exponential distribution with parameter (normalized as) *λ* = 1.\n### Amplify-and-Forward\nIn amplify-and-forward relaying, the UAV relays the information by transmitting $s\\_{\\text{r}}=y\\_{\\text{r}}/\\sqrt{P\\_{\\text{b}}g\\_{\\text{b}}|a\\_{\\text{r}}|^{2}+1}$, where the scaling factor $\\sqrt{P\\_{\\text{b}}g\\_{\\text{b}}|a\\_{\\text{r}}|^{2}+1}$ is to normalize the transmission power at the UAV to be *P*u. It was shown in that the capacity of the relay channel is given by $C\\_{\\text{AF}}=\\frac{1}{2}\\log\\_{2}\\big(1+q(P\\_{\\text{b}}g\\_{\\text{b}}|a\\_{\\text{r}}|^{2},P\\_{\\text{u}}g\\_{\\text{u}}|a\\_{\\text{u}}|^{2})\\big)$, where $q(x,y)\\triangleq xy/(x+y+1)$ and the parameter $\\frac{1}{2}$ is to capture the fact that the information requires two time slots to reach the user. The outage probability with respect to a target data rate *R* was shown to be $\\mathbb{P}\\{C\\_{\\text{AF}}<R\\}\\approx\\Big(\\frac{1}{P\\_{\\text{b}}g\\_{\\text{b}}}+\\frac{1}{P\\_{\\text{u}}g\\_{\\text{u}}}\\Big)(2^{2R}-1)^{2}$ under high snr, *i.e.*, *P*b*g*b, *P*u*g*u ≫ 1.[5](#fn5)\nTherefore, to minimize the outage probability of the relay channel, the desired UAV position can be determined as the solution to P with a cost function given as:\n \n$$\\begin{aligned}\n & f(g\\_{\\text{u}}(\\mathbf{x}),g\\_{\\text{b}}(\\mathbf{x})):=\\frac{1}{P\\_{\\text{u}}g\\_{\\text{u}}(\\mathbf{x})}+\\frac{1}{P\\_{\\text{b}}g\\_{\\text{b}}(\\mathbf{x})}.\\label{eq:outage-minimization}\\end{aligned}$$ \n### Decode-and-Forward\nIn decode-and-forward relaying, the UAV fully decodes the message *ŝ* from the receive signal *y*r, and transmits *s*r = *ŝ* to the user. The maximum capacity of such a decode-and-forward relay system can be shown to be $C\\_{\\text{DF}}=\\frac{1}{2}\\min\\{\\log\\_{2}(1+P\\_{\\text{b}}g\\_{b}|a\\_{b}|^{2}),\\log\\_{2}(1+P\\_{\\text{u}}g\\_{\\text{u}}|a\\_{\\text{u}}|^{2})\\}$. Using Jensen’s inequality E{*C*(*x*)} ≤ *C*(E{*x*}) on a concave function *C*(*x*), an upper bound of the ergodic capacity E{*C*DF} is given by $\\frac{1}{2}\\min\\{\\log\\_{2}(1+P\\_{\\text{b}}g\\_{b}),\\log\\_{2}(1+P\\_{\\text{u}}g\\_{\\text{u}})\\}$. The desired UAV position can be determined by maximizing such a capacity bound. Equivalently, the problem P can be specified by choosing the following cost function:\n \n$$\\begin{aligned}\nf(g\\_{\\text{u}}(\\mathbf{x}),g\\_{\\text{b}}(\\mathbf{x})):=\\max\\Big\\{ & -\\log\\_{2}\\big(1+P\\_{\\text{b}}g\\_{b}(\\mathbf{x})\\big),\\label{eq:drone-rate-maximization}\\\\\n & -\\log\\_{2}\\big(1+P\\_{\\text{u}}g\\_{\\text{u}}(\\mathbf{x})\\big)\\Big\\}.\\nonumber \\end{aligned}$$ \nA numerical example is given in Fig. [fig:urban-city-map], where Fig. [fig:urban-city-map] (b) simulates the received power of the UAV-user signal wrt every UAV position under a segmented propagation model with *K* = 2 segments. Fig. [fig:urban-city-map] (c) shows the corresponding relay channel capacity from the BS to the user via the UAV. It is not trivial to find the optimal UAV relay position due to the irregular propagation pattern.\n### Multiuser Clustered around a Hotspot\nSuppose that there are *N*u users clustered around a hotspot centered at $\\mathbf{x\\_{\\text{c}}}$ with radius *r*u. Let *f*(*i*)(**x**) be the cost function taking the form in ([eq:drone-rate-maximization]) for the *i*th user located at position **x**u(*i*). Specifically, the UAV-user gain *g*u(**x**; **x**u(*i*)) in ([eq:uav-user-channel-model]) is computed based on the user position **x***i*. Consider to maximize the sum rate $-\\frac{1}{N\\_{\\text{u}}}\\sum\\_{i=1}^{N\\_{\\text{u}}}f^{(i)}(\\mathbf{x})$. One may consider to simplify and approximate the cost function $\\bar{f}(\\mathbf{x})\\triangleq\\frac{1}{N\\_{\\text{u}}}\\sum\\_{i=1}^{N\\_{\\text{u}}}f^{(i)}(\\mathbf{x})$ by constructing a virtual user indexed as *i* = *N*u + 1. The virtual user is virtually placed at the hotspot center **x**c (as a similar topological model discussed in ), and the corresponding channel gain *g*u(**x**; **x**c) for is modeled using the segmented log-distance model ([eq:uav-user-channel-model]), except that the propagation segment depends on the majority vote from the *N*u actual users. Specifically, the UAV position ****x**** belongs to the *k*th propagation segment $\\tilde{\\mathcal{D}}\\_{k}(\\mathbf{x}\\_{\\text{c}})$ (for the virtual user), if the majority UAV-user links (**x**, **x**u(*i*)) belong to the *k*the segment. As a result, it is clear that the cost function $f(\\mathbf{x})\\triangleq f^{(N\\_{\\text{u}}+1)}(\\mathbf{x})$ for the virtual user is a good approximation of the average cost $\\frac{1}{N\\_{\\text{u}}}\\sum\\_{i=1}^{N\\_{\\text{u}}}f^{(i)}(\\mathbf{x})$ as long as the cluster radius *r*u is small.\nWhile such an approximation is only suboptimal, in Section [subsec:clustered-user-case], we numerically demonstrate that our proposed strategy that solves P still provide reasonably good performance up to moderate cluster radius *r*u as compared to stochastic optimization using simplified models.\nAlgorithm Designs\n=================\nIn this section, we first derive some useful insights on the optimal UAV positions, and then develop a polar representation tool. Based on that, we develop the search algorithm for the optimal UAV position.\nProperties of the Optimal UAV Position\n--------------------------------------\n[prop:Solution-Property] The optimal solution **x**\\* to P is either on the BS-user axis, or on the boundary between two propagation segments.\nSuppose there is a solution **x** which is strictly inside a propagation segment D*k* and is off the BS-user axis. Then there exists a direction $\\bm{\\delta}$, such that for a sufficiently small *ε* > 0, the new UAV position $\\mathbf{x}+\\epsilon\\bm{\\delta}$ decreases the distances to both the user and the BS, while at the same time, $\\mathbf{x}+\\epsilon\\bm{\\delta}\\in\\mathcal{D}\\_{k}$. From the nested segmented model ([eq:uav-user-channel-model]) and also the UAV-BS channel model ([eq:UAV-BS-channel-model]), smaller distances *d*u and *d*b imply larger channel gains *g*u and *g*b. Due to the monotonicity property of the cost function *f*(*g*u, *g*b), larger channel gains yield a smaller cost value, which implies that **x** is not the optimal solution. By contradiction, the proposition is therefore confirmed.\nHowever, it is still costly to search along the segment boundaries because the boundaries may have complex shapes, resulting in an unacceptably long trajectory (which could be super-linear to the radius of the search area) for the UAV as demonstrated in Fig. [fig:urban-city-map].\nPolar Representation\n--------------------\n\nTo develop a more efficient search strategy, we transform the problem into the one expressed over a polar coordinate system.\nLet *ρ* = ∥**x** − **x**u∥ be the ground projected distance from the user at (**x**u, *H*u) to the UAV at (**x**, *H*d). Let *θ* ∈ ( − *π*, *π*) be the *deviation angle* from the user-to-BS direction to the user-to-UAV direction as illustrated in Fig. [fig:polar-coordinate]. Denote $\\mathbf{u}=(u\\_{1},u\\_{2})\\triangleq\\frac{\\mathbf{x}\\_{\\text{b}}-\\mathbf{x}\\_{\\text{u}}}{\\|\\mathbf{x}\\_{\\text{b}}-\\mathbf{x}\\_{\\text{u}}\\|}$ as the normalized user-to-BS direction. The UAV position **x** ∈ R2 can be equivalently expressed by (*ρ*, *θ*) as\n \n**x**(*ρ*, *θ*) = **x**u + *ρ***M**(*θ*)**u** \nwhere\n \n$$\\mathbf{M}(\\theta)=\\left[\\begin{array}{ll}\n\\cos\\theta & -\\sin\\theta\\\\\n\\sin\\theta & \\cos\\theta\n\\end{array}\\right]\\label{eq:rotation-matrix}$$ \nis a rotation matrix, and\n \n*θ* = sign(*z*2*u*1 − *z*1*u*2) ⋅ arccos(**z**T**u**/*ρ*) \nin which $\\mathbf{z}=(z\\_{1},z\\_{2})\\triangleq\\mathbf{x}-\\mathbf{x}\\_{\\text{u}}$, sign(*x*) = 1 if *x* > 0, and sign(*x*) = − 1, otherwise.\nWe now define an alternative expression for the cost function *f*(*g*u(**x**), *g*b(**x**)).\n[Fictitious Segment Cost Function][def:Fictitious-Segment-Cost] For **x**(*ρ*, *θ*) ∈ D*k*, *k* = 1, 2, …, *K*, the cost function *f*(*g*u(**x**), *g*b(**x**)) can be written as \n*f*(*g*u(**x**), *g*b(**x**)) = *F**k*(*ρ*, *θ*) \n where\n \n$$F\\_{k}(\\rho,\\theta)\\triangleq f\\big(g\\_{\\text{u}}^{(k)}(\\mathbf{x}(\\rho,\\theta)),g\\_{\\text{b}}(\\mathbf{x}(\\rho,\\theta))\\big)\\label{eq:Fk}$$ \nand $g\\_{\\text{u}}^{(k)}(\\mathbf{x})\\triangleq\\beta\\_{k}d\\_{\\text{u}}(\\mathbf{x})^{-\\alpha\\_{k}}$ is the UAV-user channel in the *k*th segment from the channel model ([eq:uav-user-channel-model]).\nAs a result, the objective function in P is transformed into the polar domain as *F*(*ρ*, *θ*) = ∑*k* = 1*K**F**k*(*ρ*, *θ*)I{**x**(*ρ*, *θ*) ∈ D*k*}.\nThe motivation of working on the polar domain is that by fixing the deviation *θ*, increasing *ρ* only worsens the propagation condition according to the nested property ([eq:nested-segmented-model-order-condition])([eq:nested-segmented-model-nested-condition]). In addition, the overall cost function *F*(*ρ*, *θ*) is discontinuous because it contains the indicator functions, but the functions *F**k*(*ρ*, *θ*) are continuous. As result, the functions *F**k*(*ρ*, *θ*) can be used to derive search trajectories.\nSearch Trajectory Design for *K* = 2\n------------------------------------\nThe algorithm is better illustrated starting from the two segment case, where D1 corresponds to the LOS segment and D2 corresponds to the nlos segment.\n### Search on the BS-user Axis\nLet the UAV start from the BS. It first moves towards the user until it finds two critical positions (if they exist) **x***k*0 = **x**(*ρ**k*0, 0), *k* = 1, 2, which correspond to the points achieving the minimum cost over the BS-user axis in the LOS region and NLOS region, respectively. Specifically, the parameters *ρ**k*0 are the solutions that minimize the fictitious cost *F**k*(*ρ*, 0) along the BS-user axis **x**(*ρ*, 0) ∈ D*k*, for *k* = 1, 2, in the two segment case.\nFor example, when the UAV is initially in the NLOS region, it can move up to the LOS-NLOS boundary (see Section [subsec:Propagation-Segment-Detection] for a discussion on the detection method); with that, it can solve for *ρ*10 and *ρ*20 to obtain the critical points **x**10 and **x**20. On the other hand, when the UAV is initially in the LOS region, it can compute the critical position **x**10 in the LOS region, while **x**20 does not exist.\n### Search on the Right Branch\nStarting from the critical position **x**10 which minimizes the cost function on the LOS portion of the BS-user axis, the UAV first moves to **x**(*ρ*10, *δ*/*ρ*10), *i.e.*, a position just on the right of **x**10 in Fig. [fig:UAV-search-path] (a), where *δ* is a chosen step size. It then proceeds according to the following two phases; at the same time, it keeps the track record of the minimum cost value *F*min discovered and the corresponding position $\\hat{\\mathbf{x}}(\\hat{\\rho},\\hat{\\theta})$ that achieves *F*min = *F*(*ρ̂*, *θ̂*):\n* If the UAV is in the LOS region, it moves away from the user. Specifically, it moves from **x**(*ρ*, *θ*) to **x**(*ρ* + *δ*, *θ*).\n* If the UAV is in the NLOS region, it moves in the direction that maintains the same fictitious cost *F*1(*ρ*, *θ*) as it were in the LOS region, *i.e.*, contour of *F*1(*ρ*, *θ*) = *C* specified by[6](#fn6) \n$$\\frac{\\partial F\\_{1}(\\rho,\\theta)}{\\partial\\rho}d\\rho+\\frac{\\partial F\\_{1}(\\rho,\\theta)}{\\partial\\theta}d\\theta=0.$$ \n It can be shown that (Lemma [lem:partial-optimality] in Appendix [sec:app-pf-thm-convergence]), ∂*F*1(*ρ*, *θ*)/∂*θ* ≠ 0 for *θ* ≠ 0. As a result from ([eq:polar-representation-xy-plane]), we have \n$$d\\mathbf{x}=\\mathbf{M}(\\theta)\\mathbf{u}d\\rho+\\rho\\frac{d}{d\\theta}\\mathbf{M}(\\theta)\\mathbf{u}\\Big(-\\frac{\\partial F\\_{1}}{\\partial\\theta}\\Big)^{-1}\\frac{\\partial F\\_{1}}{\\partial\\rho}d\\rho.$$ \n Thus, the UAV updates its position from **x** to **x** + Δ**x**, where \n$$\\Delta\\mathbf{x}=\\gamma\\Big[\\mathbf{M}(\\theta)\\mathbf{u}+\\rho\\frac{d}{d\\theta}\\mathbf{M}(\\theta)\\mathbf{u}\\Big(-\\frac{\\partial F\\_{1}}{\\partial\\theta}\\Big)^{-1}\\frac{\\partial F\\_{1}}{\\partial\\rho}\\Big]$$ \n in which, *γ* > 0 is chosen such that the step size ∥Δ**x**∥ = *δ* and the UAV moves in the direction away from the user (see Fig. [fig:UAV-search-path]).\nThe search at this branch is completed whenever the UAV reaches a point **x**(*ρ*, *θ*) such that either ∂*F*1(*ρ*, *θ*)/∂*ρ* ≥ 0 or *ρ* ≥ *L*cos*θ*, where $L\\triangleq\\|\\mathbf{x}\\_{\\text{b}}-\\mathbf{x}\\_{\\text{u}}\\|$. The justification of the two stopping criteria will become clear in Section [subsec:global-optimality].\nChoose a step size *δ* > 0.\n1. [enu:alg-initialization] **Search on the BS-user axis:** Find the critical points *ρ**k*0, *k* = 1, 2, …, *K*, defined in ([eq:problem-critical-point-K]). Initialize *F*min = *F*1(*ρ*10, 0) and $\\hat{\\mathbf{x}}=\\mathbf{x}(\\rho\\_{1}^{0},0)$, where **x**(*ρ*, *θ*) is defined in ([eq:polar-representation-xy-plane]). Initialize *k* = 1.\n2. [enu:alg-two-phase-update] **Search on the right branch:** Set **x** ← **x**(*ρ**k*0, *δ*/*ρ**k*0).\n3. [enu:alg-two-phase-update-loop]For each iterate **x**, compute *ρ* = ∥**x** − **x**u∥ and *θ* from ([eq:theta]).\n\t1. **Search in the virtual LOS region:** If **x** ∈ ⋃*j* = 1*k*D*j*, update\n\t\n\t\n\t \n\t**x** ← **x**(*ρ* + *δ*, *θ*). \n\t\n\t\n\t\n\tWhen *F*(*ρ* + *δ*, *θ*) < *F*min, update the record *F*min ← *F*(*ρ* + *δ*, *θ*) and $\\hat{\\mathbf{x}}\\leftarrow\\mathbf{x}(\\rho+\\delta,\\theta)$.\n\t2. **Search in the virtual NLOS region:** If $\\mathbf{x}\\notin\\bigcup\\_{j=1}^{k}\\mathcal{D}\\_{j}$, update **x** ← **x** + Δ**x**, where\n\t\n\t\n\t \n\t$$\\Delta\\mathbf{x}=\\gamma\\bigg[\\mathbf{M}(\\theta)\\mathbf{u}+\\rho\\frac{d}{d\\theta}\\mathbf{M}(\\theta)\\mathbf{u}\\Big(-\\frac{\\partial F\\_{k}}{\\partial\\theta}\\Big)^{-1}\\frac{\\partial F\\_{k}}{\\partial\\rho}\\bigg]\\label{eq:alg-eq2}$$ \n\t\n\t\n\t\n\twhere *γ* > 0 is chosen such that ∥Δ**x**∥ = *δ*.Repeat this step until either (i) *ρ* ≥ *L*cos*θ* or (ii) ∂*F**k*(*ρ*, *θ*)/∂*ρ* ≥ 0.\n4. **[enu:alg-two-phase-upadte-loop-left]Search on the left branch:** Set **x** ← **x**(*ρ**k*0, − *δ*/*ρ**k*0). Repeat Step [enu:alg-two-phase-update-loop]).\n5. Let *k* ← *k* + 1. Repeat from Step [enu:alg-two-phase-update]) until *k* > *K* − 1.\n6. [enu:alg-final-step]If *F**K*(*ρ**K*0, 0) < *F*min, then *F*min ← *F**K*(*ρ**K*0, 0) and $\\hat{\\mathbf{x}}\\leftarrow\\mathbf{x}(\\rho\\_{K}^{0},0)$.\n### Search on the Left Branch\nStarting from the critical position **x**10, the UAV moves to **x**(*ρ*10, − *δ*/*ρ*10), *i.e.*, a position just on the left of **x**10 in Fig. [fig:UAV-search-path] (a). It repeats the same process as in Section [subsec:search-right-branch] until it meets the stopping criteria. When the search is completed, the track record $\\hat{\\mathbf{x}}$ that achieves the least cost will be the desired UAV position. Note that the position $\\hat{\\mathbf{x}}$, for example, point A in Fig. [fig:UAV-search-path] (a), is *not* necessarily where the search terminates.\nSearch Trajectory Design for Arbitrary *K*\n------------------------------------------\nIn the case of more than two propagation segments, one can generate *K* − 1 search trajectories following a similar procedure in Section [subsec:algorithm-K2-case], where each trajectory is computed by partitioning the whole area into two virtual propagation regions: *virtual LOS* region and *virtual NLOS* region. Specifically, for the *k*th search trajectory, the virtual LOS region is defined by grouping the first *k* propagation segments together $\\tilde{\\mathcal{D}}\\_{k}\\triangleq\\bigcup\\_{j=1}^{k}\\mathcal{D}\\_{j}$, and correspondingly, the virtual NLOS region is defined as $\\tilde{\\mathcal{D}}\\_{k}^{\\text{c}}\\triangleq\\mathbb{D}\\backslash\\tilde{\\mathcal{D}}\\_{k}=\\bigcup\\_{j=k+1}^{K}\\mathcal{D}\\_{j}$. The whole algorithm can be summarized as follows.\n* Search on BS-user axis: Let the UAV start from the BS. It first moves towards the user along the BS-user axis until it finds the *K* − 1 critical positions **x***k*0 = **x**(*ρ**k*0, 0), *k* = 1, 2, …, *K* − 1, each of which corresponds to a point achieving the minimum cost over the BS-user axis in a virtual LOS region, *i.e.*, *ρ**k*0 are the solutions to\n \n$$\\begin{aligned}\n\\underset{\\rho\\geq0}{\\text{minimize}} & \\quad F\\_{k}(\\rho,0)\\label{eq:problem-critical-point-K}\\\\\n\\text{subject to} & \\quad\\mathbf{x}(\\rho,0)\\in\\bigcup\\_{j=1}^{k}\\mathcal{D}\\_{j}\\label{eq:problem-critical-point-K-constraint}\\end{aligned}$$ \nfor *k* = 1, 2, …, *K* − 1. In addition, *ρ**K*0 is defined as the solution that minimizes ([eq:problem-critical-point-K]) subject to **x**(*ρ*, 0) ∈ D*K*.\n* Search over *K* − 1 virtual LOS/NLOS partition scenarios: In the *k*th search, *k* = 1, 2, …, *K* − 1, the UAV follows a similar procedure as that in Sections [subsec:search-right-branch] and [subsec:search-left-branch] for virtual LOS region $\\tilde{\\mathcal{D}}\\_{k}$ and virtual NLOS region $\\tilde{\\mathcal{D}}\\_{k}^{\\text{c}}$.\n* Integration: During the whole search, the UAV keeps track of the minimum achievable cost *F*min and the corresponding position $\\hat{\\mathbf{x}}$. When the algorithm terminates, $\\hat{\\mathbf{x}}$ gives the desired UAV position.\nThe entire search algorithm is summarized in Algorithm [alg:uav-positioning]. An example search trajectory under the *K* = 2 model in the polar coordinate system is visualized in Fig. [fig:UAV-search-path] (a), where the black curve represents the search trajectory and the dashed gray curves represent the contours of the cost function *F*1(*ρ*, *θ*).\nPropagation Segment Detection\n-----------------------------\nAs the parameters *a**k* and *b**k* and the statistics of *ξ**k* in ([eq:uav-user-channle-model-with-noise]) are assumed known as discussed in Section [subsec:nested-segmented-model], the propagation segment can be determined using maximum likelihood detection. Let *h**k* be the probability distribution function of the random variable *ξ**k* in propagation segment *k*. Then, the maximum likelihood estimator of the propagation segment is given by \n$$\\hat{k}=\\underset{\\scriptsize k=1,2,\\dots,K}{\\text{argmax}}\\;h\\_{k}(y-b\\_{k}+a\\_{k}\\log\\_{10}d(\\mathbf{x}))$$ \n where *y* is the channel gain *G*dB measured at the UAV location **x**. In addition, if *ξ**k* are zero mean Gaussian distributed with variance *σ**k*2, then the detection rule can be simplified as \n$$\\hat{k}=\\underset{\\scriptsize k=1,2,\\dots,K}{\\text{argmin}}\\;\\frac{1}{\\sigma\\_{k}}\\left|y-b\\_{k}+a\\_{k}\\log\\_{10}d(\\mathbf{x})\\right|.$$ \nNote that, in practice, the parameters *a**k* and *b**k* and the statistics of *ξ**k* can be estimated in a separate training phase. Alternatively, the parameter estimation can be integrated in Step 1 in Algorithm [alg:uav-positioning]. Specifically, the UAV moves along the BS-user axis (and, optionally, to a few random locations) to collect enough channel measurements. Then, a maximum likelihood estimation method can be performed to learn these parameters.\nGlobal Optimality and Linear Search Length\n==========================================\nIt turns out that Algorithm [alg:uav-positioning] can find the globally optimal solution to P for at least two types of cost functions *f*.\nAssume that the cost function *f*(*x*, *y*) in P satisfies $\\frac{\\partial^{2}f(x,y)}{\\partial x\\partial y}=0$,\n \n$$\\begin{aligned}\nx\\frac{\\partial^{2}f(x,y)}{\\partial x^{2}}+2\\frac{\\partial f(x,y)}{\\partial x} & \\geq0,\\quad y\\frac{\\partial^{2}f(x,y)}{\\partial y^{2}}+2\\frac{\\partial f(x,y)}{\\partial y}\\geq0\\label{eq:cost-function-condition-1}\\end{aligned}$$ \nfor every *x*, *y* > 0.\nIt can be easily verified that the cost function ([eq:outage-minimization]) in the outage probability minimization example in Section [subsec:example-AF] satisfies Condition 1.\nAssume that the cost function *f*(*x*, *y*) in P can be written as max{*f*1(*x*), *f*2(*y*)}, where *f*1(*x*) and *f*2(*y*) are decreasing functions.\nIt is also clear that the cost function ([eq:drone-rate-maximization]) in the rate maximization example in Section [subsec:example-DF] satisfies Condition 2.\nIn addition, we discuss optimality for continuous-time algorithm trajectory **x**(*t*), which can be obtained from Algorithm [alg:uav-positioning] using infinitesimal step size *δ* = O(*d**t*) at each infinitesimal time slot *d**t*. Specifically, the search trajectory **x**(*t*) in Algorithm [alg:uav-positioning] can be described by piece-wise continuous dynamic systems, where one replaces *δ* by *κ**d**t* in ([eq:alg-eq1]) and *γ* by *κ**γ̄**d**t* in ([eq:alg-eq2]), in which *κ* is a parameter that specifies the moving speed of the UAV. Accordingly, the continuous-time processes of the minimum cost *F*min(*t*) and the position track record $\\hat{\\mathbf{x}}(t)$ are given by *F*min(*t*) = minimize0 ≤ *τ* ≤ *t* *f*(*g*u(**x**(*τ*), *g*b(**x**(*τ*)) and $\\hat{\\mathbf{x}}(t)=\\mathbf{x}(\\hat{\\tau})$, respectively, where *τ̂* = argmin0 ≤ *τ* ≤ *t* *f*(*g*u(**x**(*τ*), *g*b(**x**(*τ*)).\nGlobal Optimality\n-----------------\nWe first present the main optimality result as follows.\n[Global Optimality][thm:convergence-two-segment] Suppose that the cost function *f* in P satisfies either Condition 1 or Condition 2. Then, $\\hat{\\mathbf{x}}(t)$ in Algorithm [alg:uav-positioning] converges to the globally optimal solution to P and *F*min(*t*) converges to the minimum cost value in finite time.\nTheorem [thm:convergence-two-segment] confirms that the globally optimal UAV position is attainable, even though the terrain topology could be arbitrarily complex.\nThe optimality result can be better understood from the polar coordinate system. From the definition of the fictitious segment cost functions *F**k*(*ρ*, *θ*) in ([eq:Fk]), problem P can be equivalently written as \n$$\\mathscr{P}':\\quad\\underset{\\rho\\geq0,-\\pi\\leq\\theta\\leq\\pi}{\\text{minimize}}\\quad F(\\rho,\\theta)\\triangleq\\sum\\_{k=1}^{K}F\\_{k}(\\rho,\\theta)\\mathbb{I}\\{(\\rho,\\theta)\\in\\mathcal{P}\\_{k}\\}$$ \n where $\\mathcal{P}\\_{k}\\triangleq\\big\\{(\\rho,\\theta):\\mathbf{x}(\\rho,\\theta)\\in\\mathcal{D}\\_{k}\\big\\}$ is the *k*th propagation segment in the polar coordinate system. The optimal solution **x**⋆ to P can be obtained as **x**⋆ = **x**(*ρ*⋆, *θ*⋆), in which (*ρ*⋆, *θ*⋆) is the optimal solution to Pʹ.\nThe following intermediate results provide some intuitions to understand Algorithm [alg:uav-positioning] and Theorem [thm:convergence-two-segment].\n[Bounded Search Region][prop:Search-region] The optimal solution **x**⋆ to P can be obtained as **x**(*ρ*⋆, *θ*⋆), where (*ρ*⋆, *θ*⋆) ∈ P and\n \n$$\\mathcal{P}=\\Big\\{(\\rho,\\theta):0\\leq\\rho\\leq L\\cos\\theta,-\\frac{\\pi}{2}\\leq\\theta\\leq\\frac{\\pi}{2}\\Big\\}\\label{eq:search-region}$$ \nin which, $L\\triangleq\\|\\mathbf{x}\\_{\\text{b}}-\\mathbf{x}\\_{\\text{u}}\\|$ is the horizontal distance from the BS to the user.\nPlease refer to Appendix [app:pf-prop-search-region].\nProposition [prop:Search-region] justifies the first stopping criterion *ρ* ≥ *L*cos*θ* in Step [enu:alg-two-phase-update-loop]) of Algorithm [alg:uav-positioning]. An intuitive explanation is that when the UAV moves outside the region P in ([eq:search-region]), one can always find a position in P that has an equal (or smaller) distances, respectively, to the BS and to the user in the same (or less obstructed) propagation segment D*k*, leading to an equal (or lower) cost to achieve. Therefore, the optimal UAV position is contained in P.\nThe following proposition justifies the second stopping criterion ∂*F**k*(*ρ*, *θ*)/∂*ρ* ≥ 0.\n[Partial Optimality][prop:Partial-Optimality] Suppose that the cost function *f* in P satisfies either Condition 1 or Condition 2. Then, *F**k*(*ρ*, *θ*) admits a unique local minimizer *ρ**k*\\*(*θ*) over *ρ* ≥ 0 for every fixed *θ*, where ∣*θ*∣ < *π*/2.\nPlease refer to Appendix [app:pf-prop-partial-optimality].\nDue to the fact that *F**k*(*ρ*, *θ*) has a unique local minimum for every *θ*, condition ∂*F**k*(*ρ*, *θ*)/∂*ρ* ≥ 0 implies that *ρ* ≥ *ρ**k*\\*(*θ*). On the other hand, Step [enu:alg-two-phase-update-loop]) of Algorithm [alg:uav-positioning] always increases *ρ* (to be formally justified in Lemma [lem:monotonicity] in Appendix [app:pf-prop-maximum-traj-length]). Therefore, it suffices to stop the search when the condition ∂*F**k*(*ρ*, *θ*)/∂*ρ* ≥ 0 is met.\nBased on Propositions [prop:Search-region] and [prop:Partial-Optimality], the proof of Theorem [thm:convergence-two-segment] is derived in Appendix [sec:app-pf-thm-convergence].\nMaximum Length of the Algorithm Trajectory\n------------------------------------------\nHere, we derive the worst-case trajectory length of Algorithm [alg:uav-positioning].\n[Maximum Trajectory Length][thm:Maximum-Trajectory-Length] The length of the search trajectory from Algorithm [alg:uav-positioning] is upper bounded by (2.4*K* − 1.4)*L*, where $L\\triangleq\\|\\mathbf{x}\\_{\\text{b}}-\\mathbf{x}\\_{\\text{u}}\\|$ is the horizontal distance from the BS to the user.\nPlease refer to Appendix [app:pf-prop-maximum-traj-length].\nTheorem [thm:Maximum-Trajectory-Length] suggests that the algorithm must terminate in a finite number of steps given a positive step size *δ* > 0. The total number of steps scales as O(*L*/*δ*). *Surprisingly*, the bound is linear in *L* and does *not* depend on the actual terrain, *i.e.*, the shapes of the propagation segments D*k*. As a benchmark, if one searches the optimal UAV position following the segment boundaries (a property from Proposition [prop:Solution-Property]), the worst-case search length is not guaranteed to be linear in *L*, depending on the actual shapes of the boundaries (see, for example, Fig. [fig:urban-city-map](c)).\nNumerical Results\n=================\nConsider a dense urban area with buildings ranging from 5–45 meter height following a uniform distribution as illustrated in Fig. [fig:urban-city-map] (a). The user is represented by a red circle and the BS locates at the top right corner denoted by a blue triangle. The height of the BS is 45 meters, and the UAV moves at 50 meter above the ground. As a result, there is always LOS propagation between the UAV and the BS. Consider two propagation scenarios, LOS and NLOS, for the UAV-to-user link; this corresponds to many existing models in the literature for a fair comparison. Correspondingly, the parameters of the UAV-BS channel in ([eq:UAV-BS-channel-model]) are chosen as (*α*0, log10*β*0) = (2.08, − 3.85); Rician fading with 20 dB K-factor is assumed according to the Rural Macro BS to UAV scenario in. The parameters of the UAV-user channel in ([eq:uav-user-channel-model]) are chosen as (*α*1, log10*β*1, *α*2, log10*β*2) = (2.14, − 3.69, 3.03, − 3.84); Rician fading with 9 dB K-factor is assumed for the LOS case and Rayleigh fading is assumed for the NLOS case according to the Urban Micro BS to UAV scenario in.\nWe consider the example capacity maximization problem in Section [subsec:example-DF] for optimal UAV positioning. The transmission powers are chosen as *P*b = 30 dBm from the BS and *P*d = 36 dBm from the UAV, and the noise power is − 80 dBm. The corresponding power map and end-to-end capacity map for every UAV position are illustrated in Fig. [fig:urban-city-map] (b) and (c).\nIn Fig. [fig:UAV-search-path] (b), the green curves show the search path from Algorithm [alg:uav-positioning]. The two curve branches correspond to the UAV searches in Step [enu:alg-two-phase-update]) and Step [enu:alg-two-phase-upadte-loop-left]) in Algorithm [alg:uav-positioning], respectively. The other curves in Fig. [fig:UAV-search-path] (b) represent the contour of the capacity map as in Fig. [fig:urban-city-map] (c). The optimal UAV position is found at the purple diamond.\nThroughput and Outage Probability for Single User Case\n------------------------------------------------------\nWe now evaluate the average throughput of UAV relay system for a single user located randomly and uniformly on the streets in Fig. [fig:urban-city-map] (a). The transmission powers are chosen as *P*b = *P*d = 33 dBm for both the BS and the UAV, and the throughput is evaluated as the absolute value of ([eq:drone-rate-maximization]). The proposed scheme places the UAV to the position obtained from Algorithm [alg:uav-positioning] under step size *δ* = 5 meters. We also consider the following baselines for UAV positioning:\n* **Probabilistic Algorithm** **:** First, obtain an empirical LOS distribution *f*LOS(*φ*) = P{LOS , *φ*} as a function of the elevation angle *φ* = *φ*(**x**, **x**u) from the user at $(\\mathbf{\\mathbf{x}\\_{\\text{u}}},0)$ to the UAV at (**x**, *H*d), by uniformly and randomly dropping 10,000 users on the streets in Fig. [fig:urban-city-map] (a) and randomly picking UAV locations within the target area. The empirical distribution function *f*LOS(*φ*) is shown in Fig. [fig:plos]. Second, given each user position **x**u, the UAV-user channel at UAV location **x** is computed as\n \n$$\\begin{aligned}\ng\\_{\\text{u}}(\\mathbf{x},\\mathbf{x}\\_{\\text{u}}) & =f\\_{\\text{LOS}}(\\varphi)\\beta\\_{1}\\|\\mathbf{x}-\\mathbf{x}\\_{\\text{u}}\\|^{-\\alpha\\_{1}}\\nonumber \\\\\n & \\qquad+\\big(1-f\\_{\\text{LOS}}(\\varphi)\\big)\\beta\\_{2}\\|\\mathbf{x}-\\mathbf{x}\\_{\\text{u}}\\|^{-\\alpha\\_{2}}.\\label{eq:offline-gu}\\end{aligned}$$ \nThe optimal UAV position is then obtained by solving P with *g*u(**x)=***g*u(**x**, **x**u) in ([eq:offline-gu]).\n* **Simple Search:** Obtain the optimal UAV position by searching only on the BS-user axis (*i.e.*, to implement only Step 1 of Algorithm [alg:uav-positioning]).\n* **Exhaustive Search:** We perform exhaustive search over the entire search region on equally-spaced grids with *δ* = 5 meter spacing. The grid point that maximizes the cost in ([eq:drone-rate-maximization]) is chosen as the UAV position. Note that such a scheme is prohibited in practice and hence it is for benchmarking only.\nThe performance on direct BS-user transmission (without UAV relaying) is evaluated using the segmented channel model ([eq:uav-user-channel-model]) by replacing *d*u(**x**) by the BS-user distance and replacing I{**x** ∈ D*k*} by the indicator of the BS-user link propagation condition.\nFig. [fig:rate-bar] compares the average capacity of the Decode-and-Forward relay system discussed in Section [subsec:example-DF] over 10,000 user positions in Fig. [fig:urban-city-map] (a). The cell edge users are recognized as those within the 20th percentile of the throughput under direct BS-user transmission, and the cell center users are recognized as those in the top 20th percentile.[7](#fn7) First, across all the three user categories shown in Fig. [fig:rate-bar] (a), the proposed scheme with the optimal UAV placement achieves the highest throughput. In particular, it realizes more than 2X throughput gain on average over an probabilistic approach. Second, the proposed scheme performs as well as the exhaustive search scheme as seen from the two overlapping cdf curves in Fig. [fig:rate-bar] (b), which verify our theoretical results on optimality.\nFig. [fig:outage] compares the outage probability reduction from the direct BS-user transmission scheme using the Amplify-and-Forward relay system discussed in Section [subsec:example-AF]. Specifically, it is defined as *f*(**x**\\*)/*f*0, where *f*(**x**) is the cost function defined in ([eq:outage-minimization]) with **x**\\* being the optimal UAV positions found by various schemes, and *f*0 = 1/(*P*b*g*0(**x**u)), in which, *g*0(**x**u) is the channel gain of the direct BS-user link. It is observed that the proposed scheme provides a significant outage probably reduction. In addition, it also confirms the globally optimality of the proposed scheme, when comparing its performance with the exhaustive search baseline.\nClustered Multiuser Case\n------------------------\nWe evaluate the proposed UAV placement strategy for the case of *N*u = 20 users clustered around a hotspot with radius *r*u meters. We implement a search trajectory from Algorithm [alg:uav-positioning] wrt a virtual user located at **x**c and evaluate the sum data rate *f̄*(**x**) as discussed in Section [subsec:multiuser]. As a benchmark, the baseline scheme minimizes the average cost *f̄*(**x**) evaluated using the probabilistic model *g*u(**x**, **x**u(*i*)) in ([eq:offline-gu]) for each user *i*. We drop 500 clusters uniformly and randomly in Fig. [fig:urban-city-map] (a), and evaluate the average throughput for the *obstructed* clusters, of which the center position **x**c is in nlos condition to the BS.\nFig. [fig:rate-radius-1] shows the average throughput performance versus the radius *r*u of the cluster. The case *r*u = 0 refers to the single user scenario, and we know that Algorithm [alg:uav-positioning] achieves the globally optimal performance. When the cluster radius increases, the average throughput drops, because the proposed search algorithm is not optimized for the multiple users away from the cluster center. Nevertheless, the system still benefits from the proposed search strategy because the users nearby still share some degree of correlation in the propagation environment. As a result, the proposed strategy does also achieve significant performance gain over the probabilistic scheme.\nConclusion\n==========\nThis paper designed algorithms to search for the optimal UAV position for establishing the best wireless relay link between a BS and a user in a dense urban area, where global topological information, such as a 3D city map, is not available. The desired UAV position, constrained at a fixed height above the ground, was defined as the minima of a general cost function. A nested segmented propagation model was proposed to model the propagation from the UAV to the ground user that is probably blocked by obstacles. A search algorithm was developed and shown to find the globally optimal UAV position. In addition, the length of the search trajectory is upper bounded by a linear function of the diameter of the target area. Significant throughput gain was found when compared to other UAV positioning approaches or direct BS-user transmissions.\nProof of Proposition [prop:Search-region]\n=========================================\nFrom the nested segmented propagation model ([eq:nested-segmented-model-order-condition])([eq:nested-segmented-model-nested-condition]), it is easy to verify that for any *ρ* and *θ*, the following holds\n \n*F**k*(*ρ*, *θ*) < *F**k* + 1(*ρ*, *θ*) \nfor *k* = 1, 2, …, *K* − 1.\nThe result of Proposition [prop:Search-region] can be proven by contradiction as follows.\nFirst, suppose that solution **x**(*ρ*ʹ, *θ*ʹ) satisfies $\\frac{\\pi}{2}<|\\theta'|<\\pi$. Then, we must have *F*1(*ρ*ʹ, *θ*ʹ) ≤ *F*(*ρ*ʹ, *θ*ʹ) from property ([eq:inequality-less-obstructed-segment-preferred]) as P*k* are mutually exclusive. Since there is LOS propagation when the UAV is on top of the outdoor user, *i.e.*, (0, 0) ∈ P1, we have \n*F*(0, 0) = *F*1(0, 0) < *F*1(*ρ*ʹ, *θ*ʹ) ≤ *F*(*ρ*ʹ, *θ*ʹ) \n where, by moving from (0, 0) to (*ρ*ʹ, *θ*ʹ), the UAV position (*ρ*ʹ, *θ*ʹ) has longer distances to both the user and the BS. Thus, by contradiction, **x**(*ρ*ʹ, *θ*ʹ) cannot be the optimal solution to P, *i.e.*, we must have $|\\theta'|\\leq\\frac{\\pi}{2}$.\nSecond, suppose that the optimal solution **x**(*ρ*ʹ, *θ*ʹ) satisfies *ρ*ʹ > *L*cos*θ*ʹ. Consider a different solution **x**(*ρ*ʺ, *θ*ʹ), where *ρ*ʺ = *L*cos*θ*ʹ − (*ρ*ʹ − *L*cos*θ*ʹ). From geometry, one can easily show that the UAV-BS distance *d*b(**x**(*ρ*ʹ, *θ*ʹ)) = *d*b(**x**(*ρ*ʺ, *θ*ʹ)) and the UAV-user distance *d*u(**x**(*ρ*ʺ, *θ*ʹ)) < *d*u(**x**(*ρ*ʹ, *θ*ʹ)). Suppose that (*ρ*ʹ, *θ*ʹ) ∈ P*k* and (*ρ*ʺ, *θ*ʹ) ∈ P*j*. From the nested segmented property ([eq:nested-segmented-model-order-condition])([eq:nested-segmented-model-nested-condition]), we must have *j* ≤ *k*. From property ([eq:inequality-less-obstructed-segment-preferred]), we have \n*F*(*ρ*ʺ, *θ*ʹ) = *F**j*(*ρ*ʺ, *θ*ʹ) < *F**k*(*ρ*ʺ, *θ*ʹ) < *F**k*(*ρ*ʹ, *θ*ʹ) = *F*(*ρ*ʹ, *θ*ʹ) \n which contradicts to the hypothesis that (*ρ*ʹ, *θ*ʹ) minimizes the objective function *F*(*ρ*, *θ*). This confirms that the optimal solution must satisfies 0 ≤ *ρ* ≤ *L*cos*θ*.\nProof of Proposition [prop:Partial-Optimality]\n==============================================\nProof under Condition I\n-----------------------\nWe first analyze the property of the composite function *f*((*g*u ∘ *d*u)(**z**), (*g*b ∘ *d*b)(**z**)).\n[lem:for-condition-1]Let *g**i*(*d**i*) = *β**i**d**i*− *α**i*, *i* = 1, 2, and *α* > 1. Let *d**i*(**z**), *i* = 1, 2, be convex functions in **z**. Suppose that *f*(*x*, *y*) satisfies Condition 1. Then, the composite function *f*((*g*1 ∘ *d*1)(**z**), (*g*2 ∘ *d*2)(**z**)) is strictly convex in **z**.\nSince ∂*f*/∂*x* < 0 and ∂*f*/∂*y* < 0, the first order partial derivative of *f*(*g*1 ∘ *d*1, *g*2 ∘ *d*2) is given by\n \n$$\\begin{aligned}\n\\frac{\\partial f}{\\partial d\\_{i}} & =\\frac{\\partial f}{\\partial g\\_{i}}\\frac{\\partial g\\_{i}}{\\partial d\\_{i}}=\\frac{\\partial f}{\\partial g\\_{i}}(-\\alpha\\_{i}\\beta\\_{i})d\\_{i}^{-\\alpha\\_{i}-1}>0\\end{aligned}$$ \nfor *i* = 1, 2, and the second order partial derivative is given by\n \n$$\\begin{aligned}\n\\frac{\\partial^{2}f}{\\partial d\\_{i}^{2}} & =\\frac{\\partial^{2}f}{\\partial g\\_{i}^{2}}\\frac{\\partial g\\_{i}}{\\partial d\\_{i}}(-\\alpha\\_{i}\\beta\\_{i})d\\_{i}^{-\\alpha\\_{i}-1}+(-\\alpha\\_{i}\\beta\\_{i})\\frac{\\partial f}{\\partial g\\_{i}}\\frac{\\partial}{\\partial d\\_{i}}d\\_{i}^{-\\alpha\\_{i}-1}\\\\\n & =\\alpha\\_{i}^{2}\\beta\\_{i}d\\_{i}^{-\\alpha\\_{i}-2}\\Big(\\beta\\_{i}d\\_{i}^{-\\alpha\\_{i}}\\frac{\\partial^{2}f}{\\partial g\\_{i}^{2}}+\\frac{\\alpha\\_{i}+1}{\\alpha\\_{i}}\\frac{\\partial f}{\\partial g\\_{i}}\\Big)\\\\\n & >\\alpha\\_{i}^{2}\\beta\\_{i}d\\_{i}^{-\\alpha\\_{i}-2}\\Big(g\\_{i}\\frac{\\partial^{2}f}{\\partial g\\_{i}^{2}}+2\\frac{\\partial f}{\\partial g\\_{i}}\\Big)\\\\\n & \\geq0\\end{aligned}$$ \nfor *i* = 1, 2, where the first inequality is due to the fact that *α**i* > 1 and $\\frac{\\partial f}{\\partial g\\_{i}}<0$.\nDefine an operator $\\nabla\\triangleq[\\frac{\\partial}{\\partial z\\_{1}}\\,\\frac{\\partial}{\\partial z\\_{2}}\\,\\dots\\,\\frac{\\partial}{\\partial z\\_{m}}]^{\\text{T}}$, where *z**i* is the *i*th entry of a vector variable **z**. From $\\nabla f=\\frac{\\partial f}{\\partial d\\_{1}}\\nabla d\\_{1}+\\frac{\\partial f}{\\partial d\\_{2}}\\nabla d\\_{2}$, the Hessian matrix of *f* is given by\n \n$$\\begin{aligned}\n\\nabla^{2}f & =\\frac{\\partial^{2}f}{\\partial d\\_{1}^{2}}\\nabla d\\_{1}\\nabla d\\_{1}^{\\text{T}}+\\frac{\\partial f}{\\partial d\\_{1}}\\nabla^{2}d\\_{1}\\\\\n & \\qquad+\\frac{\\partial^{2}f}{\\partial d\\_{2}^{2}}\\nabla d\\_{2}\\nabla d\\_{2}^{\\text{T}}+\\frac{\\partial f}{\\partial d\\_{2}}\\nabla^{2}d\\_{2}\\succeq\\mathbf{0}\\end{aligned}$$ \nsince ∇2*d**i*(**z**) ≽ **0** due to the convexity of *d**i*(**z**). Therefore, *f* is strictly convex in **z** (as *d**i*(**z**) are strictly convex).\nFrom the polar representation of the UAV-user and UAV-BS distances\n \n$$\\begin{aligned}\nd\\_{\\text{b}}(\\mathbf{x}(\\rho,\\theta)) & =\\sqrt{\\rho^{2}+L^{2}-2\\rho L\\cos\\theta+(H\\_{\\text{d}}-H\\_{\\text{b}})^{2}}\\label{eq:dist\\_b}\\\\\nd\\_{\\text{u}}(\\mathbf{x}(\\rho,\\theta)) & =\\sqrt{\\rho^{2}+(H\\_{\\text{d}}-H\\_{\\text{u}})^{2}}\\label{eq:dist\\_u}\\end{aligned}$$ \none can show that *d*b(**x**(*ρ*, *θ*)) and *d*u(**x**(*ρ*, *θ*)) are strictly convex in *ρ*.\nThen, using Lemma [lem:for-condition-1] and the definition of *F**k*(*ρ*, *θ*) in ([eq:Fk]), we can conclude that *F**k*(*ρ*, *θ*) is strictly convex in *ρ*, and therefore, *F**k*(*ρ*, *θ*) admits a unique local minima *ρ**k*\\*(*θ*) in the bounded interval *ρ* ∈ [0, *L*cos*θ*].\nProof under Condition II\n------------------------\nConsider *F**k*(*ρ*, *θ*) = max{*f*1(*g*u(*k*)(**x**(*ρ*, *θ*)), *f*2(*g*b(**x**(*ρ*, *θ*))}. From ([eq:distb])([eq:distu]), to increase *ρ*, we must have *d*u increase and *d*b decrease, and as a result, *f*1 decreases and *f*2 increases monotonically in *ρ* ∈ [0, *L*cos*θ*]. Therefore, these exists a unique local minimizer *ρ**k*\\*(*θ*) in the closed interval [0, *L*cos*θ*].\nProof of Theorem [thm:convergence-two-segment]\n==============================================\nWe first state the following lemma for the property of the fictitious segment cost functions *F**k*(*ρ*, *θ*) in ([eq:Fk]) derived in the polar domain from the original objective *f*.\n[lem:partial-optimality] It holds that\n \n∂*F**k*(*ρ*, *θ*)/∂∣*θ*∣ > 0 \nfor all *k* and *θ* ≠ 0. In addition, for any *θ*ʹ ≥ 0, the following property holds\n \n$$\\min\\_{\\rho\\geq0}F\\_{k}(\\rho,\\theta')\\quad\\leq\\quad\\min\\_{j\\geq k}\\;\\min\\_{\\rho\\geq0,\\theta'<\\theta\\leq\\frac{\\pi}{2}}F\\_{j}(\\rho,\\theta)\\label{eq:local-optimality-condition-1}$$ \nfor 1 ≤ *k* ≤ *K*. Similarly, for any *θ*ʹ < 0,\n \n$$\\min\\_{\\rho\\geq0}F\\_{k}(\\rho,\\theta')\\quad\\leq\\quad\\min\\_{j\\geq k}\\;\\min\\_{\\rho\\geq0,-\\frac{\\pi}{2}\\leq\\theta<\\theta'}F\\_{j}(\\rho,\\theta)\\label{eq:local-optimality-condition-2}$$ \nWe first note that $\\frac{\\partial}{\\partial|\\theta|}F\\_{k}(\\rho,\\theta)>0$ for all *k* = 1, 2, …, *K*, because increasing ∣*θ*∣ will increase the UAV-BS distance *d*b while the UAV-user distance *d*u = *ρ* is not affected, and hence *g*b(**x**(*ρ*, *θ*)) is decreased. As the cost function *f*(*x*, *y*) is increasing with *x* and *y*, respectively, due to conditions I or II, the cost *F**k*(*ρ*, *θ*) increases as ∣*θ*∣ increases.\nConsider that *θ*ʹ ≥ 0. For every 0 ≤ *ρ* ≤ *L*, *θ* ≥ *θ*ʹ, and *j* ≥ *k*, we must have \n*F**k*(*ρ*, *θ*ʹ) ≤ *F**j*(*ρ*, *θ*ʹ) ≤ *F**j*(*ρ*, *θ*). \n As a result, min*ρ* ≥ 0*F**k*(*ρ*, *θ*ʹ) ≤ min*ρ* ≥ 0*F**j*(*ρ*, *θ*) for every *θ* > *θ*ʹ. Hence the result ([eq:local-optimality-condition-1]) is confirmed.\nThe case of ([eq:local-optimality-condition-2]) can be shown in a similar way.\nTheorem [thm:convergence-two-segment] can be equivalently rewritten, in a more general setting, using the notions and conditions from the polar domain as follows:\nSuppose that the set of segments {P*k*} defined along with Pʹ satisfy the nested condition ([eq:nested-segmented-model-nested-condition]). In addition, assume that the globally optimal solution to Pʹ belongs to the bounded search region P defined in ([eq:search-region]), and the local minimizer *ρ**k*\\*(*θ*) of *F**k*(*ρ*, *θ*) is unique for each *k* and each *θ*, $|\\theta|<\\frac{\\pi}{2}$. Moreover, suppose that the functions *F**k*(*ρ*, *θ*) satisfy conditions ([eq:local-optimality-condition-0])([eq:local-optimality-condition-2]). Then, the polar domain trajectory (*ρ̂*(*t*), *θ̂*(*t*)) obtained from $\\hat{\\mathbf{x}}(t)$ following Algorithm [alg:uav-positioning] converges to the globally optimal solution to Pʹ.\nNote that it has been proven in Propositions [prop:Search-region] and [prop:Partial-Optimality] and Lemma [lem:partial-optimality] that the objective function *f* (which satisfies either condition I or II) in the UAV positioning problem P (and, correspondingly, the functions *F**k* in Pʹ) satisfy the conditions in Theorem 1A. The remaining part of this section thus focuses on proving Theorem 1A.\nOptimality for the Two Segment Case\n-----------------------------------\nWhen the algorithm terminates at *t* = *T*, it turns out that the cost value track record *F*min(*T*) along with the algorithm trajectory satisfies *F*min(*T*) ≤ minimize*ρ* ≥ 0, (*ρ*, *θ*) ∈ P2*F*2(*ρ*, *θ*) because\n \n$$\\begin{aligned}\nF\\_{\\min}(T) & \\leq\\underset{0\\leq\\rho\\leq L}{\\text{minimize}}\\quad F(\\rho,0)\\label{eq:thm-convergence-K2-proof-eq0}\\\\\n & =\\underset{\\rho\\geq0}{\\text{minimize}}\\quad F(\\rho,0)\\label{eq:thm-convergence-K2-proof-eq1}\\\\\n & \\leq\\underset{\\rho\\geq0}{\\text{minimize}}\\quad F\\_{2}(\\rho,0)\\label{eq:thm-convergence-K2-proof-eq1b}\\\\\n & \\leq\\underset{\\rho\\geq0,(\\rho,\\theta)\\in\\mathcal{P}\\_{2}}{\\text{minimize}}\\quad F\\_{2}(\\rho,\\theta)\\label{eq:thm-convergence-K2-proof-eq2}\\end{aligned}$$ \nwhere inequality ([eq:thm-convergence-K2-proof-eq0]) is due to Step [enu:alg-initialization]) and [enu:alg-final-step]) in Algorithm [alg:uav-positioning] and the fact that *F*min(*t*) is a non-increasing process. Equality ([eq:thm-convergence-K2-proof-eq1]) is due to the condition in Theorem 1A (proven in Proposition [prop:Search-region]), inequalities ([eq:thm-convergence-K2-proof-eq1b]) and ([eq:thm-convergence-K2-proof-eq2]) are from conditions ([eq:local-optimality-condition-1])([eq:local-optimality-condition-2]) (proven in Lemma [lem:partial-optimality]).\nIn fact, it also holds that \n*F*min(*T*) ≤ minimize*ρ* ≥ 0, (*ρ*, *θ*) ∈ P1*F*1(*ρ*, *θ*) \n as follows.\n[Optimality for *K* = 2][lem:Local-optimality] In the two segment case, the continuous trajectory (*ρ*(*t*), *θ*(*t*)) passes through (*ρ*\\*, *θ*\\*) before it completes Step [enu:alg-two-phase-update-loop]) in Algorithm [alg:uav-positioning] at *t* = *T*1, i.e., there exists *t*\\* ≤ *T*1, such that *ρ*(*t*\\*) = *ρ*\\* and *θ*(*t*\\*) = *θ*\\*, where (*ρ*\\*, *θ*\\*) is the optimal solution to the following problem\n \n$$\\underset{\\rho\\geq0,(\\rho,\\theta)\\in\\mathcal{P}\\_{1}}{\\text{minmize}}\\quad F\\_{1}(\\rho,\\theta)\\label{eq:subproblem-P1}$$ \nwhere the minimum value equals to *F*min(*T*1).\nSince *T*1 ≤ *T*, we must have *F*min(*T*) ≤ *F*min(*T*1) and the former one lower-bounds both the subproblems ([eq:thm-convergence-K2-proof-eq2]) and ([eq:subproblem-P1]). Since *F*min(*t*) represents the cost value track record along the trajectory (*ρ̂*(*t*), *θ̂*(*t*)), we conclude that (*ρ̂*(*T*), *θ̂*(*T*)) attains the globally optimal solution to Pʹ for the two segment case.\nIn the following two subsections, we first prove some preliminary properties of Algorithm [alg:uav-positioning], and then, we use these properties to prove Lemma [lem:Local-optimality].\nPreliminary Properties of Algorithm [alg:uav-positioning]\n---------------------------------------------------------\nThe following property states that, along the algorithm trajectory, the same cost *F*1(*ρ*(*t*), *θ*(*t*)) is achievable at some prior time *τ* ≤ *t*, even when the current step (*ρ*(*t*), *θ*(*t*)) is not in the LOS region.\n[lem:achievability]For every point on the algorithm trajectory (*ρ*(*t*), *θ*(*t*)), 0 ≤ *t* ≤ *T*, there exists 0 ≤ *τ* ≤ *t*, such that (*ρ*(*τ*), *θ*(*τ*)) ∈ P1 and \n*F*(*ρ*(*τ*), *θ*(*τ*)) = *F*1(*ρ*(*τ*), *θ*(*τ*)) = *F*1(*ρ*(*t*), *θ*(*t*)). \nIf (*ρ*(*t*), *θ*(*t*)) ∈ P1, we have *F*(*ρ*(*t*), *θ*(*t*)) = *F*1(*ρ*(*t*), *θ*(*t*)) and *τ* = *t*. If (*ρ*(*t*), *θ*(*t*)) ∈ P2, then the algorithm is in the loop of Step [enu:alg-two-phase-update-loop]b), where it follows the trajectory on the contour *F*1(*ρ*(*t*), *θ*(*t*)) = *C*. To trace backward, there must be a 0 ≤ *τ* < *t*, such that *F*1(*ρ*(*τ*), *θ*(*τ*)) = *F*1(*ρ*(*t*), *θ*(*t*)) and (*ρ*(*τ*), *θ*(*τ*)) ∈ P1. Note that the initial point (*ρ*10, 0) from Step [enu:alg-initialization]) is in the LOS region P1 from the definition ([eq:polar-representation-xy-plane]).\nThe following result shows that the algorithm trajectory always satisfies ∂*F*1(*ρ*, *θ*(*t*))/∂*ρ* ≤ 0 until the algorithm terminates.\n[lem:property-one-side]Let *ρ*1\\*(*θ*) minimize *F*1(*ρ*, *θ*) over all *ρ* ≥ 0 with *θ* fixed. Then, the algorithm trajectory (*ρ*(*t*), *θ*(*t*)) satisfies *ρ*(*t*) ≤ *ρ*1\\*(*θ*(*t*)) and ∂*F*1(*ρ*(*t*), *θ*(*t*))/∂*ρ* ≤ 0 for all *t* before the iteration completes Step [enu:alg-two-phase-update-loop]). Moreover, ∂*F*1(*ρ*, *θ*(*t*))/∂*ρ* ≤ 0 for all 0 ≤ *ρ* ≤ *ρ*(*t*). The same result holds for the search trajectory in Step [enu:alg-two-phase-upadte-loop-left]).\nFirst, from the definition of *ρ*10 in ([eq:problem-critical-point-K]) and the nested segmented propagation property ([eq:nested-segmented-model-order-condition]) ([eq:nested-segmented-model-nested-condition]), we have *ρ*10 ≤ *ρ*1\\*(0). This is because, we have **x**(*ρ*10, 0) ∈ D1 from ([eq:problem-critical-point-K]), and hence, all the points (*ρ*, 0), 0 ≤ *ρ* ≤ *ρ*10, are in the LOS region. As a result, if *ρ*10 > *ρ*1\\*(0), then (*ρ*1\\*(0), 0) is also in the LOS region (satisfying the constraint ([eq:problem-critical-point-K-constraint])), which implies that *ρ*1\\*(0) minimizes ([eq:problem-critical-point-K]), yielding a contradiction. Therefore, from Step [enu:alg-initialization]), we have the initial point (*ρ*(0), *θ*(0)) satisfying *θ*(0) = 0 and *ρ*(0) = *ρ*10 ≤ *ρ*1\\*(0) = *ρ*1\\*(*θ*(0)).\nSecond, as the local minimizer *ρ*1\\*(*θ*) is unique from Proposition [prop:Partial-Optimality], we must have ∂*F*1(*ρ*, *θ*)/∂*ρ* < 0 for *ρ* < *ρ*1\\*(*θ*) and ∂*F*1(*ρ*, *θ*)/∂*ρ* > 0 for *ρ* > *ρ*1\\*(*θ*). (Note that there is no saddle point either, due to the strict convexity under Condition 1 and monotonicity of *f*1 and *f*2 under Condition 2.) Once ∂*F*1(*ρ*(*t*), *θ*(*t*))/∂*ρ* ≥ 0, Step [enu:alg-two-phase-update-loop]) is completed. As a result, it holds that *ρ*(*t*) ≤ *ρ*1\\*(*θ*(*t*)).\nProof of Lemma [lem:Local-optimality]\n-------------------------------------\nIt suffices to prove for the subproblem\n \n$$\\mathscr{P}'\\_{1+}:\\quad F\\_{\\min}(T\\_{1})\\leq\\underset{\\rho\\geq0,(\\rho,\\theta)\\in\\mathcal{P}\\_{1}^{+}}{\\text{minimize}}\\quad F\\_{1}(\\rho,\\theta)\\label{eq:app-subproblem-P1-plus}$$ \nwhich is essentially solved by the iterations in Step 2) of Algorithm [alg:uav-positioning], where $\\mathcal{P}\\_{1}^{+}=\\{(\\rho,\\theta):0\\leq\\theta\\leq\\frac{\\pi}{2},(\\rho,\\theta)\\in\\mathcal{P}\\_{1}\\}$. Indeed, the counterpart subproblem over the constraint set $\\mathcal{P}\\_{1}^{-}=\\{(\\rho,\\theta):-\\frac{\\pi}{2}\\leq\\theta\\leq0,(\\rho,\\theta)\\in\\mathcal{P}\\_{1}\\}$ is solved in a similar way by Step 4). If ([eq:app-subproblem-P1-plus]) holds, then it must also hold that \n*F*min(*T*1) ≤ minimize*ρ* ≥ 0, (*ρ*, *θ*) ∈ P1−*F*1(*ρ*, *θ*) \n which confirms the result of Lemma [lem:Local-optimality].\n**Step A:** We first show that the algorithm trajectory (*ρ*(*t*), *θ*(*t*)) in the loop of Step [enu:alg-two-phase-update-loop]) can only stop at *θ*(*T*) ≥ *θ*\\*, where *T* ≤ *T*1. Therefore, as the algorithm trajectory from Step [enu:alg-two-phase-update-loop]) is continuous, we must have *θ*(*t*) = *θ*\\* for some *t* ≤ *T*.\n[lem:stopping-theta] The algorithm trajectory in Step [enu:alg-two-phase-update-loop]) (*ρ*(*t*), *θ*(*t*)) can only stop at *θ*(*T*) = *θ*ʹ ≥ *θ*\\* where (*ρ*\\*, *θ*\\*) is the optimal solution to Pʹ1 +.\nThe result can be proven by contradiction. Suppose that Step [enu:alg-two-phase-update-loop]) stops at (*ρ*(*T*), *θ*(*T*)) where *ρ*(*T*) = *ρ*ʹ and *θ*(*T*) = *θ*ʹ < *θ*\\*. As Step [enu:alg-two-phase-update-loop]) is completed, either one of the stopping criteria should have been triggered.\nFirst, suppose that the condition ∂*F*1(*ρ*ʹ, *θ*ʹ)/∂*ρ* ≥ 0 was triggered. From Lemma [lem:achievability], there exists *τ* ≤ *T*, such that *F*1(*ρ*(*τ*), *θ*(*τ*)) = *F*1(*ρ*ʹ, *θ*ʹ) and (*ρ*(*τ*), *θ*(*τ*)) ∈ P1+. From Lemma [lem:property-one-side] and the condition in Theorem 1A that corresponds to Proposition [prop:Partial-Optimality], (*ρ*ʹ, *θ*ʹ) minimizes *F*1(*ρ*, *θ*ʹ) over *ρ* ≥ 0. As a result,\n \n$$\\begin{aligned}\nF\\_{1}(\\rho(\\tau),\\theta(\\tau)) & =F\\_{1}(\\rho',\\theta')\\nonumber \\\\\n & =\\underset{\\rho\\geq0}{\\text{minimize}}\\;F\\_{1}(\\rho,\\theta')\\label{eq:app-pf-lem-stopping-eq1}\\\\\n & \\leq\\min\\_{j\\geq1}\\;\\underset{\\rho\\geq0,\\theta'<\\theta\\leq\\frac{\\pi}{2}}{\\text{minimize}}F\\_{j}(\\rho,\\theta)\\nonumber \\\\\n & \\leq\\underset{\\rho\\geq0,\\theta'<\\theta\\leq\\frac{\\pi}{2}}{\\text{minimize}}F\\_{1}(\\rho,\\theta)\\label{eq:app-pf-lem-stopping-eq2}\\\\\n & \\leq\\underset{\\rho\\geq0,\\theta'<\\theta\\leq\\frac{\\pi}{2},(\\rho,\\theta)\\in\\mathcal{P}\\_{1}^{+}}{\\text{minimize}}F\\_{1}(\\rho,\\theta)\\nonumber \\\\\n & =F\\_{1}(\\rho^{\\*},\\theta^{\\*})\\nonumber \\end{aligned}$$ \nwhere the first two inequalities are from the condition in Theorem 1A that corresponds to Lemma [lem:partial-optimality]. The third equality is by the hypothesis *θ*\\* > *θ*ʹ. However, this violates the assumption (*ρ*\\*, *θ*\\*) being the solution to subproblem Pʹ1 +, since (*ρ*(*τ*), *θ*(*τ*)) ∈ P1+ now yields a lower cost. By contradiction, the stopping criterion ∂*F*1(*ρ*ʹ, *θ*ʹ)/∂*ρ* ≥ 0 is not satisfied.\nSecond, suppose that the condition *ρ*(*T*) = *ρ*ʹ ≥ *L*cos*θ*ʹ is triggered. From the bounded search region condition in Theorem 1A (corresponding to Proposition [prop:Search-region]) and from the hypothesis *θ*(*T*3) = *θ*ʹ < *θ*\\*, we have \n*ρ*ʹ ≥ *L*cos*θ*ʹ > *L*cos*θ*\\* ≥ *ρ*\\*. \n From Lemma [lem:property-one-side], *ρ*1\\*(*θ*ʹ) > *ρ*ʹ and ∂*F*1(*ρ*, *θ*ʹ)/∂*ρ* < 0 for *ρ*ʹ ≥ *ρ* ≥ *ρ*\\*. As a result,\n \n$$\\begin{aligned}\nF\\_{1}(\\rho',\\theta')<F\\_{1}(\\rho^{\\*},\\theta') & <F\\_{1}(\\rho^{\\*},\\theta^{\\*})\\label{eq:F1-inequality}\\end{aligned}$$ \nwhere the second inequality is from Lemma [lem:partial-optimality].\nFrom Lemma [lem:achievability], there exists *τ* ≤ *T*, such that (*ρ*(*τ*), *θ*(*τ*)) ∈ P1+ and \n*F*1(*ρ*(*τ*), *θ*(*τ*)) = *F*1(*ρ*ʹ, *θ*ʹ) < *F*1(*ρ*\\*, *θ*\\*) \n which contradicts to the hypothesis (*ρ*\\*, *θ*\\*) being the solution to subproblem Pʹ1 +. Therefore, the stopping criterion *ρ*(*T*3) = *ρ*ʹ ≥ *L*cos*θ*ʹ is not satisfied either.\nTo conclude, since neither of the stopping criteria is triggered, by contradiction, the algorithm can only stop at *θ*(*T*) ≥ *θ*\\*.\n**Step B:** We then argue that when the trajectory reaches *θ*(*t*) = *θ*\\* for some *t* ≤ *T*, it must hold that (*ρ*(*t*), *θ*(*t*)) ∈ P1+ and *ρ*(*t*) ≤ *ρ*\\*.\n[lem:P1-region-stopping] The algorithm trajectory satisfies (*ρ*(*t*1), *θ*(*t*1)) ∈ P1+, where *t*1 ≥ 0 satisfies *θ*(*t*1) = *θ*\\* and *θ*(*t*) ≤ *θ*\\* for *t* < *t*1.\nWe note that *ρ*(*t*1) ≤ *ρ*\\*. This is because if *ρ*(*t*1) > *ρ*\\*, we must have *F*1(*ρ*(*t*1), *θ*\\*) < *F*1(*ρ*\\*, *θ*\\*), since ∂*F*1(*ρ*, *θ*\\*)/∂*ρ* < 0 for all *ρ*(*t*1) > *ρ* > *ρ*\\* as from Lemma [lem:property-one-side]. As a result of Lemma [lem:achievability], there exists *τ* ≤ *t*1, such that \n*F*1(*ρ*(*τ*), *θ*(*τ*)) = *F*1(*ρ*(*t*1), *θ*\\*) < *F*1(*ρ*\\*, *θ*\\*) \n and (*ρ*(*τ*), *θ*(*τ*)) ∈ P1+, which contradicts to the assumption that (*ρ*\\*, *θ*\\*) is the solution to ([eq:subproblem-P1]).\nAs (*ρ*\\*, *θ*\\*) ∈ P1+, from the nested segmented propagation property ([eq:nested-segmented-model-order-condition])([eq:nested-segmented-model-nested-condition]), we can conclude that (*ρ*(*t*1), *θ*(*t*1)) ∈ P1+.\nSince (*ρ*(*t*1), *θ*(*t*1)) ∈ P1+, *ρ*(*t*) will increase to *ρ*\\* following Step [enu:alg-two-phase-update-loop]a) in Algorithm [enu:alg-initialization]. This completes the proof that the algorithm trajectory passes through (*ρ*(*t*2), *θ*(*t*2)) = (*ρ*\\*, *θ*\\*) at time *t*2, where *t*1 ≤ *t*2 ≤ *T*3.\n**Step C:** Now, the algorithm iterate (*ρ*(*t*), *θ*(*t*)) is at the stage of Step [enu:alg-two-phase-update-loop]a) in Algorithm [alg:uav-positioning], with *θ*(*t*) = *θ*\\* and *ρ*(*t*) ≤ *ρ*\\*. It must reach (*ρ*\\*, *θ*\\*) following Step [enu:alg-two-phase-update-loop]a), where (*ρ*\\*, *θ*\\*) solves P1 +ʹ.\nWith these, we confirm the results of Lemma [lem:Local-optimality].\nOptimality for the *K* Segment Case\n-----------------------------------\nWe now extend Lemma [lem:Local-optimality] to the case of more than two segments.\n[Optimality after *k* Loops][lem:partial-opt-K] After completing the *k*th loop of Steps [enu:alg-two-phase-update])[enu:alg-two-phase-upadte-loop-left]) in Algorithm [alg:uav-positioning] at time *t* = *T**k*, the following holds,\n \n$$F\\_{\\min}(T\\_{k})\\leq\\underset{\\theta\\geq0,(\\rho,\\theta)\\in\\mathcal{P}\\_{k}}{\\text{minimize}}\\quad F\\_{k}(\\rho,\\theta)\\label{eq:per-segment-minimum}$$ \nfor all *k* = 1, 2, …, *K* − 1. Moreover, ([eq:per-segment-minimum]) holds for *k* = *K* when the entire algorithm terminates.\nFor the *k*th loop (*k* ≤ *K* − 1) of Steps [enu:alg-two-phase-update]) [enu:alg-two-phase-upadte-loop-left]) in Algorithm [alg:uav-positioning], the iteration is equivalent to that in the two segment case, *K* = 2. Specifically, the virtual propagation segment partition $(\\tilde{\\mathcal{P}}\\_{k},\\tilde{\\mathcal{P}}\\_{k}^{\\text{c}})$ in the *k*th outer loop corresponds to the LOS-NLOS partition (P1, P2) in the loop of Steps [enu:alg-two-phase-update]) [enu:alg-two-phase-upadte-loop-left]) for the *K* = 2 case. Moreover, the function *F**k* in the *k*th outer loop corresponds to the function *F*1 in the *K* = 2 case. As a result, using Lemma [lem:Local-optimality], Steps [enu:alg-two-phase-update]) [enu:alg-two-phase-upadte-loop-left]) in the *k*th loop equivalently solve\n \n$$\\mathscr{P}\\_{k}':\\quad\\underset{\\theta\\geq0,(\\rho,\\theta)\\in\\widetilde{\\mathcal{P}}\\_{k}}{\\text{minimize}}\\quad F\\_{k}(\\rho,\\theta)\\label{eq:k-th-optimization}$$ \nwhere $\\tilde{\\mathcal{P}}\\_{k}\\triangleq\\bigcup\\_{j=1}^{k}\\mathcal{P}\\_{j}$ and *k* ≤ *K* − 1.\nWe denote the minimum value of ([eq:k-th-optimization]) as *F̃*min, *k*, and the algorithm trajectory (*ρ*(*t*), *θ*(*t*)) reaches the optimal solution (*ρ̂*(*k*), *θ̂*(*k*)) at time *t**k* ≤ *T**k*. Since the constraint set $\\tilde{\\mathcal{P}}\\_{k}$ in ([eq:k-th-optimization]) contains the constraint set P*k* in ([eq:per-segment-minimum]), it must hold that \n*F̃*min, *k* ≤ minimize*θ* ≥ 0, (*ρ*, *θ*) ∈ P*k**F**k*(*ρ*, *θ*). \n Therefore, we still need to show *F*min(*T**k*) ≤ *F̃*min, *k*.\nWith such a goal, the following two cases are examined:\n(i) If (*ρ̂*(*k*), *θ̂*(*k*)) ∈ P*k*, then *F*min(*T**k*) ≤ *F*(*ρ*(*t**k*), *θ*(*t**k*)) = *F**k*(*ρ*(*t**k*), *θ*(*t**k*)) = *F̃*min, *k* due to the track record update *F*min(*t*) from Step 3a).\n(ii) If $(\\hat{\\rho}^{(k)},\\hat{\\theta}^{(k)})\\notin\\mathcal{P}\\_{k}$, then it holds that $(\\hat{\\rho}^{(k)},\\hat{\\theta}^{(k)})\\in\\widetilde{\\mathcal{P}}\\_{k}\\backslash\\mathcal{P}\\_{k}=\\bigcup\\_{i<k}\\mathcal{P}\\_{i}$. wlog, assume that (*ρ̂*(*k*), *θ̂*(*k*)) ∈ P*j* for some *j* ≤ *k* − 1. We thus have\n \n$$\\begin{aligned}\n\\tilde{F}\\_{\\min,k} & =F\\_{k}(\\hat{\\rho}^{(k)},\\hat{\\theta}^{(k)})\\nonumber \\\\\n & \\geq F\\_{j}(\\hat{\\rho}^{(k)},\\hat{\\theta}^{(k)})\\label{eq:partial-opt-K-ieq1}\\\\\n & =F(\\rho(t\\_{k}),\\theta(t\\_{k}))\\nonumber \\\\\n & \\geq F\\_{\\min}(t\\_{k})\\nonumber \\\\\n & \\geq F\\_{\\min}(T\\_{k})\\label{eq:partial-opt-K-ieq2}\\end{aligned}$$ \nwhere the inequality on the first line is from conditions ([eq:local-optimality-condition-1])([eq:local-optimality-condition-2]). The second line is due to the track record update *F*min(*t*) from Step 3a). This shows that inequality ([eq:per-segment-minimum]) is also true for *k*.\nAs a result, we have shown \n*F*min(*T**k*) ≤ *F̃*min, *k* ≤ minimize*θ* ≥ 0, (*ρ*, *θ*) ∈ P*k**F**k*(*ρ*, *θ*) \n for *k* = 1, 2, …, *K* − 1.\nFinally, the last step in Algorithm [alg:uav-positioning] yields\n \n$$\\begin{aligned}\nF\\_{\\min} & \\leq F\\_{K}(\\rho\\_{K}^{\\*}(0),0)\\nonumber \\\\\n & \\leq\\underset{\\rho\\geq0,0<\\theta\\leq\\frac{\\pi}{2}}{\\text{minimize}}\\,F\\_{K}(\\rho,\\theta)\\label{eq:partial-opt-K-ieq5}\\\\\n & \\leq\\underset{\\theta\\geq0,(\\rho,\\theta)\\in\\mathcal{P}\\_{K}}{\\text{minimize}}\\,F\\_{K}(\\rho,\\theta)\\nonumber \\end{aligned}$$ \nwhere the second inequality in ([eq:partial-opt-K-ieq5]) is from conditions ([eq:local-optimality-condition-1])([eq:local-optimality-condition-2]). The result of Lemma [lem:partial-opt-K] is thus confirmed.\nTheorem 1A is a direct result from Lemma [lem:partial-opt-K], since ([eq:per-segment-minimum]) holds for *k* = 1, 2, …, *K* and *F*min(*t*) is non-increasing, which implies that *F*min(*T**K*) is the global minimum value of Pʹ and (*ρ̂*(*T**K*), *θ̂*(*T**K*)) is the globally optimal solution to Pʹ.\nProof of Theorem [thm:Maximum-Trajectory-Length]\n================================================\nAn important property of Algorithm [alg:uav-positioning] is that the segment of search trajectory in Step [enu:alg-two-phase-update-loop]) does not “turn back”, and so does that in Step [enu:alg-two-phase-upadte-loop-left]).\n[Monotonicity][lem:monotonicity] Step [enu:alg-two-phase-update-loop]) in Algorithm [alg:uav-positioning] strictly and monotonically increases *ρ*(*t*), and it also monotonically increases ∣*θ*(*t*)∣. Similar property holds in Step [enu:alg-two-phase-upadte-loop-left]).\nIn Step [enu:alg-two-phase-update-loop]a), *ρ*(*t*) strictly and monotonically increases, while *θ*(*t*) keeps unchanged. In Step [enu:alg-two-phase-update-loop]b), we have (omitting the higher order term)\n \n$$\\begin{aligned}\n\\rho+\\Delta\\rho & =\\|\\mathbf{x}+\\Delta\\mathbf{x}-\\mathbf{x}\\_{\\text{u}}\\|\\\\\n & =\\sqrt{\\|\\mathbf{x}-\\mathbf{x}\\_{\\text{u }}\\|^{2}+2(\\mathbf{x}-\\mathbf{x}\\_{\\text{u}})^{\\text{T}}\\Delta\\mathbf{x}+\\|\\Delta\\mathbf{x}\\|^{2}}\\\\\n & =\\rho\\Big(1+\\frac{1}{\\rho^{2}}(\\mathbf{x}-\\mathbf{x}\\_{\\text{u}})^{\\text{T}}\\Delta\\mathbf{x}+o(\\|\\mathbf{x}\\|)\\Big)\\end{aligned}$$ \nand since (**x** − **x**u)/*ρ* = **M**(*θ*)**u** from ([eq:polar-representation-xy-plane]), we have\n \n$$\\begin{aligned}\n\\Delta\\rho & =\\mathbf{u}^{\\text{T}}\\mathbf{M}(\\theta)^{\\text{T}}\\Delta\\mathbf{x}\\\\\n & =\\mathbf{u}^{\\text{T}}\\mathbf{M}(\\theta)^{\\text{T}}\\mathbf{M}(\\theta)\\mathbf{u}\\gamma\\\\\n & \\qquad\\qquad+\\gamma\\rho\\mathbf{u}^{\\text{T}}\\mathbf{M}(\\theta)^{\\text{T}}\\frac{d}{d\\theta}\\mathbf{M}(\\theta)\\mathbf{u}\\Big(-\\frac{\\partial F\\_{k}}{\\partial\\theta}\\Big)^{-1}\\frac{\\partial F\\_{k}}{\\partial\\rho}\\end{aligned}$$ \nwhich equals to *γ*, being strictly positive. Therefore, Step [enu:alg-two-phase-update-loop]) strictly increases *ρ*(*t*).\nIn addition, Step [enu:alg-two-phase-update-loop]b) moves on the contour of *F**k*(*ρ*, *θ*) = *C*, whose dynamics is give by \n$$\\frac{\\partial F\\_{k}(\\rho,\\theta)}{\\partial\\rho}d\\rho+\\frac{\\partial F\\_{k}(\\rho,\\theta)}{\\partial\\theta}d\\theta=0$$ \n in which ∂*F**k*(*ρ*(*t*), *θ*(*t*))/∂*ρ* ≤ 0 according to Lemma [lem:property-one-side] (with a straight-forward generalization from *F*1 to *F**k*) and ∂*F**k*(*ρ*, *θ*)/∂∣*θ*∣ > 0 according to Lemma [lem:partial-optimality]. As a result, Step [enu:alg-two-phase-update-loop]b) monotonically increases ∣*θ*(*t*)∣.\n\nWe now show that the boundary of the search region P in ([eq:search-region]) for the *θ* > 0 branch is given by a semi-circle as illustrated in Fig. [fig:geometry-search-region] (the blue semi-circle). To see this, we first note that the boundary of P for *θ* > 0 is given by (*ρ*cos*θ*, *ρ*sin*θ*), where *ρ* = *L*cos*θ* and 0 ≤ *θ* ≤ *π*/2. Therefore, $\\overline{\\text{\\textbf{ad}}}=L\\cos\\theta$, $\\overline{\\text{\\textbf{ac}}}=L$, $\\overline{\\text{\\textbf{dc}}}=\\sqrt{(L\\cos^{2}\\theta-L)^{2}+(L\\cos\\theta\\sin\\theta-0)^{2}}$, which yields $(\\overline{\\text{\\textbf{ad}}})^{2}+(\\overline{\\text{\\textbf{dc}}})^{2}=(\\overline{\\text{\\textbf{ac}}})^{2}$ and hence point **d** is on the semi-circle with the diameter given by line segment $\\overline{\\text{\\textbf{ac}}}$.\nFrom the monotone property in Lemma [lem:monotonicity], the search trajectory is a convex curve starting from point **a** at the user and ending at the semi-circle, *e.g.*, point **d**. Note that the length of any curve $\\breve{\\text{\\textbf{ad}}}$ is less than $\\overline{\\text{\\textbf{ab}}}+\\overline{\\text{\\textbf{bd}}}=L\\cos^{2}\\theta+L\\cos\\theta\\sin\\theta$, where the maximum value over 0 ≤ *θ* < *π*/2 can be numerically evaluated to be roughly 1.2*L*.\nAlgorithm [alg:uav-positioning] consists of the search on the BS-user axis with maximum length *L*, and *K* − 1 loops for the off-BS-user axis searches, where each loop consists of searches on the left and right branches each with maximum length 1.2*L*. As a result, the total length of the search trajectory is upper bounded by (2.4*K* − 1.4)*L*.\n10 [1]#1 url@samestyle [2]#2 [2]l@#1=l@#1#2\nY. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: opportunities and challenges,” *IEEE Commun. Mag.*, vol. 54, no. 5, pp. 36–42, 2016.\nM. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Unmanned aerial vehicle with underlaid device-to-device communications: Performance and tradeoffs,” *IEEE Trans. Wireless Commun.*, vol. 15, no. 6, pp. 3949–3963, 2016.\nZ. Xiao, P. Xia, and X.-G. Xia, “Enabling UAV cellular with millimeter-wave communication: Potentials and approaches,” *IEEE Commun. Mag.*, vol. 54, no. 5, pp. 66–73, 2016.\nB. Van Der Bergh, A. Chiumento, and S. Pollin, “LTE in the sky: trading off propagation benefits with interference costs for aerial nodes,” *IEEE Commun. Mag.*, vol. 54, no. 5, pp. 44–50, 2016.\nA. Merwaday and I. Guvenc, “UAV assisted heterogeneous networks for public safety communications,” in *Proc. Wireless Commun. and Networking Conf. Workshops*, 2015, pp. 329–334.\nD.-T. Ho, P. Sujit, T. A. Johansen, and J. B. De Sousa, “Performance evaluation of cooperative relay and particle swarm optimization path planning for UAV and wireless sensor network,” in *Proc. IEEE Globecom Workshops*, 2013, pp. 1403–1408.\nV. Sharma, M. Bennis, and R. Kumar, “UAV-assisted heterogeneous networks for capacity enhancement,” *IEEE Commun. Lett.*, vol. 20, no. 6, pp. 1207–1210, 2016.\nO. Jian, Z. Yi, L. Min, and L. Jia, “Optimization of beamforming and path planning for UAV-assisted wireless relay networks,” *Chinese Journal of Aeronautics*, vol. 27, no. 2, pp. 313–320, 2014.\nY. Zeng and R. Zhang, “Energy-efficient UAV communication with trajectory optimization,” *IEEE Trans. Wireless Commun.*, vol. 16, no. 6, pp. 3747–3760, 2017.\nY. Jin, Y. D. Zhang, and B. K. Chalise, “Joint optimization of relay position and power allocation in cooperative broadcast wireless networks,” in *Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing*, 2012, pp. 2493–2496.\nD. H. Choi, B. H. Jung, and D. K. Sung, “Low-complexity maneuvering control of a UAV-based relay without location information of mobile ground nodes,” in *Proc. IEEE Symposium on Computers and Commun.*, 2014, pp. 1–6.\nA. Chamseddine, G. Charland-Arcand, O. Akhrif, S. Gagné, F. Gagnon, and D. Couillard, “Optimal position seeking for unmanned aerial vehicle communication relay using only signal strength and angle of arrival,” in *IEEE Conf. on Decision and Control*, Dec 2014, pp. 976–981.\nF. Jiang and A. L. Swindlehurst, “Optimization of UAV heading for the ground-to-air uplink,” *IEEE J. Sel. Areas Commun.*, vol. 30, no. 5, pp. 993–1005, 2012.\nM. M. Azari, F. Rosas, K.-C. Chen, and S. Pollin, “Ultra reliable UAV communication using altitude and cooperation diversity,” *IEEE Trans. Commun.*, vol. 66, no. 1, pp. 330–344, 2018.\nS. ur Rahman and Y.-Z. Cho, “UAV positioning for throughput maximization,” *EURASIP J. on Wireless Commun. and Networking*, vol. 2018, no. 1, p. 31, 2018.\nJ. Lyu, Y. Zeng, R. Zhang, and T. J. Lim, “Placement optimization of UAV-mounted mobile base stations,” *IEEE Commun. Lett.*, vol. 21, no. 3, pp. 604–607, 2017.\nA. Al-Hourani, S. Kandeepan, and A. Jamalipour, “Modeling air-to-ground path loss for low altitude platforms in urban environments,” in *Proc. IEEE Global Telecomm. Conf.*, 2014, pp. 2898–2904.\nM. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Drone small cells in the clouds: Design, deployment and performance analysis,” in *Proc. IEEE Global Telecomm. Conf.*, 2015, pp. 1–6.\nA. Hourani, K. Sithamparanathan, and S. Lardner, “Optimal LAP altitude for maximum coverage,” *IEEE Commun. Lett.*, no. 99, pp. 1–4, 2014.\nM. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Optimal transport theory for power-efficient deployment of unmanned aerial vehicles,” in *Proc. IEEE Int. Conf. Commun.*, Kuala Lumpur, Malaysia, May 2016, pp. 1–6.\n——, “Efficient deployment of multiple unmanned aerial vehicles for optimal wireless coverage,” *IEEE Commun. Lett.*, vol. 20, no. 8, 2016.\nQ. Feng, J. McGeehan, E. K. Tameh, and A. R. Nix, “Path loss models for air-to-ground radio channels in urban environments,” in *Proc. IEEE Semiannual Veh. Technol. Conf.*, vol. 6, 2006, pp. 2901–2905.\n“Study on enhanced LTE support for aerial vehicles (release 15),” 3GPP TR 36.777, Tech. Rep., 2017. [Online]. Available: <http://www.3gpp.org>\nJ. Chen and D. Gesbert, “Optimal positioning of flying relays for wireless networks: A LOS map approach,” in *Proc. IEEE Int. Conf. Commun.*, Paris, France, May 2017.\nT. Bai and R. W. Heath, “Coverage and rate analysis for millimeter-wave cellular networks,” *IEEE Trans. Wireless Commun.*, vol. 14, no. 2, pp. 1100–1114, 2015.\nJ. Chen, U. Yatnalli, and D. Gesbert, “Learning radio maps for UAV-aided wireless networks: A segmented regression approach,” in *Proc. IEEE Int. Conf. Commun.*, Paris, France, May 2017.\nJ. Chen, O. Esrafilian, D. Gesbert, and U. Mitra, “Efficient algorithms for air-to-ground channel reconstruction in UAV-aided communications,” in *Proc. IEEE Global Telecomm. Conf.*, Dec. 2017, Wi-UAV workshop.\nR. U. Nabar, H. Bolcskei, and F. W. Kneubuhler, “Fading relay channels: Performance limits and space-time signal design,” *IEEE J. Sel. Areas Commun.*, vol. 22, no. 6, pp. 1099–1109, 2004.\nJ. N. Laneman, D. N. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” *IEEE Trans. Inf. Theory*, vol. 50, no. 12, pp. 3062–3080, 2004.\nT. Wang, A. Cano, G. B. Giannakis, and J. N. Laneman, “High-performance cooperative demodulation with decode-and-forward relays,” *IEEE Trans. Commun.*, vol. 55, no. 7, pp. 1427–1438, 2007.\n---\n1. This paper has been accepted in IEEE Transactions on Wireless Communications. This research was funded, in part, by one or more of the following grants: No. ZDSYS201707251409055, No. 2017ZT07X152, No. 2018B030338001, No. 2018YFB1800800, and ERC under the European Union Horizon 2020 research and innovation program (Agreement no. 670896). This paper was presented, in part, at the IEEE International Conference on Communications, Paris, France, 2017.[↩](#fnref1)\n2. Junting Chen is with the School of Science and Engineering and the Future Network of Intelligence Institute (FNii) at the Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China (e-mail: juntingc@cuhk.edu.cn).[↩](#fnref2)\n3. David Gesbert is with Department of Communication Systems, EURECOM, 06410 Sophia-Antipolis, France (e-mail: gesbert@eurecom.fr).[↩](#fnref3)\n4. More generally, one can design the cost function *f* in terms of the UAV position **x** to capture other system characteristics, such as antenna pattern and effect of small-scale fading.[↩](#fnref4)\n5. The original problem in considered a diversity scheme that combines the signal from the relay and the signal from the BS. Such a strategy also leads to problem P1 under high snr.[↩](#fnref5)\n6. For mathematical completeness, the partial derivative is defined as $\\frac{\\partial f(x\\_{0},y\\_{0})}{\\partial x}=\\lim\\_{t\\uparrow0}\\frac{1}{t}[f(x\\_{0}+t,y\\_{0})-f(x\\_{0},y\\_{0})]$ throughout this paper.[↩](#fnref6)\n7. We found from our experiment that around 22% users have LOS condition on the BS-user link.[↩](#fnref7)\n"},"data_id":{"kind":"string","value":"arxiv_0181646"}}},{"rowIdx":45390,"cells":{"content":{"kind":"string","value":" radius 12 pc and is surrounded by 11 Bright-Rimmed clouds with embedded IRAS point sources. SFO 38, also known as IC 1396N, is located at a projected distance of ∼ 11 pc to the north of HD 206267 with the bright rim corresponding to the ionization front facing the O6.5V star. SFO 38 is located in the Cep OB2 association at a distance of 750 pc. More recent *Hipparcos* parallax based measurements estimate the distance to Cep OB2 association to be 615 pc. However in order to make our results comparable to most published results, in this paper we adopt a distance of 750 pc for SFO 38. The derived luminosities may therefore be overestimated by more than 40% and physical sizes by 20%. The region is associated with IRAS 21391+5802, a very young intermediate-mass object, with a luminosity of 235 L⊙ , which powers an extended bipolar outflow. Based on millimeter observations resolved IRAS 21391+5802 into an intermediate-mass source named BIMA 2 surrounded by two less massive and smaller objects, BIMA 1 and BIMA 3. detected **H*2*O** maser emission at 2.2 GHz towards SFO 38 which is consistent with an intermediate-mass object. used still higher angular resolution millimeter interferometric observations to reveal that the intermediate-mass protostar BIMA 2 itself consists of multiple compact sources. The gas emission surrounding IRAS 21391+5802 traces different molecular outflows. NIR images of the region have also revealed a number of small scale molecular hydrogen and Herbig-Haro (HH) flows. These observations confirm the on going star formation in the dense core of SFO 38.\n identified the mid-infrared (MIR) (*Spitzer*) counterparts of X-ray sources detected in *Chandra* observations and found that the young stellar objects (YSOs) are oriented toward the ionizing star HD 206267 and show an age gradient consistent with the RDI model for triggered star formation. Recently studied the YSO population of SFO 38 by obtaining deep J, H, K′ broadband images and deep high-angular resolution observations in the H2 narrowband filter. did not detect any clear NIR excess close to the rim, which is in contrast to the age 1 Myr of the stellar population concluded by. suggest that the YSOs closer to HD 206267 could actually be younger than the Class II objects they appear to be since their circumstellar environment were disrupted completely by intense UV radiation field from the OB association. thus suggest that in general the apparent age sequence seen close to OB associations need not necessarily be the actual evolutionary sequence and in the case of SFO38 there is no concrete evidence of star formation by the nearby OB association.\nIn this paper we address the controversial issue of the distribution of YSOs by making use of the extremely sensitive *Spitzer* IRAC (3.6 to 8.0 ) & MIPS (24 ) data in order to probe *all* embedded YSOs and protostars in SFO 38. This thus extends the work by, who had considered the 3.6–5.8 characteristics only of those MIR sources from which X-ray emission had been detected. Our approach also complements the work by which makes use of only the NIR observations to decipher any age gradient of the YSOs. In this work we shall show that most of the Class 0/I sources identified in the MIR are either barely detected (with large uncertainties in fluxes) or not detected at all in the NIR and in the X-rays. In this paper we have completed the multiwavelength overview of SFO 38 by obtaining optical broadband images in *BVRI* bands and narrow band H*α* image to understand the detailed structure of the bright-rim. Further in order to confirm the ``*youth*\" of the YSOs identified in the MIR we have made use of medium resolution spectroscopy of the optically visible YSOs in H*α* line emission. Observation details of *Spitzer* and optical *BVRI* imaging and medium resolution spectroscopy are presented in § [secobs]. We describe the selection of YSO candidates based on MIR color-color diagram (CCD) and the detailed analysis of optical–MIR data in section § [secresults]. In § [secdis] we discuss the probable star formation scenarios and in § [secconclusion] we summarize the results.\nObservations and Data Reduction [secobs]\n========================================\n*Spitzer* IRAC & MIPS observations\n----------------------------------\nWe have extracted IRAC (3.6, 4.5, 5.8, & 8 ) and MIPS (24 & 70 ) observations from the Spitzer Space Observatory archive (Program ID 30050: Star Formation in Bright Rimmed Clouds by Fazio et al.). The IRAC data were taken in the High Dynamic Range (HDR) mode using a single AOR (Astronomical Observation Request) with a five-point dither pattern. We have processed both the short (0.6 sec) and the long (12 sec) integration Basic Calibrated Data (BCD) frames in each channel using the Artifact mitigation software developed by Sean Carey [1](#fn1) and created mosaics using MOPEX. In all IRAC bands we detect point-sources down to 70 *μ*Jy. We have also created mosaics of the MIPS 24 and 70 BCDs using MOPEX. Both data sets are of reasonably good quality. The 70 *μ*m-image, due to the lower angular resolution and smaller area mapped, shows only one bright point-like source embedded in the globule.\nWe have carried out multiframe PSF photometry using the tool APEX developed by Spitzer Science Center(SSC) on all the *Spitzer* IRAC images and on the MIPS images. The long integration 3.6 and 4.5 IRAC images show signs of saturation on bright stars. For the saturated sources, the photometry derived from the short integration images were used. It is difficult to disentangle the sources in regions of strong emission from the associated Photon Dominated Regions (PDRs) from the surrounding clouds and to derive their accurate photometry. We have used a combination of automated routines and eye-inspection to detect sources and extract photometry of these sources from the IRAC and MIPS images. For sources, which APEX failed to detect automatically at one or several wavelengths, we have used the user list option in APEX to supply the coordinates for the source to successfully derive a PSF fit. This enabled us to derive photometry for every source which we could visually identify on any image. Following the IRAC and MIPS Data Handbooks we have adopted the zero-points for conversion between flux densities and magnitudes to be 280.9, 179.7, 115.0, 64.1 and 7.14 Jys in the 3.6, 4.5, 5.8, 8.0, and 24 bands, respectively.\nOptical *BVRI* Photometry\n-------------------------\nSFO 38 was observed on 17th July, 23rd August and 15th September 2009 using Bessell’s broad band filters *VR* (300 × 8 s, 300 × 8 s), *BVI* (900 × 8 s, 300 × 8 s, 150 × 8 s) and *BI* (900 × 8 s, 120 × 12 s) respectively and on 4th November 2008 using the narrow band *H*α** (900 s) with the Himalayan Faint Object Spectrograph Camera (HFOSC) mounted on the 2m Himalayan Chandra Telescope (HCT) of the Indian Astronomical Observatory, Hanle, India[2](#fn2). HFOSC uses the central 2K × 2K region of the 2K × 4K CCD in imaging mode and covers a field of 10ʹ × 10ʹ with a plate scale of 0296 *p**i**x**e**l*− 1. The nights were photometric with an average seeing of 15 to 18. photometric standard stars were observed on all the nights to calibrate the target stars. Data were reduced using standard tasks available within Image Reduction and Analysis Facility (IRAF)[3](#fn3). Bias subtracted, flat field corrected and aligned frames were combined to make the master frame for each filter. Astrometric calibration was applied to the master frames using the IDL procedure *STARASTT* of IDL Astronomy User’s Library[4](#fn4). Photometric magnitudes were calculated by aperture photometry with the optimal aperture adopted as the radius where the difference in magnitudes between two consecutive apertures is less than 1%. The limiting magnitude is defined as the magnitude at which the mean magnitude error of the star becomes 0.1 mag which implies a 10*σ* detection corresponds to a signal-to-noise ratio (S/N) of 10. The limiting magnitudes were V: 21, R: 21 on 17th July, B: 22, V: 22, I: 19.5 on 23rd August and B: 22.2, I: 20.2 on 15th September. Aperture corrections for each frame were derived from the bright and isolated stars and applied to the faint stars. The standard deviation of residuals of observed and transformed magnitudes and colors of the standard stars are within the range of 0.01-0.02 mag. We used the broadband *R* filter image for continuum subtraction from the narrow band H*α* image of SFO 38. Based on the recipe of, the point spread functions of the R-band and the narrow band H*α* images were matched and the images were scaled before subtraction to get the H*α* emission line image of SFO 38.\nMedium Resolution Spectroscopy\n------------------------------\nMedium resolution ( ∼ 7 Å) spectra were obtained for relatively bright and optically visible YSOs during July–November of both 2008 and 2009 in the wavelength range (5200–9200 Å) with the Himalayan Faint Object Spectrograph Camera (HFOSC) mounted on the Himalayan Chandra Telescope (HCT). Typical exposure time of each spectrum was 3600 s. The spectra were bias subtracted, flat fielded and the one-dimensional spectra were extracted using the standard tasks of Image Reduction and Analysis Facility (IRAF). The arc lamp spectra of FeNe were used for wavelength calibration. We further used the strong night sky emission lines e.g. [OI] *λ*5577 Å, *λ*6300 Å lines and rotational and vibrational bands of OH in the red region of the individual target spectrum to improve the wavelength calibration and achieved an accuracy of ± 0.5 Å for each target star.\nResults and Analysis [secresults]\n=================================\nMid-Infrared (3.6 to 24 ) view of SFO 38\n----------------------------------------\nFigure [figiracyso] presents the three-color image of SFO 38 using the IRAC 3.6, 8.0 and the MIPS 24 bands. The image shows that the emission from the front side of the globule is dominated by strong polyaromatic hydrocarbon emission (PAH-emission), and rather clumpy. This could in part be due to Rayleigh-Taylor instabilities at the ionization front, but it is also due to the outflows from the young stars. The cometary shape of the globule is quite apparent with the tip pointed towards the O star HD 206267 and the eastern edge appears to be more abruptly truncated to the south. This indicates that the south-eastern edge of the globule has most likely experienced intense ionizing radiation field, which has eroded the cloud material more than on the western side. Comparison with the H*α* emission line image (Fig.[figysomap]) also confirms the presence of a more pronounced ionization front to the south-east and a rather tenuous layer of ionized gas to the west of SFO 38.\nThe aim of this paper is to identify the young and embedded stellar population in SFO 38. While the 3.6 and 4.5 IRAC data is particularly sensitive, a large number of stellar objects in addition to the YSOs also detected in these bands. We have thus used all sources detected either at 5.8 or at 8.0 or at both wavelengths to create a list of MIR sources in SFO 38. Photometry for all these sources, if detected, are then extracted from the 3.6 & 4.5 images. We thus detect 98, 106, 106 and 98 sources respectively in the IRAC 3.6, 4.5, 5.8 and 8.0 images and 14 and 1 sources respectively in the MIPS 24 & 70 images. All the sources detected at 24 are found to have been detected in the IRAC bands. In all we identify a total of 110 MIR sources. We have crosscorrelated the MIR sources with the NIR sources in our mapped region from the 2MASS point source catalogue, as well as the sources detected in the NIR by and. We have used the following association radii : 1 for the IRAC images, 25 for the MIPS 24 image and 2 for NIR data. Table [tabmirsrc] gives the coordinates of the 110 sources identified in SFO 38 together with their NIR magnitudes, *Spitzer* IRAC and MIPS flux densities and a preliminary classification based on selected color-color plots and other criteria described in Sec. [secysoclass]. In Table [tabmirsrc] we have used the prefix SFO38, but throughout the paper we refer to them as MIR-nn, where nn is the number of the source. Out of the 110 sources 80 were found to have NIR counterparts. However 10 sources out of the 80 sources with NIR counterparts were found to coincide with the H2 knots identified by.\nAdditionally in order to identify the stellar and pre main sequence (PMS) stars we used photometry extracted from the IRAC 3.6 and 4.5 long integration images. The 3.6 and 4.5 images are more sensitive, have a cleaner PSF, and appear to be less affected by nebular emission than the 8 image. We identified 161 additional sources, which are detected in both the 3.6 and 4.5 wavebands. Of these sources 113 are found to have NIR counterparts, and 6 sources are coincident with H2 emission knots.\nClassification of YSOs based on Near- and Mid-infrared colors [secysoclass]\n---------------------------------------------------------------------------\nIn regions of high obscuration like SFO 38, which has a visual extinction of up to 20 mag, mid-infrared color indices provide an ideal tool for identifying YSOs and classify them according to their phases of PMS evolution. Here we use several color indices generated using the *Spitzer* IRAC and MIPS data to characterize the evolutionary stages of the MIR sources detected in SFO 38.\n demonstrated that (a) normal star-forming galaxies and narrow-line AGNs with increasing 5.8 and 8.0 and (b) broad-line AGNs with red, nonstellar SEDs, result in colors which are very similar to bona-fide YSOs. Thus prior to identifying PMS sources based on the color indices it is necessary to inspect deep IRAC images for contamination due to the extragalactic sources. have extensively discussed the criteria for identifying such extragalactic objects in the IRAC color-color diagrams. Using the criteria given in the appendix of their paper, we identify MIR-2, 17, 68, 70, 75, 81, 84, 89, 101 and 103 as AGNs and MIR-77, 81, 93 and 103 as galaxies strongly emitting in the PAH bands. Of these sources, MIR-17, 89 and 103 satisfy most of the criteria for extragalactic objects. The profiles of sources in the *R*-band is typically quite a good indicator of whether the source is actually extragalactic or stellar. However all the sources in SFO 38 which satisfy the criteria mentioned above are extremely faint so that with the exception of MIR-2, these sources though visible are not detected clearly enough in the *R*-band so that the intensity profile can be studied. Further, none of the sources is detected in the MIPS 24 image either. The *R*-band profile of MIR-2 appears to be stellar with a faint companion on one side and MIR-84 is associated with an H*α* emission star, so we do not identify both these sources as extragalactic. In the absence of any conclusive evidence we presently assume the sources identified by the empirical criteria as above to be extragalactic.\nThe 4.5 IRAC band of *Spitzer* covers many of the H2 pure rotational lines, as a result of which this band records a large excess of emission at the position of the unresolved blobs of shocked emission from high velocity outflows which interact with the molecular cloud. Using the empirical criteria outlined by here we flag the sources MIR-24, 27, 53, 65 and 98 as possible features due to shocked H2 emission. Based on 2.12 H2 integrated line emission identified as many as 97 emission features possibly created by H2 excitation by shocks driven by outflows powered by YSOs. We find that 10 out of the 110 MIR sources that we have identified are associated with such H2 knots. These include the sources MIR-20, 24, 27, 35, 53, 65, 74, 75, 98 and 105. Thus the color indices for 5 out of the 10 sources are consistent with the observed association with H2 knots. The source MIR-75 which also satisfied the criteria for being an extragalactic object is thus identified as a H2 knot.\nFigure [figiracmipscol] presents color-color diagrams of NIR, IRAC and MIPS 24 sources detected in the SFO 38 globule. We have used several criteria (shown as dashed lines and boxes in Figure [figiracmipscol]) to identify the potential YSOs using these color-color diagrams.\nThe most stringent classification scheme uses the IRAC ([3.6]–[5.8]) colors and the [8]–[24] IRAC and MIPS color. At 24 the reddening due to extinction is small and the photospheric colors are very close to zero for all spectral types. Therefore the [8]–[24] color is very sensitive to infrared excess, but of course not all young stars are bright enough to be detected at 24. Using this color-color diagram we find fourteen sources with infrared excess (Figure [figiracmipscol] (*left*)). We identify MIR-34, 36, 48, 50, 54, 55 and 59 as Class 0/I sources, while MIR-11, 31, 38, 45, 82 and 86 are within the Class II regime and MIR-29 which has a large [8.0]–[24] color excess and very small value for the [3.6]-[5.8] color is identified as a transition disk object. predicted sources with [3.6]–[4.5] > 1.0 and [8]–[24] > 4.5 to be YSOs driving molecular outflows. We find that the sources MIR-34, 48, 50, 54, and 59, all of which have been identified as Class 0/I objects also satisfy the criteria for being sources driving outflows.\nFigure [figiracmipscol] (*middle*) presents the [3.6]–[4.5] vs [5.8]–[8.0] color-color plot for the sources detected in all the IRAC bands. Sources with the colors of stellar photospheres are centered at ([3.6]–[4.5],[5.8]–[8.0])=(0,0) and include foreground and background stars as well as diskless (Class III) pre-main sequence stars. Sources which satisfy the criteria − 0.1 ≤ [3.6]–[4.5] ≤ 0.1 and 0.2 ≤ [5.8]–[8.0] ≤ 1.1 are classified as the transitional disk objects which also do not have any NIR color excess. The box outlined in Figure [figiracmipscol] (*middle*), defines the location of Class II objects, i.e. sources whose colors can be explained by young, low-mass stars surrounded by disks. have shown from their observations of young stars in the Taurus-Auriga complex that Class 0/I protostars require [3.6]–[4.5] > 0.7 and [5.8]–[8.0] > 0.7. With the exception of MIR-29, the classes derived for the sources identified in the MIPS 24 band and plotted in Fig. [figiracmipscol](*left*) are completely consistent with the classes derived from the [3.6]–[4.5] vs [5.8]–[8.0] color-color plot. We identify 10 of the MIR sources as being associated with H2 knots, this includes MIR-75, which otherwise satisfies the criteria for extragalactic objects. We also plot NIR and *Spitzer* MIR color-color diagram of the identified YSOs and transitional Disk objects in Figure [figiracmipscol] (*right*).\nTable [tabmirsrc] presents a summary of classification of the MIR sources based on the color-color diagrams presented here. We have additionally looked into the [3.6]–[5.8] vs [4.5]–[8.0] color-color diagram and found that the Classes derived from this diagram are consistent with the classes derived based on the color-color diagrams presented here. For most sources it was possible to arrive at a unique Class based on all three color-color diagrams. For all sources the Class derived based on at least two color-color diagrams match.\nUse of the more sensitive long integration 3.6 and 4.5 images in combination with 2MASS data (following a classification scheme involving the $K\\_{\\rm s}$–[3.6] and [3.6]–[4.5] colors) yields an additional 113 sources, 11 of which are PMS objects. Since the long integration 3.6 and 4.5 images go deeper and are less affected by extinction than the 2MASS survey, we also checked how many of these IRAC sources which have no 2MASS associations are likely to be PMS stars. Among the 48 remaining sources which have no NIR counterparts, following the arguments of if we use the color criterion, [3.6]-[4.5] ≥ 0.2, we obtain 8 sources. However 6 of these 8 sources are actually associated with H2 emission knots as detected by. Table [tabnirpms] lists the 13 PMS candidates identified on the basis of NIR and 3.6 and 4.5 magnitudes.\nExcluding the H2 knots and the possible extragalactic objects we identify 10 Class 0/I sources, 3 Class I sources, 13 Class I/II sources (occupying the top-left and bottom-right corners), 14 Class II sources and 19 transitional disk objects in SFO 38. Thus we find a total of 40 YSOs (Classes 0/I/II) and 13 PMS candidates based on MIR color indices. We note that 13 Class I/II objects and 19 Transitional Disk objects are ∼ 30% of the selected YSOs and show noticeable color excess only beyond 6 .\nH*α* emission of SFO 38 and spatial distribution of YSOs\n--------------------------------------------------------\n classified SFO 38 as a *B* type BRC powered by O6.5 star HD 206267. The bright rim structure is quite prominent in the continuum subtracted H*α* line image of SFO 38 in Figure [figysomap]. The H*α* emission nebulosity of the bright-rim of the globule is asymmetric with respect to the geometric axis of the globule as well as the direction of the adopted ionizing source HD 206267. The ionized rim appears to be brighter towards the east and rather diffuse towards the western edge, facing HD 206267. The H*α* intensity distribution of the diffuse material in between the bright-rim and the ionizing star also show similar kind of asymmetric distribution as that of the bright-rim. We, therefore, searched for other massive O and B type stars in the surrounding region that can produce the relatively brighter H*α* emission on the left side of the globule. We find a B0V type star HD 206773 which is situated on the symmetry axis of the left side of the bright-rim at a projected distance of ∼ 7.6 pc from SFO 38. tagged HD 206773 as a member of Cep OB2 association. The distance towards the star is between 550–1000 pc, adopting the parallax measurement and the associated error which is consistent with the distance of SFO 38. The number of ionizing photons emitted per second by a B0V star is ∼ 1048.02 and using n*H* ≃ 107 *m*− 3 for number density of hydrogen in the interior of the H ii region IC 1396, the Strömgren radius would be ∼ 8.5 pc which suggests that HD 206773 could be a potential ionizing source for SFO 38. Fig. [figysomap] shows the distribution of the YSOs and the directions toward HD 206267 and HD 206773 on the H*α* line image. We find that most of H*α* emitting Class II objects lie within the bright-rim. Class I/II objects are situated behind the Class II objects in the intermediate region between the bright-rim and the dense portion of the globule. All the Class 0/I and Class I objects are situated well within the dense core part of the globule. There is no Class 0/I to Class I/II YSO which is situated before the ionized rim. The YSOs appear to have elongated distribution to the right and left sides of the bright rim with two different axes of elongation.\nAdditional YSO candidates based on H*α* emission [secysohalpha]\n---------------------------------------------------------------\nH*α* emission surveys are very useful to identify low mass ( < 2 M$\\sun$) Classical T Tauri stars (CTTS) in young star forming regions as the H*α* emission is due to the accretion of circumstellar material in YSOs. These observations also detect a fraction of Weak-lined T Tauri stars with weak H*α* emission which is believed to arise from enhanced chromospheric activities. Based on H*α* emission, we searched for the additional YSO candidates available in the literature outside the field of view of *Spitzer* observations to complement the list of YSOs around SFO 38. Many of the candidate YSOs in and around SFO 38 were found during H*α* objective prism survey by. A total of 14 out of the 17 H*α* emission line sources are situated within the field of view of IRAC. We consider the rest of the H*α* emission line sources *viz.,*,, and as YSO candidates. We also find 4 H*α* emission line sources near SFO 38 in the catalogue of point-source H*α* emission-line objects selected from the INT/WFC Photometric H*α* Survey (IPHAS) of the northern Galactic plane. We identify as, as, as. Thus, we also consider and as YSO candidates.\nOptical and NIR variability\n---------------------------\nOut of the 115 sources (110 MIR + 5 H*α* emission sources) mentioned in Secs. [secysoclass], [secysohalpha] and [secmedspec] we detect a total of *39* sources in *B*, *46* sources in *V*, *51* source in *R* and *63* source in *I* bands. The *BVRI* magnitudes of all the sources detected in *V* band are presented in Table [taboptmag]. We do not detect the optical counterparts of the 13 PMS candidates listed in Table [tabnirpms]. Photometric variability, a very common characteristic of YSOs, can happen due to various physical processes *viz.,* variation in stellar spots or variable obscuration which affect the emission from its photosphere. Since we have *BVI* observations separated by several months, we have also looked for the variability in the *BVI* bands for all the sources. The median variability in *BVI* bands are ∼ 0.04 mag which is also indicative of the accuracy of the optical photometric measurements. We consider a source to be *variable* if the magnitude difference is ≥ 0.1 mag (Table [taboptmag]) at the two observing epochs. MIR-5, MIR-32, MIR-43, MIR-45 and MIR-49 show variation in *BVI* bands. In particular MIR-5 with W*λ*(H*α*) ∼ 80 Å shows ∼ 1 mag variability in *B* over a timescale of month.\nYSOs also show variation in NIR bands which may or may not have any correlation with the optical variation. Both the stellar photosphere and the circumstellar disk contribute to the NIR fluxes of PMS stars. Hence, NIR variability can be either due to the same physical processes responsible for optical variability or any structural variability of disk and circumstellar materials. We compare the NIR magnitudes from 2MASS and from the more recently obtained JHK magnitudes by of the sources reported in both the datasets to identify NIR variability. Sources which have shown variation of ∼ 0.1 mag in any of the NIR band are listed in the last column of Table [tabmirsrc].\nMedium resolution spectra of YSOs [secmedspec]\n----------------------------------------------\nThe H*α* emission though a very prominent feature in the spectrum of the YSO is not sufficient to identify YSOs because there also exist other kinds of H*α* emission line stars *viz.,* *Be, Mira* stars etc. Medium resolution spectroscopy is helpful to distinguish the YSOs from the *Be* and *Mira* stars. Figure [figspecplot] presents sample spectra observed with HFOSC of a few YSOs. The prominent features e.g. H*α*, CaII NIR triplet etc. are marked in the respective spectra. We have detected strong H*α* emission in most of the Class II objects except BRC3816 and therefore we do not consider it as a YSO in the remainder of the paper. Log of spectroscopic observations and H*α* equivalent widths (W*λ*) of the individual YSOs are given in Table [tabsptclass]. BRC 38 6 (MIR-31) is the brightest Class II object detected in optical band. From the multiepoch spectroscopic observations we find that MIR-31 shows variable H*α* emission (Table [tabsptclass]). We have also detected OI *λ*7773 ( blend of OI *λ*7771.96, *λ*7774.18, *λ*7775.40 ), OI *λ*8446 (blended with Pa18 *λ*8437) CaII infrared triplet i.e *λ*8498 (blended with Pa16 *λ*8502), *λ*8542 (blended with Pa15 *λ*8545) and *λ*8662 (blended with Pa13 *λ*8665) and several other Paschen lines e.g. Pa10 *λ* 9014, Pa11 *λ*8862, Pa12 *λ*8750, Pa14 *λ*8598, Pa17 *λ*8467, Pa19 *λ*8413, Pa20 *λ*8392 in MIR-31. According to, OI *λ*7773 is a very sensitive indicator of infall in the context of magnetospheric accretion scenario. LiI *λ*6708 (W*λ* ∼ 0.7 Å) is also present in the spectra of MIR-31. All these spectroscopic features indicate that MIR-31 is indeed a YSO which is going through an active accretion phase. We have detected [OI] *λ*6300 and [OI] *λ*6363 and [SII] *λ*6717 and [SII] *λ*6731 emission lines in MIR-45 which are well known signatures of outflows from YSOs. We also find a positive correlation between H*α* emission and MIR excess for YSOs with W*λ*(H*α*) ≤ 40 Å. We have detected sporadic H*α* emission in transitional disk object MIR-29 on 3rd November 2008. We observed the object in 2009 also but we do not find any further H*α* emission. We have detected weak H*α* emission from two more transitional disk objects e.g. MIR-43 ([GFG2007] 62) and MIR-76 ([GFG2007] 81). We also obtained the spectra of few more transitional disk objects e.g. MIR-1, MIR-2, MIR-9, MIR-18 and MIR-108 and do not find H*α* emission in any of these objects.\nSpectral Classification\n-----------------------\nWe have classified the observed optical spectra by manual inspection of the spectra of target stars and also by comparing it with the spectra of known spectral type. We observed several stars with spectral type of F to M from, and with HFOSC using the same set up as the target stars to minimize the effect of instrumental signatures and resolution. From preliminary analysis we find that the YSOs are of spectral type K and M.\nFor more accurate classification in the range of spectral type K–M, we use the calibration file provided in the *SPTCLASS* code for various absorption features *viz.,* Tio bands etc. The detailed information of the classification scheme can be found in the *SPTCLASS* code[5](#fn5). This scheme measures the equivalent widths or ``indices\" of the absorption features which are sensitive to T*e**f**f* using the adjacent blue and red continuum bands and calibrate the indices against the spectral type of the stars. Spectral type of the target star is determined from the weighted mean of the spectral types obtained from different absorption features of the target star excluding the apparently anomalous values. As this classification scheme uses the local continuum of the absorption feature, it is insensitive to reddening and the S/N ratio. Further, the spectral features which are used for classification are also insensitive to the luminosity class. The errors in the spectral type determination come from the measurement of indices and the standard deviation of the spectral types obtained from different absorption features. To check the applicability of the adopted calibration on *HFOSC* spectra we classify the *HFOSC* spectra of a few K and M type stars and retrieve the spectral type within ∼ 0.5 subtype. We then use the same calibration scheme to classify the YSO spectra. We further explore the Fe I *λ*6495 Å line to classify the spectra of early K type stars. The “index” or equivalent width of *λ*6495 line was calculated for spectral types F–K, using the *HFOSC* spectra of stars with known spectral type taken from. The results of our classification are summarized in Table [tabsptclass]. We also tabulate the effective temperatures of YSOs using the table of. Using the *V-I* color of a particular spectral type and our optical *BVRI* observations we calculated the $A\\_{\\rm v}$ assuming the normal interstellar extinction law as given by. The average extinction towards the globule is estimated to be $A\\_{\\rm v}$= 2.5 mag.\nRate of accretion from H*α* emission\n------------------------------------\nThe H*α* emission along with the other Balmer lines, have always been the iconic spectral diagnostics of accretion of circumstellar materials in the YSOs. Magnetospheric accretion models have provided satisfactory explanation of the observed spectral characteristics of accretion Classical T Tauri Stars (CTTS) e.g. broad asymmetric line profiles etc.. According to the magnetospheric accretion model the stellar magnetic field disrupts the circumstellar disk at several stellar radii and accreting material fall to the stellar surface along the magnetic field lines. Magnetospheric accretion models suggest that the hydrogen emission lines form in the infall zone, so that the blue shifted asymmetric emission line profiles arise due to partial obscuration of the flow by the inner part of the accretion disc and the red shifted absorption profiles result from infalling material at near free-fall velocities on the stellar surface. Strength of the H*α* emission line has been widely used to distinguish the actively accreting CTTS from their less active and evolved counterparts i.e Weak-lined T Tauri Stars (WTTS). proposed to classify a YSO as CTTS if W*λ* (H*α*) ≥ 3 Å for K0–K5 stars, W*λ* (H*α*) ≥ 10 Å for K7–M2.5 stars, W*λ* (H*α*) ≥ 20 Å for M3-M5.5 stars, and W*λ* (H*α*) ≥ 40 Å for M6-M7.5 stars. They also proposed that the stars with full width of H*α* emission line at 10 % of the peak intensity (H*α*[10%]) greater than 270 km s− 1 are accreting CTTS independent of spectral type. We identify *9* stars as CTTS and *5* stars as WTTS according to the above mentioned criteria (Table [tabsptclass]). showed that H*α*[10%] ≥ 200 km s− 1 can be used to distinguish the accretors and non accretors as well as to derive an approximate estimate of the accretion rate M*a**c* quantitatively by the following relation, logM*a**c* = -12.89 ( ± 0.3) + 9.7 ( ± 0.7) × 10− 3 H*α*[10%] \nwhere H*α*[10%] is in km s− 1 and M*a**c* is in M⊙yr− 1. In order to estimate the accretion rates for some of the well resolved H*α* line profile we fitted Gaussian profiles to the H*α* emission lines and calculate the equivalent width and FWHM of each line profile. The measured FWHM of each profile has been deconvolved assuming a Gaussian instrumental profile. We calculate the H*α*[10%] for these objects and transformed them into velocity units. We did not correct for the underlying photospheric absorption in H*α* if there is any. Table [tabsptclass] presents the accretion rates M*a**c*, calculated using the above formula along with the measured values of H*α*[10%]. Based on the observed multiepoch spectra we find variable accretion rates in BRC 38 1, MIR-5, MIR-31, MIR-32 and MIR-45.\nOptical Color-Magnitude Diagram\n-------------------------------\nFigure [figvicm] presents the extinction corrected V vs. V-I diagram of the YSOs near SFO38. We have corrected for the extinction of individual objects with known spectral types and used an average value of $A\\_{\\rm v}$=2.5 mag when the spectral type is not known. In order to estimate the mass and age of the YSOs we overplot the isochrones from assuming a distance of 750 pc. We find that most of the YSOs (Class II with(out) H*α*) have ages between 1-5 Myr and masses in the range 0.3-2.2 M⊙. We note that several transitional disk objects are located to the left of the ZAMS line. A closer examination of the position of the transitional disk objects in the MIR color-color diagram (Fig. [figiracmipscol]) reveals that 6 out of 7 transitional disk objects which are situated to the left of ZAMS line have [5.8]-[8] color excess < 0.5 and 4 out of 5 transitional disk objects which are situated to the right of ZAMS line have [5.8]-[8] color excess > 0.5. The transitional disk objects to the left of the ZAMS line are more like Class III objects with smaller values of color excess. We have used an average $A\\_{\\rm v}$=2.5 mag for extinction correction for these objects because the spectral types are not known. There are several factors that could lead the transitional disk objects to the anomalous position in Fig. [figvicm] e.g. extinction correction, uncertainties in distance and PMS evolution model used. An extinction correction with a smaller $A\\_{\\rm v}$ ( ≤ 1 mag) brings all the transitional disk objects closer to, or to the right of the ZAMS line. The distance of IC 1396 is in the range of 615 pc to 900 pc and we adopt a distance of 750 pc for SFO 38 to make our results comparable to most published results. This uncertainty in the distance corresponds to an uncertainty of ± 0.4 mag in the absolute magnitude. Adoption of a distance of 900 pc brings up the transitional disk objects somewhat closer to the ZAMS line but these objects still occupy positions below the ZAMS line. While different theoretical models differ in the details of the PMS evolution, close to the ZAMS line all these models agree with each other and with the empirical ZAMS of the young star clusters. Therefore, the anomalous positions of the transitional disk objects with respect to the ZAMS track in Fig. [figvicm] is primarily due to overcorrection of the extinction. Estimation of mass and age from the color-magnitude diagram (CMD) is strongly dependent on the adopted model. The ages can differ by as much as 10 Myr or more when two different models are used on the same dataset. The main source of error in the V vs. V-I diagram is the estimation of extinction towards the globule. In the last column of Table [tabsptclass] we list the ratios of E(V-I)/E(B-V) which are different from the standard value of 1.6 and indicate that normal extinction law may not hold good for star forming regions due to plenty of patchy obscuring materials. Although the NIR *JHK* bands suffer comparatively less extinction than the optical *BVRI* band but for the YSOs, thermal emission also contributes to *JHK* bands resulting in excess fluxes compared to the main sequence counterparts without disk. For a given model we consider the derived values of the masses and ages as representative values and the age spread of YSOs as more reliable estimate.\nSED of YSOs with MIPS 24 detection\n-----------------------------------\nIn order to interpret the observed SEDs for the YSOs detected at 24 and better characterize these sources we have explored the archive of two-dimensional (2D) axisymmetric radiative transfer models of protostars calculated for a large range of protostellar masses, accretion rates, disk masses and disk orientations created by. This archive also provides a linear regression tool which can select all model SEDs that fit the observed SED better than a specified *χ*2. Each SED is characterized by a set of model parameters, such as stellar mass, temperature, and age, envelope accretion rate, disk mass, and envelope inner radius. We have used this online tool to generate models, which fit the observed SEDs for the 14 YSOs which have been detected at 24 . We restricted the SED fitting tool to explore only distances between 650 and 850 pc. Below we present and briefly discuss the best fit models.\nFigure [figsedfits] shows results of detailed modeling of the observed SED in the mid-infrared (and sub-mm for MIR-50) for all the YSOs which were detected at 24 . A major criticism against the use of these models has been the non-uniqueness of the solutions obtained from the model library. However we find that for all sources the best fit models are distinctively better than the next few models in reproducing all the observed flux densities. Table [tabsedfits] presents the parameters corresponding to the best fit models for the fourteen sources which we fitted with the accretion disk models.\n presented a classification scheme which is essentially analogous to the Class scheme, but refers to the actual evolutionary stage of the object based on its physical properties like disk mass and envelope accretion rates rather than the slope of its near/mid-IR SED. According to the classification scheme, MIR-50 and MIR-54, are stage 0/I objects, while all the others are stage II objects. However, the YSOs MIR-34, 36, 48, 55 and 59 were identified as Class 0/I objects based on the color-color diagrams. Next we consider the fact that except for MIR-34 all the sources have rather high visual extinction ($A\\_{\\rm v}$ > 40) and the MIR color-color diagrams we had used to classify the YSOs did not correct for the reddening. Using standard interstellar extinction laws and the $A\\_{\\rm v}$ derived from the models, we find that when de-reddened, the sources MIR-48, 55 and 59 occupy the region typically occupied by Class II objects. However for MIR-34 and 36 the difference in the inferred evolutionary stage can not be explained only in terms of reddening.\nBased on the results of fitting the observed SEDs with axially symmetric accretion models we find that all the YSOs analyzed here have low to intermediate-mass central stars. Figure [figysomap] shows that consistent with our expectations the sources located within the globule have $A\\_{\\rm v}$ > 20 mag, while the sources lying at the periphery of the bright rim have $A\\_{\\rm v}$ between 3–5 mag. In Sec. [secmir5054] we discuss the two most luminous Class 0/I sources MIR-50 and MIR-54 in detail.\nThe ages estimated for these fourteen YSOs from the SED models and the location of these sources in BRC 38 show that most of the sources closer to the bright rim have ages between 5–8 Myr. The sources within the BRC and lying almost along the axis of the BRC have ages between 0.1–1 Myr, with the protostellar cluster at the position of IRAS 21391+5802 being the youngest with an age of ∼ 0.1 Myr. The age spread estimated for the YSOs (mostly Class II and above) from optical CMD diagram is of the range 1–8 Myr. However, the ages estimated from both methods are higher than the ages obtained from the NIR CMD by.\nThe protostellar cluster at IRAS 21391+5802 [secmir5054]\n--------------------------------------------------------\nInterferometric millimeter continuum images have shown that the most luminous source IRAS 21391+5802 in the IC 1396 region can be resolved into three components, named as BIMA 1, BIMA 2 and BIMA 3. Radio continuum images at 3.6 cm observed with VLA have also identified three peaks VLA 1–3, coincident with the millimeter continuum peaks. Of these three sources VLA 3 is the strongest emitter of radio continuum. Recent sub-arcsecond resolution millimeter observations have identified three components within BIMA 2 itself. The SCUBA images at 450 and 850 do not resolve the components of continuum emission associated with IRAS 21391+5802.\nWe identify MIR-50 and 54 as the mid-infrared counterparts of BIMA 2 and BIMA 3 and do not detect any source associated with BIMA 1. The source MIR-36 is located 38 away from BIMA 1 and hence is most likely not associated with BIMA 1. We have used the continuum fluxes available in literature to construct the observed SEDs for MIR-50 and MIR-54. In order to make a realistic estimate of the flux densities at 450 and 850 from each of BIMA 2 and BIMA 3, we have split up the observed total flux densities at these wavelengths according to the ratio of flux densities of these sources measured at 1.2 mm by. Fitting of the observed flux densities with the axially symmetric accretion-based models shows that MIR-50 (BIMA 2) is a protostellar object with *L* = 197 L⊙ and a mass of 5.97 M⊙. Both mass and luminosity values match reasonably well with the literature. We note however that the model fitted does not reproduce the interferometric flux densities at 1.2 and 3.1 mm well. This is consistent with the conclusions of that the interferometric observations miss a significant amount of emission. The best fit model for MIR-54 corresponds to a mass of 1.5 M⊙ and a luminosity of 33.4 L⊙. had estimated an integrated luminosity of 233 L⊙ including both BIMA 2 and BIMA 3 and here we derive a total luminosity of 230 L⊙. Based on the SED fitting performed in this paper, MIR-54 or BIMA 3 is a low mass Class 0/I protostar and MIR-50 or BIMA 2 is a Class 0/I intermediate-mass star, although have already shown BIMA 2 as having three components. The SED models derive the age of both these sources to be ∼ 105 yr.\nDiscussion: RDI mechanism at work in SFO 38 [secdis]\n====================================================\nBased on the observational evidences in the form of YSOs, an outflow-driving IRAS source and several HH objects, SFO38 is unambiguously an active star forming region. There is however considerable disagreement regarding whether the star formation is triggered or influenced by the nearby OB association. The two most recent studies of SFO38 based on Chandra X-ray and *Spitzer* Mid-IR observations and deep NIR observations arrive at diametrically opposite conclusions. Considering the spatial and temporal distribution of the fewer YSOs selected by using MID-IR color-color diagram ( [3.6]–[4.5] vs. [4.5]–[5.8] ), argued in favor of triggered star formation by Radiation Driven Implosion (RDI) model. However did not find either *(a)* enhanced star formation activity towards the southern rim of SFO38 or *(b)* systematic gradient in the evolutionary stage of the YSOs using the deep NIR imaging. Based on the absence of NIR excess from the YSOs concluded that the sources closer to the O type star appear to be more evolved primarily due to the modification of their environment by the intense UV field. Thus, according to the evolutionary sequence observed by should not necessarily be interpreted as due to triggered star formation. However also concede that NIR-only studies are inadequate to identify all the YSOs in the region.\nWe explore two possible scenarios which could have been the dominant driving mechanism to initiate the formation of stars in SFO 38 *viz.,* coeval star formation throughout the H ii region and triggered star formation by the RDI mechanism. The assumption that the stars near the globule were formed simultaneously with the other stars of the OB association suggests that the distribution of YSOs should be correlated with the distribution pattern of the other massive to low mass stars of the OB association. The observed elongated distribution of YSOs along two different axes in SFO38 is quite different from the distribution of the known members of Cep OB2 association. Moreover the ages of the outflow driving IRAS source and the associated protostellar cluster are significantly less than the dynamical age of 3 Myr of the H ii region created by the massive OB type stars. We thus rule out the possibility of coeval star formation in SFO 38 with the members of Cep OB2 association, although some of the H*α* emitting sources could have been formed prior to the arrival of the shock front. Next, we investigate the other possibility, based on the RDI model, by comparing our observations with the predicted signatures of the model. The RDI model suggests enhancement of density along the axis of the UV irradiated globule and an age sequence of YSOs with the oldest YSO lying closer to the OB association.\nSequential star formation is considered to be one of the key signatures of triggered star formation via RDI. have cautioned against the use of the spatial distribution of YSOs as a proof of triggered star formation, by arguing that intense UV fields close enough to the OB associations tend to destroy the protoplanetary disks around the YSOs and thus make them appear to be older. However we find this argument untenable for the case of SFO38, since the typical radius of influence of O stars is found to be < 1pc, while the mean projected distance of the nearest O and B -type stars close to SFO38 is ∼ 9pc. Thus the spatial distribution of the YSOs discussed in SFO38 signify the temporal evolution of the YSOs and not due to the destruction of protoplanetary atmospheres under the influence of the UV radiation from the OB association.\nThe bright-rimmed cloud SFO37, was found to have an an exemplary bow shaped geometry and show a clean evolutionary sequence of YSOs. SFO38 in contrast has a more complicated geometry. As noted earlier the H*α* emission from SFO38 shows an asymmetry relative to the axis connecting the globule to the nearest O star HD206267. Further, in SFO38 though the YSOs clearly show an age segregation, they are not distributed along a single straight line. It is however possible to identify another direction along which some of the YSOs are most likely aligned. The asymmetry of H*α* emission and the non-linearity of the distribution of YSOs suggest that the evolution of SFO38 is influenced by the nearby B star HD206773 as well. From the H*α* line image (Figure [figysomap]) HD 206773 seems to be ionizing the eastern part of the rim more efficiently than the extent to which HD 206267 has ionized the western part of the rim. Using the 13*C**O* *J* = 1–0 map of SFO 38, also found that the eastern and western wings of SFO 38 differ in velocity by about 1.5 km *s*− 1 relative to the velocity of the dense core of the globule. This suggests that HD206773 plays an active role in the present dynamics of the globule.\nWe have identified a spatio-temporal gradient along the directions of both HD206267 and HD206773. However the spatial gradient is not continuous, there is a distinguishable gap between the two classes of YSOs e.g. the projected distance of Class I/II sources from the approximate center of the distribution of Class I sources is ∼ 0.3 pc and this gap can be interpreted as due to the difference in the arrival times of the shock within the framework of the RDI model. If we consider the age spread to be ∼ a few Myr (comparing the estimated ages from SED fitting and optical CMD) and the spatial separation between the MIR-50 (Class 0/I) and MIR-29 (Class II) to be ∼ 0.6 pc, we obtain the speed of shock propagation to be 0.1–0.3km*s*− 1. The estimated shock speed is consistent with the shock velocities obtained from 3-D numerical simulations. The estimated speed of shock propagation are approximate since we use only projected distances between sources and do not consider the time lapse between the arrival of the shockwave and the triggering of star formation in the globule, which from simulations is ∼ 0.375 Myr.\nThus, our observations and subsequent estimate of the evolutionary stages of the YSOs embedded in SFO38, are consistent with a scenario of triggered star formation via the RDI mechanism powered by two different ionizing sources, HD206267 and HD206773.\nSummary [secconclusion]\n=======================\nWe have presented multiwavelength photometric study of SFO 38 from optical *BVRI* to *Spitzer* IRAC and MIPS observations along with optical spectroscopy of the selected objects. We summarize our work as follows.\n* A total of 40 YSOs (Classes 0/I/II) and 13 YSO candidates are identified based on MIR color indices and we also confirm H*α* emission from 2 YSOs which are not known in the literature. We further identify 4 additional YSOs based on H*α* emission and thus we find 44 YSOs in and around SFO 38.\n* H*α* emission line YSOs are mostly K–M type stars with an age spread of 1-8 Myr and mass range of.3-2.2 M⊙. Some of the YSOs show photometric variation in optical and NIR bands and also variable H*α* emission in the medium resolution spectra. Mass accretion rates estimated form broad H*α* line profiles are of the order of 10− 8 to 10 − 10 M⊙ yr− 1.\n* Mass, luminosity and age for the different components of the protostellar cluster at IRAS 21391+5802 were derived by SED fitting.\n* Continuum subtracted H*α* line image show asymmetric H*α* emission at the bright-rim. Two OB type stars e.g. HD 206267 (O6.5 ) and HD 206773 (B0V) are proposed as the potential ionizing sources.\n* Class II to Class 0/I objects are distributed from the rim to the core part of the globule respectively. It is possible to identify at least two different axes of elongation of the YSO distribution. The spatial gap between the two different classes of YSOs is consistent with the difference in the arrival time of a shockwave propapagating into the globule. The spatio-temporal gradient in the distribution of YSOs along two different axes that are parallel to one of the either ionizing star indicate triggered star formation due to the Radiation Driven Implosion model.\nThis work is based [in part] on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This research has made use of the SIMBAD data base, operated at CDS, Strasbourg, France. This publication makes use of the data products from the 2MASS, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. RC would like to thank Dr. P. S. Parihar and Ramya S. for useful discussions.\n[ht]\n[figiracyso]\n5.5cm \n5.5cm \n5.5cm \n[figspecplot]\n[ht]\n[figsedfits]\nlcclllrrrrrrl MIR-1 & 21:40:24.97 & 58:16:10.5 & 14.48 ± 0.04 & 13.72 ± 0.05 & 13.42 ± 0.05 & 1.67 ± 0.01& 1.10 ± 0.01& 0.96 ± 0.02& 1.03 ± 0.02& …& Trans. & **J**H**K**s** \nMIR-2 & 21:40:25.39 & 58:15:47.3 & 15.35 ± 0.06 & 14.51 ± 0.07 & 14.15 ± 0.07 & 0.94 ± 0.01& 0.63 ± 0.01& 0.68 ± 0.02& 0.98 ± 0.02& …& Trans. & … \nMIR-3 & 21:40:25.84 & 58:15:26.1 & 15.62 ± 0.07 & 14.80 ± 0.07 & 14.62 ± 0.10 & 0.52 ± 0.01& 0.31 ± 0.00& 0.23 ± 0.02& 0.07 ± 0.02& …&III/S & … \nMIR-4 & 21:40:26.84 & 58:14:51.8 & 16.26 ± 0.10 & 15.67 ± 0.15 & 15.06 ± 0.15 & 0.32 ± 0.00& 0.20 ± 0.00& 0.17 ± 0.02& 0.11 ± 0.02& …&III/S & … \nMIR-5 & 21:40:27.29 & 58:14:21.14 & 14.30 ± 0.04 & 13.30 ± 0.04 & 12.88 ± 0.04 & 6.22 ± 0.05& …& …& …& …& II & H*α*, X-ray \nMIR-6 & 21:40:27.38 & 58:16:21.3 & …& …& …& 0.07 ± 0.00& 0.03 ± 0.00& 0.51 ± 0.02& 2.00 ± 0.03& …&I/II \nMIR-7 & 21:40:27.97 & 58:15:14.1 & 14.51 ± 0.04 & 13.41 ± 0.04 & 12.94 ± 0.04 & 3.89 ± 0.02& 3.52 ± 0.02& 3.44 ± 0.03& 4.33 ± 0.03& …& II & H*α*, *J* \nMIR-8 & 21:40:28.35 & 58:16:42.3 & 11.59 ± 0.03 & 10.78 ± 0.03 & 10.43 ± 0.02 & 23.62 ± 0.07& 14.18 ± 0.04& 10.94 ± 0.06& 6.63 ± 0.04& …& III/S \nMIR-9 & 21:40:30.63 & 58:15:00.0 & 15.31 ± 0.05 & 14.90 ± 0.07 & 14.70 ± 0.11 & 0.49 ± 0.01& 0.32 ± 0.01& 0.28 ± 0.01& 0.22 ± 0.02& …& Trans. \nMIR-10 & 21:40:31.00 & 58:15:09.2 & 13.44 ± 0.03 & 12.97 ± 0.04 & 12.79 ± 0.04 & 2.61 ± 0.02& 1.63 ± 0.01& 1.29 ± 0.02& …& …& III/S \nMIR-11 & 21:40:31.56 & 58:17:55.1 & 14.03 ± 0.03 & 12.89 ± 0.03 & 12.39 ± 0.03 & 6.93 ± 0.03& 6.93 ± 0.03& 9.15 ± 0.06& 15.17 ± 0.12& 20.46 ± 0.08 & II & H*α*, X-ray, **J**H**K**s** \nMIR-12 & 21:40:31.97 & 58:14:56.0 & 16.72 ± 0.16 & 15.78 ± 0.15 & 15.41 ± 0.21 & 0.41 ± 0.00& 0.28 ± 0.00& 0.26 ± 0.01& …& …&III/S \nMIR-13 & 21:40:32.23 & 58:16:53.6 & 14.13 ± 0.03 & 12.44 ± 0.04 & 11.71 ± 0.03 & 10.82 ± 0.04& 6.39 ± 0.02& 6.54 ± 0.05& 7.36 ± 0.05& …& Trans. \nMIR-14 & 21:40:32.46 & 58:13:47.2 & 13.40 ± 0.03 & 12.83 ± 0.03 & 12.63 ± 0.03 & 2.78 ± 0.03& …& 1.30 ± 0.05& …& …& III/S \nMIR-15 & 21:40:32.71 & 58:17:23.5 & 12.19 ± 0.03 & 11.55 ± 0.03 & 11.36 ± 0.03 & 9.21 ± 0.04& 5.50 ± 0.02& 4.04 ± 0.04& 1.71 ± 0.04& …& III/S \nMIR-16 & 21:40:32.84 & 58:15:47.8 & 17.42 ± 0.01& 16.61 ± 0.01 & 16.09 ± 0.01 & 1.42 ± 0.01& 0.73 ± 0.01& 9.50 ± 0.05& 32.10 ± 0.11& …& I/II \nMIR-17 & 21:40:33.64 & 58:14:44.6 & 16.54 ± 0.14 & 15.81 ± 0.18 & 15.07 ± 0.00 & 0.21 ± 0.00& 0.13 ± 0.00& 0.11 ± 0.01& 0.16 ± 0.02& …& Galaxy \nMIR-18 & 21:40:33.69 & 58:14:59.5 & 13.06 ± 0.03 & 12.09 ± 0.03 & 11.75 ± 0.02 & 8.13 ± 0.03& 4.82 ± 0.02& 4.72 ± 0.04& 5.60 ± 0.03& …& Trans. \nMIR-19 & 21:40:34.05 & 58:18:08.1 & 15.70 ± 0.06 & 13.40 ± 0.04 & 12.30 ± 0.02 & 7.41 ± 0.03& 5.16 ± 0.02& 4.95 ± 0.04& 4.03 ± 0.04& …& Trans. \nMIR-20 & 21:40:34.30 & 58:18:23.6 & …& …& …& 0.31 ± 0.00& 0.84 ± 0.01& 1.03 ± 0.02& 1.24 ± 0.04& …& H2 \nMIR-21 & 21:40:34.59 & 58:16:11.4 & …& …& …& 0.08 ± 0.00& 0.05 ± 0.00& 1.33 ± 0.03& …& …& III/S \nMIR-22 & 21:40:34.76 & 58:15:20.9 & 17.70 ± 0.01 & 16.68 ± 0.01 & 16.04 ± 0.01 & 0.25 ± 0.01& 0.34 ± 0.01& …& …& …& III/S \nMIR-23 & 21:40:34.79 & 58:16:10.8 & …& …& …& …& …& …& 4.47 ± 0.05& …& … \nMIR-24 & 21:40:35.02 & 58:18:22.2 & 16.48 ± …& 15.72 ± …& 14.90 ± 0.21 & 0.62 ± 0.01& 2.68 ± 0.01& 1.84 ± 0.03& 1.42 ± 0.03& …& H2 \nMIR-25 & 21:40:35.11 & 58:15:30.9 & 18.21 ± 0.02 & 16.85 ± 0.01 & 16.16 ± 0.02 & 0.92 ± 0.01& 0.48 ± 0.01& 5.32 ± 0.04& 18.26 ± 0.08& …&I/II \nMIR-26 & 21:40:35.20 & 58:14:05.0 & 13.11 ± 0.03 & 12.61 ± 0.03 & 12.45 ± 0.03 & 3.55 ± 0.02& 2.40 ± 0.01& 1.59 ± 0.02& 0.84 ± 0.02& …& III/S \nMIR-27 & 21:40:35.68 & 58:18:21.3 & 16.55 ± …& 15.73 ± …& 14.79 ± 0.17 & 0.77 ± 0.01& 3.61 ± 0.02& 2.20 ± 0.03& 1.91 ± 0.03& …& H2 \nMIR-28 & 21:40:36.46 & 58:15:23.6 & 15.97 ± 0.10 & 14.73 ± 0.08 & 14.04 ± 0.07 & 2.27 ± 0.01& 1.43 ± 0.01& 6.68 ± 0.04& 19.47 ± 0.08& …& I/II \nMIR-29 & 21:40:36.52 & 58:13:45.8 & 13.51 ± 0.02 & 12.58 ± 0.03 & 12.24 ± 0.03 & 4.69 ± 0.02& 3.14 ± 0.01& 2.17 ± 0.03& 2.64 ± 0.03& 44.29 ± 0.09 & Trans. & H*α*, X-ray \nMIR-30 & 21:40:36.65 & 58:18:23.0 & 17.12 ± …& 15.18 ± 0.10 & 13.93 ± 0.07 & 1.84 ± 0.01& 1.55 ± 0.01& 1.42 ± 0.03& 1.21 ± 0.03& …& II \nMIR-31 & 21:40:36.88 & 58:14:37.8 & 11.90 ± 0.02 & 10.89 ± 0.03 & 10.23 ± 0.02 & 54.30 ± 0.14& 48.79 ± 0.10& 38.25 ± 0.12& 34.45 ± 0.10& 39.36 ± 0.07& II & H*α*, X-ray \nMIR-32 & 21:40:37.02 & 58:15:02.9 & 14.27 ± 0.03 & 13.28 ± 0.04 & 12.82 ± 0.03 & 4.01 ± 0.02& 3.07 ± 0.01& 2.86 ± 0.03& 3.87 ± 0.04& …& II & H*α*, **K**s** \nMIR-33 & 21:40:37.16 & 58:15:46.6 & 17.01 ± …& 15.80 ± 0.16 & 14.55 ± 0.10 & 1.33 ± 0.01& 1.28 ± 0.01& 0.71 ± 0.03& …& …& III/S \nMIR-34 & 21:40:37.54 & 58:15:39.3 & 18.07 ± 0.02 & 16.98 ± 0.01 & 16.24 ± 0.02 & 0.49 ± 0.01& 0.77 ± 0.01& 1.05 ± 0.03& 1.63 ± 0.04& 17.15 ± 0.06 & 0/I \nMIR-35 & 21:40:39.05 & 58:16:03.3 & …& …& …& 0.41 ± 0.01& 1.04 ± 0.01& 2.19 ± 0.03& …& …& H2 \nMIR-36 & 21:40:39.60 & 58:16:09.0 & …& > 17.6 & 15.14 ± 0.16 & 10.47 ± 0.03& 36.11 ± 0.08& 59.09 ± 0.17& 82.89 ± 0.22& 239.10 ± 0.10 & 0/I & X-ray \nMIR-37 & 21:40:39.80 & 58:18:24.9 & …& …& …& 0.53 ± 0.01& 0.76 ± 0.01& 1.15 ± 0.02& 1.39 ± 0.03& …& 0/I \nMIR-38 & 21:40:39.84 & 58:18:34.8 & 18.29 ± …& 15.30 ± 0.12 & 13.36 ± 0.04 & 6.88 ± 0.03& 8.19 ± 0.03& 9.95 ± 0.06& 13.49 ± 0.06& 30.69 ± 0.07 & II& X-ray \nMIR-39 & 21:40:40.15 & 58:16:18.2 & …& …& …& 1.36 ± 0.01& 2.65 ± 0.01& 3.24 ± 0.04& 4.25 ± 0.05& …& 0/I & \nMIR-40 & 21:40:40.43 & 58:18:26.1 & 18.28 ± …& 15.64 ± 0.15 & 13.85 ± 0.06 & 2.35 ± 0.01& 2.05 ± 0.01& 1.64 ± 0.03& 0.76 ± 0.02& …& III/S \nMIR-41 & 21:40:40.96 & 58:16:12.0 & 17.55 ± …& 15.89 ± …& 13.59 ± 0.06 & 13.04 ± 0.04& 23.41 ± 0.06& 29.43 ± 0.10& 42.25 ± 0.15& …& 0/I \nMIR-42 & 21:40:40.98 & 58:16:52.4 & 17.93 ± …& 15.34 ± 0.14 & 13.93 ± 0.06 & 1.41 ± 0.01& 1.04 ± 0.01& 0.83 ± 0.02& 0.37 ± 0.03& …& III/S \nMIR-43 & 21:40:41.07 & 58:13:58.9 & 12.96 ± 0.02 & 12.08 ± 0.03 & 11.77 ± 0.02 & 6.80 ± 0.02& 4.58 ± 0.02& 3.19 ± 0.03& 1.84 ± 0.03& …& III/S & H*α*, X-ray \nMIR-44 & 21:40:41.10 & 58:17:54.1 & 14.20 ± 0.04 & 13.46 ± 0.04 & 13.21 ± 0.03 & 1.89 ± 0.01& 1.14 ± 0.01& 0.73 ± 0.02& 0.37 ± 0.02& …& III/S \nMIR-45 & 21:40:41.14 & 58:15:11.2 & 12.97 ± 0.03 & 11.61 ± 0.04 & 10.68 ± 0.02 & 66.38 ± 0.17& 65.98 ± 0.14& 81.84 ± 0.21& 124.60 ± 0.31& 194.40 ± 0.09& II & H*α*, X-ray \nMIR-46 & 21:40:41.27 & 58:17:14.2 & …& …& …& 0.06 ± 0.00& 0.03 ± 0.00& 0.65 ± 0.02& 2.61 ± 0.03& …& I/II \nMIR-47 & 21:40:41.40 & 58:14:17.5 & 15.48 ± …& 15.50 ± 0.12 & 14.63 ± 0.00 & 0.34 ± 0.00& 0.21 ± 0.00& 0.16 ± 0.01& 0.10 ± 0.02& …& III/S \nMIR-48 & 21:40:41.43 & 58:16:37.8 & 18.30 ± …& 15.60 ± 0.15 & 12.79 ± 0.03 & 51.81 ± 0.14& 109.40 ± 0.20& 158.50 ± 0.34& 199.60 ± 0.44& 1362.00 ± 0.26 & 0/I \nMIR-49 & 21:40:41.54 & 58:14:25.5 & 13.65 ± 0.03 & 12.62 ± 0.03 & 12.17 ± 0.03 & 8.70 ± 0.03& 7.38 ± 0.03& 6.65 ± 0.04& 6.42 ± 0.04& …& II & H*α*, X-ray \nMIR-50 & 21:40:41.71 & 58:16:12.6 & …& …& …& 10.84 ± 0.04& 78.01 ± 0.16& 128.00 ± 0.29& 137.00 ± 0.34& 1871.00 ± 0.31 & 0/I & X-ray \nMIR-51 & 21:40:41.89 & 58:15:23.0 & 15.68 ± 0.09 & 14.30 ± 0.06 & 13.65 ± 0.05 & 2.88 ± 0.02& 2.30 ± 0.01& 3.75 ± 0.04& 8.33 ± 0.06& …& I/II & X-ray, **J**H**K**s** \nMIR-52 & 21:40:41.96 & 58:13:49.1 & 15.50 ± 0.06 & 15.00 ± 0.09 & 14.98 ± 0.14 & 0.40 ± 0.00& 0.27 ± 0.00& 0.19 ± 0.01& 0.10 ± 0.02& …& III/S \nMIR-53 & 21:40:42.35 & 58:18:40.1 & …& …& …& 0.48 ± 0.00& 2.16 ± 0.01& 1.99 ± 0.03& 1.26 ± 0.02& …& H2 \nMIR-54 & 21:40:42.77 & 58:16:01.1 & …& …& …& 20.16 ± 0.06& 75.86 ± 0.15& 113.50 ± 0.27& 117.60 ± 0.31& 2484.00 ± 0.38 & I & X-ray \nMIR-55 & 21:40:43.32 & 58:17:16.5 & …& > 17.6 & 15.31 ± 0.11 & 1.65 ± 0.01& 2.23 ± 0.01& 2.72 ± 0.03& 4.31 ± 0.03& 145.40 ± 0.08 & 0/I \nMIR-56 & 21:40:43.34 & 58:18:42.6 & 16.32 ± 0.13 & 14.42 ± 0.05 & 13.55 ± 0.04 & 1.93 ± 0.01& 1.39 ± 0.01& 0.91 ± 0.02& 0.26 ± 0.03& …& III/S \nMIR-57 & 21:40:43.62 & 58:16:18.8 & 17.89 ± …& 16.09 ± …& 13.51 ± 0.07 & 33.21 ± 0.09& 68.55 ± 0.14& 82.72 ± 0.21& 91.38 ± 0.24& …& 0/I & X-ray \nMIR-58 & 21:40:43.65 & 58:14:18.4 & 15.55 ± 0.08 & 15.05 ± 0.10 & 14.89 ± 0.14 & 0.41 ± 0.00& 0.28 ± 0.00& 0.19 ± 0.01& 0.14 ± 0.02& …& Trans. \nMIR-59 & 21:40:44.23 & 58:16:52.2 & …& …& …& 1.36 ± 0.01& 4.43 ± 0.02& 9.13 ± 0.05& 10.52 ± 0.06& 100.90 ± 0.07 & 0/I \nMIR-60 & 21:40:44.26 & 58:14:12.4 & 15.57 ± 0.06 & 15.00 ± 0.09 & 14.69 ± 0.12 & 0.45 ± 0.00& 0.30 ± 0.00& 0.22 ± 0.01& 0.09 ± 0.02& …& III/S \nMIR-61 & 21:40:44.31 & 58:15:13.2 & 16.05 ± 0.10 & 14.59 ± 0.07 & 13.60 ± 0.05 & 3.78 ± 0.02& 4.10 ± 0.02& 7.89 ± 0.05& 15.74 ± 0.08& …& I/II & X-ray, **J**H**K**s** \nMIR-62 & 21:40:44.64 & 58:17:04.9 & 17.83 ± …& 15.75 ± 0.15 & 14.62 ± 0.10 & 2.16 ± 0.01& 2.38 ± 0.01& 2.51 ± 0.03& 3.06 ± 0.03& …& II \nMIR-63 & 21:40:44.82 & 58:16:05.0 & 15.86 ± …& 14.28 ± 0.06 & 12.89 ± 0.05 & 7.08 ± 0.03& 7.17 ± 0.03& 6.83 ± 0.05& 7.25 ± 0.05& …& II & X-ray, **K**s** \nMIR-64 & 21:40:44.83 & 58:15:03.3 & 14.62 ± 0.04 & 13.35 ± 0.04 & 12.66 ± 0.03 & 5.64 ± 0.02& 5.23 ± 0.02& 7.28 ± 0.05& 12.94 ± 0.06& …& I/II & H*α*, X-ray \nMIR-65 & 21:40:44.89 & 58:17:47.7 & …& …& …& 0.42 ± 0.00& 2.56 ± 0.01& 2.01 ± 0.02& 2.29 ± 0.03& …& H2 \nMIR-66 & 21:40:45.11 & 58:15:24.0 & 18.46 ± 0.02 & 16.74 ± 0.01 & 15.71 ± 0.01 & 0.59 ± 0.01& 0.44 ± 0.01& 1.73 ± 0.03& 5.26 ± 0.05& …& I/II & \nMIR-67 & 21:40:45.16 & 58:15:59.6 & …& …& …& 5.73 ± 0.02& 9.26 ± 0.03& 11.46 ± 0.06& 11.78 ± 0.07& …& I& X-ray \nMIR-68 & 21:40:45.24 & 58:16:41.3 & 17.38 ± 0.01 & 16.29 ± 0.01 & 15.53 ± 0.01 & 0.43 ± 0.01& 0.42 ± 0.01& 0.41 ± 0.02& 0.68 ± 0.02& …& Galaxy \nMIR-69 & 21:40:45.29 & 58:16:53.3 & …& …& …& 0.53 ± 0.01& 0.61 ± 0.01& 0.60 ± 0.02& …& …& III/S \nMIR-70 & 21:40:45.41 & 58:16:45.5 & 20.38 ± 0.15 & 17.48 ± 0.02 & 15.96 ± 0.01 & 0.33 ± 0.00& 0.27 ± 0.01& 0.48 ± 0.02& 1.14 ± 0.02& …& Galaxy \nMIR-71 & 21:40:45.48 & 58:15:11.5 & 14.65 ± 0.05 & 13.71 ± 0.06 & 13.11 ± 0.04 & 3.17 ± 0.02& 2.61 ± 0.01& 3.41 ± 0.04& 5.73 ± 0.05& …& I/II & X-ray, **K**s** \nMIR-72 & 21:40:45.50 & 58:16:02.6 & 15.68 ± 0.09 & 13.73 ± 0.06 & 12.85 ± 0.04 & 4.13 ± 0.02& 3.15 ± 0.02& 2.74 ± 0.03& 2.88 ± 0.03& …& II & X-ray \nMIR-73 & 21:40:45.76 & 58:15:48.9 & …& …& …& 2.71 ± 0.02& 7.27 ± 0.03& 10.79 ± 0.06& 11.29 ± 0.07& …& I & X-ray \nMIR-74 & 21:40:45.82 & 58:17:43.0 & …& …& …& 0.36 ± 0.00& 0.92 ± 0.01& 1.16 ± 0.02& 1.38 ± 0.02& …& H2 \nMIR-75 & 21:40:46.22 & 58:17:40.2 & …& …& …& 0.34 ± 0.00& 0.71 ± 0.01& 1.20 ± 0.02& 1.53 ± 0.02& …& H2 \nMIR-76 & 21:40:46.46 & 58:15:23.0 & 12.81 ± 0.03 & 11.95 ± 0.03 & 11.65 ± 0.02 & 8.66 ± 0.03& 5.35 ± 0.02& 5.25 ± 0.04& 7.74 ± 0.07& …& Trans. & X-ray, H*α* \nMIR-77 & 21:40:46.86 & 58:15:33.4 & 15.30 ± 0.06 & 13.55 ± 0.05 & 12.63 ± 0.04 & 5.80 ± 0.02& 4.05 ± 0.02& 5.16 ± 0.04& 10.41 ± 0.08& …& Galaxy & X-ray \nMIR-78 & 21:40:47.35 & 58:16:37.3 & 17.08 ± …& 16.14 ± 0.24 & 14.63 ± 0.10 & 1.00 ± 0.01& 0.75 ± 0.01& 0.59 ± 0.02& 0.36 ± 0.02& …& III/S \nMIR-79 & 21:40:47.43 & 58:17:39.1 & 11.53 ± 0.03 & 9.98 ± 0.03 & 9.37 ± 0.02 & 72.47 ± 0.17& 46.36 ± 0.10& 34.42 ± 0.11& 20.21 ± 0.07& …& III/S \nMIR-80 & 21:40:48.00 & 58:15:37.6 & 13.89 ± 0.03 & 12.95 ± 0.03 & 12.67 ± 0.03 & 4.72 ± 0.02& 3.39 ± 0.02& 5.77 ± 0.04& 11.90 ± 0.06& …& I/II & H*α*, X-ray \nMIR-81 & 21:40:48.52 & 58:16:37.1 & 16.18 ± 0.09 & 14.95 ± 0.08 & 14.46 ± 0.08 & 0.71 ± 0.01& 0.46 ± 0.01& 0.38 ± 0.02& 0.62 ± 0.02& …& Galaxy& **K**s** \nMIR-82 & 21:40:48.58 & 58:16:00.9 & 15.88 ± 0.09 & 13.45 ± 0.04 & 11.88 ± 0.02 & 30.35 ± 0.08& 37.01 ± 0.08& 43.12 ± 0.13& 61.94 ± 0.17& 167.10 ± 0.09 & II& *JH* \nMIR-83 & 21:40:48.84 & 58:16:22.8 & 16.02 ± 0.12 & 15.25 ± 0.13 & 14.46 ± 0.08 & 0.61 ± 0.01& 0.43 ± 0.01& 0.25 ± 0.02& …& …& III/S \nMIR-84 & 21:40:48.87 & 58:15:10.9 & 15.64 ± 0.06 & 15.04 ± 0.08 & 14.56 ± 0.09 & 0.77 ± 0.01& 0.62 ± 0.01& 0.75 ± 0.02& 1.34 ± 0.03& …& I/II & H*α*, **J**K**s** \nMIR-85 & 21:40:49.05 & 58:17:19.4 & 15.79 ± 0.08 & 13.83 ± 0.05 & 12.92 ± 0.02 & 3.32 ± 0.02& 2.19 ± 0.01& 1.75 ± 0.02& 1.18 ± 0.02& …& Trans. & *J* \nMIR-86 & 21:40:49.06 & 58:17:09.3 & 14.14 ± 0.03 & 12.86 ± 0.04 & 12.13 ± 0.02 & 13.88 ± 0.05& 14.57 ± 0.04& 15.54 ± 0.07& 17.83 ± 0.07& 17.99 ± 0.05 & II & H*α*, X-ray, **K**s** \nMIR-87 & 21:40:49.29 & 58:14:11.3 & 13.52 ± 0.03 & 12.66 ± 0.03 & 12.35 ± 0.02 & 3.87 ± 0.02& 2.44 ± 0.01& 1.66 ± 0.02& 0.99 ± 0.02& …& III/S \nMIR-88 & 21:40:49.67 & 58:17:37.3 & 17.13 ± 0.23 & 14.99 ± 0.09 & 14.23 ± 0.07 & 1.07 ± 0.01& 0.73 ± 0.01& 0.60 ± 0.02& 0.36 ± 0.02& …& III/S \nMIR-89 & 21:40:51.07 & 58:16:24.3 & 17.80 ± 0.01 & 16.70 ± 0.01 & 16.09 ± 0.01 & 0.11 ± 0.00& 0.08 ± 0.00& 0.11 ± 0.02& 0.23 ± 0.02& …& Galaxy \nMIR-90 & 21:40:51.23 & 58:16:38.9 & 14.23 ± 0.03 & 13.76 ± 0.04 & 13.61 ± 0.04 & 1.26 ± 0.01& 0.78 ± 0.01& 0.57 ± 0.02& 0.53 ± 0.02& …& Trans. &*HK* \nMIR-91 & 21:40:51.81 & 58:16:29.1 & 15.70 ± 0.06 & 14.45 ± 0.06 & 13.93 ± 0.07 & 1.19 ± 0.01& 0.74 ± 0.01& 0.60 ± 0.02& 0.54 ± 0.02& …& Trans. &*HK* \nMIR-92 & 21:40:51.90 & 58:17:01.6 & …& …& …& 0.36 ± 0.00& 0.33 ± 0.00& 0.20 ± 0.02& …& …& III/S \nMIR-93 & 21:40:51.97 & 58:16:23.6 & 19.21 ± 0.06 & 17.47 ± 0.01 & 16.39 ± 0.01 & 0.16 ± 0.00& 0.10 ± 0.00& 0.23 ± 0.02& 0.65 ± 0.02& …&Galaxy \nMIR-94 & 21:40:52.26 & 58:17:46.0 & 15.68 ± 0.05 & 14.56 ± 0.05 & 14.06 ± 0.06 & 0.94 ± 0.01& 0.65 ± 0.01& 0.59 ± 0.02& 0.48 ± 0.03& …& Trans. \nMIR-95 & 21:40:52.98 & 58:18:01.1 & 15.29 ± 0.05 & 14.59 ± 0.06 & 14.22 ± 0.07 & 0.84 ± 0.01& 0.53 ± 0.01& 0.54 ± 0.03& 0.34 ± 0.03& …&III/S \nMIR-96 & 21:40:53.12 & 58:14:57.4 & 14.77 ± 0.06 & 14.25 ± 0.07 & 13.95 ± 0.07 & 0.84 ± 0.01& 0.54 ± 0.01& 0.38 ± 0.02& 0.37 ± 0.03& …& Trans. \nMIR-97 & 21:40:53.42 & 58:14:51.9 & 15.50 ± 0.06 & 14.88 ± 0.09 & 14.74 ± 0.11 & 0.44 ± 0.00& 0.31 ± 0.01& 0.23 ± 0.01& 0.20 ± 0.02& …& Trans. \nMIR-98 & 21:40:53.68 & 58:16:41.2 & …& …& …& 0.25 ± 0.00& 0.71 ± 0.01& 0.79 ± 0.02& 0.83 ± 0.02& …& H2 \nMIR-99 & 21:40:55.68 & 58:15:50.8 & …& …& …& 0.36 ± 0.00& 0.14 ± 0.00& 2.69 ± 0.03& 9.65 ± 0.05& …& I/II \nMIR-100 & 21:40:55.87 & 58:14:16.3 & 14.18 ± 0.03 & 13.61 ± 0.04 & 13.39 ± 0.04 & …& 0.87 ± 0.02& …& 0.45 ± 0.03& …& III/S \nMIR-101 & 21:40:55.88 & 58:15:34.8 & …& …& …& 0.25 ± 0.00& 0.32 ± 0.00& 0.41 ± 0.02& 0.71 ± 0.02& …& Galaxy \nMIR-102 & 21:40:55.93 & 58:17:07.8 & 14.45 ± 0.03 & 13.19 ± 0.04 & 12.76 ± 0.03 & 3.21 ± 0.02& 2.01 ± 0.01& 1.39 ± 0.02& 0.99 ± 0.02& …& Trans. \nMIR-103 & 21:40:57.04 & 58:16:56.8 & 16.61 ± 0.13 & 15.92 ± 0.21 & 16.22 ± 0.00 & 0.42 ± 0.01& 0.34 ± 0.00& 0.40 ± 0.02& 1.19 ± 0.03& …& Galaxy \nMIR-104 & 21:40:57.06 & 58:16:29.0 & 14.73 ± 0.03 & 13.00 ± 0.03 & 12.24 ± 0.02 & 6.10 ± 0.02& 4.11 ± 0.02& 2.87 ± 0.03& 1.96 ± 0.03& …& Trans. \nMIR-105 & 21:40:57.33 & 58:16:41.7 & 16.64 ± …& 15.41 ± …& 14.55 ± 0.13 & 0.79 ± 0.01& 1.18 ± 0.01& 2.90 ± 0.03& 6.34 ± 0.04& …& H2 \nMIR-106 & 21:40:57.50 & 58:14:43.6 & 15.00 ± 0.04 & 14.66 ± 0.07 & 14.34 ± 0.09 & …& 0.43 ± 0.01& 0.25 ± 0.03& 0.15 ± 0.02& …& III/S \nMIR-107 & 21:40:58.03 & 58:15:12.1 & 15.06 ± 0.05 & 14.50 ± 0.07 & 14.12 ± 0.08 & 0.70 ± 0.01& 0.46 ± 0.01& 0.30 ± 0.02& 0.21 ± 0.02& …& Trans. \nMIR-108 & 21:40:58.45 & 58:16:20.5 & 13.11 ± 0.03 & 12.44 ± 0.03 & 12.02 ± 0.03 & 5.93 ± 0.03& 3.77 ± 0.02& 3.13 ± 0.03& 2.52 ± 0.03& …& Trans. \nMIR-109 & 21:40:58.53 & 58:15:00.3 & 12.15 ± 0.03 & 11.26 ± 0.03 & 10.96 ± 0.02 & …& 8.34 ± 0.03& 5.73 ± 0.05& 3.46 ± 0.03& …& III/S \nMIR-110 & 21:40:59.71 & 58:17:33.3 & 12.69 ± 0.02 & 11.25 ± 0.03 & 10.59 ± 0.02 & 24.86 ± 0.17& …& 10.77 ± 0.13& …& …& III/S \nlccccccc NIR-1 & 21:40:34.36 & 58:16:19.2 & 20.88 ± …& 18.12 ± 0.03 & 16.68 ± 0.02 & 15.73 ± 0.03 & 15.38 ± 0.03 \nNIR-2 & 21:40:36.46 & 58:16:28.7 & 20.84 ± …& 19.31 ± 0.10 & 17.14 ± 0.03 & 15.23 ± 0.02 & 14.75 ± 0.02 \nNIR-3 & 21:40:37.57 & 58:16:01.2 & 20.80 ± …& 19.98 ± …& 17.35 ± …& 15.22 ± 0.03 & 14.29 ± 0.02 \nNIR-4 & 21:40:38.17 & 58:15:32.7 & 20.87 ± …& 18.32 ± 0.03 & 16.72 ± 0.02 & 14.84 ± 0.02 & 14.59 ± 0.02 \nNIR-5 & 21:40:39.53 & 58:15:46.9 & …& …& …& 14.22 ± 0.01 & 13.72 ± 0.01 \nNIR-6 & 21:40:39.65 & 58:15:06.6 & 16.59 ± 0.01 & 15.88 ± 0.01 & 15.52 ± 0.01 & 14.07 ± 0.01 & 13.82 ± 0.01 \nNIR-7 & 21:40:42.81 & 58:15:45.3 & 16.29 ± 0.00 & 15.38 ± 0.00 & 14.77 ± 0.00 & 13.91 ± 0.01 & 13.27 ± 0.01 \nNIR-8 & 21:40:44.15 & 58:15:49.6 & …& …& …& 15.78 ± 0.04 & 13.83 ± 0.01 \nNIR-9 & 21:40:44.57 & 58:15:51.6 & 15.97 ± 0.00 & 15.23 ± 0.00 & 14.71 ± 0.00 & 14.28 ± 0.02 & 13.68 ± 0.01 \nNIR-10 & 21:40:45.85 & 58:15:40.6 & 18.70 ± 0.03 & 17.09 ± 0.01 & 16.01 ± 0.01 & 14.68 ± 0.02 & 14.18 ± 0.02 \nNIR-11 & 21:40:48.28 & 58:16:18.8 & 18.17 ± 0.02 & 17.16 ± 0.01 & 16.40 ± 0.02 & 15.63 ± 0.03 & 15.24 ± 0.03 \nNIR-12 & 21:40:50.55 & 58:16:01.8 & 20.84 ± …& 19.65 ± 0.10 & 17.55 ± 0.03 & 15.65 ± 0.03 & 15.29 ± 0.03 \nNIR-13 & 21:40:51.14 & 58:17:01.5 & 20.86 ± …& 18.74 ± 0.05 & 17.23 ± 0.04 & 15.99 ± 0.03 & 15.68 ± 0.04 \nlccccc MIR-1 & 19.82 ± 0.010 & 18.24 ± 0.005 &17.18 ± 0.004 &16.00 ± 0.003 & \nMIR-2 & 20.48 ± 0.015 & 18.88 ± 0.006 &17.88 ± 0.008 &16.84 ± 0.005 & \nMIR-3 & 21.58 ± 0.021 & 19.58 ± 0.007 &18.37 ± 0.006 &17.14 ± 0.004 & \nMIR-4 & 0.00 ± 0.000 & 20.38 ± 0.016 &19.15 ± 0.007 &17.73 ± 0.006 & \nMIR-5 & 21.11 ± 0.015 & 19.32 ± 0.007 &17.77 ± 0.004 &16.37 ± 0.002 & *BVI* \nMIR-7 & 21.59 ± 0.021 & 19.39 ± 0.006 &17.90 ± 0.004 &16.38 ± 0.002 & \nMIR-8 & 17.74 ± 0.001 & 15.74 ± 0.001 &14.55 ± 0.001 &13.35 ± 0.001 & \nMIR-9 & 19.95 ± 0.017 & 18.48 ± 0.004 &17.60 ± 0.005 &16.62 ± 0.004 & \nMIR-10 & 17.28 ± 0.001 & 16.07 ± 0.001 &15.29 ± 0.001 &14.49 ± 0.001 & \nMIR-11 & 0.00 ± 0.000 & 19.52 ± 0.011 &17.94 ± 0.007 &16.19 ± 0.005 & \nMIR-14 & 17.67 ± 0.001 & 16.24 ± 0.001 &15.38 ± 0.001 &14.53 ± 0.001 & \nMIR-15 & 16.89 ± 0.001 & 15.26 ± 0.001 &14.33 ± 0.001 &13.44 ± 0.001 & \nMIR-18 & 19.62 ± 0.006 & 17.58 ± 0.003 &16.30 ± 0.002 &14.98 ± 0.009 & \nMIR-26 & 17.34 ± 0.001 & 16.03 ± 0.001 &15.20 ± 0.001 &14.30 ± 0.001 & \nMIR-29 & 20.58 ± 0.011 & 18.43 ± 0.005 &16.94 ± 0.002 &15.42 ± 0.002 & \nMIR-31 & 17.43 ± 0.002 & 15.71 ± 0.001 &14.60 ± 0.001 &13.52 ± 0.001 & \nMIR-32 & 21.15 ± 0.015 & 19.29 ± 0.006 &18.01 ± 0.007 &16.25 ± 0.002 & *BVI* \nMIR-43 & 19.41 ± 0.004 & 17.49 ± 0.002 &16.21 ± 0.001 &14.73 ± 0.001 & *BVI* \nMIR-44 & 19.36 ± 0.006 & 17.65 ± 0.003 &16.57 ± 0.001 &15.51 ± 0.002 & \nMIR-45 & 21.41 ± 0.018 & 19.45 ± 0.007 &18.10 ± 0.012 &16.33 ± 0.002 & *BVI* \nMIR-47 & 0.00 ± 0.000 & 19.86 ± 0.009 &18.76 ± 0.005 &17.61 ± 0.005 & \nMIR-49 & 20.91 ± 0.012 & 18.81 ± 0.004 &17.04 ± 0.003 &15.69 ± 0.001 & *BVI* \nMIR-52 & 19.54 ± 0.008 & 18.32 ± 0.005 &17.52 ± 0.003 &16.64 ± 0.005 & \nMIR-58 & 20.13 ± 0.007 & 18.75 ± 0.003 &17.86 ± 0.005 &16.91 ± 0.003 & *B* \nMIR-60 & 20.79 ± 0.011 & 19.13 ± 0.005 &18.07 ± 0.005 &17.06 ± 0.003 & \nMIR-64 & 0.00 ± 0.000 & 20.49 ± 0.014 &18.77 ± 0.016 &17.09 ± 0.006 & \nMIR-76 & 19.08 ± 0.006 & 17.16 ± 0.002 &15.90 ± 0.002 &14.55 ± 0.001 & \nMIR-79 & 21.41 ± 0.024 & 18.56 ± 0.004 &16.52 ± 0.002 &14.55 ± 0.001 & \nMIR-80 & 21.69 ± 0.023 & 19.81 ± 0.008 &17.90 ± 0.006 &16.00 ± 0.004 &*V* \nMIR-87 & 19.72 ± 0.005 & 17.72 ± 0.002 &16.50 ± 0.002 &15.28 ± 0.002 & \nMIR-90 & 18.15 ± 0.002 & 16.85 ± 0.002 &16.04 ± 0.001 &15.26 ± 0.002 & \nMIR-95 & 0.00 ± 0.000 & 20.16 ± 0.012 &18.71 ± 0.009 &17.34 ± 0.004 & \nMIR-96 & 19.23 ± 0.008 & 17.85 ± 0.003 &16.94 ± 0.003 &16.02 ± 0.002 & \nMIR-97 & 20.06 ± 0.007 & 18.63 ± 0.005 &17.76 ± 0.005 &16.83 ± 0.004 & \nMIR-100 & 18.31 ± 0.003 & 16.86 ± 0.002 &16.03 ± 0.001 &15.28 ± 0.002 & \nMIR-106 & 18.87 ± 0.004 & 17.75 ± 0.003 &16.96 ± 0.003 &16.21 ± 0.002 & \nMIR-107 & 19.50 ± 0.004 & 18.12 ± 0.002 &17.22 ± 0.003 &16.27 ± 0.002 & \nMIR-108 & 18.94 ± 0.005 & 17.31 ± 0.004 &16.13 ± 0.003 &14.86 ± 0.001 & \nMIR-109 & 18.17 ± 0.003 & 16.17 ± 0.001 &15.00 ± 0.001 &13.86 ± 0.001 & \nMIR-110 & 0.00 ± 0.000 & 19.78 ± 0.010 &17.76 ± 0.004 &15.76 ± 0.003 &*V* \nBRC38 1 & 0.00 ± 0.000 & 20.05 ± 0.017 &18.24 ± 0.009 &16.45 ± 0.003 & \nBRC38 17 & 0.00 ± 0.000 & 20.51 ± 0.017 &18.65 ± 0.005 &16.92 ± 0.006 & *V* \nGFG 64 & 19.01 ± 0.007 & 17.48 ± 0.003 &16.19 ± 0.001 &15.18 ± 0.001 & \nIPHAS & 0.00 ± 0.000 & 20.28 ± 0.011 &18.28 ± 0.009 &16.50 ± 0.004 & \nllllllllllllc BRC 38 1& & 5/10/08& -12.93 & M3 ± 1& 3470& CTTS& 312.23 & -9.86 & & 1.13 & 2.81 & \n& & 10/10/09& -16.24 & & & & 205.29 & -10.90 & & & & \nMIR-5 & BRC 38 2 & 5/10/08& -85.23 & M1 ± 1& 3720& CTTS& 359.79 & -9.40 & & 0.99 & 2.47 & 3.19 \n& & 10/10/09& -106.1 & & & & 244.53 & -10.52 & & & & \nMIR-7 & BRC 38 3 & 6/10/08& -19.0 & M1 ± 2& 3720& CTTS& 476.25 & -8.27 & -0.87 & 1.05 & 2.61 & 1.46 \n& & 10/10/09& -18.75 & & & & 463.50 & -8.39 & & & & \nMIR-11 & BRC 38 4 & 10/09/08& -7.30 & M1 ± 2& 3720& WTTS& & & 0.02 & 1.37 & 3.41 & \n& & 18/07/09& -11.29 & & & & & & & & & \nMIR-29 & BRC 38 5 & 6/09/08& -3.42 & M1 ± 2& 3720& WTTS& & & -1.74 & 1.05 & 2.61 & 1.57 \n& & 3/11/08& -8.84 & & & & 469.90 & -8.33 & & & & \n& & 24/08/09& -1.89 & & & & & & & & & \n& & 9/11/09& -2.74 & & & & & & & & & \nMIR-31 & BRC 38 6 & 6/09/08& -53.47 & K2 ± 1& 4900& CTTS& 384.43 & -9.16 & -1.60 & 1.18 & 2.94 & 1.42 \n& & 24/08/09& -57.32 & & & & 442.02 & -8.60 & & & & \n& & 9/10/09& -60.5 & & & & 382.26 & -9.18 & & & & \nMIR-32 & BRC 38 7 & 10/09/08& -33.12 & M2 ± 1& 3580& CTTS& 368.03 & -9.32 & -1.03 & 0.90 & 2.24 & 2.65 \n& & 12/09/09& -15.78 & & & & 220.59 & -10.75 & & & & \nMIR-43 & GFG 62 & 18/07/09& -3.516 & M1 ± 1& 3720& WTTS& & & -2.61 & 0.80 & 1.99 & 1.82 \nMIR-45 & BRC 38 9 & 6/09/08& -37.43 & K4 ± 1& 4590& CTTS& 281.09 & -10.16 & -0.18 & 1.97 & 4.91 & 2.21 \n& & 3/11/08& -41.22 & & & & 389.74 & -9.11 & & & & \n& & 24/08/09& -56.33 & & & & 273.39 & -10.24 & & & & \nMIR-49 & BRC 38 10& 4/11/08& -13.5 & M0 ± 1& 3850& CTTS& 430.70 & -8.71 & -1.37 & 1.32 & 3.29 & 1.91 \n& & 9/10/09& -30 & & & & 450.55 & -8.52 & & & & \nMIR-64 & BRC 38 11& 10/09/08& -31.46 & K5 ± 1& 4350& CTTS& 239.20 & -10.57 & 0.10 & 2.04 & 5.08 & \n& & 12/09/09& -25.06 & & & & & & & & & \nMIR-76 & GFG 81 & 18/07/09& -3.261 & K7 ± 1& 4060& WTTS& & & -1.07 & 1.01 & 2.51 & 1.87 \n& & 10/10/09& -3.135 & & & & & & & & & \nMIR-80 & BRC 38 12& 24/08/09& -8.404 & M4 ± 1& 3370& WTTS& & & 0.31 & 0.95 & 2.37 & 3.39 \nBRC 38 17& & 11/11/09& -119. & M3 ± 2& 3470& CTTS& 205.29 & -10.90 & & 1.12 & 2.79 & \ncccccccccc MIR-11& 4442 & 1.26 & 6.44 × 10− 3 & 6.39 × 10− 5 & 0.0 & 2.10 × 10− 9 &1.27 & 5.3 & 3.2 \nMIR-29& 4837 & 1.55 & 1.66 × 10− 3 & 6.13 × 10− 7 & 0.0 & 1.16 × 10− 9 &1.67 & 4.4 & 5.8 \nMIR-31& 5923 & 1.88 & 1.33 × 10− 4 & 7.04 × 10− 8 & 0.0 & 2.41 × 10− 9 &8.92 & 4.2 & 6.7 \nMIR-34& 4523 & 1.53 & 7.64 × 10− 5 & 2.56 × 10− 3 & 1.21 × 10− 8 & 1.28 × 10− 10& 5.8 & 3.2 &0.69 \nMIR-36& 5413 & 3.29 & 4.37 × 10− 3 & 1.85 × 10− 8 & 0.0 & 4.80 × 10− 8 &29.1 & 49.6 & 1.3 \nMIR-38& 7854 & 1.98 & 3.10 × 10− 5 & 2.99 × 10− 6 & 0.0 & 5.49 × 10− 11&11.9 & 23.0 & 8.2 \nMIR-45& 9960 & 2.32 & 4.23 × 10− 5 & 1.70 × 10− 8 & 0.0 & 1.37 × 10− 9 &27.0 & 8.3 & 9.4 \nMIR-48& 7554 & 5.14 & 1.71 × 10− 3 & 21.4 × 10− 2 & 2.23 × 10− 7 & 2.38 × 10− 9 & 280 &45.9 & 0.49 \nMIR-50& 4643 & 5.97 & 7.67 × 10− 2 & 40.8 & 3.82 × 10− 4 & 2.08 × 10− 8 &196.7 & 44.4 &0.1 \nMIR-54& 4290 & 1.50 & 2.46 × 10− 2 & 9.3 & 7.13 × 10− 5 & 2.98 × 10− 6 &33.4 & 19.6 &0.17 \nMIR-55& 4989 & 3.07 & 7.86 × 10− 4 & 1.44 × 10− 3 & 1.97 × 10− 9 & 1.14 × 10− 9 &14.4 & 52.3 &0.91 \nMIR-59& 5218 & 4.27 & 9.99 × 10− 3 & 7.37 × 10− 3 & 2.95 × 10− 7 & 2.00 × 10− 8 & 60.5 & 87.4 & 0.53 \nMIR-82& 11320& 2.77 & 5.85 × 10− 5 & 1.18 × 10− 6 & 0.0 & 1.40 × 10− 10 &57.2& 21.0 & 4.8 \nMIR-86& 4594 & 1.55 & 8.18 × 10− 4 & 3.63 × 10− 7 & 0.0 & 4.50 × 10− 10 &2.59 & 9.7 & 1.8 \nAllen, L. E., et al. 2004,, 154, 363\nBalog, Z., Muzerolle, J., Rieke, G. H., Su, K. Y. L., Young, E. T., & Megeath, S. T. 2007,, 660, 1532\nBeltrán, M. T., Girart, J. M., Estalella, R., & Ho, P. T. P. 2004,, 426, 941\nBeltrán, M. T., Girart, J. M., Estalella, R., Ho, P. T. P., & Palau, A. 2002,, 573, 246\nBeltrán, M. T., Massi, F., López, R., Girart, J. M., & Estalella, R. 2009,, 504, 97\nBessell, M. S., & Brett, J. M. 1988,, 100, 1134\nCaratti o Garatti, A., Giannini, T., Nisini, B., & Lorenzetti, D. 2006,, 449, 1077\nCodella, C., Bachiller, R., Nisini, B., Saraceno, P., & Testi, L. 2001,, 376, 271\nContreras, M. E., Sicilia-Aguilar, A., Muzerolle, J., Calvet, N., Berlind, P., & Hartmann, L. 2002,, 124, 1585\nCutri, R. M., et al. 2003, 2MASS All Sky Catalog of point sources. (NASA/IPAC Infrared Science Archive.)\nde Zeeuw, P. T., Hoogerwerf, R., de Bruijne, J. H. J., Brown, A. G. A., & Blaauw, A. 1999,, 117, 354\nDeharveng, L., Lefloch, B., Zavagno, A., Caplan, J., Whitworth, A. P., Nadeau, D., & Martín, S. 2003,, 408, L25\nDeharveng, L., Zavagno, A., Schuller, F., Caplan, J., Pomarès, M., & De Breuck, C. 2009,, 496, 177\nEiroa, C., et al. 2002,, 384, 1038\nElmegreen, B. G., & Lada, C. J. 1977,, 214, 725\nFang, M., van Boekel, R., Wang, W., Carmona, A., Sicilia-Aguilar, A., & Henning, T. 2009,, 504, 461\nGetman, K. V., Feigelson, E. D., Garmire, G., Broos, P., & Wang, J. 2007,, 654, 316\nGray, R. O., & Corbally, J., C. 2009, Stellar Spectral Classification (Princeton University Press)\nGutermuth, R. A., Megeath, S. T., Myers, P. C., Allen, L. E., Pipher, J. L., & Fazio, G. G. 2009,, 184, 18\nHartmann, L., Megeath, S. T., Allen, L., Luhman, K., Calvet, N., D’Alessio, P., Franco-Hernandez, R., & Fazio, G. 2005,, 629, 881\nHernández, J., Calvet, N., Briceño, C., Hartmann, L., & Berlind, P. 2004,, 127, 1682\nHernández, J., Hartmann, L., Calvet, N., Jeffries, R. D., Gutermuth, R., Muzerolle, J., & Stauffer, J. 2008,, 686, 1195\nHillenbrand, L. A., Bauermeister, A., & White, R. J. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 384, 14th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, ed. G. van Belle, 200\nHosokawa, T., & Inutsuka, S. 2006,, 648, L131\nIkeda, H., et al. 2008,, 135, 2323\nIndebetouw, R., et al. 2005,, 619, 931\nJacoby, G. H., Hunter, D. A., & Christian, C. A. 1984,, 56, 257\nJaschek, M., & Egret, D. 1982, in IAU Symposium, Vol. 98, Be Stars, ed. M. Jaschek & H.-G. Groth, 261\nJørgensen, J. K., Johnstone, D., Kirk, H., & Myers, P. C. 2007,, 656, 293\nKenyon, S. J., & Hartmann, L. 1995,, 101, 117\nKirkpatrick, J. D., Henry, T. J., & McCarthy, Jr., D. W. 1991,, 77, 417\nKoenigl, A. 1991,, 370, L39\nLandolt, A. U. 1992,, 104, 340\nMatthews, H. I. 1979,, 75, 345\nMegeath, S. T., et al. 2004,, 154, 367\nMercer, E. P., Miller, J. M., Calvet, N., Hartmann, L., Hernandez, J., Sicilia-Aguilar, A., & Gutermuth, R. 2009,, 138, 7\nMiao, J., White, G. J., Nelson, R., Thompson, M., & Morgan, L. 2006,, 369, 143\nMookerjea, B., Sandell, G., Jarrett, T. H., & McMullin, J. P. 2009,, 507, 1485\nMorgan, L. K., Thompson, M. A., Urquhart, J. S., & White, G. J. 2008,, 477, 557\nMuzerolle, J., Calvet, N., & Hartmann, L. 1998,, 492, 743\n—. 2001,, 550, 944\nMuzerolle, J., Hartmann, L., & Calvet, N. 1998,, 116, 455\nMuzerolle, J., et al. 2004,, 154, 379\nNatta, A., Testi, L., Muzerolle, J., Randich, S., Comerón, F., & Persi, P. 2004,, 424, 603\nNeri, R., et al. 2007,, 468, L33\nNisini, B., et al. 2001,, 376, 553\nOgura, K., Sugitani, K., & Pickles, A. 2002,, 123, 2597\nPatel, N. A., Goldsmith, P. F., Heyer, M. H., Snell, R. L., & Pratap, P. 1998,, 507, 241\nPatel, N. A., Goldsmith, P. F., Snell, R. L., Hezel, T., & Xie, T. 1995,, 447, 721\nReipurth, B., Armond, T., Raga, A., & Bally, J. 2003,, 593, L47\nRobitaille, T. P., Whitney, B. A., Indebetouw, R., & Wood, K. 2007,, 169, 328\nRobitaille, T. P., Whitney, B. A., Indebetouw, R., Wood, K., & Denzmore, P. 2006,, 167, 256\nSaraceno, P., et al. 1996,, 315, L293\nSiess, L., Dufour, E., & Forestini, M. 2000,, 358, 593\nStern, D., et al. 2005,, 631, 163\nSternberg, A., Hoffmann, T. L., & Pauldrach, A. W. A. 2003,, 599, 1333\nStickland, D. J. 1995, The Observatory, 115, 180\nSugitani, K., Fukui, Y., Mizuni, A., & Ohashi, N. 1989,, 342, L87\nSugitani, K., Fukui, Y., & Ogura, K. 1991,, 77, 59\nSugitani, K., Matsuo, H., Nakano, M., Tamura, M., & Ogura, K. 2000,, 119, 323\nSugitani, K., & Ogura, K. 1994,, 92, 163\nSugitani, K., Tamura, M., & Ogura, K. 1999, in Star Formation 1999, Proceedings of Star Formation 1999, held in Nagoya, Japan, June 21 - 25, 1999, Editor: T. Nakamoto, Nobeyama Radio Observatory, p. 358-364, ed. T. Nakamoto, 358–364\nSugitani, K., et al. 2002, in 8th Asian-Pacific Regional Meeting, Volume II, ed. S. Ikeuchi, J. Hearnshaw, & T. Hanawa, 213–214\nTapia, M. 1981,, 197, 949\nValdes, F., Gupta, R., Rose, J. A., Singh, H. P., & Bell, D. J. 2004,, 152, 251\nValdettaro, R., Palla, F., Brand, J., & Cesaroni, R. 2005,, 443, 535\nvan Leeuwen, F. 2007,, 474, 653\nWaller, W. H. 1990,, 102, 1217\nWhite, R. J., & Basri, G. 2003,, 582, 1109\nWitham, A. R., Knigge, C., Drew, J. E., Greimel, R., Steeghs, D., Gänsicke, B. T., Groot, P. J., & Mampaso, A. 2008,, 384, 1277\nZavagno, A., Deharveng, L., Comerón, F., Brand, J., Massi, F., Caplan, J., & Russeil, D. 2006,, 446, 171\nZavagno, A., Pomarès, M., Deharveng, L., Hosokawa, T., Russeil, D., & Caplan, J. 2007,, 472, 835\n---\n1. <http://web.ipac.caltech.edu/staff/carey/irac_artifacts/index.html>[↩](#fnref1)\n2. <http://www.iiap.res.in/centers/iao>[↩](#fnref2)\n3. The IRAF software is distributed by the National Optical Astronomy Observatory under contact with National Science Foundation. <http://iraf.noao.edu/>[↩](#fnref3)\n4. <http://idlastro.gsfc.nasa.gov/>[↩](#fnref4)\n5. [http://www.astro.lsa.umich.edu/~hernandj/SPTclass/\n sptclass.html](http://www.astro.lsa.umich.edu/~hernandj/SPTclass/\n sptclass.html)[↩](#fnref5)\n"},"data_id":{"kind":"string","value":"arxiv_0181647"}}},{"rowIdx":45391,"cells":{"content":{"kind":"string","value":"kbi)..controls +(0.1,0) and +(-0.1,0).. (km’) ;\n[xshift=-1.5cm] (kh) at (0,1.8) ; (km) at (0,1.5) ; (kb) at (0,1.2) ; (khi) at (0.5,2) ; (kbi) at (1,1) ; (kh’) at (1.5,1.8) ; (km’) at (1.5,1.5) ; (kb’) at (1.5,1.2) ; (kh)..controls +(0.1,0) and +(-0.1,0).. (kb’) ; (km)..controls +(0.1,0) and +(-0.1,0).. (khi) ; (khi)..controls +(0.1,0) and +(-0.1,0).. (kh’) ; (kb)..controls +(0.1,0) and +(-0.1,0).. (kbi) ; (kbi)..controls +(0.1,0) and +(-0.1,0).. (km’) ;\n[xshift=0cm] (kh) at (0,1.8) ; (km) at (0,1.5) ; (kb) at (0,1.2) ; (khi) at (0.5,2) ; (kbi) at (1,1) ; (kh’) at (1.5,1.8) ; (km’) at (1.5,1.5) ; (kb’) at (1.5,1.2) ; (kh)..controls +(0.1,0) and +(-0.1,0).. (kb’) ; (km)..controls +(0.1,0) and +(-0.1,0).. (khi) ; (khi)..controls +(0.1,0) and +(-0.1,0).. (kh’) ; (kb)..controls +(0.1,0) and +(-0.1,0).. (kbi) ; (kbi)..controls +(0.1,0) and +(-0.1,0).. (km’) ;\n[xshift=1.5cm] (kh) at (0,1.8) ; (km) at (0,1.5) ; (kb) at (0,1.2) ; (khi) at (0.5,2) ; (kbi) at (1,1) ; (kh’) at (1.5,1.8) ; (km’) at (1.5,1.5) ; (kb’) at (1.5,1.2) ; (kh)..controls +(0.1,0) and +(-0.1,0).. (kb’) ; (km)..controls +(0.1,0) and +(-0.1,0).. (khi) ; (khi)..controls +(0.1,0) and +(-0.1,0).. (kh’) ; (kb)..controls +(0.1,0) and +(-0.1,0).. (kbi) ; (kbi)..controls +(0.1,0) and +(-0.1,0).. (km’) ;\n(x9) at (-3,1.4) ; (x10) at (3,1.4) ; (x11) at (3,-1.4) ; (x12) at (-3,-1.4) ; (x13) at (0,-1.4) ;\n(x10)..controls +(1.7,0) and +(1.7,0).. (x11) – (x13); (x13) – (x12)..controls +(-1.7,0) and +(-1.7,0).. (x9)–(x10) ;\n(1,-1.65) node *H**h* ;\n[xshift=-2cm,rotate=0,scale=1] (yh) at (0,-0.7) ; (yb) at (0,-2.3) ; (0.3,-1) – (0.3,-0.7) – (0,-0.7); (0.3,-0.9) – (0.3,-1.15); (0,-0.7) – (-0.3,-0.7) – (-0.3,-2.3) – (0.3,-2.3) – (0.3,-2) – (0.3,-1.85) ; (0.5,-0.75) node *y* ;\n(5,1.7) node *T*(*p*, *q*) ;\n[xshift=0cm,rotate=0,scale=1] (yh) at (0,-0.7) ; (yb) at (0,-2.3) ; (0.3,-1.1) – (0.3,-0.5); (0.3,-0.3) – (0.3,0) – (-0.3,0) – (-0.3,-1.5); (-0.3,-1.5) – (-0.3,-3) – (0.3,-3) – (0.3,-1.9) ;\n(-0.65,-2.7) node *H**v* ;\n(th) at (0,3) ; (td) at (5.5,0) ; (tb) at (0,-3) ; (tg) at (-5.5,0) ;\n(2,0.4) – (2,-0.4) – (-0.2,-0.4); (-0.4,-0.4) – (-1.5,-0.4) ; (-1.5,-0.4) – (-2,-0.4) – (-2,0.4) – (2,0.4) ; (-1.5,-0.15) node *λ* ;\n[fig torus knot inside thick]\nTubular neighbourhoods of *H**h* and *H**v* have a torus *T* as a common boundary. The manifold *M**H* = *S*3 \\ (*V*(*H**v*) ∪ *V*(*H**h*)) is homeomorphic to a thickened torus *T* × [ − 1; 1]. We identify *T* ≅ *T* × {0} to the torus on which the torus knot *T*(*p*, *q*) is drawn. The space *Z* = *T* \\ *V*(*T*(*p*, *q*)) is homeomorphic to an annulus. Let *δ* be a curve that generates *π*1(*Z*). The curve *δ* is thus locally parallel to the torus knot *T*(*p*, *q*). See Figure [fig torus delta] for clarity.\n[every path/.style=string , every node/.style=transform shape, knot crossing, inner sep=1.5 pt ]\n[scale=0.8]\n[xshift=0cm,rotate=0,scale=1]\n(-2,-0.8) node *T* ;\n(ch) at (-2,-1) ; (cb) at (-2,-2) ; (ch)..controls +(0.5,0) and +(0.5,0).. (cb) ; (ch)..controls +(-0.5,0) and +(-0.5,0).. (cb) ;\n(t1) at (-3,2) ; (t2) at (3,2) ; (t3) at (-3,1) ; (t4) at (3,1) ; (t5) at (-3,-1) ; (t6) at (3,-1) ; (t7) at (-3,-2) ; (t8) at (3,-2) ; (t1) – (t2)..controls +(2.5,0) and +(2.5,0).. (t8) – (t7)..controls +(-2.5,0) and +(-2.5,0).. (t1) ; (t3) – (t4)..controls +(1,0) and +(1,0).. (t6) – (t5)..controls +(-1,0) and +(-1,0).. (t3) ;\n(k1) at (-3,1.8) ; (k2) at (3,1.8) ; (k3) at (-3,1.2) ; (k4) at (3,1.2) ; (k5) at (-3,-1.2) ; (k6) at (3,-1.2) ; (k7) at (-3,-1.8) ; (k8) at (3,-1.8) ; (k9) at (-3,1.5) ; (k10) at (3,1.5) ; (k11) at (3,-1.5) ; (k12) at (-3,-1.5) ; (k2)..controls +(2.3,0) and +(2.3,0).. (k8) – (k7)..controls +(-2.3,0) and +(-2.3,0).. (k1) ; (k4)..controls +(1.3,0) and +(1.3,0).. (k6) – (k5)..controls +(-1.3,0) and +(-1.3,0).. (k3) ; (k10)..controls +(1.8,0) and +(1.8,0).. (k11) – (k12)..controls +(-1.8,0) and +(-1.8,0).. (k9) ;\n[xshift=-3cm] (kh) at (0,1.8) ; (km) at (0,1.5) ; (kb) at (0,1.2) ; (khi) at (0.5,2) ; (kbi) at (1,1) ; (kh’) at (1.5,1.8) ; (km’) at (1.5,1.5) ; (kb’) at (1.5,1.2) ; (kh)..controls +(0.1,0) and +(-0.1,0).. (kb’) ; (km)..controls +(0.1,0) and +(-0.1,0).. (khi) ; (khi)..controls +(0.1,0) and +(-0.1,0).. (kh’) ; (kb)..controls +(0.1,0) and +(-0.1,0).. (kbi) ; (kbi)..controls +(0.1,0) and +(-0.1,0).. 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(k11’) – (k12’)..controls +(-1.7,0) and +(-1.7,0).. (k9’) ; (0,-1.31) node *δ* ;\n[fig torus delta]\n[propp,qinsidethick] The *L*2-Alexander torsion for the exterior of the 3-component link *L* = *T*(*p*, *q*) ∪ *H**v* ∪ *H**h* is non-zero for all *t* > 0 and for all admissible triples (*G**L*, (*n*1, *n*2, *n*3) ∘ *α**L*, *γ*) such that the homotopy class of the curve *δ* is sent by *γ* to an element of infinite order. One has: \n$$T^{(2)}\\_{L, (n\\_1,n\\_2,n\\_3)}(\\gamma)(t)\n\\ \\dot{=} \\ \\max(1,t)^{|pq n\\_1 + p n\\_2 + q n\\_3|}.$$ \nThe torus *T* separates *M**H* in two thickened tori *N*1 = *V*(*H**v*) \\ *H**v* and \n*N*2 = *V*(*H**h*) \\ *H**h*. Let *X* = *M**L*, *A* = *N*1 ∪ *T* \\ *V*(*T*(*p*, *q*)), *B* = *N*2 ∪ *T* \\ *V*(*T*(*p*, *q*)) and *Z* = *T* \\ *V*(*T*(*p*, *q*)), so that *X* = *A* ∪ *B* and *Z* = *A* ∩ *B*, and *X*, *A*, *B*, *Z* are path connected. We pick a base point *p**t* ∈ *Z* for all the following fundamental groups.\nThe space *Z* is an annulus, and its group *π**Z* = *π*1(*Z*) is isomorphic to Z and is generated by an element *δ* that runs between the *p* strands of *T*(*p*, *q*). The space *A* is homeomorphic to a thickened torus, by filling the missing surface lines of *V*(*T*(*p*, *q*)). Let (*y*, *λ*) be a preferred meridian-longitude system of *A*, as in Figure [fig torus knot inside thick]. Note that *λ* acts as a meridian of the unknot *H**v*. The group *π**A* = *π*1(*A*) has the presentation ⟨*y*, *λ*∣*y**λ* = *λ**y*⟩ and is isomorphic to Z2. Similarly, the space *B* is homeomorphic to a thickened torus, by filling the missing surface lines of *V*(*T*(*p*, *q*)). Let (*m*, *c*) be a preferred meridian-longitude system of *A*. Note that *m* acts as a meridian of the unknot *H**h*. The group *π**B* = *π*1(*B*) has the presentation ⟨*m*, *c*∣*m**c* = *c**m*⟩ and is isomorphic to Z2. The element *z* is sent to *λ**p**y**q* in *π**A* and to *c**p**m**q* in *π**B*. Thus, by the Seifert van Kampen theorem, the group *G**L* = *π*1(*X*) admits the presentation \n⟨*y*, *λ*, *c*, *m*∣*λ**p**y**q* = *c**p**m**q*, *y**λ* = *λ**y*, *m**c* = *c**m*⟩. \n[description/.style=fill=white,inner sep=2pt] (a)[matrix of math nodes, row sep=2em, column sep=1.5em, text height=1.5ex, text depth=0.25ex] & A \nZ & & X \n& B \n; (a-2-1) edge node[below]*I**B* (a-3-2); (a-2-1) edge node[above]*I**A* (a-1-2); (a-1-2) edge node[above]*J**A* (a-2-3); (a-3-2) edge node[below]*J**B* (a-2-3); (a-2-1) edge node[above]*I* (a-2-3);\n[xshift=6cm,rotate=0,scale=1] [description/.style=fill=white,inner sep=2pt] (a)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.5ex, text depth=0.25ex] & 1(A) \n1(Z) & & GL & & G \n& 1(B) & & 3 & \n; (a-2-1) edge node[below]*i**B* (a-3-2); (a-2-1) edge node[above]*i**A* (a-1-2); (a-1-2) edge node[above]*j**A* (a-2-3); (a-3-2) edge node[below]*j**B* (a-2-3); (a-2-1) edge node[above]*i* (a-2-3); (a-2-3) edge node[above]*γ* (a-2-5); (a-2-3) edge node[below]*α**L* (a-3-4); (a-3-4) edge node[below](*n*1, *n*2, *n*3) (a-3-5); (a-2-5) edge (a-3-5);\nLet *ϕ* denote the homomorphism (*n*1, *n*2, *n*3) ∘ *α**L*. We assume that the homotopy class of *δ* is sent by *γ* to an element of infinite order, i.e. *γ* ∘ *i*(*δ*) = *γ*(*c**p**m**q*) = *γ*(*λ**p**y**q*) has infinite order in *G*. Therefore *γ*(*π**A*) and *γ*(*π**B*) are infinite and it follows from Corollaries [torsion solid torus] and [torsion torus] that the three N(*G*)-cellular chain complexes \n*C*\\*(2)(*Z*, *ϕ* ∘ *i*, *γ* ∘ *i*, *t*), *C*\\*(2)(*A*, *ϕ* ∘ *j**A*, *γ* ∘ *j**A*, *t*), *C*\\*(2)(*B*, *ϕ* ∘ *j**B*, *γ* ∘ *j**B*, *t*) \n are weakly acyclic and of determinant class, and \n$$T^{(2)}(A,\\phi \\circ j\\_A, \\gamma \\circ j\\_A)(t) \\ \\dot{=} \\ \nT^{(2)}(B,\\phi \\circ j\\_B, \\gamma \\circ j\\_B)(t) \\ \\dot{=} \\ 1,$$ \n \n$$T^{(2)}(Z,\\phi \\circ i, \\gamma \\circ i)(t) \\ \\dot{=} \\ \\max(1,t)^{-|\\phi(c^p m^q)|}.$$ \nHence, by Proposition [prop mayer vietoris], *C*\\*(2)(*M**L*, (*n*1, *n*2, *n*3) ∘ *α**L*, *γ*, *t*) is weakly acyclic and of determinant class as well, and \n$$T^{(2)}\\_{L,(n\\_1,n\\_2,n\\_3)}(\\gamma)(t)\n\\ \\dot{=} \\ \\max(1,t)^{|\\phi(c^p m^q)|}\n= \\max(1,t)^{|pq n\\_1 + p n\\_2 + q n\\_3|}.$$ \nThe link *T*(*e**p*, *e**q*) ∪ *H**v* ∪ *H**h*\n----------------------------------------------\nWe can now compute the *L*2-Alexander torsions for a general link *L* = *T*(*e**p*, *e**q*) ∪ *H**v* ∪ *H**h* by using the fact that the torus link *T*(*e**p*, *e**q*) is a (*e*, *e**p**q*)-cable on the torus knot *K* = *T*(*p*, *q*) (one can find a proof of this result due to P. Feller in ). We refer to for the definition of a cable link.\n[every path/.style=string,black, every node/.style=transform shape, knot crossing, inner sep=3.5 pt ]\n[scale=0.9]\n[xshift=0cm, yshift=0cm,rotate=0,scale=0.8]\n[xshift=-4cm, yshift=-1.15cm,rotate=0,scale=1] (0,-0.4)–(0,-0.55); (0,-0.65)–(0,-0.9); (0.3,-0.9) node *b*1;\n[xshift=-5cm, yshift=-0.79cm,rotate=0,scale=1] (0,-0.4)–(0,-0.55); (0,-0.65)–(0,-0.9); (0.3,-0.4) node *b*2;\n(1.5,-3) node ∂(*V*(*H**K*)) ≅ *T**K* ≅ ∂(*V*(*K*)) ;\n(-5,-4) node *T**K* ;\n(-1.7,-2.2) node *H**K* ; (-3.5,-0.7) node *l**K* ;\n[xshift=-6cm, yshift=0cm,rotate=0,scale=0.7]\n[xshift=0cm, yshift=0cm,rotate=0,scale=0.5]\n[xshift=10cm, yshift=0cm,rotate=0,scale=1] (1,-8) – (-1,-8) – (-1,-3); (-1,-3) – (-1,-2) – (1,-2) – (1,-2.4); (1,-2.8) – (1,-3.6); (1,-4.2) – (1,-4.8); (1,-5.3) – (1,-8);\n(-1.2,-2.6) – (-2,-2.6) – (-2,-1.4) – (0,-1.4); (0,-1.4) – (2,-1.4) – (2,-2.6) – (-0.8,-2.6);\n(20,0)..controls +(2,0) and +(2,0).. 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(2,3) – (3,3) – (6,3);\n[xshift=4cm, yshift=-1.8cm,rotate=0,scale=1, style=dotted] (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=11cm, yshift=-0.8cm,rotate=0,scale=1, style=dotted] (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=7cm, yshift=-3.2cm,rotate=0,scale=1, style=dotted] (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=14cm, yshift=-2.2cm,rotate=0,scale=1, style=dotted] (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=1cm, yshift=-1cm,rotate=0,scale=1] (0,4) – (1,4)..controls +(0.5,0) and +(-0.5,0).. (2,3) – (3,3); (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=4cm, yshift=-2cm,rotate=0,scale=1] (0,4) – (1,4)..controls +(0.5,0) and +(-0.5,0).. (2,3) – (3,3); (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=7cm, yshift=-3cm,rotate=0,scale=1] (0,4) – (1,4)..controls +(0.5,0) and +(-0.5,0).. (2,3) – (3,3); (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=11cm, yshift=-1cm,rotate=0,scale=1] (0,4) – (1,4)..controls +(0.5,0) and +(-0.5,0).. (2,3) – (3,3); (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=14cm, yshift=-2cm,rotate=0,scale=1] (0,4) – (1,4)..controls +(0.5,0) and +(-0.5,0).. (2,3) – (3,3); (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[xshift=17cm, yshift=-3cm,rotate=0,scale=1] (0,4) – (1,4)..controls +(0.5,0) and +(-0.5,0).. (2,3) – (3,3); (0,3) – (1,3); (ca) at (1,3) ; (cc) at (1.5,3.5) ; (cb) at (2,4) ; (ca.center).. controls (ca.2 north east) and (cc.2 south west).. (cc); (cc).. controls (cc.2 north east) and (cb.2 south west).. (cb.center); (2,4) – (3,4);\n[fig t(4,2)]\nIn Figure [fig t(4,2)], we draw a torus link *T*(*e*, *e**p**q*) inside a solid torus *S*3 \\ *V*(*H**K*), the link *T*(*p*, *q*) ∪ *H**v* ∪ *H**h*, and the torus link *T*(*e**p*, *e**q*) which is a (*e*, *e**p**q*)-cable on *T*(*p*, *q*) (we did not draw *H**v* and *H**h* in the third part in order to make the figure easier to read). Here *p* = 2, *q* = 1, *e* = 2. One can see the torus *T**K* (drawn with red dotted lines) that separates *M**T*(4, 2) in the disjoint union of the exterior of the torus knot *T*(2, 1) in *S*3 and the exterior of the torus link *T*(2, 4) ∪ *H**K*. This torus *T**K* is the boundary of a tubular neighbourhood of *K* = *T*(*p*, *q*). A preferred longitude *l**K* of *K* is drawn on the figure. We identify *S*3 \\ *V*(*H**K*) to the solid torus *V*(*K*); the component *H**K* looks like a preferred meridian of *K*. Let *M* = *M**L* denote the exterior of *L* = *T*(*e**p*, *e**q*) ∪ *H**v* ∪ *H**h*, *A* = *S*3 \\ *V*(*K* ∪ *H**v* ∪ *H**h*) and *B* = *S*3 \\ *V*(*T*(*e*, *e**p**q*) ∪ *H**K*) (in Figure [fig t(4,2)], *A* is the exterior of the drawing up right and *B* of the one up left). We see that *M* is the toroidal gluing of *A* and *B* along their intersection *T**K* = *A* ∩ *B*. The following diagrams are commutative:\n[description/.style=fill=white,inner sep=2pt] (a)[matrix of math nodes, row sep=2em, column sep=1.5em, text height=1.5ex, text depth=0.25ex] & A \nTK & & M \n& B \n; (a-2-1) edge node[below]*I**B* (a-3-2); (a-2-1) edge node[above]*I**A* (a-1-2); (a-1-2) edge node[above]*J**A* (a-2-3); (a-3-2) edge node[below]*J**B* (a-2-3); (a-2-1) edge node[above]*I* (a-2-3);\n[xshift=6cm,rotate=0,scale=1] [description/.style=fill=white,inner sep=2pt] (a)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.5ex, text depth=0.25ex] & 1(A) \n1(TK) & & 1(M) = GL & G \n& 1(B) & e+2 & \n; (a-2-1) edge node[below]*i**B* (a-3-2); (a-2-1) edge node[above]*i**A* (a-1-2); (a-1-2) edge node[above]*j**A* (a-2-3); (a-3-2) edge node[below]*j**B* (a-2-3); (a-2-1) edge node[above]*i* (a-2-3); (a-2-3) edge node[above]*γ* (a-2-4); (a-2-3) edge node[right]*α**L* (a-3-3); (a-3-3) edge node[below](*n*1, …, *n**e* + 2) (a-3-4); (a-2-4) edge (a-3-4);\nAs in the previous section, let *T* be the torus on which *K* is drawn, and *δ* a simple closed curve that generates the fundamental group of *T* \\ *V*(*K*). The curve *δ* is once again locally parallel to the components of *T*(*e**p*, *e**q*).\n[propep,eqinsidethick] Let *e* ≥ 2. The *L*2-Alexander torsion for the exterior of the link *L* = *T*(*e**p*, *e**q*) ∪ *H**v* ∪ *H**h* is non-zero for all *t* > 0 and for all admissible triples (*G**L*, (*n*1, …, *n**e*, *n**e* + 1, *n**e* + 2) ∘ *α**L*, *γ*) such that the homotopy class of the curve *δ* is sent by *γ* to an element of infinite order. One has: \n$$T^{(2)}\\_{L, (n\\_1,\\ldots,n\\_{e},n\\_{e+1},n\\_{e+2})}(\\gamma)(t)\n \\ \\dot{=} \\ \\max(1,t)^{e|pq(n\\_1+ \\ldots +n\\_e) + p n\\_{e+1} + q n\\_{e+2}|}.$$ \nLet *t* > 0. Let *ϕ* denote the homomorphism (*n*1, …, *n**e* + 2) ∘ *α**L*. We assume that the homotopy class of *δ* is sent by *γ* to an element of infinite order. First, as the cabling torus *T**K* is the boundary of a tubular neighbourhood *V*(*K*) of *K* = *T*(*p*, *q*) and contains such a curve *δ*, the torus *T**K* has thus infinite image under *γ*, therefore *C*\\*(2)(*T**K*, *ϕ* ∘ *i*, *γ* ∘ *i*, *t*) is weakly acyclic and of determinant class and its *L*2-torsion is 1, by Theorem [torsion torus].\nSecondly, the curve *λ**μ**k* = *λ**μ**p**q* of Proposition [prope,ekinsidetorus] is ambient isotopic to *δ*, thus it is sent by *γ* to an element of infinite order (in Figure [fig t(4,2)], *λ* is written *l**K*, and *μ* is parallel to *H**K*), therefore *C*\\*(2)(*B*, *ϕ* ∘ *j**B*, *γ* ∘ *j**B*, *t*) is weakly acyclic and of determinant class, and\n \n$$\\begin{aligned}\n T^{(2)}(B,\\phi \\circ j\\_B, \\gamma \\circ j\\_B)(t) \n& = \n T^{(2)}\\left (M\\_{L'},(n\\_1, \\ldots, n\\_{e+2}) \\circ \\alpha\\_L \\circ j\\_B, \\gamma \\circ j\\_B\\right )(t) \\\\\n & =\n T^{(2)}\\left (M\\_{L'},(n\\_1, \\ldots, n\\_e, p n\\_{e+1} + q n\\_{e+2}) \\circ \\alpha\\_{L'}, \\gamma \\circ j\\_B\\right )(t) \\\\\n& \\ \\dot{=} \\ \\max(1,t)^{(e-1)|p n\\_{e+1} + q n\\_{e+2} + pq(n\\_1 + \\ldots + n\\_e)|},\\end{aligned}$$ \nwhere *L*ʹ = *T*(*e*, *e**p**q*) ∪ *H**K*.\nFinally, the last piece of the toroidal gluing is *A* = *M**H* \\ *V*(*K*), which corresponds to the case *e* = 1 of the previous Section [sec:tpqhh]; from the assumption on *δ*, it follows from Proposition [propp,qinsidethick] that *C*\\*(2)(*A*, *ϕ* ∘ *j**A*, *γ* ∘ *j**A*, *t*) is weakly acyclic and of determinant class, and\n \n$$\\begin{aligned}\n T^{(2)}(A,\\phi \\circ j\\_A, \\gamma \\circ j\\_A)(t) \n& = \n T^{(2)}\\left (M\\_{L''},(n\\_1, \\ldots, n\\_{e+2}) \\circ \\alpha\\_L \\circ j\\_A, \\gamma \\circ j\\_A\\right )(t) \\\\\n & =\n T^{(2)}\\left (M\\_{L''},(n\\_1+ \\ldots+ n\\_e, n\\_{e+1}, n\\_{e+2}) \\circ \\alpha\\_{L''}, \\gamma \\circ j\\_A\\right )(t) \\\\\n& \\ \\dot{=} \\ \\max(1,t)^{|p n\\_{e+1} + q n\\_{e+2} + pq(n\\_1 + \\ldots + n\\_e)|},\\end{aligned}$$ \nwhere *L*ʺ = *T*(*p*, *q*) ∪ *H**v* ∪ *H**h*. It follows then from Proposition [prop t2 JSJ] that *C*\\*(2)(*M**L*, *ϕ*, *γ*, *t*) is weakly acyclic and of determinant class, and\n \n$$\\begin{aligned}\n&T^{(2)}\\_{L, (n\\_1,\\ldots,n\\_{e},n\\_{e+1},n\\_{e+2})}(\\gamma)(t) \\\\\n&= T^{(2)}(A,\\phi \\circ j\\_A, \\gamma \\circ j\\_A)(t) \\cdot T^{(2)}(B,\\phi \\circ j\\_B, \\gamma \\circ j\\_B)(t) \\\\\n& \\ \\dot{=} \\max(1,t)^{(e-1)|p n\\_{e+1} + q n\\_{e+2} + pq(n\\_1 + \\ldots + n\\_e)|} \\cdot \\max(1,t)^{|p n\\_{e+1} + q n\\_{e+2} + pq(n\\_1 + \\ldots + n\\_e)|} \\\\\n& = \\max(1,t)^{e |p n\\_{e+1} + q n\\_{e+2} + pq(n\\_1 + \\ldots + n\\_e)|}.\\end{aligned}$$ \nThe link *T*(*e**p*, *e**q*) ∪ *H**v*\n-------------------------------------\nThe link *L* = *L*1 ∪ … ∪ *L**e* ∪ *H**v* = *T*(*e**p*, *e**q*) ∪ *H**v* is obtained from *L*ʹ = *T*(*e**p*, *e**q*) ∪ *H**v* ∪ *H**h* by deleting the component *H**h*, therefore *M**L* is obtained from *M**L*ʹ by a ∞-surgery on the boundary component of *H**h*. This helps us compute the *L*2-Alexander torsions of *L*. Let *λ**h* be the homotopy class of *H**h* in *M**L* and *δ* the simple closed curve locally parallel to the strands of *T*(*e**p*, *e**q*) as in the previous section. The epimorphism *Q* : *G**L*ʹ → *G**L* corresponds to the trivialization of the curve *λ**h*.\n[propep,eqinside] For the exterior of the link *L* = *T*(*e**p*, *e**q*) ∪ *H**v*, The *L*2-Alexander torsion is non-zero for all admissible triples (*G**L*, (*n*1, …, *n**e* + 1) ∘ *α**L*, *γ*) such that *γ*(*δ*) and *γ*(*λ**h*) have infinite order in *G* and for all *t* > 0. One has: \n*T**L*, (*n*1, …, *n**e* + 1)(2)(*γ*)(*t*) = max(1, *t*)(*e*∣*p*∣ − 1)∣*n**e* + 1 + *q*(*n*1 + … + *n**e*)∣. \nWe will use Theorem [thm surgery forget]. Here *λ**h* corresponds to the curve *λ* in the assumptions of Theorem [thm surgery forget]. Since *γ*(*δ*) has infinite order in *G*, it follows from Proposition [propep,eqinsidethick] that *C*\\*(2)(*M**L*ʹ, (*n*1, …, *n**e* + 1, 0) ∘ *α**L*ʹ, *γ* ∘ *Q*, *t*) is weakly acyclic and of determinant class, and \n$$T^{(2)}\\_{L',(n\\_1,\\ldots,n\\_{e+1},0)}(\\gamma \\circ Q)(t)\n\\ \\dot{=} \\ \\max(1,t)^{e|pq(n\\_1+ \\ldots +n\\_e) + p n\\_{e+1}|}.$$ \n Since *γ*(*λ**h*) has infinite order in *G*, then *C*\\*(2)(*M**L*, (*n*1, …, *n**e* + 1) ∘ *α**L*, *γ*, *t*) is weakly acyclic and of determinant class by Theorem [thm surgery forget], and\n \n$$\\begin{aligned}\nT^{(2)}\\_{L, (n\\_1,\\ldots,n\\_{e+1})}(\\gamma)(t) \n&\\ \\dot{=} \\ \\dfrac{T^{(2)}\\_{L',(n\\_1,\\ldots,n\\_{e+1},0)}(\\gamma \\circ Q)(t)}{\\max(1,t)^{|\\mathrm{lk}(L\\_1,H\\_h) n\\_1 + \\ldots + \\mathrm{lk}(L\\_e,H\\_h) n\\_e + \\mathrm{lk}(H\\_v,H\\_h) n\\_{e+1}|}} \\\\\n&\\ \\dot{=} \\ \\dfrac{\\max(1,t)^{e|pq(n\\_1+ \\ldots +n\\_e) + p n\\_{e+1}|}}\n{\\max(1,t)^{|q n\\_1 + \\ldots + q n\\_e + n\\_{e+1}|}} \\\\\n& = \\max(1,t)^{(e|p|-1)|n\\_{e+1}+q(n\\_1+ \\ldots +n\\_e)|}.\\end{aligned}$$ \nThe torus link *T*(*e**p*, *e**q*)\n----------------------------------\nNow we can compute *L*2-Alexander torsions for general torus links of the form *L* = *T*(*e**p*, *e**q*), where *e* ≥ 2 is an integer and *p*, *q* are relatively prime integers. The link *T*(*e**p*, *e**q*) is obtained by ∞-surgery from *T*(*e**p*, *e**q*) ∪ *H**v* on the component *H**v*. The epimorphism *Q* : *G**T*(*e**p*, *e**q*) ∪ *H**v* → *G**T*(*e**p*, *e**q*) corresponds to the trivialization of the curve *λ**v*. Let *δ* and *λ**h* be as in the previous sections, and let *λ**v* denote the homotopy class of *H**v* in *G**T*(*e**p*, *e**q*). Note that the fundamental group of the torus *T* (on which *T*(*e**p*, *e**q*) is drawn) is generated by classes of curves homotopic to *λ**h* and *λ**v*. Thus the equality *δ* = *λ**h**p**λ**v**q* stands in *G**T*(*e**p*, *e**q*). This equality and the fact that *λ**h**λ**v* = *λ**v**λ**h* imply that, for any homomorphism *γ* : *G**T*(*e**p*, *e**q*) → *G*, if two elements of {*γ*(*δ*), *γ*(*λ**h*), *γ*(*λ**v*)} are of infinite order, then the third is of infinite order as well.\n[propep,eq] The *L*2-Alexander torsion for the exterior of the torus link *L* = *T*(*e**p*, *e**q*) is non-zero for all admissible triples (*G**L*, (*n*1, …, *n**e*) ∘ *α**L*, *γ*) such that two of the three elements *γ*(*δ*), *γ*(*λ**h*), *γ*(*λ**v*) have infinite order in *G*, and for all *t* > 0. One has: \n$$T^{(2)}\\_{L, (n\\_1,\\ldots,n\\_e)}(\\gamma)(t)\n\\ \\dot{=} \\ \\max(1,t)^{|n\\_1+ \\ldots +n\\_e|\\left (e|p||q| -|p| -|q|\\right )}.$$ \nThis theorem generalises the computation of the *L*2-Alexander invariants of torus knots done in.\nHere *λ**v* corresponds to the curve *λ* in the assumptions of Theorem [thm surgery forget]. Since *γ*(*δ*), *γ*(*λ**h*) and *γ*(*λ**v*) have infinite order in *G*, the result follows from Proposition [propep,eqinside] and Theorem [thm surgery forget] exactly like in the proof of Proposition [propep,eqinside].\nGeneral cabling formulas\n------------------------\nWe can now prove a general cabling formula for *L*2-Alexander torsions as a consequence of Proposition [propep,eqinside]. Let *L* = *L*1 ∪ … ∪ *L**c* + 1 a link in *S*3, and *L*ʹ = *L*1 ∪ … ∪ *L**c* ∪ *L*ʹ*c* + 1 ∪ … ∪ *L*ʹ*c* + *e* the link obtained by cabling the component *L**c* + 1 by the torus link *T*(*e**p*, *e**q*) with *p*, *q* two relatively prime integers.\n[every path/.style=string , every node/.style=transform shape, knot crossing, inner sep=1.7 pt ]\n[scale=0.7]\n[xshift=-1cm,scale=0.7] (tl) at (-1.73,1) ; (tr) at (1.73, 1 ) ; (b) at (0,-2) ; (n) at (-2,1.2) ; (n) node[above left]*L*2 ; (-1.2,-1.7) node[above left]*L*1 ;\n(a) at (-1,-2.6) ; (c) at (-2,-2.6) ; (tl.center).. controls (tl.16 south west) and (tr.16 north west).. (tr) ; (tl).. controls (tl.32 south east) and (tr.32 north east).. (tr.center) ; (tl).. controls (tl.16 north west) and (b.16 north west).. (b.center) ; (tl.center).. controls (tl.16 north east) and (c.32 north west).. (c) ; (b).. controls (b.8 south west) and (c.4 south east).. (c) ; (b).. controls (b.16 north east) and (tr.16 south west).. (tr.center) ; (b.center).. controls (b.32 south east) and (tr.32 south east).. (tr) ; (a).. controls (a.16 south east) and (c.16 south west).. (c.center) ; (c.center).. controls (c.4 north east) and (a.4 north west).. (a) ;\n[xshift=5cm,scale=0.7] (n) at (-1,1.2) ; (m) at (-2.6,1.5) ; (0,-2) node[above left]*V* ; (O) at (0,0) ; (A) at (0.75,1.3) ; (B) at (1.25,2.17) ; (C) at (-0.75,1.3) ; (D) at (-1.25,2.17) ; (G) at (0,-1) ; (H) at (0,-3) ; (I) at (-0.15,2.1) ; (O) circle (1) ; (O) circle (3) ; (O) circle (1.5) ; (O) circle (2.5) ; (G)..controls +(-0.2,0) and +(-0.2,0).. (H) ; (G)..controls +(0.2,0) and +(0.2,0).. (H) ; (n) node[violet, above left]*P*1 ; (1,0.7) node[blue, above left]*P*2 ; (0.7,0.9) node[above left]*T*(2, 0) = ∪ ; (0.45,1.8) node[violet, above left]*P*1 ; (1.2,1.8) node[blue, above left]*P*2 ;\n(2,-3) node[above left]∂(*V*(*L*2)) ≅ ∂(*V*) ;\n[xshift=-1cm,yshift=-7cm,scale=0.7] (n) at (-1.9,2.5) ; (n) node[above left]*V*(*L*2) ; (O) at (0,0) ; (A) at (-0.87,0.5) ; (B) at (-1.73,1) ; (C) at (-1.16,1) ; (D) at (0.87,0.5) ; (E) at (1.73,1) ; (F) at (1.16,1) ; (G) at (0,3) ; (H) at (0,3.5) ; (A) – (D) – (F) – (B) ; (F)..controls +(0.5\\*1,0.5\\*3.73) and +(1,0).. (G) ; (C)..controls +(-0.5\\*1,0.5\\*3.73) and +(-1,0).. (G) ; (E)..controls +(0.6\\*1,0.6\\*3.73) and +(1,0).. (H) ; (B)..controls +(-0.6\\*1,0.6\\*3.73) and +(-1,0).. (H) ; (G)..controls +(-0.2,0) and +(-0.2,0).. (H) ; (G)..controls +(0.2,0) and +(0.2,0).. (H) ;\n[rotate=120] (O) at (0,0) ; (A) at (-0.87,0.5) ; (B) at (-1.73,1) ; (C) at (-1.16,1) ; (D) at (0.87,0.5) ; (E) at (1.73,1) ; (F) at (1.16,1) ; (G) at (0,3) ; (H) at (0,3.5) ; (A) – (D) – (F) – (B) ; (F)..controls +(0.5\\*1,0.5\\*3.73) and +(1,0).. (G) ; (C)..controls +(-0.5\\*1,0.5\\*3.73) and +(-1,0).. (G) ; (E)..controls +(0.6\\*1,0.6\\*3.73) and +(1,0).. (H) ; (B)..controls +(-0.6\\*1,0.6\\*3.73) and +(-1,0).. (H) ;\n[rotate=240] (O) at (0,0) ; (A) at (-0.87,0.5) ; (B) at (-1.73,1) ; (C) at (-1.16,1) ; (D) at (0.87,0.5) ; (E) at (1.73,1) ; (F) at (1.16,1) ; (G) at (0,3) ; (H) at (0,3.5) ; (A) – (D) – (F) – (B) ; (F)..controls +(0.5\\*1,0.5\\*3.73) and +(1,0).. (G) ; (C)..controls +(-0.5\\*1,0.5\\*3.73) and +(-1,0).. (G) ; (E)..controls +(0.6\\*1,0.6\\*3.73) and +(1,0).. (H) ; (B)..controls +(-0.6\\*1,0.6\\*3.73) and +(-1,0).. (H) ;\n(-1,-2) arc (0:290:1); (-0.95,-2) arc (0:240:1.05); (-1.05,-2) arc (0:240:0.95); (-1.5,-2) node[black, above left]*L*1 ;\n[xshift=5cm,yshift=-7cm,scale=0.8,color=blue] (n) at (-1,2) ; (g1) at (-1.63,1.18) ; (g2) at (-1.91,1.1) ; (g3) at (-1.83,0.82) ; (g4) at (-1.55,0.9) ; (d1) at (1.63, 1.18 ) ; (d2) at (1.55, 0.9 ) ; (d3) at (1.83, 0.82 ) ; (d4) at (1.91, 1.1 ) ; (b1) at (0,-1.8) ; (b2) at (-0.2,-2) ; (b3) at (0,-2.2) ; (b4) at (0.2,-2) ; (h) at (0,2.5) ; (g1).. controls (g1.16 south east) and (d1.16 north east).. (d1.center) ; (g2).. controls (g2.32 south east) and (d4.32 north east).. (d4.center) ; (g2.center).. controls (g2.32 north east) and (b3.32 south west).. (b3) ; (g3.center).. controls (g3.8 north east) and (-3,-3).. (b2) ; (g2.center) – (g1.center) ; (g3.center) – (g4.center) ; (g1)– (g4); (g2) – (g3) ; (b3.center).. controls (b3.32 south east) and (d4.32 south east).. (d4) ; (b4.center).. controls (3,-4) and (d3.8 south east).. (d3) ; (d2.center) – (d1.center) ; (d3.center) – (d4.center) ; (d1)– (d4); (d2) – (d3) ; (b1.center) – (b4.center) ; (b2.center) – (b3.center) ; (b1)– (b2); (b3) – (b4) ; (tm1) at (-0.5,1) ; (tm2) at (0.5,1) ; (g1.center).. controls (g1.4 south west) and (tm1.4 north west).. (tm1) ; (tm1).. controls (tm1.4 south east) and (tm2.4 south west).. (tm2.center) ; (tm2.center).. controls (tm2.4 north east) and (d1.4 north west).. (d1) ; (g4.center).. controls (g4.4 south west) and (tm1.4 south west).. (tm1.center) ; (tm1.center).. controls (tm1.4 north east) and (tm2.4 north west).. (tm2) ; (tm2).. controls (tm2.4 south east) and (d2.4 north west).. (d2) ; (tg1) at (-1.2,0) ; (tg2) at (-0.6,-1) ; (g4).. controls (g4.4 north west) and (tg1.4 north).. (tg1) ; (tg1).. controls (tg1.4 south) and (tg2.4 west).. (tg2.center) ; (tg2.center).. controls (tg2.4 east) and (b1.4 north west).. (b1.center) ; (g3).. controls (g3.4 north west) and (tg1.4 west).. (tg1.center) ; (tg1.center).. controls (tg1.4 east) and (tg2.4 north).. (tg2) ; (tg2).. controls (tg2.4 south) and (b2.4 north west).. (b2.center) ; (td1) at (1.2,0) ; (td2) at (0.7,-1) ; (d3.center).. controls (d3.4 south west) and (td1.4 east).. (td1) ; (td1).. controls (td1.4 west) and (td2.4 north).. (td2.center) ; (td2.center).. controls (td2.4 south) and (b4.4 north east).. (b4) ; (d2.center).. controls (d2.4 south west) and (td1.4 north).. (td1.center) ; (td1.center).. controls (td1.4 south) and (td2.4 east).. (td2) ; (td2).. controls (td2.4 west) and (b1.4 north east).. (b1) ; (-1,-2) arc (0:290:1); (-0.95,-2) arc (0:240:1.05); (-1.05,-2) arc (0:240:0.95); (-1.1,-2.3) arc (-10:-40:1) ; (-1.5,-2) node[black, above left]*L*1 ; (0,2) node[violet, above left]*L*ʹ2 ; (-0.3,1.3) node[blue, above left]*L*ʹ3 ;\n[fig cable 2,0]\nThen *M* = *M**L*ʹ = *S*3 \\ *V*(*L*ʹ) is obtained by a toroidal gluing of *A* = *M**L* = *S*3 \\ *V*(*L*) and *B* = (*S*1 × *D*2) \\ *V*(*T*(*e**p*, *e**q*)) ≅ *S*3 \\ *V*(*T*(*e**p*, *e**q*) ∪ *H**v*) between the components *L**c* + 1 and *H**v*. Let *n*1, …, *n**c* + *e* ∈ Z. Let *γ* : *G**L*ʹ → *G* be a group homomorphism such that (*G**L*ʹ, (*n*1, …, *n**c* + *e*) ∘ *α**L*ʹ, *γ*) is an admissible triple. Let *t* > 0.\n[description/.style=fill=white,inner sep=2pt] (a)[matrix of math nodes, row sep=2em, column sep=1.5em, text height=1.5ex, text depth=0.25ex] & A \nT & & M \n& B \n; (a-2-1) edge node[below]*I**B* (a-3-2); (a-2-1) edge node[above]*I**A* (a-1-2); (a-1-2) edge node[above]*J**A* (a-2-3); (a-3-2) edge node[below]*J**B* (a-2-3); (a-2-1) edge node[above]*I* (a-2-3);\n[xshift=6cm,rotate=0,scale=1] [description/.style=fill=white,inner sep=2pt] (a)[matrix of math nodes, row sep=2em, column sep=1.5em, text height=1.5ex, text depth=0.25ex] & 1(A) = GL \n1(T) & & 1(M) = GL’ & G \n& 1(B) & e+1 & \n; (a-2-1) edge node[below]*i**B* (a-3-2); (a-2-1) edge node[above]*i**A* (a-1-2); (a-1-2) edge node[above]*j**A* (a-2-3); (a-3-2) edge node[below]*j**B* (a-2-3); (a-2-1) edge node[above]*i* (a-2-3); (a-2-3) edge node[above]*γ* (a-2-4); (a-2-3) edge node[left]*α**L*ʹ (a-3-3); (a-3-3) edge node[below](*n*1, …, *n**c* + *e*) (a-3-4); (a-2-4) edge (a-3-4);\nLet *N* = *n**c* + 1 + … + *n**c* + *e* and ℓ = ∑*i* = 1*c*lk(*L**i*, *L**c* + 1)*n**i*. To clarify the notations, let us look a the example in Figure [fig cable 2,0]. The link *L* has two components (*c* = 1), *L*1 which is unknotted and *L*2 which is a trefoil, with linking number lk(*L*1, *L*2) = 1. We do a (2, 0)-cabling on *L*2 (thus *e* = 2, *p* = 1, *q* = 0), and the resulting link *L*ʹ has 3 components. We glue the tori ∂(*V*(*L*2)) and ∂(*V*) such that a meridian of *L*2 is identified with *m**V* the meridian of *V* that circles both components of *T*(2, 0). Here *N* = *n*2 + *n*3 and ℓ = *n*1.\n[thmtorsioncabling] Assume that\n* *C*\\*(2)(*M**L*, (*n*1, …, *n**c*, *p**N*) ∘ *α**L*, *γ* ∘ *j**A*, *t*) is weakly acyclic and of determinant class,\n* *C*\\*(2)(*M**T*(*e**p*, *e**q*) ∪ *H**v*, (*n**c* + 1, …, *n**c* + *e*, ℓ) ∘ *α**T*(*e**p*, *e**q*) ∪ *H**v*, *γ* ∘ *j**B*, *t*) is weakly acyclic and of determinant class,\n* *T* = ∂(*V*) ≅ ∂(*V*(*L**c* + 1)) has infinite image under *γ*,\nthen *C*\\*(2)(*M**L*ʹ, (*n*1, …, *n**c* + *e*) ∘ *α**L*ʹ, *γ*, *t*) is weakly acyclic and of determinant class and \n$$T^{(2)}\\_{L', (n\\_1, \\ldots, n\\_{c+e})}(\\gamma)(t) \n\\ \\dot{=} \\ \nT^{(2)}\\_{L, (n\\_1, \\ldots,n\\_c, p N)}(\\gamma \\circ j\\_A)(t) \\cdot\n\\max(1,t)^{(e|p|-1)|\\ell+q N|}.$$ \nThis theorem generalises the cabling formula for the *L*2-Alexander invariant of knots proven in.\nFirst let us prove that \n(*n*1, …, *n**c* + *e*) ∘ *α**L*ʹ ∘ *j**A* = (*n*1, …, *n**c*, *p**N*) ∘ *α**L* \n and that \n(*n*1, …, *n**c* + *e*) ∘ *α**L*ʹ ∘ *j**B* = (*n**c* + 1, …, *n**c* + *e*, ℓ) ∘ *α**T*(*e**p*, *e**q*) ∪ *H**v*. \n The group *G**L* = *π*1(*A*) is generated by *m*1, …*m**c* + 1, which are preferred meridians of *L*1, …*L**c* + 1 in *M**L*. One has \n((*n*1, …, *n**c* + *e*) ∘ *α**L*ʹ ∘ *j**A*)(*m**i*) = ((*n*1, …, *n**c*, *p**N*) ∘ *α**L*)(*m**i*) \n for *i* = 1, …, *c* since *L*1, …, *L**c* are the *c* first components of *L*ʹ. The identity remains true for *i* = *c* + 1, since *j**A*(*m**c* + 1) circles the *e* components *L*ʹ*c* + 1, …, *L*ʹ*c* + *e* for a total of *p* times and is unlinked with *L*1, …, *L**c*. The group *G**B* = *π*1(*B*) = *π*1(*S*1 × *D*2 \\ *V*(*T*(*e**p*, *e**q*))) is generated by *b*1, …, *b**e* (which are preferred meridians of the components of *T*(*e**p*, *e**q*)) and *λ* a longitude of the solid torus *S*1 × *D*2 ≅ ∂*V*(*L**c* + 1). Note that *j**B*(*λ*) = *j**A*(*l**c* + 1) in *M**L*ʹ where *l**c* + 1 is a preferred longitude of *L**c* + 1 in *M**L*. The identity \n(*n*1, …, *n**c* + *e*) ∘ *α**L*ʹ ∘ *j**B* = (*n**c* + 1, …, *n**c* + *e*, ℓ) ∘ *α**T*(*e**p*, *e**q*) ∪ *H**v* \n stands true when evaluated on each of the generators *b**i*, for *i* = 1…, *e* (both terms of the equality are immediately equal to *n**c* + *i*), and for *λ* the second term is equal to ℓ, and the first term is equal to\n \n$$\\begin{aligned}\n((n\\_1, \\ldots, n\\_{c+e}) \\circ \\alpha\\_{L'})( j\\_B(\\lambda))\n&=((n\\_1, \\ldots, n\\_{c+e}) \\circ \\alpha\\_{L'})( j\\_A(l\\_{c+1})) \\\\\n&= ((n\\_1, \\ldots,n\\_c, p N) \\circ \\alpha\\_L)(l\\_{c+1}) \\\\\n&= n\\_1 \\mathrm{lk}(L\\_1,L\\_{c+1}) + \\ldots + n\\_c \\mathrm{lk}(L\\_c,L\\_{c+1}) + 0\n= \\ell.\\end{aligned}$$ \nWe have established that the three different coefficents *ϕ* of the statement of the result were indeed compatible. Now, since the cabling torus *T* = ∂(*V*(*L**c* + 1)) has infinite image under *γ*, the result follows from Proposition [prop t2 JSJ] and Proposition [propep,eqinside].\n\nGluing formulas for the *L*2-Alexander torsions\n===============================================\nWe prove a Torres-like formula for the *L*2-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the *L*2-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.\nIntroduction\n============\nWhen one considers a knot invariant, one of the first questions to ask is “how can the definition be extended to links?”. The *L*2-Alexander invariant of a knot is a knot invariant taking values in the class of maps on the positive real numbers up to multiplication by monomials. It was originally constructed by W. Li and W. Zhang in from a presentation of the knot group, as an infinite-dimensional version of Fox’s construction for the Alexander polynomial (see ). J. Dubois, S. Friedl and W. Lück then generalized this invariant in with the *L*2-Alexander torsion *T*(2)(*M*, *ϕ*, *γ*)(*t*) associated to a triplet (*M*, *ϕ*, *γ*) and a positive number *t*, where *M* is a compact connected oriented 3-manifold with empty or toroidal boundary, *ϕ*: *π*1(*M*) → Z is a group homomorphism (and can be seen as a 1-cohomology class of *M*) and *γ*: *π*1(*M*) → *G* is a second homomorphism to a finitely presented group *G* such that *ϕ* factors through *γ*. One can see these *L*2-Alexander torsions as infinite-dimensional versions of the Reidemeister torsions of 3-manifolds. In particular, the *L*2-Alexander torsions associated to the exterior of a link in the three-sphere answer the previous question concerning the *L*2-Alexander invariant of knots.\nThe *L*2-Alexander torsions of links attract considerable interest for the reasons that they are continuous functions (see ) whose values give the simplicial volume and the Thurston norm associated to the link exterior (see ). Unfortunately, these torsions are difficult to compute exactly; thus, any method of computing them more efficiently for a given link is worthy of pursuit. For knots, the torsions reduce to the *L*2-Alexander invariant of Li and Zhang when *γ* is the identity. Exact computations have been made for torus knots in and then for iterated torus knots in. More generally, the *L*2-Alexander torsions of graph manifolds were computed explicitly (and reduced to the Thurston norm) in and the ones of fibered manifolds were computed partially in. In this article, we present various techniques to relate the *L*2-Alexander torsions of links related by\n* deleting a component,\n* connected sum,\n* cabling,\nand we compute explicitly the torsions associated to iterated torus links.\nThe Torres formula for the Alexander polynomial of links (see ) is a useful tool to compare the polynomials of two links differing by one component (for example two-component links and knots). The Main Theorem of this article presents a Torres-like formula for the *L*2-Alexander torsions of links. One can see the *L*2-Alexander torsions of links as functions *T**L*, (*n*1, …, *n**c*)(2)(*γ*)(*t*) of a link *L*, a finite set of integers *n**i* (one for each component of *L*), a homomorphism *γ* and a positive variable *t*; these functions are considered up to multiplication by monomials (equality up to such multiplication is denoted $\\dot{=}$).\n[Theorem [thm surgery forget]] Let *L* = *L*1 ∪ … ∪ *L**c* be a *c*-component link and *L*ʹ the link obtained by removing the last component *L**c*. Let *G**L* and *G**L*ʹ denote the fundamental groups of their exteriors and *Q*: *G**L* → *G**L*ʹ the epimorphism induced by removing of the component *L**c*. For all *t* > 0, all integers *n*1, …, *n**c* − 1 and all appropriate *γ*: *G**L*ʹ → *G*, the corresponding *L*2-Alexander torsions satisfy: \n$$T^{(2)}\\_{L',(n\\_1, \\ldots, n\\_{c-1})}(\\gamma)(t) \\ \\dot{=} \\ \n\\dfrac{T^{(2)}\\_{L,(n\\_1, \\ldots, n\\_{c-1}, 0)}(\\gamma \\circ Q)(t)}\n{\\max(1,t)^{|\\mathrm{lk}(L\\_1,L\\_c) n\\_1 + \\ldots + \\mathrm{lk}(L\\_{c-1},L\\_c) n\\_{c-1}|}}.$$ \nThe Main Theorem can be seen as a particular case of a general Dehn surgery formula for the *L*2-Alexander torsions (stated in Propositions [prop surgery] and [prop surgery links]), since removing a component of the link *L* is the same as gluing a solid torus canonically to the corresponding boundary component of the exterior of *L*. Another particular case of Dehn surgery yields an interesting connection between *L*2-Alexander torsions of the Whitehead link and the *L*2-Alexander invariants of the twist knots (see Theorem [thm surgery twist]).\nLike many of the results of this article, the surgery formulas are consequences of a general gluing formula for the *L*2-Alexander torsions. This formula is stated in Proposition [prop mayer vietoris], and can be seen as a result of an application of the multiplicativity of the classical *L*2-torsion (see ).\nAs a second class of consequences of the gluing formula of Proposition [prop mayer vietoris], we prove general formulas for computing the *L*2-Alexander torsions of connected sums of two links or general multi-component cablings of links, as summarized in the following Theorem (for simplicity *γ* is assumed to be the identity and is therefore not written):\n[Theorems [thmtorsionsum] and [thmtorsioncabling]]\n1. The *L*2-Alexander torsions are “almost multiplicative” under the connected sums of links: if *L*ʺ is a connected sum of a (*c* + 1)-component link *L* and a (*d* + 1)-component link *L*ʹ, then \n$$\\dfrac{\nT^{(2)}\\_{L'',(n\\_1, \\ldots, n\\_{c+d+1})}(t) \n}{ \\max(1,t)^{|n\\_{c+d+1}|}}\n\\ \\dot{=} \\ \nT^{(2)}\\_{L,(n\\_1, \\ldots,n\\_c, n\\_{c+d+1})}(t)\n\\cdot\nT^{(2)}\\_{L',(n\\_{c+1}, \\ldots, n\\_{c+d}, n\\_{c+d+1})}(t)\n.$$\n2. The *L*2-Alexander torsions satisfy a cabling formula: if *L* is a (*c* + 1)-component link, and *L*ʹ a (*e**p*, *e**q*)-cabling on the last component of *L*, then \n$$T^{(2)}\\_{L', (n\\_1, \\ldots, n\\_{c+e})}(t) \n\\ \\dot{=} \\ \nT^{(2)}\\_{L, (n\\_1, \\ldots,n\\_c, p N)}(t) \\cdot\n\\max(1,t)^{(e|p|-1)|\\ell+q N|},$$ \n where *N* = *n**c* + 1 + … + *n**c* + *e* and ℓ = ∑*i* = 1*c*lk(*L**i*, *L**c* + 1)*n**i*.\nFormulas (1) and (2) are generalizations of Theorems 3.2 and 4.3 of to the case of links. Unlike where we used Fox calculus as the main tool, here we use methods of computation that rely on CW-complexes structures and the gluing formula of Proposition [prop mayer vietoris]. We hope that and the present article will provide *L*2-Alexander enthusiasts with a wide array of methods of computations.\nThe proof of the previous Theorem uses the gluing formula of Proposition [prop mayer vietoris] and the fact that in making a connected sum of two links or cabling one link one has to glue specific Seifert-fibered pieces to produce the corresponding link exteriors. We are thus naturally led to explicitly compute all *L*2-Alexander torsions of links whose exteriors are Seifert-fibered. This is the third and final class of applications of the gluing formula of Proposition [prop mayer vietoris] and they are interesting in their own right. These computations are summarized in the following Theorem (for simplicity *γ* is assumed to be the identity and is therefore not written):\n[All Propositions of Section [sec:splicing]] Let *L* be a link in *S*3 with Seifert-fibered exterior. Then *L* can be seen as a torus link either in *S*3, or in a solid torus or else in a thickened torus. Moreover:\n* If *L* = *T*(*e**p*, *e**q*), i.e. *L* is an *e*-component torus link with *p*, *q* coprime numbers, then \n$$T^{(2)}\\_{L, (n\\_1, \\ldots,n\\_e)}(t) \\ \\dot{=} \\\n\\max(1,t)^{(e|pq| - |p| - |q|)|n\\_1 + \\ldots +n\\_e|}.$$\n* If *L* = *T*(*e**p*, *e**q*) ∪ *L**e* + 1, i.e. *L* is an *e*-component torus link in a solid torus (seen as the exterior of the unknot *L**e* + 1), then \n$$T^{(2)}\\_{L, (n\\_1, \\ldots,n\\_e, n\\_{e+1})}(t) \\ \\dot{=} \\\n\\max(1,t)^{(e|p|-1)|q(n\\_1 + \\ldots +n\\_e)+ n\\_{e+1}|}.$$\n* If *L* = *T*(*e**p*, *e**q*) ∪ *L**e* + 1 ∪ *L**e* + 2, i.e. *L* is an *e*-component torus link in a thickened torus (seen as the exterior of the Hopf link *L**e* + 1 ∪ *L**e* + 2), then \n$$T^{(2)}\\_{L, (n\\_1, \\ldots,n\\_e, n\\_{e+1}, n\\_{e+2})}(t) \n\\ \\dot{=} \\\n\\max(1,t)^{e|pq(n\\_1 + \\ldots +n\\_e) + p n\\_{e+1} + q n\\_{e+2}|}.$$\nSince the *L*2-Alexander torsions of a Seifert-fibered manifold are known to be (*t* ↦ max(1, *t*)) to the power the Thurston norm of the cohomology class *ϕ* (see ), then the previous Theorem offers a new way of computing the Thurston norm for all links with Seifert-fibered exteriors. We hope this work can be of use to fellow topologists interested in Thurston norms of links.\nMost of the results of this article come from the author’s PhD thesis. The article is organized as follows: Section [sec:def] reviews some well-known facts about knots, groups, and *L*2-invariants, and can be skimmed by the experienced reader; Section [sec:first] reviews some important basic formulas satisfied by the *L*2-Alexander torsions, notably the gluing formula; Section [sec:dehn] presents the various Dehn surgery formulas and the Main Theorem; finally, Section [sec:splicing] tackles the computation of the torsions of Seifert-fibered link exteriors as well as the connected sum and cabling formulas.\nAcknowledgements\n================\nI would like to thank my PhD advisor Jérôme Dubois, for his teachings and great advice. This article is based on work supported by the *Ministère de l’Enseignement Supérieur et de la Recherche* at the Université Paris Diderot and by the *Swiss National Science Foundation*, subsidy 200021\\_162431, at the Université de Genève.\nPreliminaries\n=============\nKnots and links\n---------------\nHere we follow mostly. We choose an orientation for the three-sphere *S*3. A *link* with *c* ∈ N components is an embedding of a disjoint union of *c* circles ⊔ *i* = 1*c**S*1 into *S*3; we will assume that all links have ordered oriented components. We consider links up to ambient isotopies in *S*3 that preserve the order and the orientation of the components, unless precised otherwise. For *L* = *L*1 ∪ … ∪ *L**c* an link in *S*3, let *V*(*L*) denote an open tubular neighbourhood of *L*, and *M**L* = *S*3 \\ *V*(*L*) denote the *exterior of *L**, which is a compact 3-manifold with toroidal boundary. The orientation of *M**L* comes from the one of *S*3, and does not depend on the orientation of *L*. Each boundary torus ∂*M**L**i* is oriented with the convention that vectors normal to the boundary point outside of *M**L*. A *split link* is a link *L* ⊂ *S*3 such that there exists a 2-sphere Σ ⊂ *S*3, *L* = *L*ʹ ⊔ *L*ʺ with *L*ʹ and *L*ʺ sub-links, and *L*ʹ and *L*ʺ are contained in different connected components of *S*3 \\ Σ. Most of the time we will assume that links are non-split. The *group* of a link *L* is the fundamental group of its exterior and is denoted *G**L* = *π*1(*M**L*). We denote $\\alpha\\_L\\colon G\\_L \\twoheadrightarrow {\\mathbb{Z}}^c$ the abelianization homomorphism, where *c* is the number of components of *L*. The linking number between two components *L**i*, *L**j* of a link *L* is denoted lk(*L**i*, *L**j*).\nA link with one component is called a *knot*. When *K* is an oriented knot, there exists, up to isotopy, a unique pair of simple closed curves *μ**K* and *λ**K* on the 2-torus ∂*M**K* = ∂*V*(*K*) such that *μ**K* bounds a disk in *V*(*K*) and *λ**K* is homologous to *K* in *V*(*K*). We choose an orientation for these two curves such that the linking number between *μ**K* and *K* and the intersection number between *μ**K* and *λ**K* are both + 1. The pair (*μ**K*, *λ**K*) is called a *preferred meridian-longitude pair for *K**. Any such *μ**K* is called a *meridian curve*. Here we have used the notations and definitions of.\n*L*2-invariants\n---------------\nWe follow and for the rest of Section [sec:def]. Given a countable discrete group *G*, the completion of the algebra C[*G*] endowed with the scalar product $ \\left \\langle \\sum\\_{g \\in G} \\lambda\\_g g, \\sum\\_{g \\in G} \\mu\\_g g \\right \\rangle:= \\sum\\_{g \\in G} \\lambda\\_g \\overline{\\mu\\_g}$ is the Hilbert space \nℓ2(*G*) : = {∑*g* ∈ *G**λ**g**g* ∣ *λ**g* ∈ C, ∑*g* ∈ *G*∣*λ**g*∣2 < ∞}, \n of square-summable complex functions on *G*. We denote by *B*(ℓ2(*G*)) the algebra of operators on ℓ2(*G*) that are bounded with respect to the operator norm.\nGiven *h* ∈ *G*, we define the corresponding *left-* and *right-multiplication operators* *L**h* and *R**h* in *B*(ℓ2(*G*)) as extensions of the automorphisms (*g* ↦ *h**g*) and (*g* ↦ *g**h*) of *G*. One can extend the operators *R**h* C-linearly to an operator *R**w*: ℓ2(*G*) → ℓ2(*G*) for any *w* ∈ C[*G*]. Moreover, if ℓ2(*G*)*n* is endowed with its usual Hilbert space structure and *A* = (*a**i*, *j*) ∈ *M**p*, *q*(C[*G*]) is a C[*G*]-valued *p* × *q* matrix, then the right multiplication \n*R**A* : = (*R**a**i*, *j*)1 ≤ *i* ≤ *p*, 1 ≤ *j* ≤ *q* \n provides a bounded operator ℓ2(*G*)*q* → ℓ2(*G*)*p*. Note that we shall consider elements of ℓ2(*G*)*n* as *column vectors* and suppose that matrices with coefficients in *B*(ℓ2(*G*)) act on the *left* (even if the coefficients are themselves *right*-multiplication operators).\nThe *von Neumann algebra* N(*G*) of the group *G* is the sub-algebra of *B*(ℓ2(*G*)) made up of *G*-equivariant operators (i.e. operators that commute with all left multiplications *L**h*). A *finitely generated Hilbert N(*G*)-module* consists in a Hilbert space *V* together with a left *G*-action by isometries such that there exists a positive integer *m* and an embedding *φ* of *V* into ℓ2(*G*)*m*. A *morphism of finitely generated Hilbert N(*G*)-modules* *f*: *U* → *V* is a linear bounded map which is *G*-equivariant.\nDenoting by *e* the neutral element of *G*, the von Neuman algebra of *G* is endowed with the *trace* trN(*G*): N(*G*) → C, *ϕ* ↦ ⟨*ϕ*(*e*), *e*⟩ which extends to \ntrN(*G*): *M**n*, *n*(N(*G*)) → C by summing up the traces of the diagonal elements.\nThe *von Neumann dimension* of a finitely generated Hilbert N(*G*)-module *V* is defined as \ndimN(*G*)(*V*) : = trN(*G*)(pr*φ*(*V*)) ∈ R≥ 0, \n where pr*φ*(*V*): ℓ2(*G*)*m* → ℓ2(*G*)*m* is the orthogonal projection onto *φ*(*V*).\nThe von Neumann dimension does not depend on the embedding of *V* into the finite direct sum of copies of ℓ2(*G*).\nThe Fuglede-Kadison determinant\n-------------------------------\nThe *spectral density* *F*(*f*) of a morphism *f*: *U* → *V* of finitely generated Hilbert N(*G*)-modules maps *λ* ∈ R≥ 0 to \n*F*(*f*)(*λ*) : = sup{dimN(*G*)(*L*)∣*L* ∈ L(*f*, *λ*)}, \n where L(*f*, *λ*) is the set of finitely generated Hilbert N(*G*)-submodules of *U* on which the restriction of *f* has a norm smaller or equal to *λ*. Since *F*(*f*)(*λ*) is monotonous and right-continuous, it defines a measure *d**F*(*f*) on the Borel set of R≥ 0 solely determined by the equation *d**F*(*f*)(]*a*, *b*]) = *F*(*f*)(*b*) − *F*(*f*)(*a*) for all *a* < *b*.\n[def detFK] The *Fuglede-Kadison determinant of *f** is defined by:\n \n*d**e**t*N(*G*)(*f*) : = exp(∫0+∞ln(*λ*) *d**F*(*f*)(*λ*)) \nif ∫0+∞ln(*λ*) *d**F*(*f*)(*λ*) > − ∞; if not, *d**e**t*N(*G*)(*f*) = 0.\nWhen ∫0+∞ln(*λ*) *d**F*(*f*)(*λ*) > − ∞, we say that **f* is of determinant class*.\nHere are two properties of the determinant we will use in the rest of this article (see for more details and proofs).\n[prop operations det] Let *G* be a countable discrete group.\n1. For all *f*, *g* morphisms of finitely generated Hilbert N(*G*)-modules, \n$$det\\_{\\mathcal{N}(G)} \\left (\n\\begin{pmatrix}\nf & 0 \\\\ 0 & g\n\\end{pmatrix}\n\\right ) = det\\_{\\mathcal{N}(G)}(f) \\cdot det\\_{\\mathcal{N}(G)}(g).$$\n2. For all *t* ∈ C, if *g* ∈ *G* has infinite order, then *I**d* − *t**R**g* is injective and \n*d**e**t*N(*G*)(*I**d* − *t**R**g*) = max(1, ∣*t*∣).\n*L*2-torsion\n------------\nA *finite Hilbert N(*G*)-chain complex* *C*\\* is a sequence of morphisms of finitely generated Hilbert N(*G*)-modules \n$$C\\_\\* = \\left (0 \\to C\\_n \\overset{\\partial\\_n}{\\longrightarrow} C\\_{n-1} \n\\overset{\\partial\\_{n-1}}{\\longrightarrow} \\ldots\n\\overset{\\partial\\_2}{\\longrightarrow} C\\_1 \\overset{\\partial\\_1}{\\longrightarrow} C\\_0 \\to 0\\right )$$ \n such that ∂*p* ∘ ∂*p* + 1 = 0 for all *p*. The **p*-th *L*2-homology* of such a chain complex *C*\\* is the finitely generated Hilbert N(*G*)-module \n$$H\\_p^{(2)}(C\\_\\*) := \\textrm{Ker}(\\partial\\_p) / \\overline{\\textrm{Im}(\\partial\\_{p+1})}.$$ \n The **p*-th *L*2-Betti number of *C*\\** is defined as *b**p*(2)(*C*\\*) : = dimN(*G*)(*H**p*(2)(*C*\\*)). A finite Hilbert N(*G*)-chain complex *C*\\* is *weakly acyclic* if its *L*2-homology is trivial (i.e. if all its *L*2-Betti numbers vanish) and of *determinant class* if all the operators ∂*p* are of determinant class.\nLet *C*\\* be a finite Hilbert N(*G*)-chain complex as above. We define its **L*2-torsion* by \n*T*(2)(*C*\\*) : = ∏*i* = 1*n*detN(*G*)(∂*i*)( − 1)*i* ∈ R> 0 \n when *C*\\* is weakly acyclic and of determinant class, and by *T*(2)(*C*\\*) = 0 otherwise.\nThe following proposition will be useful for computations of *L*2-torsions. Compare with and.\n[prop tau chaine] Let $C\\_\\* = \\left (0 \\to \n\\ell^2(G)^k\n\\overset{\\partial\\_2}{\\longrightarrow} \n\\ell^2(G)^{k+l}\n \\overset{\\partial\\_1}{\\longrightarrow} \n\\ell^2(G)^l\n \\to 0\\right )$ be a 2-dimensional finite Hilbert N(*G*)-chain complex and let *J* ⊂ {1, …, *k* + *l*} be a subset of {1, …, *k* + *l*} of size *l*. For *i* = 1, 2, ∂*i* is naturally written as a matrix with coefficients operators in *B*(ℓ2(*G*)). We write ∂1(*J*) : ℓ2(*G*)*l* → ℓ2(*G*)*l* the operator composed of the columns of ∂1 indexed by *J*, and ∂2(*J*) : ℓ2(*G*)*k* → ℓ2(*G*)*k* the operator obtained from ∂2 by deleting the rows indexed by *J*. If ∂2(*J*) and ∂1(*J*) are injective and of determinant class, then *C*\\* is weakly acyclic and of determinant class, and \n$$T^{(2)}(C\\_\\*) = \\dfrac{\\det\\_{{\\mathcal{N}}(G)}(\\partial\\_2)}{\\det\\_{{\\mathcal{N}}(G)}(\\partial\\_1)} = \\dfrac{\\det\\_{{\\mathcal{N}}(G)}(\\partial\\_2(J))}{\\det\\_{{\\mathcal{N}}(G)}(\\partial\\_1(J))}.$$ \nThere exists an immediate generalisation of Proposition [prop tau chaine] to any dimension that mirrors the formula of.\n*L*2-Alexander torsions\n-----------------------\nWe follow the definitions and notations of. Let *π* be a group and *ϕ*: *π* → Z, *γ*: *π* → *G* two group homomorphisms. We say that (*π*, *ϕ*, *γ*) *forms an admissible triple* if *ϕ*: *π* → Z factors through *γ* (i.e. there is a group homomorphism *ψ*: *G* → Z such that *ϕ* = *ψ* ∘ *γ*, see the diagram below).\n[xshift=7cm,rotate=0,scale=1] [description/.style=fill=white,inner sep=2pt] (a)[matrix of math nodes, row sep=1em, column sep=3.5em, text height=1.5ex, text depth=0.25ex] & G \n & \n; (a-1-1) edge node[above]*γ* (a-1-2); (a-1-1) edge node[below]*ϕ* (a-2-2); (a-1-2) edge (a-2-2);\nFor *X* a CW-complex, we say that (*X*, *ϕ*: *π*1(*X*) → Z, *γ*: *π*1(*X*) → *G*) *forms an admissible triple* if (*π*1(*X*), *ϕ*, *γ*) forms one. Let (*X*, *ϕ*, *γ*) be such an admissible triple, *π* = *π*1(*X*) and *t* > 0. We define a ring homomorphism \n$$\\kappa(\\pi, \\phi, \\gamma, t)\\colon \\begin{pmatrix}\n & {\\mathbb{Z}}[\\pi]& \\longrightarrow & {\\mathbb{R}}[G] \\\\\n & \\sum\\_{j=1}^r m\\_j g\\_j & \\longmapsto & \\sum\\_{j=1}^r m\\_j t^{\\phi(g\\_j)} \\gamma(g\\_j)\n\\end{pmatrix}$$ \n and we also denote *κ*(*π*, *ϕ*, *γ*, *t*) its induction over the *M**p*, *q*(Z[*π*]).\nAssume *X* is compact. The cellular chain complex of $\\widetilde{X}$ denoted $C\\_\\*(\\widetilde{X},{\\mathbb{Z}}) = $ \n$\\left (\\ldots \\to \\bigoplus\\_i {\\mathbb{Z}}[\\pi] \\widetilde{e}\\_i^k \\to \\ldots\\right )$ is a chain complex of left Z[*π*]-modules. Here the $\\widetilde{e}\\_i^k$ are lifts of the cells *e**i**k* of *X*. The group *π* acts on the right on ℓ2(*G*) by *g* ↦ *R**κ*(*π*, *ϕ*, *γ*, *t*)(*g*), an action which induces a structure of right Z[*π*]-module on ℓ2(*G*). Let \n$$C\\_\\*^{(2)}(X,\\phi,\\gamma,t) = \\ell^2(G) \\otimes\\_{{\\mathbb{Z}}[\\pi]} C\\_\\*(\\widetilde{X},{\\mathbb{Z}})$$ \n denote the finite Hilbert N(*G*)-chain complex obtained by the tensor product associated to these left- and right-actions; we call *C*\\*(2)(*X*, *ϕ*, *γ*, *t*) *a N(*G*)-cellular chain complex of *X**.\nIf *C*\\*(2)(*X*, *ϕ*, *γ*, *t*) is a N(*G*)-cellular chain complex of *X*, then denote \n*T*(2)(*X*, *ϕ*, *γ*)(*t*) = *T*(2)(*C*\\*(2)(*X*, *ϕ*, *γ*, *t*)) \n the **L*2-Alexander torsion of (*X*, *ϕ*, *γ*) at *t* > 0*. It is non-zero if and only if *C*\\*(2)(*X*, *ϕ*, *γ*, *t*) is weakly acyclic and of determinant class\nAs a consequence of, if *X* is a compact connected oriented irreducible 3-manifold with empty or toroidal boundary and infinite fundamental group *π* = *π*1(*X*), then *C*\\*(2)(*X*, *ϕ*, *i**d*, *t*) is weakly acyclic and of determinant class for all *ϕ* and all *t*. In particular, most of the statements in this article concern torsions of exteriors of non-split links in the three-sphere, and thus one can skip the assumptions on weak acyclicity and determinant class for *γ* = *i**d* in such statements.\nWe will consider the equivalence class of (*t* ↦ *T*(2)(*X*, *ϕ*, *γ*, *t*)) up to multiplication by the (*t* ↦ *t**m*), *m* ∈ Z, which does not depend on the CW-structure chosen on *X* (the technical details can be found in ). For two maps *f*, *g*: R> 0 → R> 0, we write \n$$f \\ \\dot{=} \\ g \\ \\Longleftrightarrow \\ \\exists m \\in {\\mathbb{Z}}, \\forall t>0, f(t) = t^m g(t).$$ \n Note that for *t* > 0 and any integer *k*, $\\max(1,t^k) = t^{\\frac{k-|k|}{2}} \\max(1,t)^{|k|}$. Thus $\\left (t \\mapsto \\max(1,t^k)\\right ) \\ \\dot{=} \\ \\left (t \\mapsto \\max(1,t)^{|k|}\\right )$.\nFor *X* a CW-complex, its **L*2-torsion* is defined as *T*(2)(*X*) : = *T*(2)(*X*, 0, *i**d*)(1) when *C*\\*(2)(*X*, 0, *i**d*, 1) is weakly acyclic and of determinant class. Of course, historically, *L*2-torsions came before *L*2-Alexander torsions (see ). The following astonishing theorem of W. Lück and T. Schick (see ) states that the *L*2-torsion of an irreducible 3-manifold gives precisely the simplicial volume of this manifold. Recall that a compact connected orientable 3-manifold *M* is called *hyperbolic* if its interior admits a complete Riemannian metric whose sectional curvature is constant equal to − 1, and *Seifert* if it admits a foliation by circles.\n[, Theorem 4.3] [thm t2 volume] Let *M* be a compact connected orientable irreducible 3-manifold with infinite fundamental group and empty or incompressible toroidal boundary. According to Thurston-Perelman’s Geometrization theorem, *M* splits along disjoint incompressible tori into pieces that are Seifert manifolds or hyperbolic manifolds with finite volume. Moreover, *C*\\*(2)(*M*, 0, *i**d*, 1) is weakly acyclic and of determinant class, and \n$$T^{(2)}(M) = \\exp\\left (\\dfrac{\\textrm{vol}(M)}{6 \\pi}\\right )$$ \n where vol(*M*) = ∑*i* = 1*h*volhyp(*M**i*) is the simplicial volume of *M*, defined as the sum of the hyperbolic volumes of the hyperbolic pieces *M*1, …, *M**h*.\n*L*2-Alexander torsions of links\n--------------------------------\nLet *L* = *L*1 ∪ … ∪ *L**c* be a link in *S*3, *M**L* its exterior and *α**L*: *G**L* → Z*c* the abelianization of its group. Any homomorphism *ϕ*: *G**L* → Z factors through *α**L* and thus is written *ϕ* = (*n*1, …, *n**c*) ∘ *α**L* where *n*1, …, *n**c* ∈ Z. Any admissible triple (*M**L*, *ϕ*, *γ*) can thus be written (*M**L*, (*n*1, …, *n**c*) ∘ *α**L*, *γ*), and we will therefore denote \n*T**L*, (*n*1, …, *n**c*)(2)(*γ*)(*t*) : = *T*(2)(*M**L*, (*n*1, …, *n**c*) ∘ *α**L*, *γ*)(*t*) \n the **L*2-Alexander torsion associated to the multi-link (*L*, (*n*1, …, *n**c*)) and the morphism *γ* at the value *t**, and *T**L*, (*n*1, …, *n**c*)(2) : = *T*(2)(*M**L*, (*n*1, …, *n**c*) ∘ *α**L*, *i**d*) the *full *L*2-Alexander torsion function associated to the multi-link (*L*, (*n*1, …, *n**c*))*.\nNote that if *L* = *K* is a knot and *n* ∈ Z, then the full *L*2-Alexander torsion of *K* reduces to its *L*2-Alexander invariant Δ*K*(2) (see for details): \n$$T^{(2)}\\_{K,n}(t) \\ \\dot{=} \\ \n\\dfrac{\\Delta\\_{K}^{(2)}(t^n)}{\\max(1,t)^{|n|}}.$$ \nFirst formulas\n==============\nWe present three useful formulas, which are direct generalizations of known properties of the *L*2-torsions. The details of the proofs of Propositions [prop L2 torsion simple homotopy] and [prop mayer vietoris] can be found in.\nSimple homotopy equivalence\n---------------------------\nThe following proposition states that the *L*2-Alexander torsions are invariant by simple homotopy equivalence. Note that this result is a direct generalization of the formulas of and, and was announced in. The technical details of the proof can be found in.\n[, Theorem 2.12] [prop L2 torsion simple homotopy] Let *f*: *X* → *Y* be a simple homotopy equivalence between two finite CW-complexes that induces the group isomorphism *f*\\*: *π*1(*X*) → *π*1(*Y*). The triple (*Y*, *ϕ*, *γ*) is an admissible triple if and only if (*X*, *ϕ* ∘ *f*\\*, *γ* ∘ *f*\\*) is one, the N(*G*)-cellular chain complex *C*\\*(2)(*X*, *ϕ* ∘ *f*\\*, *γ* ∘ *f*\\*, *t*) is weakly acyclic and of determinant class if and only if *C*\\*(2)(*Y*, *ϕ*, *γ*, *t*) is, and \n$$T^{(2)}(X, \\phi \\circ f\\_\\*, \\gamma \\circ f\\_\\*)(t) \\ \\dot{=} \\\nT^{(2)}(Y, \\phi, \\gamma)(t).$$ \nIf *N* is a compact smooth 3-manifold, then it follows from theorems due to Chapman and Cohen (see ) that any two CW-structures on *N* are simple homotopy equivalent. Proposition [prop L2 torsion simple homotopy] thus implies that for any admissible triple (*π*1(*N*), *ϕ*, *γ*), *T*(2)(*N*, *ϕ*, *γ*) is a well-defined topological invariant of *N*.\nCW-complexes of the form *W* × *S*1\n-----------------------------------\nLet *W* be a finite CW-complex, *S*1 the one-dimensional circle with its classical CW-complex structure (one 0-cell and one 1-cell). Let *X* = *W* × *S*1 be the product space, whose CW-structure is induced by the direct product. We prove how to relate the *L*2-Alexander torsions of *X* to the Euler characteristic *χ*(*W*) of *W*, as a generalisation of. Let *π**W* = *π*1(*W*), *T* a fixed generator of *π*1(*S*1) ≅ Z and *π**X* = *π*1(*X*) ≅ *π**W* × Z. We consider the natural inclusion $S^1 \\hookrightarrow S^1 \\times \\{p\\_W\\} \\subset X$ (where *p**W* is the chosen basepoint on *W*) and *i*: *π*1(*S*1) → *π**X* the induced group homomorphism.\n[prop WS1] Let *t* > 0 and let *ϕ* : *π**X* → Z and *γ* : *π**X* → *G* such that (*π**X*, *ϕ*, *γ*) is an admissible triple. If (*γ* ∘ *i*)(*T*) has infinite order in *G*, then *C*\\*(2)(*X*, *ϕ*, *γ*, *t*) is weakly acyclic and of determinant class, and \n$$T^{(2)}(X,\\phi, \\gamma)(t)\n\\ \\dot{=} \\ \\max(1,t)^{- \\chi(W) |(\\phi \\circ i)(T)|}.$$ \nLet *n* be the dimension of the CW-complex *W*, and *c**k* the number of *k*-cells of *W* of dimension *k*, for 0 ≤ *k* ≤ *n*. One has immediately *χ*(*W*) = ∑*k* = 0*n*( − 1)*k**c**k*. Let us denote *e**i**k* the *k*-cells of *W*, and *P*, *a* the 0-cell and the 1-cell of *S*1. We consider the cellular chain complex of Z[*π**X*]-modules $C\\_\\*(\\widetilde{X})$. We fix lifts $\\widetilde{e^k\\_i \\times P}$ and $\\widetilde{e^{k-1}\\_j \\times a}$ as bases of the Z[*π**X*]-modules (with 0 ≤ *i* ≤ *c**k*, 0 ≤ *j* ≤ *c**k* − 1).\nIf we denote the boundary of the cells in $C\\_\\*(\\widetilde{W})$ in the following way: \n$$\\partial \\left (\\widetilde{e^k\\_i}\\right ) = \\sum\\_{j=1}^{c\\_{k-1}} \\lambda\\_{k,i,j} \\ g\\_{k,i,j} \\ \\widetilde{e^{k-1}\\_j}$$ \n where *λ**k*, *i*, *j* ∈ C, *g**k*, *i*, *j* ∈ *π**W*, then the boundary operators in $C\\_\\*(\\widetilde{X})$ act as: \n$$\\partial \\left (\\widetilde{e^k\\_i \\times P}\\right ) = \\sum\\_{j=1}^{c\\_{k-1}} \\lambda\\_{k,i,j} \\ g\\_{k,i,j} \\ \\widetilde{e^{k-1}\\_j \\times P},$$ \n \n$$\\partial \\left (\\widetilde{e^k\\_i \\times a}\\right ) =\n(i(T)-1) \\widetilde{e^k\\_i \\times P} +\n \\sum\\_{j=1}^{c\\_{k-1}} \\lambda\\_{k,i,j} \\ g\\_{k,i,j} \\ \\widetilde{e^{k-1}\\_j \\times a}.$$ \n Thus the boundary operators are of the matricial form: \n$$\\kbordermatrix{\n & \\ \\ \\ \\ & \\widetilde{e^k\\_i \\times P} \\ \\ \\ \\ & \\vrule & & \\widetilde{e^{k-1}\\_j \\times a} & \\\\\n & & & \\vrule & i(T)-1 & & 0 \\\\\n\\widetilde{e^{k-1}\\_j \\times P} & & \\* \\ \\ \\ & \\vrule & & \\ddots & \\\\\n & & & \\vrule & 0 & & i(T)-1 \\\\\n \\hline\n & & & \\vrule & & & \\\\\n\\widetilde{e^{k-2}\\_l \\times a} & & 0 & \\vrule & & \\* & \\\\\n & & & \\vrule & & & \n},$$ \n where the square upper right block is of size *c**k* − 1.\nFrom the generalisation of Proposition [prop tau chaine] to any dimension, one can compute *T*(2)(*X*, *ϕ*, *γ*)(*t*) from the Fuglede-Kadison determinants of the corresponding upper right block operators"},"data_id":{"kind":"string","value":"arxiv_0181651"}}},{"rowIdx":45392,"cells":{"content":{"kind":"string","value":"Topological insulators and superconductors: ten-fold way and dimensional hierarchy\n==================================================================================\nIt has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a Z or a Z2 topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension *d*, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via “dimensional reduction” by compactifying one or more spatial dimensions (in “Kaluza-Klein”-like fashion). For Z-topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The Z2-topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent Z-topological insulators in the same class, from which they inherit their topological properties. The 8-fold periodicity in dimension *d* that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle-hole symmetries) is a reflection of the 8-fold periodicity of the spinor representations of the orthogonal groups SO(*N*) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern-Simons invariant. For lower dimensional cases, this formula relates the winding number to the electric polarization (*d* = 1 spatial dimensions), or to the magnetoelectric polarizability (*d* = 3 spatial dimensions). Finally, we also discuss topological field theories describing the space time theory of linear responses in topological insulators (superconductors), and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).\nIntroduction\n============\nTopological insulators (superconductors) are gapped phases of non-interacting fermions which exhibit topologically protected boundary modes. These boundary states are gapless, with extended wavefunctions, and protected against arbitrary deformations (or perturbations) of the Hamiltonian, as long as the generic symmetries (such as e.g. time-reversal symmetry) of the Hamiltonian are preserved and the bulk gap is not closed [1](#fn1). The different phases within a given topological insulator (superconductor) are characterized by a topological invariant, which is either an integer Chern/winding number or a Z2 quantity. The integer quantum Hall effect (IQHE) is the best known example of such a phase. In the IQHE, the transverse (Hall) electrical conductance *σ**x**y* carried by edge states is quantized and proportional to the topologically invariant Chern number [2](#fn2). As a consequence of the quantization of the Hall conductance *σ**x**y*, two different quantum Hall states with different values of *σ**x**y* cannot be adiabatically connected without closing the energy gap in the bulk. Therefore they represent distinct “topologically ordered” phases that are separated by a quantum phase transition (figure [topinsulators]a).\nOther examples of gapped topological phases are the (spinless) chiral *p**x* + i*p**y* superconductor, which breaks time-reversal symmetry (T), and the quantum spin Hall (QSH) states, which are time-reversal invariant. In the former case, the topological features in question are those of the fermionic quasiparticle excitations deep in the superconducting phase, whose dynamics is described by the Bogoliubov-de Gennes (BdG) Hamiltonian [3](#fn3). The BdG Hamiltonian of any superconductor has, by construction, a “built-in” charge-conjugation (or: particle-hole) symmetry (C). In analogy to the IQHE, the different topological phases of the *p**x* + i*p**y* superconductor can be labeled by an integer (Z) topological invariant. The corresponding quantized conductance is the transverse (Hall) thermal, or “Leduc-Righi” conductance (divided by temperature). The quantum spin Hall effect (QSHE), on the other hand, which occurs in two-dimensional and three-dimensional time-reversal invariant insulators, is characterized by a Z2 topological number *ν*0. This binary quantity distinguishes the non-trivial state (*ν*0 = 1), whose boundary modes consist of an odd number of Dirac fermion modes (i.e., odd number of Kramers’ pairs), from the trivial state (*ν*0 = 0), which is characterized by an even number of Kramers’ pairs of boundary (surface or edge) states (figure [topinsulators]b). A physical realization of the QSHE was theoretically predicted and experimentally observed in HgTe/(Hg,Ce)Te semiconductor quantum wells. Subsequently, based both on theoretical considerations and experimental measurements, the three-dimensional Z2 topological insulator phase was shown to occur in Bismuth-related materials, such as BiSb alloys, Bi2Se3 and Bi2Te3.\nThe key difference among the IQHE, the QSHE and three-dimensional Z2 topological insulator, and the chiral $p\\_x+\\rmi p\\_y$ superconductor lies in their generic symmetry properties (to be discussed in more detail below). All these states are topological in the sense that they cannot be continuously deformed into a trivial insulating state (i.e., a state without any gapless boundary modes) while keeping the generic symmetries of the system intact and without closing the bulk energy gap. The key property of the QSH states, for example, is the spin-orbit interaction in combination with time-reversal symmetry (T). The latter protects the boundary states from acquiring a gap or becoming localized in the presence of disorder (= terms added to the Hamiltonian which break translational symmetry). The chiral $p\\_x+\\rmi p\\_y$ superconductor, on the other hand, is characterized by C but the lack of T.\n [topinsulators]\nThe aforementioned examples of “symmetry protected” topological states are part of a larger scheme[4](#fn4) that fully classifies all topological insulators (superconductors) in terms of symmetry and spatial dimension. This classification scheme is summarized in table [periodic table]. The first column in this table provides the complete list of all possible “symmetry classes” of single-particle Hamiltonians. There are precisely ten such “symmetry classes”, a fundamental result due to Zirnbauer, and Altland and Zirnbauer. These authors recognized that there is a one-to-one correspondence between single-particle Hamiltonians and the set of (“large”) symmetric spaces, of which there are precisely ten, a classic result obtained in 1926 by the mathematician Élie Cartan. The work of Altland and Zirnbauer extends, indeed completes, the much earlier work by Wigner and Dyson most well known from the context of random matrix theory as the three “unitary, orthogonal, and symplectic” symmetry classes. Before discussing in more detail the result of the classification in table [periodic table], we will first briefly explain the notion of “symmetry class” and the ten-fold list of Hamiltonians, which is the framework within which the classification scheme in table [periodic table] is formulated. The reader familiar with this topic may skip this subsection.\nReview: Classification of generic Hamiltonians – “The Ten-Fold Way”\n-------------------------------------------------------------------\nConsider the gapped first-quantized Hamiltonian describing the topological insulator (superconductor) in the bulk (*d* spatial dimensions). The topological features of this Hamiltonian, which we are interested in characterizing, are (by definition) not changed if we modify (deform) the Hamiltonian by adding terms (or perturbations) to it which break any translational symmetry that may be present. Thus the topological features must be properties of a translationally *not* invariant Hamiltonian. Seeking a classification scheme, one must have some framework within which to classify such translationally *not* invariant Hamiltonians. Such a classification cannot involve the notion of ordinary symmetries, i.e., of *unitary* operators which commute with the Hamiltonian. Consider, e.g., (Pauli-) spin rotation symmetry, ubiquitous in condensed matter systems. The notion of spin-rotation invariance can be eliminated by writing the Hamiltonian in block form, and by focussing attention on a block of the Hamiltonian. Such a block decomposition can be performed for any symmetry which commutes with the Hamiltonian. The notion of “symmetry classes” mentioned above refers to the properties of these blocks of symmetry-less “irreducible” Hamiltonians: as it turns out, there are only ten of them. The basic idea why this is the case is simple to understand. The only properties that the blocks can satisfy are certain reality conditions that follow from the “extremely generic symmetries” of time-reversal and charge-conjugation (or: particle-hole symmetry). These are not symmetries in the above sense, since they are both represented by *anti-*unitary operators, when acting on the single-particle Hilbert space. Specifically, consider a general system of non-interacting fermions described by a “second-quantized” Hamiltonian *H*. For a non-superconducting system, e.g., this reads\n \n$$\\begin{aligned}\nH\n= \\sum\\_{A,B} \\, {\\psi}^\\dagger\\_{A} \\, \\mathcal{H}^{\\ }\\_{A,B} \\, {\\psi}^{\\ }\\_{B},\\end{aligned}$$ \nwith fermion creation and annihilation operators satisfying canonical anti-commutation relations\n \n$$\\begin{aligned}\n\\{{\\psi}^{\\ }\\_{A},{\\psi}^\\dagger\\_{B} \\} = \\delta\\_{A,B}. \\end{aligned}$$ \nHere we imagine for convenience of notation that we have “regularized” the system on a lattice, and *A*, *B* are combined labels for the lattice sites *i*, *j*, and if relevant, of additional quantum numbers such as e.g., a Pauli-spin quantum number (e.g., *A* = (*i*, *a*) with *a*, *b* = ± 1/2). Then ${\\cal H}\\_{A,B}$ is an *N* × *N* matrix, the “first quantized” Hamiltonian. (Similarly, a superconducting system is described by a BdG Hamiltonian for which we use the Nambu-spinor instead of complex fermion operators *ψ**A*, *ψ**A*† and whose first quantized form is again a matrix ${\\cal H}$ when discretized on a lattice.)\nNow, time reversal symmetry can be expressed in terms of ${\\cal H}$: the system is invariant under time reversal symmetry if and only if the complex conjugate of the first quantized Hamiltonian ${\\cal H}^\\*$ is equal to ${\\cal H}$ up to a unitary rotation *U**T*, i.e.\n \n$$\\label{DEFTimeReversal}\n\\mathcal{T}: \n\\quad \nU\\_{T}^\\dagger\n \\ {\\cal H}^{\\*} \\ \nU\\_{T} = + {\\cal H }.$$ \nMoreover, the system is invariant under charge-conjugation (or: particle-hole) symmetry if and only if the complex conjugate of the Hamiltonian ${\\cal H}^\\*={\\cal H}^T$ is equal to *minus* ${\\cal H}$ up to a unitary rotation *U**C*, i.e.\n \n$$\\label{DEFChargeConjugation}\n\\mathcal{C}: \n\\quad \nU\\_{C}^\\dagger\n \\ {\\cal H}^{\\*} \\ \nU\\_{C} = - {\\cal H}.$$ \n(This property may be less familiar, but it is easy to check that it is a characterization of charge-conjugation (particle-hole) symmetry for non-interacting systems of fermions). A look at equations ([DEFTimeReversal],[DEFChargeConjugation]) reveals that T and C, when acting on the single particle Hilbert space, are not unitary symmetries, but rather reality conditions on the Hamiltonian ${\\cal H}$ *modulo* unitary rotations [5](#fn5) [6](#fn6).\nNow it is easy to see that there are only ten possible ways for a system to respond to time-reversal and charge-conjugation (particle-hole symmetry) operations. As for time-reversal symmetry (T), the Hamiltonian can either be (i): not time-reversal invariant, in which case we write T = 0, or (ii): it may be time-reversal invariant and the anti-unitary time-reversal symmetry operator T squares to plus the identity operator, in which case we write T = + 1, or (iii): it may be time-reversal invariant and the anti-unitary time-reversal symmetry operator T squares to minus the identity, in which case we write T = − 1. Similarly, there are three possible ways for the Hamiltonian ${\\cal H}$ to respond to charge-conjugation (particle-hole symmetry) C (again, C may square to plus or minus the identity operator). For these three possibilities we write C = 0, + 1, − 1. Hence, there are 3 × 3 = 9 possible ways for a Hamiltonian ${\\cal H}$ to respond to time-reversal and charge-conjugation (particle-hole transformation). These are not yet all ten cases because it is also necessary to consider the behavior of the Hamiltonian under the product S = T ⋅ C which is a unitary operation. A moment’s thought (see table [TableSymmetryClassesTwo]) shows that for 8 of the 9 possibilities the behavior of the Hamiltonian under the product S = T ⋅ C is uniquely fixed [7](#fn7). (We write S = 0 if the operation S is not a symmetry of the Hamiltonian, and S = 1 if it is.) The only case when the behavior under the combined transformation S = T ⋅ C is not determined by the behavior under T and C is the case where T = 0 and C = 0. In this case either S = 0 or S = 1 is possible. This then yields (3 × 3 − 1) + 2 = 10 possible behaviors of the Hamiltonian.\nThe list of ten possible behaviors of the first quantized Hamiltonian under T, C and S is listed in table [TableSymmetryClassesTwo]. These are the ten generic symmetry classes (the “ten-fold way”) which are the framework within which the classification scheme of topological insulators (superconductors) is formulated.\nLet us first point out a very general structure seen in table [TableSymmetryClassesTwo]. This is listed in the column entitled “Hamiltonian”. When the first quantized Hamiltonian ${\\cal H}$ is “regularized” (or: “put on”) a finite lattice, it becomes an *N* × *N* matrix (as discussed above). The entries in the column “Hamiltonian” specify the type of *N* × *N* matrix that the quantum mechanical time-evolution operator $\\exp( \\mathrm{i} t {\\cal H} )$ is. For example, for systems which have no time-reversal or charge-conjugation symmetry properties at all, i.e., for which T = 0, C = 0, S = 0, which are listed in the first row of the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus ${\\cal H}$ is a generic Hermitian matrix and the time-evolution operator is a generic unitary matrix, so that $\\exp( \\mathrm{i} t {\\cal H} )$ is an element of the unitary group U(*N*) of unitary *N* × *N* matrices. By imposing time-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g., spin), there exists a basis in which ${\\cal H}$ is represented by a real symmetric *N* × *N* matrix. This, in turn, can be expressed as saying that the time-evolution operator is an element of the coset of groups, U(*N*)/O(*N*). All other entries of the column “Hamiltonian” can be obtained from analogous considerations [8](#fn8). What is interesting about this column is that its entries run precisely over what is known as the complete set of ten (“large”) symmetric spaces, classified in 1926 in fundamental work by the mathematician Élie Cartan. Thus, as the first quantized Hamiltonian runs over all ten possible symmetry classes, the corresponding quantum mechanical time-evolution operator runs over all ten symmetric spaces. Thus, the appearance of the Cartan symmetric spaces is a reflection of fundamental aspects of (single-particle) quantum mechanics. We will discuss the last column entitled “*G*/*H* (ferm. NL*σ*M)” in the following subsection.\nc||c|c|c||c||c & T&C &S & Hamiltonian & *G*/*H* (ferm. NL*σ*M) \nA (unitary) &0 &0&0&U(*N*)&U(2*n*)/U(*n*) × U(*n*) \nAI (orthogonal) & + 1&0&0&U(*N*)/O(*N*)& Sp(2*n*)/Sp(*n*) × Sp(*n*) \nAII (symplectic) & − 1&0&0&U(2*N*)/Sp(2*N*)& O(2*n*)/O(*n*) × O(*n*) \nAIII (ch. unit.) &0&0&1&U(*N* + *M*)/U(*N*) × U(*M*) &U(*n*) \nBDI (ch. orth.) & + 1& + 1&1&O(*N* + *M*)/O(*N*) × O(*M*)& U(2*n*)/Sp(2*n*) \nCII (ch. sympl.) & − 1& − 1&1&Sp(*N* + *M*)/Sp(*N*) × Sp(*M*)& U(2*n*)/O(2*n*) \nD (BdG) &0 & + 1&0 &SO(2*N*) &O(2*n*)/U(*n*) \nC (BdG) &0 & − 1&0 &Sp(2*N*) & Sp(2*n*)/U(*n*) \nDIII (BdG) & − 1& + 1&1 &SO(2*N*)/U(*N*) & O(2*n*) \nCI (BdG) & + 1& − 1&1 &Sp(2*N*)/U(*N*) & Sp(2*n*) \nReview: Classification of Topological Insulators (Superconductors)\n------------------------------------------------------------------\nThe approach used in Ref. to classify all possible topological insulators (superconductors) rested on the existence of protected extended degrees of freedom at the system’s boundary. These are, in particular, also protected in the presence of arbitrarily strong perturbations at the boundary which break translational symmetry (commonly referred to as “random”, or “disordered”). The existence of extended, gapless degrees of freedom even in strongly random fermionic systems is highly unusual, because of the phenomenon of Anderson localization [9](#fn9). Thus, the degrees of freedom at the boundary of topological insulators (superconductors) must be of a very special kind, in that they entirely evade the phenomenon of Anderson localization. Our approach consists in classifying precisely all those problems of (non-interacting) fermionic systems which completely evade the phenomenon of Anderson localization. We have completed this task in Ref. . Thus, we have reduced the problem of classifying all topological insulators (superconductors) in *d* spatial bulk dimensions to a classification problem of Anderson localization at the (*d* − 1) dimensional boundary. By solving the mentioned problem of Anderson localization, we have thereby solved the problem of classifying all topological insulators (superconductors).\nWe will now focus attention on the above-mentioned problem of Anderson localization at the (*d* − 1) dimensional boundary of a *d* dimensional topological insulator (superconductor). Specifically, we will now review a solution of this problem which allows us to see directly the dependence on dimensionality *d* of the classification of topological insulators (superconductors). In the later, main part of this article, we will present another point of view of this dependence on dimensionality *d* (using “dimensional reduction”).\nThe theoretical description of problems of Anderson localization is well known to be very systematic and geometrical. A problem of Anderson localization is in general described by a random Hamiltonian (i.e., one that lacks translational symmetry). That Hamiltonian will be in one of the ten symmetry classes listed in table [TableSymmetryClassesTwo] and we are currently focusing on Hamiltonians describing the boundary of the topological insulator (superconductor). Now, as it turns out, at long length scales (much larger than the “mean free path”) a description in terms of a “non-linear-sigma-model” (NL*σ*M) emerges. A NL*σ*M is a system like that describing the classical statistical mechanics of a Heisenberg magnet. The only difference is that while the magnet is formulated in terms of unit vector spins, pointing to the surface of a two-dimensional sphere, for a general NL*σ*M that spin is replaced [10](#fn10) by an element of one of the ten symmetric spaces listed in the last column of table [TableSymmetryClassesTwo], called the “target space” of the NL*σ*M, denoted by *G*/*H* [11](#fn11). For a given symmetry class of the original Hamiltonian, whose time-evolution operator is characterized by the penultimate column, the specific “target space” that needs to be used for the NL*σ*M describing Anderson localization in this symmetry class is listed in the last column of table [TableSymmetryClassesTwo] (see also appendix [subsec:RMT]).\nNow, the NL*σ*M on the (*d* − 1) dimensional boundary of the *d* dimensional topological insulator (superconductor) completely evades Anderson localization if a certain extra *term of topological origin* can be added to the action of the NL*σ*M which has *no adjustable parameter*. Whether such an extra term is allowed depends on (i): the “target space” of the NL*σ*M in the symmetry class in question[12](#fn12), and (ii): the dimensionality *d̄* : = (*d* − 1) of the boundary on which the NL*σ*M is defined. There are only three *terms of topological origin* which can possibly be added to the action of the NL*σ*M: these are a *θ*-term (Pruisken term), a Z2 topological term, or a Wess-Zumino-Witten (WZW) term. It is the homotopy groups of the NL*σ*M target spaces *G*/*H* which determine whether it is possible to add such a topological term to a given NL*σ*M action (see table [homotopy]). Specifically, a *θ*-term (Pruisken term) can appear when *π**d̄*(*G*/*H*) = Z. Similarly, a Z2 topological term is allowed when *π**d̄*(*G*/*H*) = Z2, and a WZW term can be included [13](#fn13) when *π**d̄* + 1(*G*/*H*) = Z. Note that, on the one hand, one obtains, upon addition of a *θ*-term (Pruisken term), a one-parameter family of theories (on the boundary) depending on the value of *θ*, all of which reside in the same symmetry class. On the other hand, however, it is known that only for a special value of the parameter *θ* Anderson localization is avoided; for generic values of *θ*, this is not the case. It is for this reason that the ability to add a *θ*-term (Pruisken term) is not of interest to the question we are asking. One is therefore left only with a Z2 topological term or a WZW term as the only terms of topological origin which have no freely adjustable parameter, and which are thus of relevance here.\nThe homotopy groups for all ten NL*σ*M target spaces *G*/*H* listed in the last column of table [TableSymmetryClassesTwo] are well known from the literature (see, e.g., Ref. for a summary) and this information is summarized in table [homotopy] for the convenience of the reader. In order to make a certain regular structure of this table apparent, the rows of table [TableSymmetryClassesTwo] have been re-ordered in a specific way. Part of this re-ordering is a subdivision of all symmetry classes into what is called “*complex case*” and “*real case*” in table [homotopy]. The physical origin of this subdivision is simple to understand. In the category “*complex case*” appear precisely those symmetry classes in which there is no reality condition (T or C) whatsoever imposed on the Hamiltonian. We see from table [TableSymmetryClassesTwo] that these are the classes which carry Cartan label A and AIII. The Hamiltonians in these symmetry classes are therefore “complex” (in this sense). The other category called “*real case*” in table [homotopy] consists precisely of all the other eight symmetry classes which have the property that there is at least one reality condition [T or C – cf. equations ([DEFTimeReversal],[DEFChargeConjugation])] imposed on the Hamiltonian (see table [TableSymmetryClassesTwo]). In this sense the Hamiltonians in these symmetry classes are therefore “real”.\n[h]\n#### complex case:\nccccccccccccAZ & *G*/*H* & & *d̄* = 0 & *d̄* = 1 & *d̄* = 2 & *d̄* = 3 & *d̄* = 4 & *d̄* = 5 & *d̄* = 6 & *d̄* = 7 & \nA & $\\scriptstyle \\mathrm{U}(N+M)/\\mathrm{U}(N)\\times \\mathrm{U}(M)$ & & & & & \n & $\\scriptstyle \\mathrm{U}(N)$ & & & & & & \n#### real case:\nccccccccccccAZ & *G*/*H* & & *d̄* = 0 & *d̄* = 1 & *d̄* = 2 & *d̄* = 3 & *d̄* = 4 & *d̄* = 5 & *d̄* = 6 & *d̄* = 7 & \nAI &\n$\\scriptstyle \\mathrm{Sp}(N+M)/\\mathrm{Sp}(N)\\times \\mathrm{Sp}(M)$\n& & 0& 0 & & & & \n & $\\scriptstyle \\mathrm{U}(2N)/\\mathrm{Sp}(2N)$ & & & 0 & 0 & & & & \nD & $\\scriptstyle \\mathrm{O}(2N)/\\mathrm{U}(N)$ & & & & 0 & 0 & & & \n & $\\scriptstyle \\mathrm{O}(N)$ & & & & & 0 & 0 & & \nAII & $\\scriptstyle \\mathrm{O}(N+M)/\\mathrm{O}(N)\\times \\mathrm{O}(M)$ & & & & & 0 & 0 & \n & $\\scriptstyle \\mathrm{U}(N)/\\mathrm{O}(N)$ & & & & & & 0 & 0 & \nC & $\\scriptstyle \\mathrm{Sp}(2N)/\\mathrm{U}(N)$ & & 0 & & & & & 0 & \n & $\\scriptstyle \\mathrm{Sp}(2N)$ & & 0 & 0 & & & & \nTable [homotopy] now tells us directly in which symmetry class there exist topological insulators or superconductors (and of which type, Z2 or Z): according to the above discussion we know that a topological insulator (superconductor) exists in a given symmetry class in *d* = *d̄* + 1 spatial dimensions if and only if the target space of the NL*σ*M on the *d̄*-dimensional boundary allows either (a): for Z2 topological term, which is the case when *π**d̄*(*G*/*H*) = *π**d* − 1(*G*/*H*) = Z2, or (b): allows for a WZW term, which is the case when *π**d*(*G*/*H*) = *π**d̄* + 1(*G*/*H*) = Z. By using this rule in conjunction with table [homotopy] of homotopy groups, we arrive with the help of table [TableSymmetryClassesTwo] at the table [periodic table] of topological insulators and superconductors [14](#fn14).\nA look at table [periodic table] reveals that in each spatial dimension there exist five distinct classes of topological insulators (superconductors), three of which are characterized by an integral (Z) topological number, while the remaining two possess a binary (Z2) topological quantity [15](#fn15).\n[t!]\n#### complex case:\n| Cartan ∖ *d* | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | ⋯ |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| A | Z | 0 | Z | 0 | Z | 0 | Z | 0 | Z | 0 | Z | 0 | ⋯ |\n| AIII | 0 | Z | 0 | Z | 0 | Z | 0 | Z | 0 | Z | 0 | Z | ⋯ |\n#### real case:\n| Cartan ∖ *d* | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | ⋯ |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| AI | Z | 0 | 0 | 0 | 2Z | 0 | Z2 | Z2 | Z | 0 | 0 | 0 | ⋯ |\n| BDI | Z2 | Z | 0 | 0 | 0 | 2Z | 0 | Z2 | Z2 | Z | 0 | 0 | ⋯ |\n| D | Z2 | Z2 | Z | 0 | 0 | 0 | 2Z | 0 | Z2 | Z2 | Z | 0 | ⋯ |\n| DIII | 0 | Z2 | Z2 | Z | 0 | 0 | 0 | 2Z | 0 | Z2 | Z2 | Z | ⋯ |\n| AII | 2Z | 0 | Z2 | Z2 | Z | 0 | 0 | 0 | 2Z | 0 | Z2 | Z2 | ⋯ |\n| CII | 0 | 2Z | 0 | Z2 | Z2 | Z | 0 | 0 | 0 | 2Z | 0 | Z2 | ⋯ |\n| C | 0 | 0 | 2Z | 0 | Z2 | Z2 | Z | 0 | 0 | 0 | 2Z | 0 | ⋯ |\n| CI | 0 | 0 | 0 | 2Z | 0 | Z2 | Z2 | Z | 0 | 0 | 0 | 2Z | ⋯ |\n[periodic table]\nThe topological insulators (superconductors) appear in table [periodic table] along diagonal lines. That is, as the spatial dimension is increased by one, locations where the topological insulators (superconductors) appear in the table shift down by one column. This entire pattern descends directly from an analogous one appearing in the table [homotopy] of homotopy groups.\nIn the main part of the present paper we will give another derivation of this periodicity structure of table [periodic table], by “dimensional reduction” and by considering the Dirac Hamiltonian representatives in each of the symmetry classes in which there exist topological insulators (superconductors) in a given spatial dimension. By running the argument backwards, this can then, in turn, also be viewed as another route to obtain the table [homotopy] of homotopy groups.\nLet us pause for a moment to point out a few interesting aspects relating to table [periodic table]. First, it is interesting to note, that the periodic structure of table [periodic table] is quite analogous to the dimensional hierarchy of anomalies (“anomaly ladder”) known in gauge theories, e.g., the (3+1)-dimensional Abelian anomaly, the (2+1)-dimensional parity anomaly, and the (1+1)-dimensional non-Abelian anomaly. This may not appear to be completely unexpected, considering that the IQHE is known as a condensed matter realization of the parity “anomaly” [16](#fn16).\nSecond, we want to draw the reader’s attention to the fact that for a given symmetry class in table [periodic table] any Z2 topological insulator (superconductor) always appears as part of a “triplet” that consists of a *d*-dimensional Z topological insulator (superconductor) and two Z2 topological insulators (superconductors) in (*d* − 1) and (*d* − 2) dimensions. This suggests that the topological characteristics of the members of such a “triplet” are closely related and indeed, it was shown in Ref. that the Z2 classification in symmetry class AII in *d* = 2 and 3 can be derived from the four-dimensional Z topological insulator by a process of “dimensional reduction”. The same procedure has been applied to derive Z2 classifications in those symmetry classes that do not possess a form of chiral symmetry (i.e., for which S = 0) – there are five of them. In section [sec:Z2dimreduction] we will extend this approach to the Z2 topological insulators (superconductors) in the remaining five symmetry classes which possess a form of chiral symmetry (i.e., for which S = 1).\nLet us now discuss explicit forms of the “terms of topological origin” which can be added to the action of the NL*σ*M with “target space” *G*/*H*, at the boundary of the topological insulator (superconductor).\n### Terms of WZW type.\nHere there is a pattern, alternating in the dimension *d̄* of the boundary, for the way these topological terms can be constructed. (i): For a *d̄* = odd-dimensional boundary, and for the NL*σ*Ms which describe the Anderson-delocalized boundary of the topological bulk, the field (denoted by *Q* ∈ *G*/*H* below – cf. last column of table [TableSymmetryClassesTwo]) of the NL*σ*M field theory is a Hermitean matrix field which satisfies certain constraints. The WZW term responsible for the lack of Anderson localization takes on the form of the *d̄*-dimensional integral over the boundary of the *d̄*-dimensional Chern-Simons form (compare equation of the main text below). Alternatively, it can be written also as an integral over a *d̄* + 1-dimensional region whose boundary coincides with the physical *d̄*-dimensional space, by smoothly interpolating the NL*σ*M field configuration *Q* into one dimension higher. The fact that *π**d̄* + 1(*G*/*H*) = Z guarantees that different ways of interpolations do not matter, and the action depends only on the physical field configurations in *d̄*-dimensional space. (ii): For a *d̄* = even-dimensional boundary, and for the NL*σ*Ms which describe the Anderson-delocalized boundary of the topological bulk, the field in the NL*σ*M field theory is either a group element or a unitary matrix field (denoted by *g* ∈ *G*/*H* below – cf. last column of table [TableSymmetryClassesTwo]) which may, depending on the case, be subject to certain constraints. The WZW term responsible for the lack of Anderson localization takes on the form of a *d̄* + 1-dimensional integral of the winding number density, as defined in equation of the main text below.\nTo illustrate these terms, we will now give explicit examples for them in low dimensionalities:\n* For the *d* = 2-dimensional topological insulator of the IQHE, the *d̄* = 1-dimensional NL*σ*M describing the edge states is the one on U(2*n*)/U(*n*) × U(*n*). The field *Q* of the NL*σ*M field theory can be parameterized as *Q* = *U*†Λ*U* ∈ U(2*n*)/U(*n*) × U(*n*) where *U* ∈ U(2*n*) and Λ = diag (*I**n*, − *I**n*). The term of relevance for the absence of Anderson localization at the edges of the integer quantum Hall insulator is ∝ *σ**x**y*∫*d**x* tr[*A**x*] where *A**x* : = *U*†∂*x**U*; this is a *d̄* = 1 dimensional analogue of the three-dimensional Chern-Simons term. This term can also be rewritten as a WZW type two-dimensional integral ∝ *σ**x**y*∫*D**d**x**d**u* *ε**μ**ν*tr [*Q*∂*μ**Q*∂*ν**Q*] (*μ*, *ν* = *x*, *u*). Here, we have extended the original *d̄* = 1 dimensional space to a two-dimensional region *D* by adding a fictitious space direction, parameterized by *u* ∈ [0, 1]. The boundary ∂*D* of the two-dimensional region coincides with the original *d̄* = 1 dimensional space. Accordingly, the original NL*σ*M field *Q*(*x*) is smoothly extended to *Q*(*x*, *u*) such that it coincides with *Q*(*x*) when *u* = 0. Since *π*2[U(2*n*)/U(*n*) × U(*n*)] = Z the two-dimensional integral turns out to depend only on the field configuration on ∂*D*. Note that when we specialize to the case of *n* = 1, all these expressions are well-known from the coherent state path integral of an SU(2) quantum spin whose path integral is described by a *d* = 1 dimensional NL*σ*M on the target space O(3) = U(2)/U(1) × U(1).\n* : For *d* = 3 topological insulators (superconductors) in classes AIII, DIII and CI, the relevant NL*σ*Ms at their *d̄* = 2-dimensional boundary (surface) are the ones with a group manifold as target manifold [U(*n*), O(*n*), Sp(*n*) for classes AIII, DIII, and CI, respectively]. The relevant WZW type term can be written as a three-dimensional integral, ∫*D**d*2*x* *d**u* *ε**μ**ν**λ*tr [(*g*− 1∂*μ**g*)(*g*− 1∂*ν**g*)(*g*− 1∂*λ**g*)], where *g* ∈ U(*n*), O(*n*), Sp(*n*). Here again, the original *d̄* = 2 dimensional space is smoothly extended to the three-dimensional region *D*, in such a way that ∂*D* coincides with the physical *d̄* = 2 dimensional space. Accordingly, the field configuration *g*(*x*, *y*) defined on the *d̄* = 2 dimensional space is smoothly extended to the three-dimensional one *g*(*x*, *y*, *u*) such that *g*(*x*, *y*, *u* = 0) = *g*(*x*, *y*).\n* : For the *d* = 4 topological insulators in classes A, AII, and AI, the relevant *d̄* = 3 NL*σ*Ms have target spaces *G*/*H* = G(2*n*)/G(*n*) × G(*n*) with G = U, O, and Sp respectively. The field of the NL*σ*M can be parameterized by *Q* = *U*†Λ*U* where *U* ∈ G(2*n*). The relevant “term of topological origin” is the Chern-Simons term ∝ ∫*d*3*x* *ε**μ**ν**λ*tr [*A**μ*∂*ν**A**λ* + (2/3)*A**μ**A**ν**A**λ*] where *A**μ* = *U*†∂*μ**U* (*μ* = *x*, *y*, *z*). The appearance of this term at the boundary of the *d* = 4 topological insulators in class AII was discussed in Ref. .\n* : For the *d* = 5 topological insulators in symmetry classes AIII, BDI, CII, the *d̄* = 4 dimensional NL*σ*Ms have target spaces with a matrix field *g* which is an element of *G*/*H* = U(*n*), U(2*n*)/Sp(2*n*), U(2*n*)/O(2*n*), respectively. For these spaces the WZW term takes the form of the *d̄* + 1 = 5 dimensional integral (with boundary) of the winding number density defined in equation of the main text below.\nNote that those entries in table [homotopy] of homotopy groups which are integers, *π**d̄*(*G*/*H*) = Z, have a periodicity in *d̄* equal to four. Therefore, the forms of the “terms of topological origin” of WZW type listed in the above low-dimensional cases will repeat, with the appropriate replacement of *d̄*.\n.5cm\n### Terms of topological origin of Z2 type.\nThe Z2 topological term for the NL*σ*M at the *d̄* = 2 dimensional surface of the *d* = 3-dimensional Z2 topological insulators in symmetry class AII was discussed in Refs. . Similarly, at the *d̄* = 2 dimensional surface of the *d* = 3-dimensional Z2 topological insulators in symmetry class CII, the Z2 topological term is added to the NL*σ*M.\n.5cm\n### Classifying space.\nThere is a third way in which the set of Cartan symmetric spaces listed in table [TableSymmetryClassesTwo] appear in the context of the classification of topological insulators (superconductors). This arises from Fermi statistics which allows for a distinction between those states of the (first quantized) Hamiltonian ${\\cal H}$ with energies which lie above, and those that lie below the Fermi energy *E**F*. In a topological insulator (superconductor) there is always an energy gap between the Fermi energy *E**F* and the eigenstates of ${\\cal H}$ lying above as well as those lying below *E**F*. Therefore one may, *without closing the bulk gap*, *continuously deform* the Hamiltonian ${\\cal H}$ into one in which all eigenstates below the Fermi energy *E**F* have the same energy (say) *E* = − 1, and all eigenstates above the Fermi energy have the same energy (say) *E* = + 1. By construction, the resulting “simplified Hamiltonian” [17](#fn17), which we often denote by *Q*, has the same topological properties as the original Hamiltonian. The structure of *Q* for all symmetry classes is listed in table III (second column) of the first article of Ref. for systems with translational symmetry. (*Q* will also be used again in the present article.) If we were to consider a “zero-dimensional” topological insulator (superconductor) where “space consisted of a single point”, then for each of the ten symmetry classes we obtain[18](#fn18) a matrix *Q* of a certain kind. It turns out that the so-obtained matrix *Q* for each symmetry class is again an element of one of the ten symmetric spaces listed in table [TableSymmetryClassesTwo], except that the order in which these spaces appear is different from the order in the two columns of table [TableSymmetryClassesTwo]. The resulting assignment is listed[19](#fn19) in the last column in table [RMT, NLsM]. The symmetric spaces to which the matrices *Q* belong are what is called the “classifying space” in the classification scheme of Ref. employing K-Theory [20](#fn20). In appendix [subsec:RMT] we collect a number of interesting relationships between these three lists of the ten symmetric spaces: (i) the unitary time evolution operator, (ii) the NL*σ*M target space, and (iii) the classifying space.\nWe end this subsection by commenting on additional information that can be read off from our main results, listed in Table [periodic table]. This pertains to the existence of so-called *weak* topological insulators (superconductors), as well as of zero-energy or extended modes localized on topological defects in topological insulators (superconductors).\n### Weak topological insulators and superconductors.\nTable [periodic table] classifies topological features of gapped free fermion Hamiltonians that do not depend on the presence of translational symmetries of a crystal lattice [21](#fn21). In particular, these properties are not destroyed when translation symmetry is broken, e.g., by the introduction of positional disorder. Systems which exhibit such robust topological properties are often also referred to as *strong* topological insulators (superconductors). This is to contrast them with so-called *weak* topological insulators (superconductors) which only possess topological features when translational symmetry is present. As soon as translational symmetry is broken, such *weak* topological features are no longer guaranteed to exist, and the system is allowed to become topologically trivial. For example, systems defined on a *d*-dimensional lattice whose momentum space is the *d*-dimensional torus *T**d*, allow for weak topological insulators which are not strong topological insulators. Such systems are topologically equivalent to parallel stacks of lower dimensional strong topological insulators (superconductors). Specifically, the three-dimensional “weak” integer quantum Hall insulator on a lattice, discussed in Ref. , is essentially a layered version of the two-dimensional integer quantum Hall insulator. Hence, it is characterized by a triplet of Chern numbers, each describing the winding of a map from the two-dimensional torus *T*2, which is a subspace of the three-dimensional momentum space *T*3, onto the complex Grassmannian, *G**m*, *m* + *n*(C). Similarly, in symmetry class AII, there exists a three-dimensional weak topological insulator, which consists of layered two-dimensional Z2 topological quantum states, and whose weak topological features are described by a triplet of Z2 invariants.\nThe three-dimensional ``weak“ integer quantum Hall insulators, as well as the weak topological insulators of Refs. are both examples of *d* = 3-dimensional weak topological insulators of ``codimension” one, i.e., they can be viewed as one-dimensional arrays of *d* = 2-dimensional strong topological insulators. The existence of these states can be read off from the *d* = 2-column and the rows of Table [periodic table] labeled A and AII, respectively. In general, *d*-dimensional weak topological states can be of any ``codimension“ *k*, with 0 < *k* ≤ *d*. The existence of weak topological insulators and superconductors in *d* spatial dimensions and for a given symmetry class can be readily inferred from table [periodic table]. That is, *d*-dimensional weak topological insulators (superconductors) of ``codimension” *k* can occur whenever there exists a strong topological state in the same symmetry class, but in *d* − *k* dimensions. Specifically, the topological order (characterized by elements in Z or in Z2) can be specified for each of the $d \\choose k$ “orientations” of the submanifold. Moreover, it is also possible that in addition ``strong\" topological order, i.e., in the full space dimension *d*, exists. In the K-Theory description of Ref. the presence of weak topological features is described by additional summands in the Abelian group of topological invariants (i.e., Z or Z2) which appears when homotopy classes of maps from the sphere (strong topological insulators) are replaced by maps from the torus *T**d* (weak and strong topological insulators).\n### Zero modes localized on topological defects.\nAnother interesting application of table [periodic table] is to determine whether *r*-dimensional topological defects in topological insulators (superconductors) can support localized (isolated) zero modes. For example, let us first consider a point-like defect (i.e., *r* = 0), which is embedded in a *d*-dimensional system (*d* ≥ 1) of any symmetry class. Furthermore, let us assume that at the defect time-reversal symmetry is broken (T = 0) but particle-hole symmetry is preserved with C = + 1, i.e., we are dealing with symmetry class D. From table [periodic table] we infer that in symmetry class D in *d* = *r* + 1 = 1 dimensions there is a Z2 classification, which implies that the (*r* = 0)-dimensional boundary can support gapless states, and, therefore, a point-like defect with the symmetries of class D can bind zero modes. This situation is realized, e.g., for a vortex core in a (*p* ± i*p*)-wave superconductor, which can bind isolated Majorana modes. In general, whether it is possible for a *r*-dimensional topological defect of a given symmetry class to support gapless states or not is determined by the entry of table [periodic table] at the intersection of column *d* = *r* + 1 and the row of the given symmetry class. An application of this general statement can for example be found in the work of Ref. , where a dislocation line (*r* = 1) in a *d* = 3-dimensional lattice hosting a weak topological insulator in symmetry class AII was found to bind an extended gapless zero mode. (This corresponds in table [periodic table] to the intersection of the row denoting symmetry class AII with the column *d* = *r* + 1 = 1 + 1 = 2, the latter representing the *d* = 3-dimensional weak topological insulator, which corresponds to a strong topological insulator in ``codimension“ *k* = 1.) To illustrate the use of table [periodic table], we can for example predict that a similar binding of extended zero modes to lattice dislocation lines (*r* = 1) can also occur in weak *d* = 3-dimensional topological insulators of ``codimension” *k* = 1 in symmetry classes A, D, DIII, and C.\nOutline of the present article\n------------------------------\nIn this paper we discuss in detail the mechanism behind the dimensional periodicity and shift property appearing in table [periodic table]. We first demonstrate that there exists, for all five symmetry classes of topological insulators or superconductors, and in all dimensions *d*, a representative of the Hamiltonian in this class which has the form of a Dirac Hamiltonian, by constructing explicitly such a representative. We use these Dirac Hamiltonian representatives in the five symmetry classes with topological insulators (superconductors) to obtain relationships between these five symmetry classes in different dimensions. First we construct dimensional hierarchies (“dimensional ladders”) relating Z topological insulators (superconductors) in different dimensions and symmetry classes (see sections [sec: dimensional ladder: complex case] and [sec: dimensional ladder: real case]). This is done by using a process of “dimensional reduction” in which spatial dimensions are compactified (in “Kaluza-Klein like” fashion), thus relating higher to lower dimensional theories. By employing the same dimensional reduction process, we derive in section [sec:Z2dimreduction] the topological classification of the Z2 topological insulators (superconductors) from their higher-dimensional parent Z topological insulators (superconductors) in the same symmetry class. Sections [sec: dimensional ladder: complex case], [sec: dimensional ladder: real case], [sec:Z2dimreduction] taken together provide a complete, independent derivation of table [periodic table] of topological insulators (superconductors).\nIn section [class AIII gen diss] we establish a connection between the topological invariant (winding number) defined for topological insulators and superconductors with chiral symmetry and yet another topological invariant, the Chern-Simons invariant, for general dimensions [equation ([eq: master formula winding num and Wilson loop])]. For lower dimensional cases, the formula ([eq: master formula winding num and Wilson loop]) relates the winding number to the electric polarization (*d* = 1 spatial dimensions), and to the the magnetoelectric polarizability (*d* = 3 spatial dimensions).\nIn section [fieldtheory] we list, for topological insulators in symmetry classes A and AIII the topological field theory describing the space time theory of linear responses in all space time dimensions *D* = *d* + 1. For symmetry class A this is the Chern-Simons action, and in class AIII it is the theta term with theta angle *θ* = *π*. (Generalizations to topological singlet superconductors with SU(2) spin-rotation symmetry are mentioned in section [Discussion]).\nFinally, as already mentioned in subsection [Review-Classification-TopIns], we have discussed for all dimensionalities the “terms of topological origin” of Wess-Zumino-Witten type which can be added to NL*σ*M field theories on symmetric spaces. In particular, in odd dimensions, these are the Chern-Simons terms.\nIn appendix [subsec:RMT] we collect a number of interesting relationships between the Cartan classification of generic Hamiltonians due to Altland and Zirnbauer, the NL*σ*M target space of Anderson localization, and the classifying space appearing in the K-theory approach to topological insulators. The representation theory of the spinor representations of the orthogonal groups SO(*N*) is briefly reviewed in appendix [app: spinors]. Furthermore, we study in appendix [secParity] the influence of inversion symmetry on the classification of topological insulators (superconductors), combined with either time-reversal or charge-conjugation (particle-hole) symmetry.\nDimensional hierarchy: complex case (A → AIII)\n==============================================\nIn this section, we consider the relationship between 2*n*-dimensional class A topological insulator and (2*n* − 1)-dimensional class AIII topological insulator. (Recall from table [TableSymmetryClassesTwo] that Hamiltonians in symmetry classes A and AIII are not invariant under time-reversal and charge-conjugation symmetries (T and C), but Hamiltonians in class AIII possess “chiral symmetry” called S in the table, i.e., they *anti-*commute with a unitary operator.) First we present a general discussion about class A and AIII topological insulators (sections [class A gen discuss] and [class AIII gen diss]) and then we will focus on Dirac Hamiltonian representatives in these symmetry classes (sections [dirac insul and dim reduc], [example eins], and [example zwei]). The insights gained from the study of these Dirac Hamiltonian examples are applicable to a wider class of insulators. Finally, in section [fieldtheory] we discuss effective topological field theories describing topological response functions of class A and AIII topological insulators.\nClass A in *d* = 2*n* + 2 dimensions\n------------------------------------\nThe class of problems that we will consider below is non-interacting fermionic systems with translation invariance. For such systems, the eigenvalue problem at each momentum *k* in the Brillouin zone (BZ) is described by\n \n$$\\begin{aligned}\n\\mathcal{H}(k)\n|u^{\\ }\\_{a}(k) \\rangle\n=\nE\\_{a} (k)|u^{\\ }\\_{a}(k) \\rangle,\n\\quad\na = 1,\\ldots, N\\_{\\mathrm{tot}}. \\end{aligned}$$ \nHere, H(*k*) is a *N*tot × *N*tot single-particle Hamiltonian in momentum space, and ∣*u**a*(*k*)⟩ is the *a*-th Bloch wavefunction with energy *E**a*(*k*). We assume that there is a finite gap at the Fermi level, and therefore, we obtain a unique ground state by filling all states below the Fermi level. (In this paper, we always adjust *E**a*(*k*) in such a way that the Fermi level is at zero energy.)\nWe assume there are *N*− (*N*+) occupied (unoccupied) Bloch wavefunctions for each *k* with *N*+ + *N*− = *N*tot. We call the set of filled Bloch wavefunctions $\n\\{\n|u^{-}\\_{\\hat a}(k) \\rangle\n\\}\n$, where hatted indices *â* = 1, …, *N*− labels the occupied bands only. We introduce the spectral projector onto the filled Bloch states and the “*Q*-matrix” by\n \n$$\\begin{aligned}\nP(k)\n=\n\\sum\\_{\\hat{a}}\n|u^{-}\\_{\\hat a}(k) \\rangle\n\\langle u^{-}\\_{\\hat a}(k) |, \n\\qquad\nQ(k) = 1 - 2P(k).\n\\label{P and Q}\\end{aligned}$$ \nThe ground state (filled Fermi sea) is characterized at each *k* by the set of normalized vectors $\n\\{\n|u^{-}\\_{\\hat a}(k) \\rangle\n\\}\n$, which is a member of U(*N*tot), or, equivalently, by *Q*(*k*), up to basis transformations within occupied and unoccupied bands. Thus *Q*(*k*) can be viewed as an element of the complex Grassmannian *G**N*+, *N*+ + *N*−(C) = U(*N*+ + *N*−)/[U(*N*+) × U(*N*−)]. For a given system, *Q*(*k*) defines a map from BZ into the complex Grassmannian, and hence classifying topological classes of band insulators is equivalent to counting how many distinct classes there are for the space of all such mappings. The answer to this question is given by the homotopy group *π**d*(*G**m*, *m* + *n*(C)), which is nontrivial in even dimensions *d* = 2*n* + 2.\nDefine for the occupied bands, the non-Abelian Berry connection \n \n$$\\begin{aligned}\n\\fl\\qquad\n\\mathcal{A}^{\\hat a \\hat b}(k)\n=\nA^{\\hat a \\hat b}\\_{\\mu}(k)\\rmd k\\_{\\mu} \n=\n\\langle u^{-}\\_{\\hat a}(k) |\\rmd u^{-}\\_{\\hat b}(k) \\rangle,\n\\quad \n\\mu=1,\\cdots,d, \n\\quad \n\\hat a, \\hat b\n=1,\\cdots, N\\_-,\\end{aligned}$$ \nwhere $A^{\\hat a \\hat b}\\_{\\mu}\n=\n-(A^{\\hat b \\hat a}\\_{\\mu})^\\*$. The Berry curvature is defined by\n \n$$\\begin{aligned}\n\\mathcal{F}^{\\hat a\\hat b} (k) \n=\n\\rmd\\mathcal{A}^{\\hat a\\hat b}\n+\n\\left(\n\\mathcal{A}^2\n\\right)^{\\hat a\\hat b}\n=\n\\frac{1}{2} \nF^{\\hat a\\hat b}\\_{\\mu\\nu}(k) \\rmd k\\_{\\mu} \\wedge\n \\rmd k\\_{\\nu}. \n\\label{eq: suN gauge}\\end{aligned}$$ \nClass A insulators in even spatial dimensions *d* = 2*n* + 2 (*n* = 0, 1, 2, …) can be characterized by the Chern form of the Berry connection in momentum space. The (*n* + 1)-th Chern character is\n \n$$\\begin{aligned}\n\\mathrm{ch}\\_{n+1} \n(\\mathcal{F})\n=\n\\frac{1}{(n+1)!}\n\\tr\n\\left(\n\\frac{ \\rmi\\mathcal{F}}{2\\pi}\n\\right)^{n+1}.\\end{aligned}$$ \nThe integral of the Chern character in *d* = 2*n* + 2 dimensions is an integer, the (*n* + 1)st Chern number,\n \n$$\\begin{aligned}\n\\label{Chern\\_number}\n\\mathrm{Ch}\\_{n+1}[\\mathcal{F}]\n=\n\\int\\_{\\mathrm{BZ}^{d=2n+2}}\n\\mathrm{ch}\\_{n+1} \n(\\mathcal{F})\n=\n\\int\\_{\\mathrm{BZ}^{d=2n+2}}\n\\frac{1}{(n+1)!}\n\\tr\n\\left(\n\\frac{ \\rmi\\mathcal{F}}{2\\pi}\n\\right)^{n+1}\n\\in \n\\mathbb{Z}.\\end{aligned}$$ \nHere, ∫BZ*d* denotes the integration over *d*-dimensional *k*-space. For lattice models, it can be taken as a Wigner-Seitz cell in the reciprocal lattice space. On the other hand, for continuum models, it can be taken as R*d*. Assuming that the asymptotic behavior of the Bloch wavefunctions approach a *k*-independent value as ∣*k*∣ → ∞, the domain of integration can be regarded as *S**d* [22](#fn22).\nWhen, *d* = 2 (*n* = 0), Ch1[F] is the TKNN integer,\n \n$$\\begin{aligned}\n\\mathrm{Ch}\\_{1}[\\mathcal{F}]\n=\n\\frac{\\rmi}{2\\pi}\n\\int\\_{\\mathrm{BZ}^{d=2}}\n\\tr\n\\left(\n\\mathcal{F}\n\\right)\n=\n\\frac{\\rmi}{2\\pi}\n\\int\n\\rmd^2 k \n\\tr\n\\left(\nF\\_{1 2}\n\\right),\\end{aligned}$$ \nwhich is nothing but the quantized Hall conductance *σ**x**y* in units of (*e*2/*h*). The second Chern-number (*n* = 1), Ch2[F],\n \n$$\\begin{aligned}\n\\mathrm{Ch}\\_{2}[\\mathcal{F}]\n=\n\\frac{-1}{8\\pi^2}\n\\int\\_{\\mathrm{BZ}^{d=4}}\n\\tr\n\\left(\n\\mathcal{F}^2\n\\right)\n=\n\\frac{-1}{32\\pi^2}\n\\int \\rmd^4 k\n\\epsilon^{\\kappa\\lambda \\mu\\nu}\n\\tr\n\\left(\nF\\_{\\kappa\\lambda}\nF\\_{\\mu\\nu} \n\\right),\\end{aligned}$$ \ncan be used to describe a topological insulator in *d* = 4, or, alternatively, a certain adiabatic “pumping process” in lower spatial dimension.\nThe Chern character ch*n* + 1 can be written in terms of its Chern-Simons form,\n \n$$\\begin{aligned}\n\\mathrm{ch}\\_{n+1}(\\mathcal{F})\n=\n\\rmd Q\\_{2(n+1)-1}(\\mathcal{A},\\mathcal{F}).\\end{aligned}$$ \nHere the Chern-Simons form is defined as\n \n$$\\begin{aligned}\nQ\\_{2n+1}(\\mathcal{A},\\mathcal{F})\n:=\n\\frac{1}{n!}\n\\left(\n\\frac{\\rmi}{2 \\pi}\n\\right)^{n+1}\n\\int^1\\_0 \\rmd t\\, \n\\tr\n\\left(\n\\mathcal{A} \\mathcal{F}^{n}\\_t \n\\right),\n\\label{eq: def CS form}\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n\\mathcal{F}\\_t\n=\nt \\rmd\\mathcal{A} + t^2 \\mathcal{A}^2 \n=\nt \\mathcal{F} \n+ (t^2-t) \\mathcal{A}^2.\\end{aligned}$$ \nFor example,\n \n$$\\begin{aligned}\nQ\\_1\\left(\\mathcal{A},\\mathcal{F}\\right)\n=\n\\frac{\\rmi}{2\\pi} \\tr\n\\mathcal{A},\n\\quad\nQ\\_3\\left(\\mathcal{A},\\mathcal{F}\\right)\n=\n\\frac{ -1}{8\\pi^2} \n\\tr\n\\left(\n\\mathcal{A} \\rmd \\mathcal{A}\n+\n\\frac{2}{3} \\mathcal{A}^3\n\\right).\\end{aligned}$$ \nClass AIII in *d* = 2*n* + 1 dimensions\n---------------------------------------\n### Winding number.\nWe now discuss band insulators in symmetry class AIII. By definition, for all class AIII band insulators, we can find a unitary matrix Γ which anticommutes with the Hamiltonians,\n \n$$\\begin{aligned}\n\\left\\{\n\\mathcal{H}(k), \\Gamma\n\\right\\}\n= \n0,\n\\quad\n\\Gamma^2=1.\n\\label{chiral symmetry}\\end{aligned}$$ \nIt follows that the spectrum is symmetric with respect to zero energy and *N*+ = *N*− = : *N*. As a consequence of the chiral symmetry ([chiral symmetry]), all class AIII Hamiltonians, as well as their *Q*-matrix ([P and Q]), can be brought into block off-diagonal form,\n \n$$\\begin{aligned}\nQ(k)\n=\n\\left(\n\\begin{array}{cc}\n0 & q(k) \\\\\nq^{\\dag}(k) & 0 \n\\end{array}\n\\right),\n\\quad\nq \\in \\mathrm{U}(N), \n\\label{eq: off diagonal Q}\\end{aligned}$$ \nin the basis in which Γ is diagonal. The off-diagonal component *q*(*k*) defines a map from BZ onto U(*N*), and classifying class AIII topological insulators reduces to considering the homotopy group *π**d*(U(*N*)). The homotopy group is nontrivial in odd spatial dimensions *d* = 2*n* + 1, while it is trivial in even spatial dimensions, i.e., there is no non-trivial topological insulator in class AIII in even spatial dimensions.\nIn odd spatial dimensions *d* = 2*n* + 1, class AIII topological insulators are characterized by the winding number \n \n$$\\begin{aligned}\n\\label{winding no}\n\\nu\\_{2n+1}[q]\n: =\n\\int\\_{\\mathrm{BZ}^{d=2n+1}}\n\\omega\\_{2n+1}[q],\\end{aligned}$$ \nwhere the winding number density is given by\n \n$$\\begin{aligned}\n \\label{winding no density}\n\\omega\\_{2n+1}[q]\n&:=&\n\\frac{(-1)^n n!}{(2n+1)!}\n\\left(\\frac{\\rmi}{2\\pi}\\right)^{n+1}\n\\tr\n\\left[(q^{-1} \\rmd q)^{2n+1}\\right]\n\\\\ \n&=&\n\\frac{(-1)^n n!}{(2n+1)!}\n\\left(\\frac{\\rmi}{2\\pi}\\right)^{n+1}\n\\epsilon^{\\alpha\\_1\\alpha\\_2\\cdots}\n\\mathrm{tr} \\left[\n q^{-1} \\partial\\_{\\alpha\\_1} q \\cdot\n q^{-1} \\partial\\_{\\alpha\\_2} q\n\\cdots\\right]\n\\rmd^{2n+1} k. \n\\nonumber\\end{aligned}$$ \n### Chern-Simons invariant.\nWhile derived from the Chern form, the Chern-Simons forms, introduced in equation ([eq: def CS form]), themselves define a characteristic class for an odd-dimensional manifold. This suggests that the Chern-Simons forms can be used to characterize also AIII topological insulators in *d* = 2*n* + 1 dimensions. Integrating the Chern-Simons form over the BZ, we introduce\n \n$$\\begin{aligned}\n\\label{IntegralOfChernSimonsForm}\n\\mathrm{CS}\\_{2n+1} [\\mathcal{A},\\mathcal{F}] \n:=\n\\int\\_{\\mathrm{BZ}^{2n+1}}\nQ\\_{2n+1}(\\mathcal{A},\\mathcal{F}). \\end{aligned}$$ \nUnlike the winding number, CS2*n* + 1[A, F] is defined without assuming chiral symmetry and can be used for non-chiral topological insulators (superconductors).\nBefore discussing how useful Chern-Simons forms are, however, observe that they are not gauge invariant. Neither are the integral of the Chern-Simons forms over the BZ. However, for two different choices of gauge A and Aʹ, which are connected by a gauge transformation\n \n$$\\begin{aligned}\n\\mathcal{A}'\n= g^{-1}\\mathcal{A} g + g^{-1} \\rmd g,\n\\quad\n\\mathcal{F}'\n= g^{-1}\\mathcal{F} g,\\end{aligned}$$ \nwe have\n \n$$\\begin{aligned}\nQ\\_{2n+1}(\\mathcal{A}', \\mathcal{F}')\n-\nQ\\_{2n+1}(\\mathcal{A}, \\mathcal{F})\n=\nQ\\_{2n+1}(g^{-1}\\rmd g, 0)\n+\n\\rmd\\alpha\\_{2n},\n\\label{eq: Cartan homotopy}\\end{aligned}$$ \nwhere *α*2*n* is some 2*n* form. We note that\n \n$$\\begin{aligned}\nQ\\_{2n+1}\\left(g^{-1}\\rmd g, 0 \\right)\n&=\n\\frac{1}{n!}\n\\left(\n\\frac{\\rmi}{2 \\pi}\n\\right)^{n+1}\n\\int^1\\_0 \\rmd t \\,\n\\tr\n\\left(\n\\mathcal{A} (t^2-t)^n \\mathcal{A}^{2n} \n\\right)\n\\nonumber \\\\\n&=\n\\frac{1}{n!}\n\\left(\n\\frac{\\rmi}{2 \\pi}\n\\right)^{n+1}\n\\tr\n\\left[\n(g^{-1} \\rmd g)^{2n+1} \n\\right]\n\\int^1\\_0 \\rmd t\\, \n(t^2-t)^n \n\\nonumber \\\\\n&=\n\\omega\\_{2n+1}[g]. \n\\label{eq: CS form when pure gauge}\\end{aligned}$$ \nThis is nothing but the winding number density, and its integral\n \n$$\\begin{aligned}\n\\nu\\_{2n+1}[g]\n:=\n\\int\\_{\\mathrm{BZ}^{2n+1}}\nQ\\_{2n+1}\\left(g^{-1}\\rmd g, 0 \\right)\n=\n\\int\\_{\\mathrm{BZ}^{2n+1}}\n\\omega\\_{2n+1}[g]\\end{aligned}$$ \nis an integer, which counts the non-trivial winding of the map *g*(*k*) : BZ2*n* + 1 → U(*N*). Note that *π*2*n* + 1[U(*N*)] = Z (for large enough *N*). We thus conclude\n \n$$\\begin{aligned}\n\\mathrm{CS}\\_{2n+1}\\left[\\mathcal{A}',\\mathcal{F}'\\right]\n-\n\\mathrm{CS}\\_{2n+1}\\left[\\mathcal{A},\\mathcal{F}\\right]\n=\n\\mbox{integer}, \n\\label{eq: CS invariant upto integer}\\end{aligned}$$ \nand hence the exponential\n \n$$\\begin{aligned}\nW\\_{2n+1}\n: =\n\\exp\\{\n 2\\pi \\rmi\n\\mathrm{CS}\\_{2n+1}\n\\left[\\mathcal{A},\\mathcal{F}\\right]\n\\}\\end{aligned}$$ \nis a well-defined, gauge invariant quantity, although it is not necessarily quantized.\nThe discussion so far has been general. In particular, it is *not* restricted to *chiral* topological insulators. We now compute the Chern-Simons invariant for class AIII topological insulators in *d* = 2*n* + 1. We first explicitly write down the Berry connection for chiral symmetric Hamiltonians. To this end, we observe that for unoccupied and occupied bands, the Bloch wavefucntions satisfy\n \n$$\\begin{aligned}\nQ (k) |u^+\\_{\\hat{a}}(k)\\rangle\n= \n+|u^+\\_{\\hat{a}}(k)\\rangle,\n\\qquad\nQ (k) |u^-\\_{\\hat{a}}(k)\\rangle\n= \n-|u^-\\_{\\hat{a}}(k)\\rangle,\n\\label{eigenequations}\\end{aligned}$$ \nrespectively. Introducing\n \n$$\\begin{aligned}\n|u^\\epsilon\\_{\\hat{a}}(k)\\rangle\n=\\frac{1}{\\sqrt2}\n\\left(\n\\begin{array}{c}\n\\chi^\\epsilon\\_{\\hat{a}}(k)\\\\\n\\eta^\\epsilon\\_{\\hat{a}}(k)\n\\end{array}\n\\right),\n\\qquad\n\\epsilon=\\pm,\\end{aligned}$$ \nwe rewrite ([eigenequations]) as\n \n$$\\begin{aligned}\n\\left(\n\\begin{array}{cc}\n0 & q(k) \\\\\nq^{\\dag}(k) & 0 \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\chi^{\\pm}\\_{\\hat{a}}(k)\\\\\n\\eta^{\\pm}\\_{\\hat{a}}(k)\n\\end{array}\n\\right)\n=\n\\pm \n\\left(\n\\begin{array}{c}\n\\chi^{\\pm}\\_{\\hat{a}}(k)\\\\\n\\eta^{\\pm}\\_{\\hat{a}}(k)\n\\end{array}\n\\right).\\end{aligned}$$ \nWe can then construct a set of eigen Bloch functions as\n \n$$\\begin{aligned}\n|u^{\\epsilon}\\_{\\hat{a}}(k)\\rangle\n=\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{c}\n \\chi^{\\epsilon}\\_{\\hat{a}} \\\\\n\\epsilon q^{\\dag}(k)\\chi^{\\epsilon}\\_{\\hat{a}}\n\\end{array}\n\\right).\\end{aligned}$$ \nThe *N*-dimesional space spanned by the occupied states {∣*u**â*−(*k*)⟩} can be obtained by first choosing *N* independent orthonormal vectors *n**â**ε* which are *k*-independent, and then from *n**â**ε*,\n \n$$\\begin{aligned}\n|u^{\\epsilon}\\_{\\hat{a}}(k)\\rangle\\_{\\mathrm{N}}\n=\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{c}\nn^{\\epsilon}\\_{\\hat{a}} \\\\\n\\epsilon q^{\\dag}(k) n^{\\epsilon}\\_{\\hat{a}}\n\\end{array}\n\\right).\n\\label{u\\_N}\\end{aligned}$$ \nFrom these Bloch functions, the Berry connection is computed as\n \n$$\\begin{aligned}\n{ }\\_{\\mathrm{N}}\\langle u^{\\epsilon}\\_{\\hat{a}}(k) |\n\\partial\\_{\\mu} u^{\\epsilon}\\_{\\hat{b}} (k)\n\\rangle\\_{\\mathrm{N}}\n\\rmd k\\_{\\mu} \n&=&\n\\frac{1}{2}\n\\left[\n\\langle n^{\\epsilon}\\_{\\hat{a}} |\n\\partial\\_{\\mu} |n^{\\epsilon}\\_{\\hat{b}} \\rangle \n+\n\\epsilon^{2}\n\\langle n^{\\epsilon}\\_{\\hat{a}} q(k)|\n\\partial\\_{\\mu} |q^{\\dag}(k) n^{\\epsilon}\\_{\\hat{b}} \\rangle \n\\right]\n\\rmd k\\_{\\mu} \n\\nonumber \\\\ \n&=&\n\\frac{1}{2}\n\\left[\n0\n+\n\\epsilon^{2}\n\\langle n^{\\epsilon}\\_{\\hat{a}} q(k)|\n\\partial\\_{\\mu} |q^{\\dag}(k) n^{\\epsilon}\\_{\\hat{b}} \\rangle \n\\right]\n\\rmd k\\_{\\mu} \n\\nonumber \\\\ \n&=&\n\\frac{1}{2}\n[q(k)\\partial\\_{\\mu} q^{\\dag}(k) ]\\_{\\hat{a}\\hat{b}}\n\\rmd k\\_{\\mu} \n=:\n\\mathcal{A}^{\\mathrm{N}}\\_{\\hat{a}\\hat{b}},\\end{aligned}$$ \nwhere we have made a convenient choice (*n**â**ε*)*b̂* = *δ**â**b̂*. While this looks *almost* like a pure gauge, it is not exactly so because of the factor 1/2. Note also that this calculation shows that the Berry connection for the occupied and unoccupied bands are identical. With equations ([eigenequations]) through ([uN]), we have succeeded in constructing eigen Bloch wavefuctions of the “*Q*-matrix”, equation ([eq: off diagonal Q]), in block-off diagonal basis, that are free from any singularity. I.e., we have explicitly demonstrated that there is no obstruction to constructing eigen wavefuctions globally. We emphasize that this applies to all symmetry classes with chiral symmetry (AIII, BDI, DIII, CII, CI) for any spatial dimension (*d* = 2*n* as well as *d* = 2*n* + 1).\nAlternatively, one can construct another set of eigen Bloch functions\n \n$$\\begin{aligned}\n|u^{\\epsilon}\\_{\\hat{a}}(k)\\rangle\\_{\\mathrm{S}}\n=\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{c}\n\\epsilon q(k) n^{\\epsilon}\\_{\\hat{a}} \\\\\nn^{\\epsilon}\\_{\\hat{a}}\n\\end{array}\n\\right).\n\\label{u\\_S}\\end{aligned}$$ \nFor these Bloch functions, the Berry connection is computed as\n \n$$\\begin{aligned}\n{ }\\_{\\mathrm{S}} \\langle u^{\\epsilon}\\_{\\hat{a}}(k) |\n\\partial\\_{\\mu} u^{\\epsilon}\\_{\\hat{b}} (k)\\rangle\\_{\\mathrm{S}}\n\\rmd k\\_{\\mu} \n&=&\n\\frac{1}{2}\n\\left[\n\\epsilon^2\n\\langle n^{\\epsilon}\\_{\\hat{a}} q^{\\dag}(k)|\n\\partial\\_{\\mu} | q(k) n^{\\epsilon}\\_{\\hat{b}} \\rangle \n+\n\\langle n^{\\epsilon}\\_{\\hat{a}} |\n\\partial\\_{\\mu} | n^{\\epsilon}\\_{\\hat{b}} \\rangle \n\\right]\n\\rmd k\\_{\\mu} \n\\nonumber \\\\ \n&=&\n\\frac{1}{2}\n\\left[\n\\epsilon^{2}\n\\langle n^{\\epsilon}\\_{\\hat{a}} q^{\\dag}(k)|\n\\partial\\_{\\mu} |q(k) n^{\\epsilon}\\_{\\hat{b}} \\rangle \n+\n0\n\\right] \n\\rmd k\\_{\\mu} \n\\nonumber \\\\ \n&=&\n\\frac{1}{2}\n[q^{\\dag}(k)\\partial\\_{\\mu} q(k) ]\\_{\\hat{a}\\hat{b}}\n\\rmd k\\_{\\mu} \n=:\n\\mathcal{A}^{\\mathrm{S}}\\_{\\hat{a}\\hat{b}},\\end{aligned}$$ \nwhere we have made a convenient choice (*n**â**ε*)*b̂* = *δ**â**b̂*. The two gauges are related to each other by\n \n$$\\begin{aligned}\n\\mathcal{A}^{\\mathrm{S}}\n=\ng^{-1} \\mathcal{A}^{\\mathrm{N}} g\n+\ng^{-1} \\rmd g,\\end{aligned}$$ \nwhere the transition function *g* is the block off-diagonal projector, *g* = *q*:\n \n$$\\begin{aligned}\n\\fl\nq^{\\dag}(k) \n\\left[\n\\frac{1}{2}q(k) \\partial\\_{\\mu} q^{\\dag}(k)\n\\right]\nq(k) \n+\nq^{\\dag}(k) \n\\partial\\_{\\mu} \nq(k) \n&=\n\\frac{1}{2} \n\\left[\\partial\\_{\\mu} q^{\\dag}(k)\\right] q(k) \n+\nq^{\\dag}(k) \n\\partial\\_{\\mu} \nq(k) \n\\nonumber \\\\ \n&=\n-\\frac{1}{2} \nq^{\\dag}(k) \\partial\\_{\\mu} q(k) \n+\nq^{\\dag}(k) \n\\partial\\_{\\mu} \nq(k)\n\\nonumber\\\\\n&=\n\\frac{1}{2} \nq^{\\dag}(k) \n\\partial\\_{\\mu} \nq(k). \\end{aligned}$$ \nWe now compute the Chern-Simons invariant for class AIII topological insulators in *d* = 2*n* + 1. In the gauge where $\\mathcal{A}=\\mathcal{A}^\\mathrm{S}= (1/2) (q^{-1} \\rmd q)$, we have\n \n$$\\begin{aligned}\n\\rmd\\mathcal{A}\n&=\n\\frac{1}{2} \n\\rmd(q^{-1} \\rmd q)\n=\n-\\frac{1}{2}\n(q^{-1} \\rmd q) (q^{-1} \\rmd q),\n\\nonumber \\\\ \n\\mathcal{F}\\_t\n&=\nt\\, \\rmd\\mathcal{A}\n+\nt^2 \\mathcal{A}^2\n=\n\\left(\n-\\frac{t}{2}\n+\\frac{t^2}{4}\n\\right)\n(q^{-1} \\rmd q)^2.\\end{aligned}$$ \nThen,\n \n$$\\begin{aligned}\nQ\\_{2n+1}(\\mathcal{A},\\mathcal{F})\n&=\n\\frac{1}{n!}\n\\left(\n\\frac{\\rmi}{2 \\pi}\n\\right)^{n+1}\n\\int^1\\_0 \\rmd t\\, \n\\tr\n\\left(\n\\mathcal{A} \\mathcal{F}^{n}\\_t \n\\right)\n\\nonumber \\\\ \n&=\n\\frac{1}{n!}\n\\left(\n\\frac{\\rmi}{2 \\pi}\n\\right)^{n+1}\n\\int^1\\_0 \\left(\n-\\frac{t}{2}\n+\\frac{t^2}{4}\n\\right)^n \n\\frac{\\rmd t}{2}\n\\tr\n\\left[\n (q^{-1} \\rmd q)^{2n+1}\n\\right]\n\\nonumber \\\\\n&=\n\\frac{1}{n!}\n\\left(\n\\frac{\\rmi}{2 \\pi}\n\\right)^{n+1}\n\\frac{1}{2}\\int^{1}\\_0 \nu^n(u-1)^n\n \\rmd u\\, \n\\tr\n\\left[\n (q^{-1} \\rmd q)^{2n+1}\n\\right],\\end{aligned}$$ \nwhich is a half of the winding number density,\n \n$$\\begin{aligned}\nQ\\_{2n+1}(\\mathcal{A}^{\\mathrm{S}},\n\\mathcal{F}^{\\mathrm{S}})\n=\n\\frac{1}{2}\n\\omega\\_{2n+1}[q].\\end{aligned}$$ \nHence we conclude\n \n$$\\begin{aligned}\n\\mathrm{CS}\\_{2n+1}\n\\left[\\mathcal{A}^{\\mathrm{S}},\n\\mathcal{F}^{\\mathrm{S}}\\right]\n=\n\\int\\_{\\mathrm{BZ}^{d=2n+1}}\nQ\\_{2n+1}(\\mathcal{A}^{\\mathrm{S}},\\mathcal{F}^{\\mathrm{S}})\n=\n\\frac{1}{2} \\nu\\_{2n+1} [q].\\end{aligned}$$ \nAs a corollary,\n \n$$\\begin{aligned}\nW\\_{2n+1}\n =\n\\exp\\{\n 2\\pi \\rmi \\,\n\\mathrm{CS}\\_{2n+1} \\left[\\mathcal{A},\\mathcal{F}\\right]\n\\}\n=\n\\exp\\{ \\pi \\rmi \\, \\nu\\_{2n+1}[q]\\}.\n\\label{eq: master formula winding num and Wilson loop}\\end{aligned}$$ \nThus, while in general the quantity *W*2*n* + 1 is not quantized, for class AIII Hamiltonians in *d* = 2*n* + 1, *W*2*n* + 1 can take only two values, *W*2*n* + 1 = ± 1.\nPhysically, CS2*n* + 1[A, F] in *d* = 1 (*n* = 0) spatial dimension takes the form of the U(1) Wilson loop defined for BZ*d* = 1 ≃ *S*1. It is quantized for chiral symmetric systems. Also, the logarithm of *W*1 represents the electric polarization. For a lattice system, the non-invariance of CS1[A, F] [i.e., it can change under a gauge transformation, CS1[A, F] → CS1[A, F] + (integer)] has a clear meaning: since the system is periodic, the displacement of electron coordinates has a meaning only within a unit cell, and two electron coordinates that differ by an integer multiple of the lattice constant should be identified.\nIn *d* = 3 (*n* = 1) spatial dimensions CS3 represents the magnetoelectric polarizability. While this was discussed originally for three-dimensional Z2 topological insulators in symplectic symmetry (class AII), the magnetoelectric polarizability is also quantized for class AIII.\nDirac insulators and dimensional reduction\n------------------------------------------\nWe start from a *d* = 2*n* + 3-dimensional *gapless* Dirac Hamiltonian defined in momentum space,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2n+3}\\_{(2n+3)}(k)\n= \n\\sum\\_{a=1}^{d=2n+3} k\\_a \\Gamma^a\\_{(2n+3)}.\\end{aligned}$$ \nHere, *k**a* = 1, …, *d* are *d*-dimensional momenta, and Γ(2*n* + 3)*a* = 1, …, 2*n* + 3 are (2*n* + 1 × 2*n* + 1)-dimensional Hermitian matrices that satisfy {Γ(2*n* + 3)*a*, Γ(2*n* + 3)*b*} = 2*δ**a*, *b*. Some results for the gamma matrices, on which the following discussion is based, are summarized in appendix [app: spinors]. The massless Dirac Hamiltonian H(2*n* + 3)*d* = 2*n* + 3(*k*) cannot be realized on a lattice in a naive way, because of the fermion doubling problem.\nBy replacing *k*2*n* + 3 by a mass term, we obtain a *d* = 2*n* + 2 dimensional class A topological Dirac insulator,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2n+2}\\_{(2n+3)}(k,m)\n= \n\\sum\\_{a=1}^{d=2n+2} k\\_a \\Gamma^a\\_{(2n+3)}\n+m \\Gamma^{2n+3}\\_{(2n+3)}.\n\\label{eq: class A massive dirac}\\end{aligned}$$ \nThe topological character of this Dirac insulator will be further discussed below.\nBy setting in addition *k*2*n* + 2 = 0, we obtain an insulator in one dimension lower,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2n+1}\\_{(2n+3)}(k,m)\n= \n\\sum\\_{a=1}^{d=2n+1} k\\_a \\Gamma^a\\_{(2n+3)}\n+m \\Gamma^{2n+3}\\_{(2n+3)}.\n\\label{eq: class AIII massive dirac}\\end{aligned}$$ \nBy construction, H(2*n* + 3)*d* = 2*n* + 1(*k*, *m*) anticommutes with Γ(2*n* + 3)2*n* + 2,\n \n$$\\begin{aligned}\n\\left\\{\n\\mathcal{H}^{d=2n+1}\\_{(2n+3)}(k,m),\\, \n\\Gamma^{2n+2}\\_{(2n+3)}\n\\right\\}\n=0, \\end{aligned}$$ \nand hence it is a member of class AIII.\nThis construction of the lower-dimensional models from their higher-dimensional “parent” is an example of the Kaluza-Klein dimensional reduction. To obtain H(2*n* + 3)*d* = 2*n* + 1(*k*, *m*) from H(2*n* + 3)*d* = 2*n* + 2(*k*, *m*), one can first compactify the (2*n* + 2)-th spatial direction into a circle *S*1. The (2*n* + 2)-th component of the momentum is then quantized, *k*2*n* + 2 = 2*π**N*2*n* + 2/ℓ, where *N*2*n* + 2 ∈ Z, and ℓ is the radius of *S*1. The energy eigenvalues now carry the integral label *N*2*n* + 2, in addition to the continuous label, *k**i* = 1, …, 2*n* + 1. By making the radius of the circle very small, all levels with *N*2*n* + 2 ≠ 0 have very large energies while the levels with *N*2*n* + 2 = 0 are not affected by the small radius limit. In this way, in the limit ℓ → 0, there is a separation of energy scales, and we only keep the states with *N*2*n* + 2 = 0, while neglecting all states with *N*2*n* + 2 ≠ 0 (the Kaluza-Klein modes).\nSimilarly, H(2*n* + 3)*d* = 2*n* + 2(*k*, *m*) is naturally obtained from H(2*n* + 3)*d* = 2*n* + 3(*k*) by dimensional reduction, if we also include a U(1) gauge field. We can introduce a (external, electromagnetic) U(1) gauge field *a**i* = 1, ⋯, 2*n* + 3 by minimal coupling, H(2*n* + 3)*d* = 2*n* + 3(*k*) → H(2*n* + 3)*d* = 2*n* + 3(*k* + *a*). In the *S*1 compactification of the (2*n* + 3)-th spatial coordinate, we again keep modes with *k*2*n* + 3 = 0 only, while neglecting all the Kaluza-Klein modes with *k*2*n* + 3 ≠ 0. The vector field (gauge field) *a**i* = 1, …2*n* + 3 in *d* = 2*n* + 3 dimensions can be decomposed into a vector field (*a**i* = 1, …2*n* + 2) and a scalar field (*a*2*n* + 3 ≡ *φ*) in *d* = 2*n* + 2, leading to\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2n+2}\\_{(2n+3)}(k+a,\\varphi)\n= \n\\sum\\_{i=1}^{d=2n+2} (k\\_i+a\\_i) \\Gamma^i\\_{(2n+3)}\n+\n\\varphi \\Gamma^{2n+3}\\_{(2n+3)}.\n\\label{eq: Dirac + external gauge}\\end{aligned}$$ \nSwitching off the *d* = 2*n* + 2-dimensional gauge field, and assuming the scalar field is constant *φ* = *m*, we obtain H(2*n* + 3)*d* = 2*n* + 2(*k*, *m*).\nFor the massive Dirac insulator ([eq: class A massive dirac]) in class A in 2*n* + 2 dimensions, the *n*-th Chern number Ch*n* + 1 is nonvanishing. Accordingly, for the massive Dirac insulator ([eq: class AIII massive dirac]) in class AIII in 2*n* + 1 dimensions, the winding number and the Chern-Simons form are non-zero.\nAs the Chern-Simons form can be derived from its higher-dimensional parent, the Chern form, one may wonder if there is a connection between $\\mathrm{CS}\\_{2n+1}=\\frac{1}{2}\\nu\\_{2n+1}$ of a class AIII Dirac topological insulator in *d* = 2*n* + 1 and Ch*n* + 1 of a class A Dirac topological insulator in *d* = 2*n* + 2. We now try to answer this question.\nLet us start from\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2n+2}\\_{(2n+3)}(k,m)\n= \n\\sum\\_{a=1}^{d=2n+2} k\\_a \\Gamma^a\\_{(2n+3)}\n+m(k) \\Gamma^{2n+3}\\_{(2n+3)}.\\end{aligned}$$ \nHere we have regularized the Dirac spectrum by making the mass term *k* dependent,\n \n$$\\begin{aligned}\nm(k) = m - C k^2,\n\\quad \n\\mbox{with}\n\\quad\n\\mathrm{sgn} (C) = \\mathrm{sgn} (m).\n\\label{eq: regularized mass}\\end{aligned}$$ \nWith this regularization, as ∣*k*∣ → ∞, H(2*n* + 3)*d* = 2*n* + 2(*k*, *m*) ∝ Γ(2*n* + 3)2*n* + 3, and hence the wavefunctions become *k*-independent (i.e., the BZ is topologically *S**d*). The sign choice of the constant *C*, sgn(*C*) = sgn(*m*), makes the Chern invariant for the Dirac insulator non-zero.\nBy dropping *k*2*n* + 2, we get the (regularized) Dirac insulator in one dimension lower,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2n+1}\\_{(2n+3)}( \\tilde{k},m)\n= \n\\sum\\_{a=1}^{d=2n+1} k\\_a \\Gamma^a\\_{(2n+3)}\n+m(\\tilde{k},0) \\Gamma^{2n+3}\\_{(2n+3)},\\end{aligned}$$ \nwhere *k̃* = (*k*1, …, *k*2*n* + 1). As mentioned before, H(2*n* + 3)*d* = 2*n* + 1(*k̃*, *m*) is chiral symmetric, {H(2*n* + 3)*d* = 2*n* + 1(*k̃*, *m*), Γ(2*n* + 3)2*n* + 2} = 0. We work in the basis where Γ(2*n* + 3)2*n* + 2 is diagonal. In this basis, H(2*n* + 3)*d* = 2*n* + 1 is block-off diagonal,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2n+1}\\_{(2n+3)}(\\tilde{k},m)\n=\n\\left(\n\\begin{array}{cc}\n0 & \\tilde{D}(\\tilde{k}) \\\\\n\\tilde{D}^{\\dag}(\\tilde{k}) & 0\n\\end{array}\n\\right).\n\\label{eq: chiral H, block off diagonal}\\end{aligned}$$ \nCorrespondingly,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2n+2}\\_{(2n+3)}(k)\n=\n\\left(\n\\begin{array}{cc}\n\\Delta(k\\_{2n+2}) & D(k) \\\\\nD^{\\dag}(k) & -\\Delta(k\\_{2n+2})\n\\end{array}\n\\right),\\end{aligned}$$ \nwhere Δ(*k*2*n* + 2) = *k*2*n* + 2, and *D*(*k*) = *D*(*k̃*, *k*2*n* + 2) satisfies\n \n$$\\begin{aligned}\nD(\\tilde{k},k\\_{2n+2}) = \\tilde{D}(\\tilde{k}),\n\\quad\n\\mbox{when} \n\\quad \nk\\_{2n+2}=0. \\end{aligned}$$ \nWe now look for eigenfunctions of H(2*n* + 3)*d* = 2*n* + 2 with negative eigenvalues,\n \n$$\\begin{aligned}\n\\left(\n\\begin{array}{cc}\n\\Delta(k\\_{2n+2}) & D(k) \\\\\nD^{\\dag}(k) & -\\Delta(k\\_{2n+2}) \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\chi \\\\\n\\eta\n\\end{array}\n\\right)\n=\n-\\lambda(k)\n\\left(\n\\begin{array}{c}\n\\chi \\\\\n\\eta\n\\end{array}\n\\right).\\end{aligned}$$ \nTo this end, we first solve the following auxiliary eigenvalue problem\n \n$$\\begin{aligned}\n\\left(\n\\begin{array}{cc}\n0 & D(k) \\\\\nD^{\\dag}(k) & 0\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\n\\chi \\\\\n\\eta\n\\end{array}\n\\right)\n=\n-\\tilde{\\lambda}(k)\n\\left(\n\\begin{array}{c}\n\\chi \\\\\n\\eta\n\\end{array}\n\\right),\\end{aligned}$$ \nwhere $\\tilde{\\lambda}=\\sqrt{\\tilde{k}^2+[m(k)]^2}$. We see from ([uN]) and ([uS]) that the solutions are given by\n \n$$\\begin{aligned}\n\\frac{1}{\\sqrt2}\n\\left(\\begin{array}{c}\n-n\\_{\\hat a} \\\\\nq^\\dag(k) n\\_{\\hat a}\n\\end{array}\n\\right)\n\\quad\n\\mbox{or}\n\\quad\n\\frac{1}{\\sqrt2}\n\\left(\\begin{array}{c}\nq(k) n\\_{\\hat a} \\\\\n-n\\_{\\hat a}\n\\end{array}\n\\right),\n\\label{auxiliary\\_solution}\\end{aligned}$$ \nwhere *q*(*k*) and *q*†(*k*) are the off-diagonal blocks of the *Q*-matrix derived from the auxiliary Hamiltonian. If we specialize to the case where *k* = (*k̃*, 0), the projector of H(2*n* + 3)*d* = 2*n* + 1(*k̃*) is given in terms of *q*(*k̃*) and *q*†(*k̃*). We then construct two different sets of normalized eigenfunctions of H(2*n* + 3)*d* = 2*n* + 2\n \n$$\\begin{aligned}\n|u^{-}\\_{\\hat a} (k)\\rangle\\_{\\mathrm{N}}\n&=&\n\\frac{1}{\n\\sqrt{ 2\\lambda (\\lambda+ \\Delta) }\n}\n\\left(\n\\begin{array}{c}\n-\\tilde{\\lambda} n\\_{\\hat a} \\\\\n( \\lambda+\\Delta) q^{\\dag} n\\_{\\hat a}\n\\end{array}\n\\right), \\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n|u^{-}\\_{\\hat a} (k)\\rangle\\_{\\mathrm{S}}\n&=&\n\\frac{1}{\n\\sqrt{ 2 \\lambda (\\lambda -\\Delta) }\n}\n\\left(\n\\begin{array}{c}\n(\\Delta-\\lambda) q n\\_{\\hat a} \\\\\n\\tilde{\\lambda} n\\_{\\hat a}\n\\end{array}\n\\right),\\end{aligned}$$ \nwith the eigenvalue $-\\lambda=-\\sqrt{\\tilde{\\lambda}^2+\\Delta^2}$. When specialized to *k* = (*k̃*, 0), these wavefunctions yield the eigenfunctions ([auxiliarysolution]) of the chiral Dirac Hamiltonian H(2*n* + 3)*d* = 2*n* + 1.\nSince the Hamiltonian H*d* = 2*n* + 2(*k*) is characterized by the non-zero Chern number, Ch*n* + 1[F] ≠ 0, it is not possible to define wavefunctions globally. We thus need to split the BZ (BZ*d* = 2*n* + 2) into two parts, BZN*d* = 2*n* + 2 and BZS*d* = 2*n* + 2. We choose the interface of these two patches as ∂BZN*d* = 2*n* + 2 = − ∂BZS*d* = 2*n* + 2 = BZ*d* = 2*n* + 1, on which a lower dimensional Hamiltonian ``lives\". For each patch, BZN, S*d* = 2*n* + 2, we can choose a global gauge. Observe that the wavefunction $|u^{-}\\_{\\hat a} (k)\\rangle\\_\\mathrm{N}$ is well-defined for *k*2*n* + 2 > 0, and has a singularity at $(\\tilde{k},k\\_{2n+2})=(0,-\\sqrt{m/C})$. Similarly, $|u^{-}\\_{\\hat a} (k)\\rangle\\_\\mathrm{S}$ is well-defined for *k*2*n* + 2 < 0 and singular at $(\\tilde{k},k\\_{2n+2})=(0,\\sqrt{m/C})$.\n [sphere]\nThe two gauges are thus complementary and at the boundary ∂BZN*d* = 2*n* + 2 = − ∂BZS*d* = 2*n* + 2 they are glued by the transition function\n \n$$\\begin{aligned}\n\\mathcal{A}^{\\mathrm{S}} \n&=& g^{-1} \n\\mathcal{A}^{\\mathrm{N}} g + g^{-1} \\rmd g,\\end{aligned}$$ \nwhere *g*(*k̃*) ∈ U(*N*−) with *k̃* ∈ ∂BZN*d* = 2*n* + 2. Then,\n \n$$\\begin{aligned}\n\\mathrm{Ch}\\_{n+1}\\left[\\mathcal{F}\\right]\n&=&\n\\int\\_{\\mathrm{BZ}^{d=2n+2}}\n\\mathrm{ch}\\_{n+1} \n(\\mathcal{F})\n=\n\\int\\_{\\mathrm{BZ}^{d=2n+2}}\n\\rmd Q\\_{2n+1}(\\mathcal{A},\\mathcal{F})\n\\nonumber \\\\%&=&\n\\int\\_{\\mathrm{BZ}\\_{\\mathrm{N}}^{d=2n+2}}\n\\rmd Q\\_{2n+1}(\\mathcal{A}^{\\mathrm{N}},\\mathcal{F})\n+\n\\int\\_{\\mathrm{BZ}\\_{\\mathrm{S}}^{d=2n+2}}\n\\rmd Q\\_{2n+1}(\\mathcal{A}^{\\mathrm{S}},\\mathcal{F})\n\\nonumber \\\\%&=&\n\\int\\_{\\partial \\mathrm{BZ}\\_{\\mathrm{N}}^{d=2n+2}}\nQ\\_{2n+1}(\\mathcal{A}^{\\mathrm{N}},\\mathcal{F})\n-\n\\int\\_{\\partial \\mathrm{BZ}\\_{\\mathrm{N}}^{d=2n+2}}\nQ\\_{2n+1}(\\mathcal{A}^{\\mathrm{S}},\\mathcal{F}),\\end{aligned}$$ \nwhere we have used Stokes theorem. From formula ([eq: Cartan homotopy]), it follows that\n \n$$\\begin{aligned}\n\\mathrm{Ch}\\_{n+1}\\left[\\mathcal{F}\\right]\n&=&\n\\int\\_{\\partial \\mathrm{BZ}^{d=2n+1}\\_{\\mathrm{N}}}\nQ\\_{2n+1}\\left(g^{-1}\\rmd g, 0 \\right). \\end{aligned}$$ \nBy use of equation ([eq: CS form when pure gauge]) we conclude that the RHS of the above equation is nothing but the winding number. Thus, the Chern number can be expressed as a winding number of the transition function *g*. For the case we are interested in, i.e., when a *d* = 2*n* + 1 topological insulator “lives” on ∂BZN*d* = 2*n* + 1, the transition function is given by the off-diagonal block of the projector, *g*(*k̃*) = *q*(*k̃*). This is how two topological insulators in *d* = 2*n* + 2 and *d* = 2*n* + 1 are related.\nExample: *d* = 3 → 2 → 1\n------------------------\nConsider three mutually anticommuting, Hermitian matrices\n \n$$\\begin{aligned}\n\\Gamma^{a=1,2,3}\\_{(3)}=\n\\{\n\\sigma\\_{x},\n\\,\n\\sigma\\_{y},\n\\,\n\\sigma\\_{z}\n\\},\\end{aligned}$$ \nwhere *σ**x*, *y*, *z* are the 2 × 2 Pauli matrices. These matrices can be used to construct a *d* = 3 gapless (Weyl) chiral fermion\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=3}\\_{(3)}(k)\n=\n\\sum\\_{a=1}^3\nk\\_a \\Gamma^a\\_{(3)}. \\end{aligned}$$ \nBy replacing the third component of the momemtum by a mass term, *k**a* = 3 → *m*, we get a *d* = 2 dimensional Dirac Hamiltonian,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=2}\\_{(3)}(k,m)\n&=\n\\sum\\_{a=1}^2\nk\\_a \\Gamma^a\\_{(3)}\n+\nm \\Gamma^3\\_{(3)}\n\\nonumber \\\\%&=\nk\\_x \\sigma\\_x \n+\nk\\_y \\sigma\\_y\n+\nm \\sigma\\_z.\n\\label{QHE Dirac Hamiltonian}\\end{aligned}$$ \nThis is a class A Hamiltonian. The Bloch wavefunctions are given by [$\\lambda(k) := \\sqrt{ k^2 + m^2 }$]\n \n$$\\begin{aligned}\n|u^+ (k) \\rangle\n&=\n\\frac{1}{\\sqrt{2 \\lambda \\left(\\lambda - m\\right) }}\n\\left(\n\\begin{array}{c}\nk\\_x - \\rmi k\\_y \\\\\n\\lambda - m\n\\end{array}\n\\right),\n\\nonumber \\\\%|u^- (k) \\rangle\n&=\n\\frac{1}{\\sqrt{2 \\lambda \\left(\\lambda + m\\right) }}\n\\left(\n\\begin{array}{c}\n-k\\_x + \\rmi k\\_y \\\\\n\\lambda + m \\\\\n\\end{array}\n\\right).\n\\label{eq: wfns}\\end{aligned}$$ \nWe note that ∣*u*−(*k*)⟩ is well-defined for all *k* when *m* > 0. (When *m* < 0, by choosing a gauge properly, we can obtain a similar well-defined wavefunction.) Assuming *m* > 0, we adopt ∣*u*−(*k*)⟩, for which we obtain the Berry connection\n \n$$\\begin{aligned}\nA\\_x (k,m) =\n+\\frac{\\mathrm{i} k\\_y }{2 \\lambda \\left(\\lambda +m\\right)},\n\\qquad\nA\\_y (k,m) =\n-\\frac{ \\mathrm{i} k\\_x }{2 \\lambda \\left(\\lambda +m\\right)},\\end{aligned}$$ \nand the Berry curvature\n \n$$\\begin{aligned}\nF\\_{xy}\\left(k,m \\right) =\n\\partial\\_{k\\_x} A\\_y - \\partial\\_{k\\_y} A\\_x\n= -\\frac{\\rmi m }{ 2 \\lambda^3 }.\\end{aligned}$$ \nThe Chern number is non-zero,\n \n$$\\begin{aligned}\n\\mathrm{Ch}\\_{1}\n[\\mathcal{F}]\n=\n\\frac{\\mathrm{i}}{2 \\pi } \\int \\rmd^2 k F\\_{xy}\n=\n\\frac{\\rmi}{2 \\pi} \\int \\rmd^2 k \n\\frac{-\\rmi m }{ 2 \\lambda^3 } \n=\n\\frac{1}{2}\n\\frac{m}{|m|}, \\end{aligned}$$ \nwhich is nothing but the Hall conductance *σ**x**y*. This model can be realized, at low energies, in the honeycomb lattice model introduced by Haldane.\nFinally, setting *k*2 = 0, we get a *d* = 1 dimensional Dirac Hamiltonian,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=1}\\_{(3)}(k,m)\n&=&\nk\\_x \\sigma\\_x \n+\nm \\sigma\\_z.\n\\label{eq: 1d AIII massive Dirac}\\end{aligned}$$ \nThis is nothing but the chiral topological Dirac insulator in class AIII, as it anticommutes with *σ**y*. A topological invariant can be defined for the *d* = 1 dimensional Dirac Hamiltonian as follows. We first go to the canonical form by (*σ**x*, *σ**y*, *σ**z*) → (*σ**x*, *σ**z*, − *σ**y*), H(3)*d* = 1(*k*, *m*) → *k**x**σ**x* − *m**σ**y*. This *d* = 1 dimensional Hamiltonian describes the physics of polyacetylene (see, for example, Refs. .) The Bloch wavefunction with negative eigenvalue is\n \n$$\\begin{aligned}\n|u^- (k\\_x) \\rangle\n&=&\n\\frac{1}{\\sqrt{2 (k^2\\_x+m^2) }}\n\\left(\n\\begin{array}{c}\n-k\\_x - \\rmi m \\\\\n\\lambda \\\\\n\\end{array}\n\\right).\n\\label{WF in d=1}\\end{aligned}$$ \nThe off-diagonal block of the projection operator *Q*(*k*) in this basis is\n \n$$\\begin{aligned}\nq(k) &=&\n-\\frac{k\\_x + \\rmi m}{\\sqrt{k^2\\_x + m^2}},\\end{aligned}$$ \nand the winding number is given by\n \n$$\\begin{aligned}\n\\nu\\_1 [q]\n=\n\\frac{\\rmi}{2\\pi}\n\\int\\_{\\mathrm{BZ}}\nq^{-1} \\rmd q\n=\n\\frac{\\rmi}{2\\pi}\n\\int^{\\infty}\\_{-\\infty} \n\\rmd k\\_x \\,\n\\frac{ -\\rmi m}{ k^2\\_x + m^2}\n=\n\\frac{1}{2} \\frac{m}{|m|}. \n\\label{eq: winding number nu1 for AIII dirac}\\end{aligned}$$ \nFrom equation ([WF in d=1]) the Berry connection is obtained,\n \n$$\\begin{aligned}\n\\mathcal{A}(k\\_x) \n=\n\\langle u^-(k\\_x)|\\rmd u^-(k\\_x)\\rangle \n=\n\\frac{1}{2}\n\\frac{ -\\rmi m }{ k^2\\_x + m^2}\n\\rmd k\\_x.\\end{aligned}$$ \nThen, we find\n \n$$\\begin{aligned}\n\\mathrm{CS}\\_1\\left[\\mathcal{A},\\mathcal{F}\\right]\n=\n\\frac{\\rmi}{2\\pi}\n\\int\\_{\\mathrm{BZ}} \\mathrm{tr}\\, \\mathcal{A} \n=\n\\frac{\\rmi}{2\\pi}\n\\int^{\\infty}\\_{-\\infty} \\rmd k\\_x\n\\frac{1}{2}\n\\frac{ -\\rmi m }{ k^2\\_x + m^2}\n=\n\\frac{1}{4}\n\\frac{m}{|m|}.\\end{aligned}$$ \nThis is a half of the winding number *ν*1[*q*] as expected. Hence, the Wilson “loop” is given by\n \n$$\\begin{aligned}\nW\\_1 &=&\n\\exp \\left\\{\n2\\pi \\mathrm{i} \\mathrm{CS}\\_1\\left[\\mathcal{A},\\mathcal{F}\\right]\n\\right\\}\n=\n\\exp \\int\\_{\\mathrm{BZ}} \\mathrm{tr}\\, \\mathcal{A} \n= \ne^{ \\pm \\pi \\rmi /2 }.\\end{aligned}$$ \nHere, we mention, however, that there is a subtlety in computing the Wilson loop. In the basis where the Hamiltonian in momentum space is real, H(3)*d* = 1(*k*, *m*) = *k**x**σ**x* + *m**σ**z* ([eq: 1d AIII massive Dirac]), the Bloch wavefunction is real for a given *k*, and as a consequence the Berry connection A(*k**x*) vanishes identically. One would then conclude *W*1 = 1, not *W*1 = − 1, although $\\nu\\_1= \\frac{1}{2} \\mathrm{sgn}(m)$. This puzzle can be solved by properly regularizing the Dirac insulator. For simplicity, let us replace *k**x* by sin*k**x*, and *m* by *m* − 1 + cos(*k**x*). With such a regularization, the BZ is topologically *S*1, and one finds a singularity in the wavefunction at *k**x* = *π*, where the phase of the wavefunction jumps by *π* (i.e., there is a Dirac string at *k**x* = *π*.). If, on the other hand, we use a different basis ([WF in d=1]), we can avoid to have a Dirac string in the BZ, and the Wilson loop can be obtained by integrating the Berry connection over the BZ.\nExample: *d* = 5 → 4 → 3\n------------------------\n[subsec: A-AIII examples in d=5,4,3]\nConsider five mutually anticommuting, Hermitian matrices\n \n$$\\begin{aligned}\n\\Gamma^{a=1,\\ldots,5}\\_{(5)}=\n\\{\n\\alpha\\_{x},\n\\,\n\\alpha\\_{y},\n\\,\n\\alpha\\_{z},\n\\,\n\\beta,\n\\, \n-\\rmi \\beta \\gamma^5\n\\},\\end{aligned}$$ \nwhere we are using the Dirac representation,\n \n$$\\begin{aligned}\n\\alpha\\_i\n=\\left(\\begin{array}{cc}\n0 & \\sigma\\_i\\\\\n\\sigma\\_i & 0\n\\end{array}\\right),\n\\qquad\n\\beta=\\left(\\begin{array}{cc}\n1 & 0\\\\\n0 & -1\\end{array}\\right),\n\\qquad\n\\gamma^{5}=\\left(\\begin{array}{cc}\n0 & 1\\\\\n1 & 0\\end{array}\n\\right).\\end{aligned}$$ \nObserve that $\n\\alpha\\_x\n\\alpha\\_y\n\\alpha\\_z\n\\beta \n= -\\rmi \\beta\\gamma\\_5\n$. These matrices can be used to construct a (*d* = 5)-dimensional gapless chiral fermion\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=5}\\_{(5)}(k)\n=\n\\sum\\_{a=1}^5\nk\\_a \\Gamma^a\\_{(5)}. \\end{aligned}$$ \nBy replacing the fifth component of the momemtum by a mass term, *k**a* = 5 → *m*5, we obtain a *d* = 4 dimensional Dirac Hamiltonian,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=4}\\_{(5)}(k,m\\_5)\n=\n\\sum\\_{a=1}^4\nk\\_a \\Gamma^a\\_{(5)}\n+\nm\\_5 \\Gamma^5\\_{(5)}.\\end{aligned}$$ \nIt is known that for this gapped Hamiltonian, the second Chern number is non-zero.\n [surfacemodesAIII] (a) Winding number *ν*3 for Hamiltonian as a function of *m*5. (b) Two-dimensional energy spectrum of the surface states of model with mass *m*5 = + 0.5. There are two inequivalent surface modes in agreement with the winding number *ν*3(*m*5 = + 0.5) = − 2. \nFinally, setting *k*4 = 0, we get a *d* = 3 dimensional Dirac Hamiltonian,\n \n$$\\begin{aligned}\n\\mathcal{H}^{d=3}\\_{(5)}(k,m\\_5)\n&=&\n\\sum\\_{a=1}^3\nk\\_a \\Gamma^a\\_{(5)}\n+\nm\\_5 \\Gamma^5\\_{(5)}\n=\n\\sum\\_{a=1}^3\nk\\_a \\alpha\\_a\n-\nm\\_5 \\rmi \\beta \\gamma^5.\\end{aligned}$$ \nThis is nothing but the chiral topological Dirac insulator in class AIII discussed in Ref. . The tight-binding version of this model for a simple cubic lattice is given by\n \n$$\\begin{aligned}\n \\label{matrixAIII}\n\\mathcal{H}^{d=3}\\_{(5)}(k,m\\_5)\n&=&\n\\sum\\_{a=1}^3 \\sin k\\_a \\Gamma^a\\_{(5)} + \n\\left( m\\_5 + \\sum\\_{a=1}^3 \\cos k\\_a \\right) \\Gamma^5\\_{(5)}.\n\\label{tight-binding}\\end{aligned}$$ \nBy taking open boundary conditions in the *z* direction and periodic boundary conditions in the *x* and *y* directions, we can study the surface states of this model. When the winding number *ν*3 is non-vanishing, there are ∣*ν*3∣ surface Dirac states, which cross the bulk band gap (see figure [surfacemodesAIII]).\nWe now study the Berry connection of the above class A (*d* = 4) and class AIII (*d* = 3) Dirac insulators. To treat these two cases in a unified fashion, let us consider the following *d* = 4 Dirac Hamiltonian\n \n$$\\begin{aligned}\n\\mathcal{H}(k,m\\_0) \n&=&\n\\sum\\_{a=1}^3 k\\_a \\alpha\\_a\n+\nm\\_0 \\beta \n+\nk\\_w (- \\rmi \\beta \\gamma^5).\\end{aligned}$$ \nUpon identification *k**w* → *m*5 it can be also viewed as a *d* = 3 dimensional Dirac Hamiltonian with two masses. The four eigenvalues are\n \n$$\\begin{aligned}\nE(k) \n&=&\n\\pm \\lambda(k),\n\\quad\n\\lambda(k)\n=\n\\sqrt{k^2 + m^2\\_0}.\\end{aligned}$$ \nThe two normalized negative energy eigenstates with *E*(*k*) = − *λ*(*k*) are\n \n$$\\begin{aligned}\n|u^{-}\\_1(k)\\rangle\n&=&\n\\frac{1}{\n\\sqrt{\n2\\lambda (\\lambda + m\\_0)\n}\n}\n\\left(\n\\begin{array}{c}\n-k\\_x + \\rmi k\\_y \\\\\n\\rmi k\\_w + k\\_z \\\\\n0 \\\\\n\\lambda+ m\\_0\n\\end{array}\n\\right),\n\\nonumber \\\\%|u^{-}\\_2(k)\\rangle\n&=&\n\\frac{1}{\n\\sqrt{\n2\\lambda (\\lambda + m\\_0)\n}\n}\n\\left(\n\\begin{array}{c}\n-k\\_z + \\rmi k\\_w \\\\\n-k\\_x - \\rmi k\\_y \\\\\n\\lambda+m\\_0 \\\\\n0 \\\\\n\\end{array}\n\\right).\n\\label{eq: wfn 4x4 occupied}\\end{aligned}$$ \nThe two normalized positive energy eigenstates with *E*(*k*) = + *λ*(*k*) are\n \n$$\\begin{aligned}\n|u^{+}\\_1(k)\\rangle\n&=&\n\\frac{1}{\n\\sqrt{\n2\\lambda (\\lambda - m\\_0)\n}\n}\n\\left(\n\\begin{array}{c}\nk\\_x - \\rmi k\\_y \\\\\n- \\rmi k\\_w -k\\_z \\\\\n0 \\\\\n\\lambda-m\\_0\n\\end{array}\n\\right)\n,\n\\nonumber \\\\%|u^{+}\\_2(k)\\rangle\n&=&\n\\frac{1}{\n\\sqrt{\n2\\lambda (\\lambda - m\\_0)\n}\n}\n\\left(\n\\begin{array}{c}\n-\\rmi k\\_w+k\\_z \\\\\nk\\_x + \\rmi k\\_y \\\\\n\\lambda-m\\_0 \\\\\n0 \n\\end{array}\n\\right).\n\\label{eq: wfn 4x4 unoccupied}\\end{aligned}$$ \nNote that if *m*0 > 0, ∣*u*1, 2+(*k*)⟩ are not well-defined at *λ*(*k*) = *m*0 (i.e, *k* = 0). Observe that when *m*0 > 0, ∣*u*1, 2−(*k*)⟩ have the form of wavefunctions expected from the general consideration (corresponding to the AN gauge choice).\nWhen *m*0 = 0, the Hamiltonian anticommutes with *β* and is block off-diagonal. The off-diagonal component of the *Q*-matrix is given by\n \n$$\\begin{aligned}\nq(k) = \n\\frac{-1}{\\lambda}\\left(\nk\\cdot \\sigma -\\rmi m\\_5\n\\right)\\end{aligned}$$ \nBy noting (*μ* = 1, 2, 3)\n \n$$\\begin{aligned}\n\\partial\\_{\\mu}q \n=\n\\frac{k\\_{\\mu}}{\\lambda^3}\n\\left(k\\cdot \\sigma -\\rmi m\\_5 \\right)\n-\n\\frac{1}{\\lambda}\\sigma\\_{\\mu},\n\\quad \nq^{\\dag}\\partial\\_{\\mu}q \n=\n\\frac{-1}{\\lambda^2}\n\\left(\nk\\_{\\mu}-k\\cdot \\sigma \\sigma\\_{\\mu} -\\rmi m \\sigma\\_{\\mu}\n\\right), \\end{aligned}$$ \nand also,\n \n$$\\begin{aligned}\n&\n\\int \\rmd^3 k\\, \n\\epsilon^{\\mu\\nu\\rho}\n\\tr\\left[\nq^{\\dag}\\partial\\_{\\mu}q\nq^{\\dag}\\partial\\_{\\nu}q\nq^{\\dag}\\partial\\_{\\rho}q\n\\right]\n\\nonumber \\\\%&=\n\\int \\rmd^3 k\\, \n\\frac{-\\epsilon^{\\mu\\nu\\rho}}{\\lambda^6}\n\\tr\n\\left[\n-3 \\rmi m\\_5 (k\\cdot \\sigma) \\sigma\\_{\\mu} (k\\cdot \\sigma)\n \\sigma\\_{\\nu} \\sigma\\_{\\rho}\n+\\rmi m^3\\_5 \\sigma\\_{\\mu} \\sigma\\_{\\nu} \\sigma\\_{\\rho}\n\\right]\n\\nonumber \\\\%&=\n(-3 \\rmi m\\_5)\n\\epsilon\\_{\\mu\\nu\\rho}\n\\frac{2 \\rmi}{3}\n\\int \\rmd^3 k\\,\n\\frac{\\epsilon^{\\mu\\nu\\rho} k^2}{(k^2+m^2\\_5)^3}\n+\n\\rmi m^3\\_5 \n2\\rmi\\epsilon\\_{\\mu\\nu\\rho}\n\\int \\rmd^3 k\\,\n\\frac{-\\epsilon^{\\mu\\nu\\rho}}{(k^2+m^2\\_5)^3}\n\\nonumber \\\\%&=\n12m\\_5\n\\int \\rmd^3k\n\\frac{1}{(k^2+m^2\\_5)^2}, \\end{aligned}$$ \nthe winding number is given by\n \n$$\\begin{aligned}\n\\nu\\_3[q] \n=\n\\frac{12}{24 \\pi^2}\nm\\_5\n\\int^{\\infty}\\_0 \\rmd k \n\\frac{4\\pi k^2}{(k^2+m^2\\_5)^2}\n=\n\\frac{1}{2\\pi^2}\n4\\pi\n\\times \n\\frac{\\pi}{4}\n\\frac{m\\_5}{|m\\_5|} \n=\n\\frac{1}{2}\n\\frac{m\\_5}{|m\\_5|}.\\end{aligned}$$ \nWe now compute the Chern-Simons invariant (magnetoelectric polarizability) for the massive Dirac Hamiltonian. For the lower two occupied bands, we can introduce a U(2) gauge field by $\nA\\_{\\mu}^{\\hat{a}\\hat{b}}(k)\n\\rmd k\\_{\\mu} =\n\\langle u\\_{\\hat{a}}^-(k)|\\rmd u\\_{\\hat{b}}^-(k)\\rangle\n$ (*â*, *b̂* = 1, 2). The U(2) gauge field is given by, in the matrix notation,\n \n$$\\begin{aligned}\nA\\_x \n&=&\n\\frac{\\rmi}{2\\lambda (\\lambda +m\\_0 )}\n\\left(\n\\begin{array}{cc}\n+k\\_y & k\\_w +\\rmi k\\_z\\\\\nk\\_w -\\rmi k\\_z & -k\\_y \\\\\n\\end{array}\n\\right),\n\\nonumber \\\\%A\\_y\n&=&\n\\frac{\\rmi}{2\\lambda (\\lambda +m\\_0 )}\n\\left(\n\\begin{array}{cc}\n-k\\_x & \\rmi k\\_w - k\\_z\\\\\n-\\rmi k\\_w - k\\_z & +k\\_x \\\\\n\\end{array}\n\\right),\n\\nonumber \\\\%A\\_z \n&=&\n\\frac{\\rmi}{2\\lambda (\\lambda +m\\_0 )}\n\\left(\n\\begin{array}{cc}\n-k\\_w & -\\rmi k\\_x + k\\_y \\\\\n\\rmi k\\_x + k\\_y & +k\\_w \\\\\n\\end{array}\n\\right).\\end{aligned}$$ \nThe gauge field can be decomposed into U(1) (*a*0) and SU(2) (*a**j* = *x*, *y*, *z*) parts as\n \n$$\\begin{aligned}\nA\\_{\\mu}^{\\ }(k)=\na\\_{\\mu}^{0}(k)\\frac{\\sigma\\_{0}}{2}+a\\_{\\mu}^{j}(k)\\frac{\\sigma\\_{j}}{2}.\n\\label{eq: decomposition of U(2) into U(1)xSU(2)}\\end{aligned}$$ \nThe U(1) part of the Berry connection is trivial, whereas the SU(2) part is given by\n \n$$\\begin{aligned}\n&&\na^{x}\\_{x} = \n\\rmi\\frac{k\\_w}{\\lambda(\\lambda+ m\\_0)},\n\\qquad\na^{y}\\_{x} =\n\\rmi\\frac{-k\\_z}{\\lambda(\\lambda+ m\\_0)},\n\\qquad\na^{z}\\_{x} =\n\\rmi\\frac{+k\\_y}{\\lambda(\\lambda+ m\\_0)},\n\\nonumber \\\\%&&\na^{x}\\_{y} = \n\\rmi\\frac{-k\\_z}{\\lambda(\\lambda+ m\\_0)},\n\\qquad\na^{y}\\_{y} = \n\\rmi\\frac{-k\\_w}{\\lambda(\\lambda+ m\\_0)},\n\\qquad\na^{z}\\_{y} =\n\\rmi\\frac{-k\\_x}{\\lambda(\\lambda+ m\\_0)},\n\\nonumber \\\\%&&\na^{x}\\_{z} = \n\\rmi\\frac{+k\\_y}{\\lambda(\\lambda+ m\\_0)},\n\\qquad\na^{y}\\_{z} = \n\\rmi\\frac{+k\\_x}{\\lambda(\\lambda+ m\\_0)},\n\\qquad\na^{z}\\_{z} = \n\\rmi\\frac{-k\\_w}{\\lambda(\\lambda+ m\\_0)}.\\end{aligned}$$ \nThe Chern-Simons form can be computed as\n \n$$\\begin{aligned}\n\\mathrm{CS}\\_3\n&=&\n\\frac{-1}{8\\pi^2}\n\\int \\rmd^3 k\n\\epsilon^{\\mu\\nu\\rho}\n\\mathrm{tr}\\,\\left(\nA\\_{\\mu} \\partial\\_{\\nu} A\\_{\\rho}\n+\n\\frac{2}{3}\nA\\_{\\mu} A\\_{\\nu} A\\_{\\rho}\n\\right)\n\\nonumber \\\\%&=&\n\\frac{1}{8\\pi^2}\n\\int \\rmd^3 k\n\\frac{m\\_5 \n(2\\lambda + m\\_0 )\n}{\n\\lambda^3 (\\lambda + m\\_0 )^2\n}\n\\nonumber \\\\%\\!\\!&=&\\!\\! \n\\frac{1}{\\pi}\n\\frac{m\\_5}{|m\\_5|}\n\\mathrm{arctan}\n\\left[\n\\frac{\n(m^2\\_5 + m^2\\_0)^{1/2} - m\\_0}\n{\n(m^2\\_5 + m^2\\_0)^{1/2} + m\\_0\n}\n\\right]^{1/2}. \\end{aligned}$$ \nWhen *m*0 = 0,\n \n$$\\begin{aligned}\n\\mathrm{CS}\\_3\n=\n\\frac{1}{4}\\frac{m\\_{5}}{|m\\_{5}|}.\\end{aligned}$$ \nAs the CS term is determined only modulo 1, the fractional part $\\frac{1}{4}\\mathrm{sgn}(m\\_5)$ is an intrinsic property. On a lattice, as there are two-copies of the 4 × 4 Dirac Hamiltonians, the total Chern-Simons invariant is given by $\\mathrm{CS}\\_3=\\frac{1}{2}\\mathrm{sgn}(m\\_{5})$, and $e^{2\\pi\\rmi\\mathrm{CS}\\_3}=-1$.\nThere is a subtlety in computing CS3 and *W*3, which is similar to the case of the Wilson loop *W*1. When *m*0 ≠ 0 and *m*5 = 0, the Chern-Simons form is zero identically, and one would conclude *W*3 = 1, while chiral symmetry for the case of *m*0 ≠ 0 and *m*5 = 0 allows us to define *ν*3, and $\\nu\\_3= \\frac{1}{2} \\mathrm{sgn}(m)$. As before, this puzzle can be solved by properly regularizing the Dirac insulator. With such a regularization, the BZ is topologically *S*3, and one finds a singularity in the wavefunctions. If, on the other hand, we use a different basis, we can avoid to have a Dirac string in the BZ.\nDescription in terms of field theory for the linear responses in *D* = *d* + 1 space-time dimensions\n----------------------------------------------------------------------------------------------------\nFor class A and AIII topological insulators or superconductors, there is always a conserved U(1) quantity; either electric charge or one component (the *z*-component, say) of spin is conserved. Transport properties of these conserved quantities in these topological insulators or superconductors in *d* spatial dimensions can be described in terms of an effective field theory for the linear responses in *D* = *d* + 1 space-time dimensions, which can be obtained by coupling an external space-time dependent gauge field *a**μ* to the topological insulator or superconductor (we will choose the Dirac representative; see the discussion near ([eq: Dirac + external gauge])), and by integrating out the gapped fermions. One can then read off the theory of the linear responses from the so-obtained (classical) effective action for *a**μ*.\nThe theory for the linear responses of a topological insulator in symmetry class A and in *d* spatial dimensions is then given by the Chern-Simons action in *D* = *d* + 1 space-time dimensions,\n \n$$\\begin{aligned}\nS\\_{\\mathrm{CS}}^{D=2n+1} [a]\n=\n2\\pi \\rmi \nk\n\\int\\_{\\mathbb{R}^{D=2n+1}} Q\\_{2n+1}[a],\n\\quad\nk = \\mathrm{Ch}\\_{n}.\n\\label{eq: CS action}\\end{aligned}$$ \nHere, *Q*[*a*] is the Chern-Simons form of the U(1) gauge field *a**μ* defined in *D* = 2*n* + 1 dimensional (Euclidean) spacetime, and the level *k* of the Chern-Simons term is given in terms of Ch*n*, which is the Chern invariant computed from the Bloch wavefunctions in *d* = 2*n* space dimensions.\nOn the other hand, the theory of linear responses of a topological insulator in symmetry class AIII in *d* = 2*n* − 1 spatial dimensions is given by a theta term in *D* = 2*n* space-time dimensions,\n \n$$\\begin{aligned}\nS\\_{\\theta }^{D=2n} [a]\n=\n\\rmi \\theta \\int\\_{\\mathbb{R}^{D=2n}} \n\\mathrm{ch}\\_n[f],\n\\quad\n\\theta = \\pi \\nu\\_{2n-1},\n\\label{eq: theta term}\\end{aligned}$$ \nwhere *f* is the field strength of the external gauge field *a**μ*, and the theta angle *θ* is related to the winding number *ν*2*n* − 1, which can be computed from the Bloch wavefunctions. The integration of fermions in *D* = 2*n* dimensions is trickier than in *D* = 2*n* + 1 dimensions; while a simple derivative expansion of the fermion determinant would appear to yield the Chern-Simons action ([eq: CS action]), a more careful evaluation yields in fact the theta term, which is obtained from the Fujikawa Jacobian.\nThe theta angle in the theta term ([eq: theta term]) can take on a priori any value, if there is no discrete symmetry that pins its value. In class AII Z2 topological insulators in *D* = 4*n* dimensions, time-reversal symmetry demands *θ* to be an integer multiple of *π*. Similarly, in class AIII, chiral symmetry pins *θ* to be an integer multiple of *π*. This can be seen from the symmetry properties of the Chern-Simons and theta terms,\n \n$$\\begin{aligned}\n\\mathcal{T}: S\\_{\\mathrm{CS}}^{D=2n+1} \\to (-1)^n \nS\\_{\\mathrm{CS}}^{D=2n+1}, \n\\quad\nS\\_{\\theta }^{D=2n} \\to (-1)^{n+1} S\\_{\\theta }^{D=2n},\n\\nonumber \\\\%\\mathcal{C}: \n"},"data_id":{"kind":"string","value":"arxiv_0181660"}}},{"rowIdx":45393,"cells":{"content":{"kind":"string","value":"A quadratically convergent semismooth Newton method for nonlinear semidefinite programming without the subdifferential regularity\n=================================================================================================================================\n### This version: 2024/2/21\nWe introduce a quadratically convergent semismooth Newton method for nonlinear semidefinite programming that eliminates the need for the subdifferential regularity, a common yet stringent requirement in existing approaches. Our strategy revolves around identifying a single nonsingular element within the B(ouligand)-subdifferential, thus avoiding the standard requirement for nonsingularity across the entire subdifferential set, which is often too restrictive for practical applications. The theoretical framework is supported by the introduction of the weak second order condition (W-SOC) and the weak strict Robinson constraint qualification (W-SRCQ). These conditions not only guarantee the existence of a nonsingular element in the subdifferential but also forge a primal-dual connection in linearly constrained convex quadratic programming. The theoretical advancements further lay the foundation for the algorithmic designing of a novel semismooth Newton method, which integrates a correction step to address degenerate issues. Particularly, this correction step ensures the local convergence as well as a superlinear/quadratic convergence rate of the proposed method. Preliminary numerical experiments corroborate our theoretical findings and underscore the practical effectiveness of our method. 10 true pt\nSemismooth Newton method, singualrity, subdifferential, nonlinear semidefinite programming, second order conditions, constraint qualifications. 10 true pt\nIntroduction\n============\nLet X and Y be two Euclidean spaces each equipped with an inner product ⟨ ⋅ , ⋅ ⟩ and its induced norm ∥ ⋅ ∥. Consider the following optimization problem (OP):\n \n$$\\begin{aligned}\n \\begin{aligned}\n \\min\\quad & f(x) \\\\\n \\mbox{s.t.}\\quad & G(x) \\in K,\n \\end{aligned}\n \\label{prog:P}\\end{aligned}$$ \nwhere *K* ⊆ Y is a nonempty closed convex set and *f* : X → R, *G* : X → Y are twice continuously differentiable.\nLet $L:{\\mathbb X}\\times{\\mathbb Y}\\to\\mathbb{R}$ be the Lagrangian function of problem defined by\n \n$$L(x;y):=f(x)+ \\langle y, { G}(x) \\rangle, \\quad (x,y)\\in{\\mathbb X}\\times{\\mathbb Y}.$$ \nFor any $y\\in{\\mathbb Y}$, denote the derivative of *L*( ⋅ ; *y*) at *x* ∈ X by *L*ʹ*x*(*x*; *y*) and denote the adjoint of *L*ʹ*x*(*x*; *y*) by ∇*x**L*(*x*; *y*). The Karush-Kuhn-Tucker (KKT) optimality conditions for problem takes the following form:\n \n$$\\begin{aligned}\n \\left\\{\n \\begin{aligned}\n &\\nabla\\_x {L}(x,y) = 0, \\\\\n &y \\in \\mathcal{N}\\_{K}(G(x)),\n \\end{aligned}\n \\right. \\label{KKT:P}\\end{aligned}$$ \nwhere N*K*(*G*(*x*)) is the normal cone to *K* at *G*(*x*) in the context of convex analysis. We say (*x*, *y*) ∈ Z : = X × Y is a KKT point of if it solves. The set of Lagrange multipliers associated with a feasible point *x* is defined by *M*(*x*) = {*y* ∣ (*x*, *y*)is a KKT point}. For a given feasible point $\\overline{x}$, if $M(\\overline{x}) \\neq \\emptyset$, then we call it a stationary point of. It is well-known that $(\\overline{x}, \\overline{y})$ is a KKT point of if and only if it is a solution of the nonsmooth system\n \n$$\\label{eq:natural-mapping}\n F(x,y) := \\begin{bmatrix}\n \\nabla\\_x L(x,y) \\\\\n -G(x) + \\Pi\\_{K}(G(x) + y)\n \\end{bmatrix}=0,$$ \nwhere Π*K* : Y → Y is the metric projection operator onto *K*, i.e., $\\Pi\\_K(z) =\\displaystyle{ {\\rm arg}\\min\\_{y' \\in K} \\frac{1}{2}\\| z - y'\\|^2}$ for *z* ∈ Y.\nWhen solving *F*(*x*, *y*) = 0, one may encounter challenges related to the nonsmoothness of *F*, primarily arising from the nonsmooth nature of the metric projection operator Π*K*( ⋅ ), especially when *K* is not a subspace. Fortunately, for various problems, including nonlinear programming (NLP), nonlinear second-order cone programming, and nonlinear semidefinite programming (NLSDP), the corresponding nonsmooth map *F* exhibits the so-called (strong) semismoothness (see Definition [def:semismooth]). This property is conducive to the use of nonsmooth Newton methods. Qi and Sun introduced a semismooth Newton method, which takes the following semismooth Newton step in each iteration: \n*Z**k* + 1 = *Z**k* − (*U**k*)− 1*F*(*Z**k*), \n with *U**k* being selected from the B(ouligand)-subdifferential ∂*B**F*(*Z**k*) (see for definition) or the Clarke-subdifferential ∂*C**F*(*Z**k*) (see for definition). The semismooth Newton method, renowned for its high efficiency, has found widespread applications across various domains. It has been effectively utilized for tackling variational inequalities and constrained optimization problems in function spaces, infinite-dimensional linear complementarity problems and computing the nearest correlation matrix. Notably, its efficacy has been particularly pronounced in solving augmented Lagrangian subproblems, whose versatility is underscored by its implementation in several influential software packages, including SDPNAL, SDPNAL+, QSDPNAL, and SSNAL. For an in-depth exploration, one may refer to.\nIt is well-known (see e.g., ) that the (quadratic) superlinear convergence of semismooth Newton methods hinge on the (strong) g-semismoothness (see Definition [def:semismooth]) of *F* and the uniform boundedness of {(*U**k*)− 1}. To guarantee such uniform boundedness, typically, one has to assume the KKT nonsmooth mapping *F* satisfies certain subdifferential regularity at the solution point $(\\overline{x},\\overline{y})$, for example, all elements in some subdifferential sets of *F* at $(\\overline{x},\\overline{y})$ are nonsingular. Particularly, one may assume that every element in $ \\partial\\_B F(\\overline{x},\\overline{y}) $ or even $ \\partial\\_C F(\\overline{x},\\overline{y}) $ is nonsingular, known as the BD-regularity or CD-regularity, respectively. However, this condition is frequently demanding and may not always be fulfilled in practical applications. Thus, the following important question arises:\n> \n> *Can we design a locally superlinear or quadratic convergent semismooth Newton method for solving without assuming the subdifferential regularity of *F*?* \n> \n> \n> \nIn this paper, we provide an affirmative answer to this question. Specifically, for the NLSDP, we design a semismooth Newton method with a novel correction step to avoid the requirement of nonsingularity across the entire subdifferential of *F*, and prove its fast local convergence rates, thus advancing the traditional paradigms. Take (more details can be found in Example [example:4]) as an illustrative example. It can be shown that for this special example, the corresponding B-subdifferential $\\partial\\_B F(\\overline{x},\\overline{y})$ at the KKT point $(\\overline{x},\\overline{y})$ contains singular matrices. Figure [fig:ex31] provides an illustration of the performance of our newly proposed semismooth Newton method with a correction step for solving this degenerate problem. Clearly, even in the absence of the BD-regularity, our method still exhibits a local quadratic convergence.\n [fig:ex31]\nFor the optimization problem, the characterization of the subdifferential regularity for the nonsmooth function *F* is closely intertwined with perturbation theory, a fundamental topic in optimization. Indeed, the perturbation theory is crucial for both theoretical insights and practical algorithm design. Over the past four decades, significant advancements in this field has been well-documented in the literature, especially concerning with *K* being a polyhedral set (i.e., *K* is the intersection of a finite number of half-spaces). However, less exploration has been done regarding non-polyhedral cases, such as *K* being the set of positive semidefinite matrices.\nIn the realm of nonlinear semidefinite programming, Sun established that under the Robinson constraint qualification (RCQ) (Definition [def:RCQ]), the nonsingularity of $\\partial\\_C F(\\overline x,\\overline y)$ (i.e., the CD-regularity of *F*) at a local optimal solution $\\overline{x}$ and a associated multiplier $\\overline{y}$ is equivalent to the strong second-order sufficient condition (S-SOSC) (Definition [def:S-SOSC]) and the constraint nondegeneracy (Definition [def:constraint nondegeneracy]), and is also equivalent with the strong regularity introduced by Robinson of the solution to the KKT system. For linear SDP problems, Chan and Sun provided deeper insights into the strong regularity, demonstrating that under RCQ, primal and dual constraint nondegeneracies, the nonsingularity of $\\partial\\_B F(\\overline x,\\overline y)$ (i.e., the BD-regularity of *F*), CD-regularity and the strong regularity at $(\\overline x,\\overline y)$ are all equivalent. This comprehensive understanding underscores that the conditions for the nonsingularity of all elements in the subdifferentials of *F* at $(\\overline x,\\overline y)$ may be stringent and not consistently be satisfied in practice. For instance, the S-SOSC condition presupposes the isolatedness of a local optimal solution, a premise that is often violated in statistical optimization applications such as LASSO. Consequently, it is crucial to develop a new framework of semismooth Newton methods that can bypass the subdifferential regularity and still achieve a local superlinear or quadratic convergence rate.\nIn this paper, we propose to relax the stringent subdifferential regularity conditions by identifying a single nonsingular element within the B-subdifferential. However, characterizing the nonsingularity of even a single element within the subdifferentials of *F* at a KKT point $(\\overline{x},\\overline{y})$, especially in non-polyhedral cases, remains an unresolved challenge. When *K* is polyhedral, Izmailov and Solodov indicated that the weak strict Robinson constraint qualification (W-SRCQ) and the weak second-order condition (W-SOC) (see Definitions [def:W-SRCQ] and [def:W-SOC], respectively) guarantee the existence of a nonsingular element in the parametric-subdifferential. Here the parametric-subdifferential, defined as differentials of all possible parametric systems, contains $\\partial\\_B F(\\overline x,\\overline y)$. Indeed, a straightforward deduction from their findings indicates that the W-SRCQ and W-SOC also guarantee the presence of a nonsingular element in $\\partial\\_B F(\\overline x,\\overline y)$. However, these conditions are inadequate when applied to NLSDP. Specifically, the W-SRCQ and W-SOC are insufficient to ensure the existence of a nonsingular element in $\\partial\\_B F(\\overline x,\\overline y)$, as demonstrated by our counterexample (Example [example:non-SRCQ]). Meanwhile, in NLSDP, the presence of a nonsingular subdifferential element, as in Example [example:4], does not necessarily imply calmness, an important property described in. The lack of calmness may lead to slow progress of algorithms for solving the KKT system, as the associated solution mapping exhibits heightened sensitivity to small perturbations.\nIn this paper, we show that the existence of a nonsingular element in $\\partial\\_B F(\\overline x,\\overline y)$ can be ensured under the W-SOC with constraint nondegeneracy or the S-SOSC with the W-SRCQ at a KKT point $(\\overline{x},\\overline{y})$. Additionally, our results indicate that the W-SRCQ and W-SOC are sufficient for the existence of a nonsingular element in $\\partial\\_C F(\\overline x,\\overline y)$, and are also necessary for convex cases. More interestingly, we uncover a profound primal-dual connection between the W-SOC and W-SRCQ for linearly constrained convex quadratic semidefinite programming. These connections mirror those found between the strict Robinson constraint qualification (SRCQ) (Definition [def:SRCQ]) and the second-order sufficient condition (SOSC) (Definition [def:SOSC]) for linearly constrained quadratic semidefinite programming, as well as those between the constraint nondegeneracy and the S-SOSC in the setting of linear semidefinite programming. We summarize our results and connections in Figure [figure:con]. Based on the theoretical findings, we are able to design an inexact semismooth Newton method with a correction step. Each iteration commences with a semismooth Newton step, followed by a correction step designed to ensure the nonsingularity. Under the W-SOC and constraint nondegeneracy (or the S-SOSC and W-SRCQ), we show the local convergence and superliner convergence of the designed algorithm. Furthermore, if the function is twice Lipschitz continuously differentiable, the convergence rate can be further improved to quadratic.\n![Connections of second order conditions, constraint qualifications and nonsingularity of subdifferentials. Here the black arrow holds for ; the orange arrow holds for NLSDP; the green arrow holds for CSDP; and the violet arrow holds for QSDP. Moreover, primal-dual means that the starting point of the arrow for the primal problem implies the arriving point of the arrow for the dual problem.](connection.png \"fig:\") [figure:con]\nThe remaining parts of this paper are organized as follows. Section [section:pre] provides some basic definitions and preliminary results concerning the variational properties of the metric projection projector over the positive semidefinite cone. Section [section:nonsingularity] delves into the sufficient and necessray conditions for the existence of one nonsingular element in the subdifferentials of *F* for NLSDP. The primal-dual connections of the W-SOC and the W-SRCQ are explored in Section [section:cha] for convex quadratic semidefinite programming. Then, we present a semismooth Newton method with a correction step in Section [section:5], and some preliminary numerical results in Section [section:numerical]. Finally, we conclude this paper and highlight potential directions for future research in Section [section:conclusion].\nPreliminaries\n=============\nWe first give the following notation that will be used throughout the paper. Let X and Y be two real Euclidean spaces. Given a set *S* ⊆ X and a point *x* ∈ X, the distance from *x* to *S* is denoted by \n$${\\rm dist}(x,S):=\\mbox{inf}\\{\\|y-x\\|\\,|\\ y\\in S\\}.$$ \n For a linear operator ${\\cal A}: \\mathbb{X}\\rightarrow \\mathbb{Y}$, ${\\cal A}^\\*$ denotes the adjoint of the linear operator ${\\cal A}$. We denote by *A**T* the transpose of a given matrix *A*. For a mapping Φ : X → Y and *x* ∈ X, Φʹ(*x*) stands for the classical derivative or the Jacobian of Φ at *x* and ∇Φ(*x*) the adjoint of the Jacobian. We use S*n* to denote the linear space of all *n* × *n* real symmetric matrices and use S+*n* to denote the closed convex cone of all *n* × *n* positive semidefinite matrices in S*n*. For a function *g* : X → R, we denote *g*+(*x*) : = max{0, *g*(*x*)} and if it is vector-valued then the maximum is taken componentwise. Some other notation are listed below:\n* Let ${\\cal O}^{n}$ be the set of all *n* × *n* orthogonal matrices.\n* For any *Z* ∈ R*m* × *n*, we denote by *Z**i**j* the (*i*, *j*)-th entry of *Z*.\n* For any *Z* ∈ R*m* × *n* and a given index set ${\\cal J}\\subseteq \\{1,\\ldots, n\\}$, we use $ Z\\_{{\\cal J}}$ to denote the sub-matrix of *Z* obtained by removing all the columns of *Z* not in ${\\cal J}$. In particular, we use *Z**j* to represent the *j*-th column of *Z*, *j* = 1, …, *n*.\n* Let ${\\cal I}\\subseteq \\{1,\\ldots, m\\}$ and ${\\cal J}\\subseteq \\{1,\\ldots, n\\}$ be two index sets. For any *Z* ∈ R*m* × *n*, we use $Z\\_{{\\cal I}{\\cal J}}$ to denote the $|{\\cal I}|\\times|{\\cal J}|$ sub-matrix of *Z* obtained by removing all the rows of *Z* not in ${\\cal I}$ and all the columns of *Z* not in ${\\cal J}$.\n* We use $``\\circ\"$ to denote the Hardamard product between matrices, i.e., for any two matrices *A* and *B* in R*m* × *n* the (*i*, *j*)-th entry of *Z* : = *A* ∘ *B* ∈ R*m* × *n* is *Z**i**j* = *A**i**j**B**i**j*.\n* Let ${\\rm Diag}(\\cdot):\\mathbb{R}^n\\to \\mathbb{S}^n$ be the linear map such that for any *x* ∈ R*n*, ${\\rm Diag}(x)$ denotes the diagonal matrix whose *i*-th diagonal entry is *x**i*, *i* = 1, …, *n*.\n* Let ${{\\rm lin}}(C)$ be the linearity space of a closed convex set *C* ⊆ X, i.e., the largest subspace in *C*. Meanwhile, let ${{\\rm aff}}(C)$ be the affine hull of *C*, i.e., the smallest subspace containing *C*.\n* For a map Φ : X → Y, define ${{\\rm Im}}(\\Phi):=\\{\\Phi(x)\\ |\\ x\\in{\\mathbb{X}}\\}$ and ${{\\rm Null}}(\\Phi):=\\{x\\in{\\mathbb{X}}\\ |\\ \\Phi(x)=0\\}$.\n* The supprot function of a set *S* ⊆ X is defined by *σ*(*x*, *S*) = sup*s* ∈ *S*⟨*x*, *s*⟩ for *x* ∈ X.\n* For *r* > 0 and $\\overline X\\in{\\mathbb{X}}$, the closed ball centered at $\\overline X$ with radius *r* is denote by $\\mathbb{B}({\\overline{X}},r)=\\{X\\in{\\mathbb{X}}\\ |\\ \\|X-\\overline X\\|\\le r\\}$.\n* For *f*, *g* : X → R, *f*(*x*) = Θ(*g*(*x*)) for $x\\to \\overline x$ means that *f*(*x*) = *O*(*g*(*x*)) and *g*(*x*) = *O*(*f*(*x*)).\nFor a given map Φ : X → Y, the directional differential of Φ at point *x* ∈ X along a direction *d* ∈ X is defined by \n$$\\Phi'(x;d)=\\lim\\_{t\\downarrow 0}\\frac{F(x+td)-F(x)}{t}\\quad \\mbox{if exists.}$$ \n Φ is directionally differentiable at *x* ∈ X if Φʹ(*x*; *d*) exists for all *d* ∈ X. Suppose that Φ : *N* ⊆ X → Y is a locally Lipschitz continuous function on the open set *N*. Then, according to Rademacher’s theorem, Φ is almost everywhere F-differentiable in *N*. Let *D*Φ be the set of points in *N* where Φ is differentiable. Let Φ′(*x*) be the derivative of Φ at *x* ∈ *D*Φ. Then the B-subdifferential of Φ at *x* ∈ *N* is denoted by :\n \n$$\\label{eq:partialB}\n \\partial\\_{B}\\Phi(x):=\\big\\{ \\lim\\_{D\\_{\\Phi}\\ni x^{k}\\to x} \\Phi'(x^{k}) \\big\\}\\,,$$ \nand the Clarke-subdifferential of Φ at *x* ∈ *N* takes the form:\n \n$$\\begin{aligned}\n\\label{eq:partialC}\n \\partial\\_C \\Phi(x)={\\rm conv}\\big\\{\\partial\\_{B} \\Phi(x)\\big\\},\n \\end{aligned}$$ \nwhere “${\\rm conv}$” stands for the convex hull in the usual sense of convex analysis. The following g-semismoothness of a locally Lipschitz continuous function is an extension of the semismoothness (originally defined by Mifflin for real-valued functions and adopted by Qi and Sun to the vector case.)\n[def:semismooth] Let Φ : *N* ⊆ X → Y be a locally Lipschitz continuous function on the open set *N*. The function Φ is said to be g-semismooth at a point *x* ∈ *N* if for any *y* → *x* and *V* ∈ ∂*C*Φ(*y*), \nΦ(*y*) − Φ(*x*) − *V*(*y* − *x*) = *o*(∣∣*y* − *x*∣∣) . \n The function Φ is said to be strongly g-semismooth at *x* if for any *y* → *x* and *V* ∈ ∂*C*Φ(*y*), \nΦ(*y*) − Φ(*x*) − *V*(*y* − *x*) = *O*(∣∣*y* − *x*∣∣2) . \n Furthermore, the function Φ is said to be (strongly) semismooth at *x* ∈ *N* if (i) Φ is directional differentiable at *x*; and (ii) Φ is (strongly) g-semismooth.\nBackground in variational analysis\n----------------------------------\nIn this subsection, we introduce some variational properties related to problem, which will be useful in our subsequent discussions.\nFor the closed convex set *K* ⊆ Y, the tangent cone to *K* at *y* ∈ Y is defined by\n \n$$\\label{eq:def-Tangent-g}\n {\\mathcal{T}}\\_K(y)=\\{d\\in{\\mathbb{Y}}\\ |\\ \\exists\\, t^k\\downarrow 0,\\ {{{\\rm dist}}}(y+t^kd,K)=o(t^k)\\}.$$ \nLet $\\overline{x}$ be a given feasible solution to. It is well-known that for the feasible solution $\\overline{x}$, the following Robinson constraint qualification is equivalent to the nonemptyness and boundedness of the corresponding Lagrange multiplier set $M(\\overline{x})$.\n[def:RCQ] The Robinson constraint qualification (RCQ) is said to hold at $\\overline x$ if\n \n$$G'(\\overline x){\\mathbb{X}}+{\\mathcal{T}}\\_K(G(\\overline x))={\\mathbb{Y}}.$$ \nFor a stationary solution $\\overline{x}$ of, the strict Robinson constraint qualification condition is defined as follows.\n[def:SRCQ] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of with a Lagrange multiplier $\\overline{y}\\in M(\\overline x)$. The strict Robinson constraint qualification (SRCQ) is said to hold at $\\overline x$ with respect to $\\overline y$ if\n \n$$G'(\\overline x){\\mathbb{X}}+{\\mathcal{T}}\\_K(G(\\overline x))\\cap \\overline y^{\\perp}={\\mathbb{Y}},$$ \nwhere *y*⊥ : = {*s* ∈ Y ∣ ⟨*s*, *y*⟩ = 0} for any vector *y* ∈ Y.\nIt follows from that the set of Lagrange multipliers $M(\\overline x)$ is a singleton if the SRCQ hold.\n \n$$\\label{eq:critical-1}\n{\\mathcal{C}}(\\overline x)=\\{d\\in{\\mathbb{X}}\\ |\\ G'(\\overline x)d\\in {\\mathcal{T}}\\_K(G(\\overline x)),\\ f'(\\overline x)d=0\\}.$$ \n \n$$\\label{eq:critical-2}\n{\\mathcal{C}}(\\overline x)=\\{d\\in{\\mathbb{X}}\\ |\\ G'(\\overline x)d\\in{\\mathcal{T}}\\_K(G(\\overline x))\\cap \\overline y^{\\perp}\\}.$$ \nThe following definition on the constraint nondegeneracy of, introduced by Robinson, is stronger than the SRCQ since ${{\\rm lin}}({\\mathcal{T}}\\_K(G(\\overline x)))\\subseteq {\\mathcal{T}}\\_K(G(\\overline x))\\cap \\overline y^{\\perp}$ (cf. ).\n[def:constraint nondegeneracy] The constraint nondegeneracy of is said to hold at $\\overline x$ if\n \n$$G'(\\overline x){\\mathbb{X}}+{{\\rm lin}}({\\mathcal{T}}\\_K(G(\\overline x)))={\\mathbb{Y}}.$$ \nRecall that the inner and outer second order tangent sets () to the given closed set *K* ∈ Y in the direction *d* ∈ Y can be defined, respectively, by\n \n$${\\mathcal{T}}^{i,2}\\_K(y,d):=\\{w\\in {\\mathbb{Y}}\\mid {{\\rm dist}}(y+td+\\frac{1}{2}t^2w,C)=o(t^2),\\ t\\ge 0\\}$$ \nand\n \n$${\\mathcal{T}}^{2}\\_K(y,d):=\\{w\\in {\\mathbb{Y}}\\mid \\exists\\,t^k\\downarrow 0,\\ {{\\rm dist}}(y+t^kd+\\frac{1}{2}(t^k)^2w,C)=o((t^k)^2)\\}.{{\\nonumber}}$$ \nSuppose that *K* in is *C*2-cone reducible. The following second order sufficient condition is adopted from.\n[def:SOSC]\n \n$$\\begin{aligned}\n\\label{eq:SOSC-P}\n{\\langle}d,\\nabla^2\\_{xx} L(\\overline x,\\overline y)d{\\rangle}-\\sigma(\\overline y,\\mathcal{T}^2\\_{K}(G(\\overline x),G'(\\overline x)d))>0 \\quad \\forall\\, d\\in{\\mathcal{C}}(\\overline x)\\backslash\\{0\\}.\\end{aligned}$$ \nThe following definition of strong second order sufficient condition for is an extension of for NLSDP. It is worth to note when $M(\\overline{x})$ is a singleton, both of them are the same.\n[def:S-SOSC] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of with a Lagrange multiplier $\\overline{y}\\in M(\\overline x)$. The strong second order sufficient condition (S-SOSC) is said to hold at $(\\overline x,\\overline y)$ if\n \n$${\\langle}d,\\nabla^2\\_{xx} L(\\overline x,\\overline y)d{\\rangle}-\\sigma(\\overline y,\\mathcal{T}^2\\_{K}(G(\\overline x),G'(\\overline x)d))>0\\quad \\forall\\, d\\in{{\\rm app}}(\\overline x,\\overline y)\\backslash\\{0\\},$$ \nwhere ${{\\rm app}}(\\overline x,\\overline y)$ is the out approximation of the affine hull of the critical cone ${\\cal C}(\\overline{x})$ of with respect to $(\\overline{x},\\overline{y})$, i.e.,\n \n$$\\label{eq:def-app}\n {{\\rm app}}(\\overline x,\\overline y):=\\{d\\in\\mathbb{X}\\mid G'(\\overline x)d\\in{{\\rm aff}}({\\mathcal{T}}\\_K(G(\\overline x))\\cap \\overline y^{\\perp})\\}.$$ \nEigenvalue decomposition of symmetric matrices\n----------------------------------------------\nIn this subsection, we introduce some useful preliminary results on eigenvalue decompositions of real symmetric matrices and the differentiability of the metric projectors over the positive semidefinite matrix cone S+*n*.\nLet *A* ∈ S*n* be given. We use *λ*1(*A*) ≥ *λ*2(*A*) ≥ … ≥ *λ**n*(*A*) to denote the eigenvalues of *A* (all real and counting multiplicity) arranging in nonincreasing order and use *λ*(*A*) to denote the vector of the ordered eigenvalues of *A*. Let $\\Lambda(A):= {\\rm Diag}(\\lambda(A))$. Consider the eigenvalue decomposition of *A*, i.e., *A* = *P*Λ(*A*)*P**T*, where ${P}\\in{\\cal O}^{n}$. By considering the index sets of positive, zero, and negative eigenvalues of *A*, we are able to write *A* in the following form\n \n$$\\label{eq:eig-de}\n A= \\left[\\begin{array}{ccc}\n {P}\\_{\\alpha} & {P}\\_{\\beta} & {P}\\_{\\gamma}\n \\end{array}\\right] \\left[\\begin{array}{ccc}\n \\Lambda(A)\\_{\\alpha\\alpha} & 0 & 0 \\\\ [3pt]\n 0 & 0 & 0\\\\ [3pt]\n 0 & 0 & \\Lambda(A)\\_{\\gamma\\gamma}\n \\end{array}\\right]\\left[\\begin{array}{c}\n {P}\\_{\\alpha}^T \\\\ [3pt]\n {P}\\_{\\beta}^T \\\\\n {P}\\_{\\gamma}^T\n \\end{array}\\right],$$ \nwhere *α*, *β* and *γ* are index sets defined by\n \n*α* : = {*i* ∣ *λ**i* > 0}, *β* : = {*i* ∣ *λ**i* = 0} and *γ* : = {*i* ∣ *λ**i* < 0}. \nWe use ${\\cal O}^{n}(A)$ to denote the set of all orthogonal matrices $P\\in{\\cal O}^{n}$ satisfying.\nThe next proposition, which was stated in, illustrates that the eigenvectors of symmetric matrices, though not continuous, are upper Lipschitz continuous.\n[prop:dis-P] Given *A* ∈ S*n*, for any *H* ∈ S*n*, let *P* be an orthogonal matrix such that *X* + *H* = *P*Λ(*A* + *H*)*P**T*. Then, as *H* → 0, we have\n \n$$\\begin{aligned}\n{{\\nonumber}}{{\\rm dist}}(P,{\\mathcal{O}}^n(A))=O(\\| H\\| ).\n \\end{aligned}$$ \nLet ΠS+*n* : S*n* → S*n* be the metric projection operator over the positive semidefinite matrix cone S+*n*, i.e.,\n \n$$\\label{eq:Pi-SDP}\n\\Pi\\_{\\mathbb{S}^n\\_+}(A):={\\rm arg}\\min\\_{X\\in\\mathbb{S}^n\\_+}\\left\\{\\frac{1}{2}\\|X-A\\|^2\\right\\},\\quad A\\in\\mathbb{S}^n.$$ \nIt is clear that ΠS+*n*( ⋅ ) is globally Lipschtiz continuous. Furthermore, it follows from (see also ) that ΠS+*n*( ⋅ ) is directionally differentiable at any *A* ∈ S*n* and the directional derivative at *A* along direction *H* ∈ S*n* is given by\n \n$$\\label{eq:dd-projection}\n \\Pi\\_{\\mathbb{S}\\_+^n} ^{\\prime}(A;H) = {P} \\left[\n \\begin{array}{ccc}\n {\\widetilde H}\\_ {\\alpha\\alpha} & {\\widetilde H}\\_{\\alpha\\beta} &\\Xi\\_{\\alpha\\gamma} \\circ {\\widetilde H}\\_{\\alpha\\gamma}\\\\\n \\widetilde{H}\\_{\\alpha\\beta}^T & \\Pi\\_{{\\cal S}\\_+^{\\rvert\\beta\\lvert} }({\\widetilde H}\\_{\\beta\\beta}) &0\\\\\n \\Xi\\_{\\alpha\\gamma}^T \\circ {\\widetilde H}\\_{\\alpha\\gamma}^T & 0 & 0\n \\end{array}\n \\right] {P}^T,$$ \nwith *P* ∈ O*n*(*A*), ${\\widetilde H}:={P}^T H{P}$ and Ξ ∈ S*n* is defined by\n \n$$\\label{eq:Xi}\n \\Xi\\_{ij}:=\\frac{\\max\\{\\lambda\\_{i}(A),0\\}-\\max\\{\\lambda\\_{j}(A),0\\}}{\\lambda\\_{i}(A)-\\lambda\\_{j}(A)},\\quad i,j=1,\\ldots,n,$$ \nwhere 0/0 is defined to be 1. Meanwhile, it is well-known that the metric projection operator ΠS+*n*( ⋅ ) is strongly semismooth everywhere.\nThe following result on the characterizations of the B-subdifferential and Clarke-subdifferential of ΠS+*n*( ⋅ ) is taken from.\n[prop:dif-B-F] Given *A* ∈ S*n*. ${\\cal V}\\in\\partial\\_B\\Pi\\_{{\\mathbb{S}}^n\\_+}(A)$ (∂*C*ΠS+*n*(*A*)) if and only if there is ${\\cal W}\\in\\partial\\_B\\Pi\\_{{\\mathbb{S}}^{|\\beta|}\\_+}(0)$ (∂*C*ΠS+∣*β*∣(0)) such that for any *H* ∈ S*n*, \n$${\\cal V}(H)=P\n\\begin{bmatrix}\n\\widetilde{H}\\_{\\alpha\\alpha}&\\widetilde{H}\\_{\\alpha\\beta}&\\Xi\\_{\\alpha\\gamma}\\circ\\widetilde{H}\\_{\\alpha\\gamma} \\\\\n\\widetilde{H}\\_{\\beta\\alpha}&{\\cal W}(\\widetilde{H}\\_{\\beta\\beta})&0 \\\\\n\\Xi\\_{\\gamma\\alpha}\\circ\\widetilde{H}\\_{\\gamma\\alpha}&0&0 \n\\end{bmatrix}\nP^T,$$ \n where $P\\in{\\cal O}^n(A)$ and $\\widetilde{H}=P^THP$. In particular, since both the zero mapping ${\\cal W}\\equiv 0$ and the identity mapping ${\\cal W}\\equiv {\\cal I}$ from S∣*β*∣ → S∣*β*∣ are elements of ∂*B*ΠS+∣*β*∣(0), the following two mappings ${\\cal V}\\_0$ and ${\\cal V}\\_{{\\mathcal{I}}}$ defined by\n \n$$\\label{eq:exactV0}\n {\\cal V}\\_0(H)=P\n \\begin{bmatrix}\n \\widetilde{H}\\_{\\alpha\\alpha}&\\widetilde{H}\\_{\\alpha\\beta}&\\Xi\\_{\\alpha\\gamma}\\circ\\widetilde{H}\\_{\\alpha\\gamma} \\\\\n \\widetilde{H}\\_{\\beta\\alpha}&0&0 \\\\\n \\Xi\\_{\\gamma\\alpha}\\circ\\widetilde{H}\\_{\\gamma\\alpha}&0&0 \n \\end{bmatrix}\n P^T,\\quad H\\in {\\mathbb{S}}^n$$ \nand\n \n$$\\label{eq:exactV1}\n {\\cal V}\\_{{\\mathcal{I}}}(H)=P\n \\begin{bmatrix}\n \\widetilde{H}\\_{\\alpha\\alpha}&\\widetilde{H}\\_{\\alpha\\beta}&\\Xi\\_{\\alpha\\gamma}\\circ\\widetilde{H}\\_{\\alpha\\gamma} \\\\\n \\widetilde{H}\\_{\\beta\\alpha}&\\widetilde{H}\\_{\\beta\\beta}&0 \\\\\n \\Xi\\_{\\gamma\\alpha}\\circ\\widetilde{H}\\_{\\gamma\\alpha}&0&0 \n \\end{bmatrix}\n P^T,\\quad H\\in {\\mathbb{S}}^n$$ \nare elements of ∂*B*ΠS+*n*(*A*).\nExistence of nonsingular elements of the subdifferentials\n=========================================================\nAs mentioned in the introduction, our preliminary objective is to explore the existence of nonsingular elements within the subdifferentials of *F* defined in. This investigation sets the stage for the subsequent development of a semismooth Newton method that achieves quadratic convergence without the BD-regularity. To commence, we introduce the concept of the weak strict Robinson constraint qualification (W-SRCQ), which serves as an extension to the SRCQ condtion (Definition [def:SRCQ]).\n[def:W-SRCQ] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of with a Lagrange multiplier $\\overline{y}\\in M(\\overline{x})$. The weak strict Robinson constraint qualification (W-SRCQ) is said to hold at $\\overline x$ with respect to $\\overline{y}$ if \n$$G'(\\overline x){\\mathbb{X}}+{{\\rm aff}}({\\mathcal{T}}\\_K(G(\\overline x))\\cap \\overline{y}^{\\perp})={\\mathbb{Y}}.$$ \nThe W-SRCQ condition is referred to as the *weak linear independence constraint qualification* (W-LICQ) in the context of nonlinear variational inequalities, as explored in. Nevertheless, we choose not to adopt the W-LICQ terminology in this context. This is because the W-SRCQ is not simply a weaker form of the classical LICQ; rather, it is a distinct concept that arises by incorporating the affine hull into the SRCQ.\nAs can be inferred directly from the definitions (Definitions [def:SRCQ] and [def:W-SRCQ]), the W-SRCQ is weaker than the SRCQ. This implies that the W-SRCQ is also milder than the constraint non-degeneracy condition as outlined in Definition [def:constraint nondegeneracy]. Moreover, the example provided below demonstrates that W-SRCQ is, in fact, even weaker than the RCQ defined by Definition [def:RCQ].\n[example:1] Consider the following quadratic SDP problem:\n \n$$\\begin{aligned}\n\\label{eq:prob\\_example1}\n \\begin{aligned}\n \\min\\quad &\\frac{1}{2}\\| X\\_{11}\\| ^2-\\frac{1}{2}\\| X\\_{22}\\| ^2 \\\\\n \\mbox{s.t.}\\quad & X\\_{12}=0, \\quad X\\_{22}=0, \\\\\n & X\\in {\\mathbb{S}}^n\\_+, \n \\end{aligned}\n \\end{aligned}$$ \nwhere $X=\\begin{bmatrix} X\\_{11} &X\\_{12}\\\\X\\_{21}&X\\_{22}\\end{bmatrix}\\in \\mathbb{S}^n$ with *X*11 ∈ R*l*1 × *l*1 and *X*22 ∈ R*l*2 × *l*2. It is easy to verify that $\\overline{X}=0$ is a local optimal solution with the following Lagrange multiplier set \n$$M(\\overline{X})=\\left\\{(\\xi\\_{12},\\xi\\_{22},\\Gamma)\\in\\mathbb{R}^{l\\_1\\times l\\_2}\\times\\mathbb{R}^{l\\_2\\times l\\_2}\\times \\mathbb{S}^n \\mid \\begin{bmatrix} 0 &\\xi\\_{12}\\\\ \\xi\\_{12}^T&\\xi\\_{22} \\end{bmatrix}+\\Gamma=0\\right\\}.$$ \n Thus, $(\\overline{\\xi}\\_{12},\\overline{\\xi}\\_{22},\\overline{\\Gamma})=(0,0,0)\\in M(\\overline{X})$. Consider the KKT point $(\\overline{X},\\overline \\xi\\_{12},\\overline\\xi\\_{22},\\overline{\\Gamma})=(0,0,0,0)$. We know that the index sets defined by are *α* = *γ* = ∅ and *β* = {1, …, *n*}. It then can be checked directly by Definition [def:W-SRCQ] that the W-SRCQ holds at $\\overline{X}$ with respect to $(\\overline{\\xi}\\_{12},\\overline{\\xi}\\_{22},\\overline{\\Gamma})$. Meanwhile, since the multiplier set $M(\\overline{X})$ is unbounded, we know that the RCQ does not hold at $\\overline{X}$.\nNext, we shall introduce the concept of the weak second order condition for a KKT solution $(\\overline{x},\\overline{y})$ of, which is a generalization of the SOSC (Definition [def:SOSC]).\n[def:W-SOC]\n \n$$\\label{eq:the W-SOC-P}\n{\\langle}d,\\nabla^2\\_{xx} L(\\overline x,\\overline y)d{\\rangle}-\\sigma(\\overline y,\\mathcal{T}^2\\_{K}(G(\\overline x),G'(\\overline x)d))>0 \\quad \\forall\\, d\\in{{\\rm appl}}(\\overline x,\\overline y)\\backslash\\{0\\},$$ \nwhere ${{\\rm appl}}(\\overline x,\\overline y)$ is defined by\n \n$$\\label{eq:def-appl}\n{{\\rm appl}}(\\overline x,\\overline y):=\\left\\{d\\in\\mathbb{X}\\mid G'(\\overline x)d\\in{{\\rm lin}}({\\mathcal{T}}\\_K(G(\\overline x))\\cap \\overline y^{\\perp})\\right\\}.$$ \nThe W-SOC is referred to as the *second order condition* for nonlinear variational inequalities in. Moreover, it can be seen from the definitions that at a KKT point, the following implications hold: \nS-SOSC ⇒ SOSC ⇒ W-SOC. \n Furthermore, it is important to note that the W-SOC does not suffice to ensure local optimality for general optimization problems. Additionally, within the domain of convex problems, the W-SOC alone does not necessarily lead to the isolation of the optimal solution.\nHenceforth, our primary focus will be on the following nonlinear semidefinite programming (NLSDP) problem:\n \n$$\\label{prog:NLSDP}\n \\begin{array}{cl}\n \\min & f(x) \\\\ [3pt]\n \\mbox{s.t.} & h(x)=0, \\\\ [3pt]\n & g(x)\\in {\\mathbb{S}}^n\\_+.\n \\end{array}$$ \nIt should be noted that the results derived herein can be similarly extended through straightforward generalizations to cases where *K* in represents the Cartesian product of a finite number of positive semidefinite cones and zero vectors. Furthermore, considering that R*n* can be represented as the Cartesian product of *n* one-dimensional positive semidefinite cones S1, our analysis encompasses the classical nonlinear programming (NLP) as well. In addiction, we call the NLSDP the convex semidefinite programming (CSDP) if *f* is convex, *h* is affine and *g* is S+*n*-convex, i.e., for any *x*, *y* ∈ X and *t* ∈ (0, 1), \n*g*(*t**x* + (1 − *t*)*y*) − *t**g*(*x*) − (1 − *t*)*g*(*y*) ∈ S+*n*. \nFor the NLSDP, the nonsmooth system which is equivalent to the KKT optimality condition, takes the following form:\n \n$$\\label{eq:SDP-natural-mapping}\nF(x,\\xi,\\Gamma)=\\begin{bmatrix}\n&\\nabla\\_xL(x,\\xi,\\Gamma) \\\\\n& h(x) \\\\\n&-g(x)+\\Pi\\_{{\\mathbb{S}}^n\\_+}(g(x)+\\Gamma) \n\\end{bmatrix} = 0,$$ \nwhere the associated Lagrange function is given by *L*(*x*, *ξ*, Γ) = *f*(*x*) + ⟨*ξ*, *h*(*x*)⟩ + ⟨Γ, *g*(*x*)⟩ and ΠS+*n*( ⋅ ) is the metric projection operator over S+*n* defined by.\nLet $\\overline x$ be a stationery solution of with the Lagrange multiplers $(\\overline{\\xi},\\overline{\\Gamma})\\in M(\\overline{x})$. Denote $\\overline A=g(\\overline{x})+\\overline{\\Gamma}$. Consider the corresponding eigenvalue decomposition of $\\overline A$ with the index sets *α*, *β* and *γ* defined by, i.e.,\n \n$$\\label{eq:eig-decomp-SDP}\nA=g(\\overline x)+\\overline\\Gamma =\\overline{P}\n\\left[\\begin{array}{ccc}\n\\overline\\Lambda\\_{\\alpha\\alpha} & 0 & 0 \\\\ [3pt]\n0 & 0 & 0 \\\\ [3pt]\n0 & 0 & \\overline\\Lambda\\_{\\gamma\\gamma}\n\\end{array}\\right]\n\\overline{P}^T,$$ \nwhere $\\overline\\Lambda=\\Lambda(A)={{\\rm Diag}}(\\overline\\lambda\\_i)$ with $\\overline{\\lambda}:=\\lambda(A)$ and $\\overline{P}\\in {\\mathcal{O}}^n(A)$. Thus, the tangent cone ${\\cal T}\\_{\\mathbb{S}^n\\_+}(g(\\overline{x}))$ of S+*n* defined by has the following explicit expression (cf. e.g., ):\n \n$$\\label{eq:tangent-SDP}\n{\\mathcal{T}}\\_{{\\mathbb{S}}^n\\_+}(g(\\overline x))=\\left\\{\\overline{P}B\\overline{P}^T\\in{\\mathbb{S}}^n\\mid\\begin{bmatrix} B\\_{\\beta\\beta}& B\\_{\\beta\\gamma}\\\\B\\_{\\gamma\\beta}&B\\_{\\gamma\\gamma}\\end{bmatrix}\\succeq 0\\right\\}.$$ \nMoreover, by simply calculations, we have\n \n$$\\begin{aligned}\n{\\mathcal{T}}\\_{{\\mathbb{S}}^n\\_+}(g(\\overline x))\\cap \\overline\\Gamma^{\\perp}&=&\\left\\{\\overline{P}B\\overline{P}^T\\in{\\mathbb{S}}^n\\mid B\\_{\\beta\\beta}\\succeq 0,\\ B\\_{\\beta\\gamma}=0,\\ B\\_{\\gamma\\gamma}=0\\right\\}, \\label{eq:critical-cone-SDP} \\\\ [3pt] \n{{\\rm lin}}({\\mathcal{T}}\\_{{\\mathbb{S}}^n\\_+}(g(\\overline x)))&=&\\left\\{\\overline{P}B\\overline{P}^T\\in{\\mathbb{S}}^n\\mid B\\_{\\beta\\beta}= 0,\\ B\\_{\\beta\\gamma}=0,\\ B\\_{\\gamma\\gamma}=0\\right\\}, \\label{eq:lin-tangent-SDP} \\\\ [3pt]\n{{\\rm aff}}({\\mathcal{T}}\\_{{\\mathbb{S}}^n\\_+}(g(\\overline x))\\cap \\overline\\Gamma^{\\perp})&=&\\left\\{\\overline{P}B\\overline{P}^T\\in{\\mathbb{S}}^n\\mid B\\_{\\beta\\gamma}=0,\\ B\\_{\\gamma\\gamma}=0\\right\\}. \\label{eq:aff-critical-cone-SDP}\\end{aligned}$$ \nFor the notationally simplicity, we further define the linear operator $\\overline{{\\mathcal{B}}}:{\\mathbb{X}}\\to{\\mathbb{S}}^n$ by\n \n$$\\label{eq:def-B-map}\n \\overline{\\mathcal{B}}(d)=\\overline{P}^Tg'(\\overline x)d\\overline{P},\\quad d\\in\\mathbb{X}.$$ \nThus, together with, we know from that the critical cone ${\\mathcal{C}}(\\overline x)$ of the NLSDP at the stationary point $\\overline x$ takes the following form:\n \n$$\\label{eq:C-set-SDP}\n {\\mathcal{C}}(\\overline x)=\\left\\{d\\in\\mathbb{X}\\mid h'(\\overline x)d=0,\\ \\overline{\\mathcal{B}}(d)\\_{\\beta\\beta}\\succeq 0,\\ \\overline{\\mathcal{B}}(d)\\_{\\beta\\gamma}=0,\\ \\overline{\\mathcal{B}}(d)\\_{\\gamma\\gamma}=0\\right\\}.$$ \nIt follows from and that for the stationary point $\\overline{x}\\in\\mathbb{X}$ of with a Lagrange multiplier $(\\overline\\xi,\\overline\\Gamma)\\in M(\\overline{x})$, we have\n \n$$\\label{eq:affC} \n {{\\rm app}}(\\overline x,\\overline\\xi,\\overline\\Gamma)=\\left\\{d\\in\\mathbb{X}\\mid h'(\\overline x)d=0,\\ \\overline{\\mathcal{B}}(d)\\_{\\beta\\gamma}= 0,\\ \\overline{\\mathcal{B}}(d)\\_{\\gamma\\gamma}= 0\\right\\}$$ \nand\n \n$$\\label{eq:linC}\n {{\\rm appl}}(\\overline x,\\overline\\xi,\\overline\\Gamma)=\\left\\{d\\in\\mathbb{X}\\mid h'(\\overline x)d=0,\\ \\overline{\\mathcal{B}}(d)\\_{\\beta\\beta}= 0,\\ \\overline{\\mathcal{B}}(d)\\_{\\beta\\gamma}= 0,\\ \\overline{\\mathcal{B}}(d)\\_{\\gamma\\gamma}= 0\\right\\},$$ \nwhere the sets ${{\\rm app}}(\\overline x,\\overline\\xi,\\overline\\Gamma)$ and ${{\\rm appl}}(\\overline x,\\overline\\xi,\\overline\\Gamma)$ are defined by and, respectively. Moreover, by employing the explicit formulas and, we obtain the following useful characterizations of the W-SRCQ (Definition [def:W-SRCQ]) and constarint nondegeneracy (Definition [def:constraint nondegeneracy]) for the NLSDP. For simplicity, we omit the detail proof here.\n[lemma:W-SRCQ-and-CN] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of with a Lagrange multiplier $(\\overline\\xi,\\overline\\Gamma)\\in M(\\overline{x})$.\n* The W-SRCQ holds at $\\overline x$ with respect to $(\\overline\\xi,\\overline\\Gamma)$ if and only if\n \n$$\\label{eq:equal:W-SRCQ}\n \\left\\{\n \\begin{array}{l}\n h'(\\overline x)^\\* {\\Delta \\xi}+\\overline{{\\mathcal{B}}}^\\*{\\Delta \\Gamma}=0,\\\\ [3pt]\n {\\Delta \\Gamma}\\_{\\alpha\\alpha}=0,\\ {\\Delta \\Gamma}\\_{\\alpha\\beta}=0, \\ {\\Delta \\Gamma}\\_{\\alpha\\gamma}=0, \\ {\\Delta \\Gamma}\\_{\\beta\\beta}=0\n \\end{array}\n \\right.\n \\quad\\Longrightarrow\\quad\n ({\\Delta \\xi},{\\Delta \\Gamma})=0.$$\n* The constraint nondegeneracy holds at $\\overline{x}$ if and only if\n \n$$\\label{eq:equal:CNCQ}\n \\left\\{\n \\begin{array}{l}\n h'(\\overline x)^\\* {\\Delta \\xi}+\\overline{{\\mathcal{B}}}^\\*{\\Delta \\Gamma}=0,\\\\ [3pt]\n {\\Delta \\Gamma}\\_{\\alpha\\alpha}=0,\\ {\\Delta \\Gamma}\\_{\\alpha\\beta}=0,\\ {\\Delta \\Gamma}\\_{\\alpha\\gamma}=0 \n \\end{array}\n \\right.\n \\quad\\Longrightarrow\\quad\n ({\\Delta \\xi},{\\Delta \\Gamma})=0.$$\nFor the stationary point $\\overline{x}\\in\\mathbb{X}$ of the NLSDP with a Lagrange multiplier $(\\overline{\\xi},\\overline{\\Gamma})\\in M(\\overline{x})$, the support function of the second order tangent set in the second order condtions (Definitions [def:SOSC], [def:S-SOSC] and [def:W-SOC]) has the following explicit characterization, which takes from :\n \n$$\\label{eq:sigma}\n\\sigma(\\overline y,\\mathcal{T}^2\\_{K}(G(\\overline x),G'(\\overline x)d))\n=\\sigma(\\overline\\Gamma,\\mathcal{T}^2\\_{{\\mathbb{S}}^n\\_+}(g(\\overline x),g'(\\overline x)d))\n=2\\sum\\_{i\\in\\alpha}\\sum\\_{j\\in \\gamma}\\frac{\\overline\\lambda\\_j}{\\overline\\lambda\\_i}\\overline{\\mathcal{B}}(d)\\_{ij}^2,\\quad d\\in \\mathbb{X},$$ \nwhere *K* = {0} × S+*n*, *G*(*x*) = (*h*(*x*), *g*(*x*)) and $\\overline{y}=(\\overline{\\xi},\\overline{\\Gamma})$.\nDefine Z : = X × R*m* × S*n*. Let $(\\overline{x},\\overline{\\xi},\\overline{\\Gamma})$ be a KKT solution of the NLSDP. Consider the B-subdifferential $\\partial\\_BF(\\overline x,\\overline\\xi,\\overline\\Gamma)$ and Clarke-subdifferential $\\partial\\_CF(\\overline x,\\overline\\xi,\\overline\\Gamma)$ of the KKT nonsmooth mapping *F* given by. It follows from (see also ) that $\\overline{\\cal U}\\in \\partial\\_BF(\\overline x,\\overline\\xi,\\overline\\Gamma)$ ($\\partial\\_CF(\\overline x,\\overline\\xi,\\overline\\Gamma)$) if and only if there exists $\\overline{\\cal V}\\in \\partial\\_B\\Pi\\_{{\\mathbb{S}}^m\\_+}(g(\\overline x)+\\overline\\Gamma)$ ($\\partial\\_C\\Pi\\_{{\\mathbb{S}}^m\\_+}(g(\\overline x)+\\overline\\Gamma)$) such that for any (Δ*x*, Δ*ξ*, ΔΓ) ∈ Z,\n \n$$\\label{eq:UandV}\n\\overline{\\cal U}({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma})=\n\\begin{bmatrix}\n\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}+h'(\\overline x)^\\*{\\Delta \\xi}+g'(\\overline x)^\\*{\\Delta \\Gamma}\\\\\nh'(\\overline x){\\Delta x}\\\\\n-g'(\\overline x){\\Delta x}+\\overline{\\cal V}(g'(\\overline x){\\Delta x}+{\\Delta \\Gamma}) \n\\end{bmatrix}.$$ \nIn particular, let $\\overline{\\cal V}\\_{0}$ and $\\overline{\\cal V}\\_{{\\mathcal{I}}}$ be two mappings defined by and for $A=g(\\overline x)+\\overline\\Gamma$, respectively. Define the following two elements $\\overline{\\cal U}\\_{0}$ and $\\overline{\\cal U}\\_{{\\mathcal{I}}}$ in $\\partial\\_BF(\\overline x,\\overline\\xi,\\overline\\Gamma)$ with $\\overline{\\cal V}\\_{0}$ and $\\overline{\\cal V}\\_{{\\mathcal{I}}}$ by, respectively, i.e., for any (Δ*x*, Δ*ξ*, ΔΓ) ∈ Z,\n \n$$\\label{eq:U0}\n \\overline{\\cal U}\\_{0}({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma})=\n \\begin{bmatrix}\n \\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}+h'(\\overline x)^\\*{\\Delta \\xi}+g'(\\overline x)^\\*{\\Delta \\Gamma}\\\\\n h'(\\overline x){\\Delta x}\\\\\n -g'(\\overline x){\\Delta x}+\\overline{\\cal V}\\_0(g'(\\overline x){\\Delta x}+{\\Delta \\Gamma})\n \\end{bmatrix}$$ \nand\n \n$$\\label{eq:U1}\n \\overline{\\cal U}\\_{\\cal I}({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma})=\n \\begin{bmatrix}\n \\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}+h'(\\overline x)^\\*{\\Delta \\xi}+g'(\\overline x)^\\*{\\Delta \\Gamma}\\\\\n h'(\\overline x){\\Delta x}\\\\\n -g'(\\overline x){\\Delta x}+\\overline{\\cal V}\\_{{\\mathcal{I}}}(g'(\\overline x){\\Delta x}+{\\Delta \\Gamma})\n \\end{bmatrix}.$$ \nIn the following two subsections, we shall study the existence of nonsingularity elements in subdifferentials at a KKT point $(\\overline x,\\overline \\xi,\\overline\\Gamma)$.\nSufficient conditions for the existence of nonsingularity elements in subdifferentials\n--------------------------------------------------------------------------------------\nIn this subsection, we study the sufficient conditions to ensure the existence of nonsingularity elements in subdifferentials of the KKT nonsmooth mapping *F*. The following results illustrate that under the W-SOC and constraint nondegeneracy (or the S-SOSC and W-SRCQ), the mapping $\\overline{\\cal U}\\_{0}$ (or $\\overline{\\cal U}\\_{{\\mathcal{I}}}$) defined in (or ) is indeed a nonsingular element of the B-subdifferential of *F*.\n[theorem:bar-N-SDP] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of with a Lagrange multiplier $(\\overline\\xi,\\overline\\Gamma)\\in M(\\overline{x})$.\n* If the W-SOC and constraint nondegeneracy hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$, then $\\overline{\\cal U}\\_{0}\\in \\partial\\_B F(\\overline x,\\overline \\xi,\\overline\\Gamma)$ in is nonsingular.\n* If the S-SOSC and W-SRCQ hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$, then $\\overline{\\cal U}\\_{{\\mathcal{I}}}\\in \\partial\\_B F(\\overline x,\\overline \\xi,\\overline\\Gamma)$ in is nonsingular.\nLet $A=g(\\overline{x})+\\overline{\\Gamma}$ satisfy the eigenvalue decomposition and $\\overline{P}\\in{\\cal O}^n(A)$. We only show (i) holds, as (ii) can be established through a similar argument.\nSuppose that $\\overline{\\cal U}\\_{0}\\in \\partial\\_B F(\\overline x,\\overline \\xi,\\overline\\Gamma)$ is singular, i.e., there is 0 ≠ (Δ*x*, Δ*ξ*, ΔΓ) ∈ Z such that $\\overline{\\cal U}\\_{0}({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma})=0$. It then follows from and Proposition [prop:dif-B-F] that\n \n$$\\begin{aligned}\n \\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}+h'(\\overline x)^\\*{\\Delta \\xi}+g'(\\overline x)^\\*{\\Delta \\Gamma}=0,&\\label{eq:Vz1-Vzero}\\\\\n h'(\\overline x){\\Delta x}=0,& \\label{eq:h-part} \\\\\n \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\alpha}=0,\\quad \\widetilde{\\Delta \\Gamma}\\_{\\alpha\\beta}=0,&\n \\label{theo:fo-LI-1}\\\\\n \\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\beta}=0,\\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\gamma}=0,\\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\gamma\\gamma}=0,& \\label{theo:fo-linNLCQ-1}\\\\\n (E\\_{\\alpha\\gamma}-\\overline \\Xi\\_{\\alpha\\gamma} )\\circ \\overline{\\mathcal{B}}({\\Delta x})\\_{\\alpha\\gamma}-\\overline \\Xi\\_{\\alpha\\gamma} \\circ \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\gamma}=0, & \\label{theo:fo-U-1}\n \\end{aligned}$$ \nwhere $\\widetilde{\\Delta\\Gamma}:= \\overline{P}^T\\Delta\\Gamma\\overline{P}$, the linear operator $\\overline{\\cal B}:\\mathbb{X}\\to \\mathbb{S}^n$ is defined by, the matrix $\\overline\\Xi\\in \\mathbb{S}^n$ is given by for *A* and *E* ∈ S*n* denotes the matrix whose elements are all ones. Then, by - and, we obtain that\n \n$$\\begin{aligned}\n 0&=\\left{\\langle}{\\Delta x},\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma) {\\Delta x}+h'(\\overline x) ^\\*{\\Delta \\xi}+g'(\\overline x)^\\*{\\Delta \\Gamma}\\right{\\rangle}{{\\nonumber}}\\\\\n &=\\left{\\langle}{\\Delta x},\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma) {\\Delta x}\\right{\\rangle}+\\left{\\langle}h'(\\overline x) {\\Delta x},{\\Delta \\xi}\\right{\\rangle}+\\left{\\langle}\\overline{\\mathcal{B}}({\\Delta x}), \\widetilde{\\Delta \\Gamma}\\right{\\rangle}{{\\nonumber}}\\\\\n &={\\langle}{\\Delta x},\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma) {\\Delta x}{\\rangle}+2\\sum\\_{i\\in\\alpha}\\sum\\_{j\\in \\gamma}\\frac{-\\overline\\lambda\\_j}{\\overline\\lambda\\_i}(\\overline{\\mathcal{B}}({\\Delta x})\\_{ij})^2 {{\\nonumber}}\\\\\n &={\\langle}{\\Delta x},\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma) {\\Delta x}{\\rangle}- \\sigma(\\overline\\Gamma,\\mathcal{T}^2\\_{{\\mathbb{S}}^n\\_+}(g(\\overline x),g'(\\overline x){\\Delta x})). \\label{eq:cont-SOSC-bar}\\end{aligned}$$ \nMeanwhile, by and, we know from that ${\\Delta x}\\in{{\\rm appl}}(\\overline x,\\overline \\xi,\\overline\\Gamma)$. Since the W-SOC holds at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$, we know from that Δ*x* = 0. It then follows from that $\\widetilde{\\Delta\\Gamma}\\_{\\alpha\\gamma}=0$. This, together with and, yields that\n \n$$\\left\\{\n\\begin{array}{l}\nh'(\\overline x)^\\*{\\Delta \\xi}+\\overline{\\mathcal{B}}^\\*\\widetilde{\\Delta\\Gamma}=0,\\\\\n\\widetilde{\\Delta\\Gamma}\\_{\\alpha\\alpha}=0,\\quad \\widetilde{\\Delta \\Gamma}\\_{\\alpha\\beta}=0,\\quad \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\gamma}=0.\n\\end{array}\n\\right.$$ \nSince the constraint nondegeneracy hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$, we know from in Lemma [lemma:W-SRCQ-and-CN] (ii) that (Δ*ξ*, ΔΓ) = 0, which contradicts with (Δ*x*, Δ*ξ*, ΔΓ) ≠ 0. The proof is then completed. $\\hfill \\Box$\nIt should be noted that Theorem [theorem:bar-N-SDP] can be easiliy extended to cases where *K* in is a Cartesian product of a finite number of positive semidefinite cones and zero vectors. By regarding a polyhedron as a Cartesian product of a finite number of one dimensional positive semidefinite cones and zero vectors, the results derived in Theorem [theorem:bar-N-SDP] are consistent with the findings presented in.\nIt is natural to question whether the W-SRCQ and W-SOC are sufficient for the existence of nonsingular elements in the B-subdifferential $\\partial\\_B F(\\overline x,\\overline \\xi,\\overline\\Gamma)$. We address this question with the following proposition, which confirms the sufficiency under the condition that ∣*β*∣ ≤ 1, i.e., the number of zero eigenvalues of $g(\\overline x)+\\overline\\Gamma$ is less than or equal to one.\n[prop:beta<=1] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of with a Lagrange multiplier $(\\overline\\xi,\\overline\\Gamma)\\in M(\\overline{x})$. Suppose that ∣*β*∣ ≤ 1. If the W-SOC and W-SRCQ hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$, then at least one of $\\{\\overline{\\cal U}\\_{0},\\overline{\\cal U}\\_{{\\mathcal{I}}}\\}$ is nonsingular.\nConsider the following two cases.\n: ∣*β*∣ = 0. In this case, from Lemma [lemma:W-SRCQ-and-CN], we know that the W-SRCQ is equivalent to the constraint nondegeneracy. Moreover, and imply that ${{\\rm app}}(\\overline x, \\overline \\xi, \\overline\\Gamma) = {{\\rm appl}}(\\overline x, \\overline \\xi, \\overline\\Gamma)$, and thus further imply that the W-SOC is equivalent to the S-SOSC. We also have from Proposition [prop:dif-B-F] that $\\partial\\_B F(\\overline x,\\overline \\xi,\\overline\\Gamma)=\\{ F'(\\overline x,\\overline \\xi,\\overline\\Gamma)\\}=\\{\\overline {\\cal U}\\_{0}\\}=\\{\\overline {\\cal U}\\_{{\\mathcal{I}}}\\}$. The desired result follows directly from Theorem [theorem:bar-N-SDP].\n: ∣*β*∣ = 1. Since the W-SOC holds at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$, we know from (i) of Theorem [theorem:bar-N-SDP] that if the constraint nondegeneracy hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$, then $\\overline{\\cal U}\\_{0}$ is nonsingular. Suppose that the constraint nondegeneracy does not hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$. Since the W-SRCQ holds at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$, we know from Lemma [lemma:W-SRCQ-and-CN] that there exists (Δ*ξ*, ΔΓ) ≠ 0 such that\n \n$$\\left\\{\n \\begin{array}{l}\n h'(\\overline x)^\\* {\\Delta \\xi}+\\overline{{\\mathcal{B}}}^\\*{\\Delta \\Gamma}=0,\\\\ [3pt]\n {\\Delta \\Gamma}\\_{\\alpha\\alpha}=0,\\ {\\Delta \\Gamma}\\_{\\alpha\\beta}=0,\\ {\\Delta \\Gamma}\\_{\\alpha\\gamma}=0, \\\\\n {\\Delta \\Gamma}\\_{\\beta\\beta}\\neq 0.\n \\end{array} \n \\right.$$ \nNext, we show that ${{\\rm app}}(\\overline x,\\overline\\xi,\\overline\\Gamma)={{\\rm appl}}(\\overline x,\\overline\\xi,\\overline\\Gamma)$ with their explicit expressions given in and, respectively. Clearly, ${{\\rm appl}}(\\overline x,\\overline \\xi,\\overline\\Gamma)\\subseteq {{\\rm app}}(\\overline x,\\overline \\xi,\\overline\\Gamma)$. For any $d\\in{{\\rm app}}(\\overline x,\\overline \\xi,\\overline\\Gamma)$, we have\n \n$$\\begin{aligned}\n 0=&{\\langle}d,h'(\\overline x)^\\* {\\Delta \\xi}+\\overline{{\\mathcal{B}}}^\\*{\\Delta \\Gamma}{\\rangle}={\\langle}h'(\\overline x)d, {\\Delta \\xi}{\\rangle}+{\\langle}\\overline{{\\mathcal{B}}}(d),{\\Delta \\Gamma}{\\rangle}={\\langle}\\overline{{\\mathcal{B}}}(d)\\_{\\beta\\beta},{\\Delta \\Gamma}\\_{\\beta\\beta}{\\rangle},\n \\end{aligned}$$ \nwhere the last equation follows from. Since ∣*β*∣ = 1, by noting that ΔΓ*β**β* ≠ 0, we know that $\\overline{{\\mathcal{B}}}(d)\\_{\\beta\\beta}=0$. Therefore, we obtain from that ${{\\rm appl}}(\\overline x,\\overline \\xi,\\overline\\Gamma)\\subseteq {{\\rm app}}(\\overline x,\\overline \\xi,\\overline\\Gamma)$. Then, by Definitions [def:W-SOC] and [def:S-SOSC], the W-SOC and S-SOSC are equivalent. Therefore, we know from (ii) of Theorem [theorem:bar-N-SDP] that $\\overline{\\cal U}\\_{{\\mathcal{I}}}$ is nonsingular. The proof is then completed. $\\hfill \\Box$\n[rema:w-w-NLP-B] It is worth noting that the condition ∣*β*∣ ≤ 1 is always satisfied for classic nonlinear programming (NLP). Actually, as we mentioned before, the positive orthant R+*n* can be represented as the Cartesian product of *n* one-dimensional positive semidefinite cones. Consequently, based on Proposition [prop:beta<=1], the existence of a nonsingular element in the Bouligand subdifferential $\\partial\\_B F(\\overline x,\\overline \\xi,\\overline\\Gamma)$ is ensured under the W-SOC and the W-SRCQ for the conventional NLP, which recovers the corresponding result obtained by Izmailov and Solodov.\nHowever, the following simple example illustrates that for the NLSDP, if the number of zero eigenvalues of $g(\\overline x)+\\overline\\Gamma$ is greater than one, i.e., ∣*β*∣ > 1, the W-SRCQ and W-SOC are not sufficient to guarantee the existence of nonsingular elements in $\\partial\\_B F(\\overline x,\\overline \\xi,\\overline\\Gamma)$. This showcases the fundamental differences between the NLSDP and the NLP.\n[example:non-SRCQ] Consider the following SDP problem:\n \n$$\\begin{aligned}\n{{\\nonumber}}\\begin{aligned}\n\\min\\quad & 0 \\\\\n\\mbox{s.t.}\\quad & x\\_{12}=0, \\\\\n& X\\in {\\mathbb{S}}^2\\_+, \n\\end{aligned}\\end{aligned}$$ \nwhere $X=\\begin{bmatrix} x\\_{11} &x\\_{12}\\\\x\\_{21}&x\\_{22}\\end{bmatrix}\\in{\\mathbb{S}}^2$. It is clear that $(\\overline X,\\overline \\xi, \\overline\\Gamma)=(0,0,0)$ is a KKT point. We have *β* = {1, 2} and *α* = *γ* = ∅. It is easy to verify that the W-SRCQ and W-SOC hold at $(\\overline X,\\overline \\xi,\\overline\\Gamma)$. Moreover, it follows from that ${\\cal U}\\in\\partial\\_B F(\\overline X,\\overline \\xi,\\overline\\Gamma)$ if and only if there exists Ω ∈ *R* such that for any (Δ*X*, Δ*ξ*, ΔΓ) ∈ S2 × R × S2,\n \n$$\\label{eq:def-U-exmaple}\n {\\cal U}({\\Delta X},{\\Delta \\xi},{\\Delta \\Gamma})=\n \\begin{bmatrix}\n \\begin{bmatrix}\n 0 & 0 \\\\\n 0 &0 \n \\end{bmatrix}+\n \\begin{bmatrix}\n 0 & {\\Delta \\xi}\\\\\n {\\Delta \\xi}&0\n \\end{bmatrix}+{\\Delta \\Gamma}\\\\\n {\\Delta X}\\_{12}\\\\\n - {\\Delta X}+\\Omega\\circ({\\Delta X}+{\\Delta \\Gamma})\n \\end{bmatrix},$$ \nwhere *R* ∈ S*n* is the set of matrices defined by \n$$R:=\\left\\{\\begin{bmatrix} 0 &0\\\\0&0\\end{bmatrix},\\begin{bmatrix} 1 &1\\\\1&1\\end{bmatrix} \\right\\}\\bigcup\\left\\{\\begin{bmatrix} 0 &t\\\\t&1\\end{bmatrix}\\mid t\\in[0,1]\\right\\}\\bigcup\\left\\{\\begin{bmatrix} 1 &t\\\\t&0\\end{bmatrix}\\mid t\\in[0,1]\\right\\}.$$ \n It is easy to see that all elements in $\\partial\\_B F(\\overline X,\\overline \\xi,\\overline\\Gamma)$ are singular. Indeed, let ${\\cal U}\\in\\partial\\_B F(\\overline X,\\overline \\xi,\\overline\\Gamma)$ be arbitrarily given. Then,\nif $\\Omega=\\begin{bmatrix} 0 &0\\\\0&0\\end{bmatrix}$, then ${\\Delta X}=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 &0 \n \\end{bmatrix}$, Δ*ξ* = − 1 and $\n {\\Delta \\Gamma}=\\begin{bmatrix}\n 0 & 1 \\\\\n 1 &0 \n \\end{bmatrix}$ is a nonzero solution of ${\\cal U}({\\Delta X},{\\Delta \\xi},{\\Delta \\Gamma})=0$;\nif $\\Omega=\\begin{bmatrix} 1 &1\\\\1&1\\end{bmatrix}$, then ${\\Delta X}=\\begin{bmatrix}\n 1 & 0 \\\\\n 0 &1 \n\\end{bmatrix}$, Δ*ξ* = 0 and $\n{\\Delta \\Gamma}=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 &0 \n\\end{bmatrix}$ is a nonzero solution of ${\\cal U}({\\Delta X},{\\Delta \\xi},{\\Delta \\Gamma})=0$;\nif $\\Omega=\\begin{bmatrix} 0 &t\\\\t&1\\end{bmatrix}$ with *t* ∈ [0, 1], then ${\\Delta X}=\\begin{bmatrix}\n 1 & 0 \\\\\n 0 &0 \n\\end{bmatrix}$, Δ*ξ* = 0 and $\n{\\Delta \\Gamma}=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 &0 \n\\end{bmatrix}$ is a nonzero solution of ${\\cal U}({\\Delta X},{\\Delta \\xi},{\\Delta \\Gamma})=0$;\nif $\\Omega=\\begin{bmatrix} 1 &t\\\\t&0\\end{bmatrix}$ with *t* ∈ [0, 1], then ${\\Delta X}=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 &1 \n\\end{bmatrix}$, Δ*ξ* = 0 and $\n{\\Delta \\Gamma}=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 &0 \n\\end{bmatrix}$ is a nonzero solution of ${\\cal U}({\\Delta X},{\\Delta \\xi},{\\Delta \\Gamma})=0$.\nNext, let us consider the Clarke-subdifferential defined by of the KKT nonsmooth mapping. Interestingly, one may observe that for Example [example:non-SRCQ], the convex combination $t {\\cal U}\\_0+(1-t){\\cal U}\\_{\\cal I}\\in \\partial\\_C F(\\overline X,\\overline \\xi,\\overline\\Gamma)$ for any *t* ∈ (0, 1) is actually nonsingular, where ${\\cal U}\\_0$ and ${\\cal U}\\_{\\cal I}$ are elements in $\\partial\\_B F(\\overline X,\\overline \\xi,\\overline\\Gamma)$ defined by with $\\Omega=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 &0 \n \\end{bmatrix}$ and $\\begin{bmatrix}\n 1 & 1 \\\\\n 1 & 1 \n \\end{bmatrix}$, respectively. In fact, we have the following general results, which demonstrates that the W-SOC and W-SRCQ guarantee the existence of a nonsingular element in the Clarke-subdifferential.\n[prop:pC]\n* There exists *t* ∈ (0, 1) such that\n \n$$t\\overline{\\cal U}\\_{0}+(1-t)\\overline{\\cal U}\\_{{\\mathcal{I}}}\\in\\partial\\_C F(\\overline x,\\overline \\xi,\\overline\\Gamma) \\text{ is nonsingular.}$$\n* If, additionally, the NLSDP is convex, that is, it becomes a CSDP, then for any *t* ∈ (0, 1),\n \n$$t\\overline{\\cal U}\\_{0}+(1-t)\\overline{\\cal U}\\_{{\\mathcal{I}}}\\in\\partial\\_C F(\\overline x,\\overline \\xi,\\overline\\Gamma) \\text{ is nonsingular.}$$\nLet $A=g(\\overline{x})+\\overline{\\Gamma}$ satisfy the eigenvalue decomposition and $\\overline{P}\\in{\\cal O}^n(A)$ with the index sets *α*, *β* and *γ* defined by. Recall the linear operator $\\overline{\\cal B}:\\mathbb{X}\\to \\mathbb{S}^n$ defined in.\nAssume that for any *t* ∈ (0, 1), $t\\overline{\\cal U}\\_{0}+(1-t)\\overline{\\cal U}\\_{{\\mathcal{I}}}\\in\\partial\\_C F(\\overline x,\\overline \\xi,\\overline\\Gamma)$ is singular. Define a sequence of elements $\\{\\overline{\\cal U}^k\\}$ as $\\overline{\\cal U}^k=\\frac{k}{k+1}\\overline{\\cal U}\\_{0}+\\frac{1}{k+1}\\overline{\\cal U}\\_{{\\mathcal{I}}}\\in\\partial\\_C F(\\overline x,\\overline \\xi,\\overline\\Gamma)$ for each *k*. Since $\\overline{\\cal U}^k$ is singular, we know that for each *k*, there are 0 ≠ (Δ*x**k*, Δ*ξ**k*, ΔΓ*k*) ∈ Z such that $\\overline{\\cal U}^k({\\Delta x}^k,{\\Delta \\xi}^k,{\\Delta \\Gamma}^k)=0$. Thus, for each *k*, we know from and Proposition [prop:dif-B-F] that\n \n$$\\begin{aligned}\n\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}^k+h'(\\overline x)^\\*{\\Delta \\xi}^k+g'(\\overline x)^\\*{\\Delta \\Gamma}^k=0,&\\label{eq:Uk=0-1}\\\\\nh'(\\overline x){\\Delta x}^k =0,& \\label{eq:Uk=0-2}\\\\\n\\widetilde{\\Delta\\Gamma}^k\\_{\\alpha\\alpha}=0,\\quad \\widetilde{\\Delta \\Gamma}^k\\_{\\alpha\\beta}=0,&\n\\label{eq:Uk=0-3}\\\\\n\\overline{{\\mathcal{B}}}({\\Delta x}^k)\\_{\\beta\\gamma}=0,\\quad \\overline{{\\mathcal{B}}}({\\Delta x}^k)\\_{\\gamma\\gamma}=0,& \\label{eq:Uk=0-4}\\\\\n(E\\_{\\alpha\\gamma}-\\overline{\\Xi}\\_{\\alpha\\gamma} )\\circ \\overline{{\\mathcal{B}}}({\\Delta x}^k)\\_{\\alpha\\gamma}-\\overline{\\Xi}\\_{\\alpha\\gamma} \\circ \\widetilde{\\Delta\\Gamma}^k\\_{\\alpha\\gamma}=0, & \\label{eq:Uk=0-5}\\\\\nk\\overline{{\\mathcal{B}}}({\\Delta x}^k)\\_{\\beta\\beta}-\\widetilde{\\Delta\\Gamma}^k\\_{\\beta\\beta}=0, & \\label{eq:Uk=0-6}\\end{aligned}$$ \nwhere $\\widetilde{\\Delta\\Gamma}^k =\\overline{P}^T\\Delta\\Gamma^k\\overline{P}$ with $\\overline{P}\\in\\mathcal{O}^n(A)$ and $A=g(\\overline{x})+\\overline{\\Gamma}$, the matrix $\\overline\\Xi\\in \\mathbb{S}^n$ is given by and *E* ∈ S*n* denotes the matrix whose elements are all ones. We then conclude that for each *k*, Δ*x**k* ≠ 0. Indeed, if there exists some *k̄* such that Δ*x**k̄* = 0, then by, and, we obtain that\n \n$$\\label{eq:pC-dx0-1}\n\\widetilde{\\Delta\\Gamma}^{\\bar{k}}\\_{\\alpha\\alpha}=0,\\quad \\widetilde{\\Delta\\Gamma}^{\\bar{k}}\\_{\\alpha\\beta}=0, \\quad \\widetilde{\\Delta\\Gamma}^{\\bar{k}}\\_{\\beta\\beta}=0 \\quad {\\rm and} \\quad\\widetilde{\\Delta\\Gamma}^{\\bar{k}}\\_{\\alpha\\gamma}=0.$$ \nThus, combining with, we know that\n \n$$\\label{eq:pC-dx0-2}\n\\left\\{\n\\begin{array}{l}\nh'(\\overline x)^\\*{\\Delta \\xi}^{\\bar{k}}+\\overline{\\mathcal{B}}^\\*{\\Delta\\widetilde\\Gamma}^{\\bar{k}}=0, \\\\ [3pt]\n\\widetilde{\\Delta\\Gamma}^{\\bar{k}}\\_{\\alpha\\alpha}=0,\\quad \\widetilde{\\Delta\\Gamma}^{\\bar{k}}\\_{\\alpha\\beta}=0,\\quad \\widetilde{\\Delta\\Gamma}^{\\bar{k}}\\_{\\beta\\beta}=0,\\quad \\widetilde{\\Delta\\Gamma}^{\\bar{k}}\\_{\\alpha\\gamma}=0.\n\\end{array}\n\\right.$$ \nIt then follows from in Lemma [lemma:W-SRCQ-and-CN] (i) that (Δ*ξ**k̄*, ΔΓ*k̄*) = 0, which, together with $\\Delta x^{\\bar k} = 0$, contradicts with the assumption. Now, since Δ*x**k* ≠ 0 for all *k*, without loss of generality, we may assume that ∥Δ*x**k*∥ = 1 and Δ*x**k* → Δ*x*∞ ≠ 0 as *k* → ∞. By taking the inner product with Δ*x**k* in, we obtain from - that for each *k*,\n \n$$\\begin{aligned}\n{\\langle}{\\Delta x}^k,\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma) {\\Delta x}^k {\\rangle}+2\\sum\\_{i\\in\\alpha}\\sum\\_{j\\in \\gamma}\\frac{-\\overline\\lambda\\_j }{\\overline\\lambda\\_i }\\overline{\\mathcal{B}}({\\Delta x}^k)\\_{ij}^2+\\sum\\_{i\\in\\beta}\\sum\\_{j\\in\\beta}k\\overline{\\mathcal{B}}({\\Delta x}^k)\\_{ij}^2 =0. \\label{prop-eq:Vz-1-dx}\\end{aligned}$$ \nLet Q : X → X be the linear map such that\n \n$$\\begin{aligned}\n{{\\nonumber}}{\\langle}d,{\\mathcal{Q}}(d){\\rangle}={\\langle}d,\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma) d {\\rangle}+2\\sum\\_{i\\in\\alpha}\\sum\\_{j\\in \\gamma}\\frac{-\\overline\\lambda\\_j }{\\overline\\lambda\\_i }\\overline{\\mathcal{B}}(d)\\_{ij}^2, \\quad \\forall \\, d\\in {\\mathbb{X}}.\\end{aligned}$$ \nThen there exists *q* > 0 that ∣Q(*d*)∣ ≤ *q*∥*d*∥2 for any *d* ∈ X. Therefore, we have from that for each *k*,\n \n$$\\begin{aligned}\n{{\\nonumber}}|\\overline{\\mathcal{B}}({\\Delta x}^k)\\_{ij}|^2\\le \\frac{1}{k} q,\\quad (i,j)\\in \\beta\\times\\beta.\\end{aligned}$$ \nAs a result, $\\overline{\\mathcal{B}}({\\Delta x}^{\\infty})\\_{\\beta\\beta}=0$, which, together with, implies ${\\Delta x}^{\\infty}\\in{{\\rm appl}}(\\overline x,\\overline \\xi,\\overline\\Gamma)$. Further note that implies that\n \n$$\\begin{aligned}\n\\nonumber\n \\langle {\\Delta x}^k, {\\mathcal{Q}}{\\Delta x}^k\\rangle \\le 0.\n \\end{aligned}$$ \nTaking the limit as *k* → + ∞, we obtain ⟨Δ*x*∞, QΔ*x*∞⟩ ≤ 0, which contradicts the W-SOC given that $0 \\neq {\\Delta x}^{\\infty} \\in {{\\rm appl}}(\\overline x, \\overline \\xi, \\overline\\Gamma)$. Therefore, there exists a nonsingular $\\overline{\\cal U}^k$.\nFor the CSDP, if there is *t* ∈ (0, 1) that $\\overline{\\cal U}\\_t=t\\overline {\\cal U}\\_{0}+(1-t)\\overline {\\cal U}\\_{{\\mathcal{I}}}$ is singular. Then, there is (Δ*x*, Δ*ξ*, ΔΓ) ≠ 0 such that $\\overline{\\cal U}\\_t({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma})=0$. Similarly as the proof of (i), we obtain from - that Δ*x* ≠ 0 and\n \n$$\\begin{aligned}\n\\label{prop-eq:Vz-1-dx:22}\n{\\langle}{\\Delta x},\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma) {\\Delta x}{\\rangle}+2\\sum\\_{i\\in\\alpha}\\sum\\_{j\\in \\gamma}\\frac{-\\overline\\lambda\\_j }{\\overline\\lambda\\_i }\\overline{\\mathcal{B}}({\\Delta x})\\_{ij}^2+\\sum\\_{i\\in\\beta}\\sum\\_{j\\in\\beta}\\frac{t}{1-t}\\overline{\\mathcal{B}}({\\Delta x})\\_{ij}^2 =0. \\end{aligned}$$ \nBy convexity, we know that $\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma)$ is positive semidefinite. Thus, $\\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\beta}=0$. This, together with and, yields $0\\neq {\\Delta x}\\in{{\\rm appl}}(\\overline x,\\overline \\xi,\\overline\\Gamma)$, and futher implies the contradiction between the W-SOC. Therefore, we conclude that $\\overline{\\cal U}\\_t=t\\overline {\\cal U}\\_{0}+(1-t)\\overline {\\cal U}\\_{{\\mathcal{I}}}$ is nonsingular for any *t* ∈ (0, 1). $\\hfill \\Box$\n[rema:w-w-SDP-C] According to the proof of part (i) of Proposition [prop:pC], there exists only a finite number of singular elements on the line segment connecting $\\overline{\\cal U}\\_0$ and $\\overline{\\cal U}\\_{{\\mathcal{I}}}$. In fact, by considering the matrix representations of $\\overline{\\cal U}\\_0$ and $\\overline{\\cal U}\\_{{\\mathcal{I}}}$ and invoking the fundamental theorem of algebra, we are able to deduce that the non-Zero polynomial ${\\rm det}(t\\overline{\\cal U}\\_{0}+(1-t)\\overline{\\cal U}\\_{{\\mathcal{I}}})$ has a number of roots that does not exceed its degree.\nNecessary conditions for the existence of nonsingularity elements in subdifferentials\n-------------------------------------------------------------------------------------\nWe first show that when the NLSDP is convex, i.e., for the CSDP, the conditions of Theorem [theorem:bar-N-SDP] are actually necessary. In fact, we have the following characterizations of the nonsingularity of $\\overline{\\cal U}\\_0$ and $\\overline{\\cal U}\\_{\\cal I}$ defined by and, respectively.\n[prop:pd:LSDP-V01]\n* The W-SOC and constraint nondegeneracy hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$ if and only if $\\overline{\\cal U}\\_0$ defined in is nonsingular;\n* the S-SOSC and W-SRCQ hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$ if and only if $\\overline{\\cal U}\\_{\\cal I}$ defined in is nonsingular.\nLet $A=g(\\overline{x})+\\overline{\\Gamma}$ satisfy the eigenvalue decomposition and $\\overline{P}\\in{\\cal O}^n(A)$ with the index sets *α*, *β* and *γ* defined by. Recall the linear operator $\\overline{\\cal B}:\\mathbb{X}\\to \\mathbb{S}^n$ defined by. We only show (i), as (ii) can be established through a similar argument.\nBy Theorem [theorem:bar-N-SDP], we only need to show that the nonsingularity of $\\overline{\\cal U}\\_0$ implies the W-SOC and constraint nondegeneracy. Suppose $\\overline{\\cal U}\\_0({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma})=0$. We know from and Proposition [prop:dif-B-F] that\n \n$$\\begin{aligned}\n\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}+h'(\\overline x)^\\*{\\Delta \\xi}+g'(\\overline x)^\\*{\\Delta \\Gamma}=0,&\\label{eq:U0=1-1}\\\\\nh'(\\overline x){\\Delta x}=0,\\label{eq:U0=1-2}& \\\\\n\\widetilde{\\Delta\\Gamma}\\_{\\alpha\\alpha}=0,\\quad \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\beta}=0,&\n\\label{eq:U0=1-3}\\\\\n\\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\beta}=0,\\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\gamma}=0,\\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\gamma\\gamma}=0,& \\label{eq:U0=1-4}\\\\\n(E\\_{\\alpha\\gamma}-\\overline\\Xi\\_{\\alpha\\gamma} )\\circ \\overline{\\mathcal{B}}({\\Delta x})\\_{\\alpha\\gamma}-\\overline \\Xi\\_{\\alpha\\gamma} \\circ \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\gamma}=0, & \\label{eq:U0=1-5}\\end{aligned}$$ \nwhere $\\widetilde{\\Delta\\Gamma}=\\overline{P}^T{\\Delta \\Gamma}\\overline{P} $, $\\overline\\Xi$ is defined in for *A* and *E* represents the matrix consisting of all ones in S*n*. Suppose that the constraint nondegeneracy does not hold. Then, by (ii) of Lemma [lemma:W-SRCQ-and-CN], we know that there exist 0 ≠ (Δ*ξ*, ΔΓ) such that\n \n$$\\begin{aligned}\n{{\\nonumber}}\\left\\{\n \\begin{array}{l}\n h'(\\overline x)^\\*{\\Delta \\xi}+\\overline{\\mathcal{B}}^\\*\\widetilde{\\Delta\\Gamma}=0,\\\\\n \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\alpha}=0,\\quad \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\beta}=0, \\quad \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\gamma}=0.\n \\end{array}\n \\right.\\end{aligned}$$ \nThus, it follows from - that (Δ*x*, Δ*ξ*, ΔΓ) = (0, Δ*ξ*, ΔΓ) ≠ 0 satisfy $\\overline{\\cal U}\\_0({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma})=0$, which contradicts with the nonsingularity of $\\overline{\\cal U}\\_0$. Thus, the constraint nondegeneracy holds at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$.\nOn the other hand, suppose that the W-SOC does not hold at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$. Then, there exists *d* ≠ 0 that\n \n$$\\begin{aligned}\n{{\\nonumber}}\\left\\{\n\\begin{aligned}\n&{\\langle}d,\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma)d{\\rangle}+2\\sum\\_{i\\in\\alpha}\\sum\\_{j\\in \\gamma}\\frac{-\\overline\\lambda\\_j }{\\overline\\lambda\\_i }\\overline{\\mathcal{B}}(d)\\_{ij}^2=0, \\\\\n&h'(\\overline x) d=0,\\\\\n&\\overline{\\mathcal{B}}(d)\\_{\\beta\\beta}= 0,\\quad \\overline{\\mathcal{B}}(d)\\_{\\beta\\gamma}= 0,\\quad \\overline{\\mathcal{B}}(d)\\_{\\gamma\\gamma}= 0.\n\\end{aligned}\n\\right.\\end{aligned}$$ \nBy convexity, we know that $\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma)$ is positive semidefinite. Together with $\\overline\\lambda\\_j<0$ for *j* ∈ *γ* and $\\overline\\lambda\\_i>0$ for *i* ∈ *α*, we conclude that\n \n$$\\begin{aligned}\n\\label{eq:the W-SOC-convex}\n\\left\\{\n\\begin{aligned}\n&\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma)d=0,\\\\\n&h'(\\overline x) d=0,\\\\\n&\\overline{\\mathcal{B}}(d)\\_{\\alpha\\gamma}=0, \\quad \\overline{\\mathcal{B}}(d)\\_{\\beta\\beta}= 0,\\quad \\overline{\\mathcal{B}}(d)\\_{\\beta\\gamma}= 0,\\quad \\overline{\\mathcal{B}}(d)\\_{\\gamma\\gamma}= 0,\n\\end{aligned}\n\\right.\\end{aligned}$$ \nThis, yields that $\\overline{\\cal U}\\_0(d,0,0)=0$, which again contradicts with the nonsingularity of $\\overline{\\cal U}\\_0$. The proof of part (i) has been established. $\\hfill \\Box$\nIn the context of linear semidefinite programming, a particular instance of the CSDP, Chan and Sun demonstrated that the nonsingularity of $\\overline{\\cal U}\\_{0}$ ($\\overline{\\cal U}\\_{\\cal I}$) implies primal (dual) constraint nondegeneracy. In contrast, Proposition [prop:pd:LSDP-V01] in the current work provides a complete characterization of the nonsingularity of $\\overline{\\cal U}\\_{0}$ and $\\overline{\\cal U}\\_{\\cal I}$ for more general CSDP.\nFor the general NLSDP, one can still use the first part of the proof to show that the the nonsingularity of $\\overline{\\cal U}\\_0$ (or $\\overline{\\cal U}\\_{{\\mathcal{I}}}$) implies the constraint nondegeneracy (or the W-SRCQ).\nFinally, we shall establish in the forthcoming proposition that, for general NLSDP, the W-SRCQ is necessary to ensure the existence of nonsingular elements in the Clarke subdifferential. Additionally, for the CSDP, both the W-SRCQ and the W-SOC are requisite.\n* \n* \nLet $A=g(\\overline{x})+\\overline{\\Gamma}$ satisfy the eigenvalue decomposition and $\\overline{P}\\in{\\cal O}^n(A)$ with the index sets *α*, *β* and *γ* defined by. Recall the linear operator $\\overline{\\cal B}:\\mathbb{X}\\to \\mathbb{S}^n$ defined by.\nLet (Δ*ξ*, ΔΓ) ∈ R*m* × S*n* be a solution to the following system:\n \n$$\\left\\{\n \\begin{aligned}\n &h'(\\overline x)^\\*{\\Delta \\xi}+\\overline{\\mathcal{B}}^\\*\\widetilde{\\Delta\\Gamma}=0,\\\\\n &\\widetilde{\\Delta\\Gamma}\\_{\\beta\\beta}=0,\\quad \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\alpha}=0, \\quad \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\beta}=0,\\quad \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\gamma}=0,\n \\end{aligned}\n \\right.$$ \nwhere $\\widetilde{\\Delta\\Gamma}=\\overline{P}^T\\Delta\\Gamma\\overline{P}$. Then, we know from and Proposition [prop:dif-B-F] that\n \n$$\\overline{\\cal U}(0,{\\Delta \\xi},{\\Delta \\Gamma})=\n \\begin{bmatrix}\n h'(\\overline x)^\\*{\\Delta \\xi}+\\overline{\\mathcal{B}}^\\*\\widetilde{\\Delta\\Gamma} \\\\\n 0\\\\\n \\overline{P} \n \\begin{bmatrix}\n \\widetilde{\\Delta\\Gamma}\\_{\\alpha\\alpha}&\\widetilde{\\Delta\\Gamma}\\_{\\alpha\\beta}&\\overline\\Xi\\_{\\alpha\\gamma}\\circ\\widetilde{\\Delta\\Gamma}\\_{\\alpha\\gamma} \\\\\n \\widetilde{\\Delta\\Gamma}\\_{\\beta\\alpha}&{\\cal W}(\\widetilde{\\Delta\\Gamma}\\_{\\beta\\beta})&0 \\\\\n \\overline\\Xi\\_{\\gamma\\alpha}\\circ\\widetilde{\\Delta\\Gamma}\\_{\\gamma\\alpha}&0&0 \n \\end{bmatrix}\n \\overline{P}^T\n \\end{bmatrix}=0,$$ \nwhere ${\\cal W}\\in\\partial\\_C\\Pi\\_{{\\mathbb{S}}^{|\\beta|}\\_+}(0)$. Since $\\overline{\\cal U}$ is nonsingular, we obtain that (Δ*ξ*, ΔΓ) = 0. Therefore, we know from in Lemma [lemma:W-SRCQ-and-CN] (i) that the W-SRCQ holds at $(\\overline x,\\overline \\xi,\\overline \\Gamma)$.\nWe only need to show that the W-SOC (Definition ) holds at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$. Suppose on the contrary that the W-SOC does not hold. Then, we know from and that there exists 0 ≠ Δ*x* such that \n$$\\left\\{\\begin{array}{l}\n {\\langle}{\\Delta x},\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}{\\rangle}+2\\displaystyle\\sum\\_{i\\in \\alpha}\\sum\\_{j\\in\\beta}\\frac{-\\overline\\lambda\\_j }{\\overline\\lambda\\_i }\\overline{\\mathcal{B}}({\\Delta x})\\_{ij}^2=0, \\\\ [3pt]\n h'(\\overline x) {\\Delta x}=0,\\\\ [3pt]\n \\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\beta}= 0,\\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\gamma}= 0,\\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\gamma\\gamma}= 0.\n\\end{array}\\right.$$ \n Since $\\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma)$ is positive semidefinite and $\\displaystyle\\frac{-\\overline\\lambda\\_j }{\\overline\\lambda\\_i }>0$ for any *i* ∈ *α* and *j* ∈ *γ*, we have\n \n$$\\left\\{\n \\begin{array}{l}\n \\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}=0 \\\\ [3pt]\n h'(\\overline x) {\\Delta x}=0,\\\\ [3pt]\n \\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\beta}= 0,\\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\beta\\gamma}= 0,\\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\gamma\\gamma}= 0, \\quad \\overline{\\mathcal{B}}({\\Delta x})\\_{\\alpha\\gamma}= 0.\n \\end{array}\n \\right.$$ \nThis, together with and Proposition [prop:dif-B-F], implies that\n \n$$\\begin{aligned}\n{{\\nonumber}}\\overline{\\cal U}({\\Delta x},0,0)=\n \\begin{bmatrix}\n \\nabla\\_{xx}^2L(\\overline x,\\overline \\xi,\\overline\\Gamma){\\Delta x}\\\\\n h'(\\overline x){\\Delta x}\\\\\n -\\overline{\\mathcal{B}}{\\Delta x}+\n \\begin{bmatrix}\n (\\overline{\\mathcal{B}}{\\Delta x})\\_{\\alpha\\alpha}&(\\overline{\\mathcal{B}}{\\Delta x})\\_{\\alpha\\beta}&\\overline\\Xi\\_{\\alpha\\gamma}\\circ(\\overline{\\mathcal{B}}{\\Delta x})\\_{\\alpha\\gamma} \\\\\n (\\overline{\\mathcal{B}}{\\Delta x})\\_{\\beta\\alpha}&W((\\overline{\\mathcal{B}}{\\Delta x})\\_{\\beta\\beta})&0 \\\\\n \\overline\\Xi\\_{\\gamma\\alpha}\\circ(\\overline{\\mathcal{B}}{\\Delta x})\\_{\\gamma\\alpha}&0&0 \n \\end{bmatrix}\n \\end{bmatrix}=0,\\end{aligned}$$ \nwhich contricts with the nonsingularity of $\\overline{\\cal U}$. The proof is then completed. $\\hfil \\Box$\nThe primal-dual characterizations for convex quadratic semidefinite programming\n===============================================================================\nIn this section, we shall study the primal-dual characterizations of the W-SRCQ (Definition [def:W-SRCQ]) and W-SOC (Definition [def:W-SOC]) for the following convex quadratic semidefinite programming (QSDP):\n \n$$\\label{prog:P-CP}\n\\begin{array}{cl}\n\\min\\quad & \\displaystyle\\frac{1}{2}{\\langle}x,{\\mathcal{Q}}x{\\rangle}+{\\langle}c,x{\\rangle}\\\\\n\\mbox{s.t.}\\quad & {\\mathcal{H}}x-p=0, \\\\\n& {\\mathcal{G}}x-q\\in{\\mathbb{S}}^n\\_+, \n\\end{array}$$ \nwhere *c* ∈ X, *p* ∈ R*m*, *q* ∈ S*n*, the self-adjoint linear operator Q : X → X is positive semi-definite, and H : X → R*m* and G : X → S*n* are two given linear operators. The dual problem (in the restricted Wolfe sense ) of the QSDP takes the form:\n \n$$\\label{prog:D-CP}\n \\begin{array}{cl}\n \\displaystyle\\min\\_{\\omega\\in {\\mathbb{W}},\\xi\\in{\\mathbb{R}}^m,\\Gamma\\in{\\mathbb{S}}^n}\\quad& \\displaystyle\\frac{1}{2}{\\langle}\\omega,{\\mathcal{Q}}\\omega{\\rangle}+{\\langle}p,\\xi{\\rangle}+{\\langle}q,\\Gamma{\\rangle}\\\\\n \\mbox{s.t.}\\quad & -{\\mathcal{Q}}\\omega-c-{\\mathcal{H}}^\\*\\xi-{\\mathcal{G}}^\\*\\Gamma=0, \\\\\n & \\Gamma\\in {\\mathbb{S}}^n\\_-, \n \\end{array}$$ \nwhere W ⊆ X is a linear subspace containing ${{\\rm Im}}({\\mathcal{Q}})$. Thus, the KKT system of is given by\n \n$$\\begin{aligned}\n\\left\\{ \n\\begin{aligned}\n&\\begin{bmatrix}\n{\\mathcal{Q}}\\omega \\\\\np \\\\\nq\n\\end{bmatrix}+\n\\begin{bmatrix}\n-{\\mathcal{Q}}\\\\\n-{\\mathcal{H}}\\\\\n-{\\mathcal{G}}\\end{bmatrix}\\tau+\n\\begin{bmatrix}\n0 \\\\\n0\\\\\n{\\mathcal{I}}\\end{bmatrix}\\Upsilon=0, \\\\\n&{\\mathcal{Q}}\\omega+c+{\\mathcal{H}}^\\*\\xi+{\\mathcal{G}}^\\*\\Gamma=0, \\\\\n&\\Upsilon\\in \\mathcal{N}\\_{{\\mathbb{S}}^n\\_-}(\\Gamma), \\\\\n&\\omega\\in {\\mathbb{W}},\\quad \\xi\\in{\\mathbb{R}}^m,\\quad \\Gamma\\in{\\mathbb{S}}^n,\\quad \\tau\\in{\\mathbb{X}},\\quad \\Upsilon\\in{\\mathbb{S}}^n.\\\\\n\\end{aligned}\n\\right. \\label{KKT:prog:D-CP}\\end{aligned}$$ \n[prop:pd-opc] Consider the convex quadratic semidefinite programming and its dual problem.\n* Suppose that W = X. Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of the convex quadratic semidefinite programming with a Lagrange multiplier $(\\overline\\xi,\\overline\\Gamma)\\in M(\\overline{x})$. The primal W-SOC holds at $(\\overline x,\\overline \\xi,\\overline\\Gamma)$ if and only if the dual W-SRCQ holds at $(\\overline x,\\overline \\xi,\\overline\\Gamma,\\overline x,{\\mathcal{G}}\\overline x-q)$.\n* Suppose that ${\\mathbb{W}}={{\\rm Im}}({\\mathcal{Q}})$. Let $(\\overline \\omega,\\overline \\xi,\\overline \\Gamma)\\in{\\mathbb{W}}\\times{\\mathbb{R}}^m\\times{\\mathbb{S}}^n$ be a stationary point of the dual problem with a Lagrange multiplier $(\\overline x,\\overline\\Upsilon)\\in M(\\overline \\omega,\\overline \\xi,\\overline \\Gamma)$. The dual W-SOC holds at $(\\overline \\omega,\\overline \\xi,\\overline \\Gamma,\\overline x,\\overline\\Upsilon)$ if and only if the primal W-SRCQ holds at $(\\overline x,\\overline \\xi,\\overline \\Gamma)$.\nWe demonstrate (i) here, as (ii) follows from it with only minor modifications.\nLet $A=g(\\overline{x})+\\overline{\\Gamma} = {\\cal G}\\overline x - q + \\overline\\Gamma$ satisfy the eigenvalue decomposition with $\\overline{P}\\in{\\cal O}^n(A)$, the index sets *α*, *β* and *γ* defined by, and $\\overline{\\cal B}:\\mathbb{X}\\to \\mathbb{S}^n$ defined by. By the convexity, we know that the primal W-SOC holds at $(\\overline x,\\overline \\xi,\\overline{\\Gamma})$ if and only if\n \n$$\\begin{aligned}\n\\left\\{\n\\begin{aligned}\n&{\\mathcal{Q}}d=0, \\quad {\\mathcal{H}}d=0,\\\\\n&\\overline{\\mathcal{B}}(d)\\_{\\alpha\\gamma}=0, \\quad \\overline{\\mathcal{B}}(d)\\_{\\beta\\beta}= 0,\\quad \\overline{\\mathcal{B}}(d)\\_{\\beta\\gamma}= 0, \\quad \\overline{\\mathcal{B}}(d)\\_{\\gamma\\gamma}= 0 \n\\end{aligned}\n\\right.\n\\quad \\Longrightarrow \\quad \nd=0. \\label{eq:pd-prithe W-SOC}\\end{aligned}$$ \nOn the other side, it is obvious that $(\\overline x,\\overline \\xi,\\overline\\Gamma,\\overline x,{\\mathcal{G}}\\overline x-q)$ is a KKT point of by. By Definition [def:W-SRCQ], we know that the W-SRCQ of the dual problem at $(\\overline x,\\overline \\xi,\\overline\\Gamma,\\overline x,{\\mathcal{G}}\\overline x-q)$ takes the form:\n \n$$\\begin{aligned}\n\\label{eq:d-w-srcq}\n\\begin{bmatrix}\n {\\mathcal{Q}}&{\\mathcal{H}}^\\* & {\\mathcal{G}}^\\* \\\\\n0 & 0 & {\\mathcal{I}}\\end{bmatrix}\n\\begin{bmatrix}\n{\\mathbb{X}}\\\\ {\\mathbb{R}}^m \\\\ {\\mathbb{S}}^n\n\\end{bmatrix}+\n\\begin{bmatrix}\n\\{0\\} \\\\ \n{{\\rm aff}}({\\mathcal{T}}\\_{{\\mathbb{S}}^n\\_-}(\\overline\\Gamma)\\cap [{\\mathcal{G}}\\overline x-q]^{\\perp})\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n{\\mathbb{X}}\\\\\n{\\mathbb{S}}^n\n\\end{bmatrix}.\\end{aligned}$$ \nSince ${{\\rm aff}}({\\mathcal{T}}\\_{{\\mathbb{S}}^n\\_-}(\\overline\\Gamma)\\cap [{\\mathcal{G}}\\overline x-q]^{\\perp})=\\{\\overline{P}B\\overline{P}^T\\in{\\mathbb{S}}^n\\ |\\ B\\_{\\alpha\\beta}=0,\\ B\\_{\\alpha\\alpha}= 0\\}$, we know that is equivalent to\n \n$$\\begin{aligned}\n\\label{eq:pd-dualWSRCQ}\n{{\\rm Null}}\\Big(\\begin{bmatrix}\n{\\mathcal{Q}}& 0\\\\ {\\mathcal{H}}& 0\\\\ \\overline{\\mathcal{B}}&{\\mathcal{I}}\\end{bmatrix} \\Big)\n\\bigcap\n\\begin{bmatrix}\n{\\mathbb{X}}\\\\\n\\{B\\in{\\mathbb{S}}^n\\ |\\ B\\_{\\alpha\\beta}=0,\\ B\\_{\\alpha\\alpha}= 0\\}^{\\perp}\n\\end{bmatrix}\n=\\{0\\}.\\end{aligned}$$ \nMeanwhile, it is not difficult to observe that and are equivalent. Consequently, the primal W-SOC is satisfied at $(\\overline x, \\overline \\xi,\\overline\\Gamma)$ if and only if the dual W-SRCQ is satisfied at $(\\overline x, \\overline \\xi,\\overline\\Gamma,\\overline x, {\\mathcal{G}}\\overline x - q)$. This completes the proof. $\\hfill \\Box$\nBy employing the similar argument, we are able to obtain the primal-dual characterizations of the S-SOSC (Definition [def:S-SOSC]) and the constraint nondegeneracy (Definition [def:constraint nondegeneracy]) for the QSDP. This extends the corresponding results obtained in Chan and Sun for the linear SDP. Also, for the QSDP, a primal-dual characterizations of the SRCQ (Definition [def:SRCQ]) and SOSC (Definition [def:SOSC]) is established by.\nA semismooth Newton method with correction\n==========================================\nIn this section, based on the theoretical results obtained in previous sections, we introduce a new semismooth Newton method with a correction step, which enjoys the locally quadratically convergence rate even without the subdifferential regularity.\nDenote Z : = X × R*m* × S*n*. Let *δ* > 0 be some given constant. Define the correction mapping ${\\cal P}\\_{\\delta}: \\mathbb{Z}\\to \\mathbb{Z}$ by\n \n$$\\label{eq:def-pi-mapping}\n{\\cal P}\\_{\\delta}(Z):=\\big({x},{\\xi},{\\Gamma}-\\sum\\_{i\\in \\theta} \\lambda\\_i({A}){P}\\_{i}{P}\\_i^T\\big), \\quad Z=({x},{\\xi},{\\Gamma})\\in\\mathbb{Z},$$ \nwhere *A* : = *g*(*x*) + Γ has the eigenvalue decomposition with ${P}\\in{\\cal O}^n({A})$ and *θ* : = {*i* ∣ ∣*λ**i*(*A*)∣ ≤ *δ*}.\nGiven *Z* = (*x*, *ξ*, Γ) ∈ Z, define two elements ${\\cal U}\\_0$ and ${\\cal U}\\_{\\cal I}\\in \\partial\\_BF(Z)$ at *Z* similarly as those in and, i.e., for any (Δ*x*, Δ*ξ*, ΔΓ) ∈ Z,\n \n$$\\begin{aligned}\n\\label{eq:U0k}\n{\\cal U}\\_{0}({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma}) =\n\\begin{bmatrix}\n\\nabla^2\\_{xx}L(x,\\xi,\\Gamma){\\Delta x}+ h'(x)^\\*{\\Delta \\xi}+ g'(x)^\\*{\\Delta \\Gamma}\\\\\n h'(x){\\Delta x}\\\\\n- g'(x){\\Delta x}+{\\cal V}\\_{0}(g'(x){\\Delta x}+{\\Delta \\Gamma})\n\\end{bmatrix} \\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n\\label{eq:U1k}\n {\\cal U}\\_{\\cal I}({\\Delta x},{\\Delta \\xi},{\\Delta \\Gamma}) =\n \\begin{bmatrix}\n \\nabla^2\\_{xx}L(x,\\xi,\\Gamma){\\Delta x}+ h'(x)^\\*{\\Delta \\xi}+ g'(x)^\\*{\\Delta \\Gamma}\\\\\n h'(x){\\Delta x}\\\\\n - g'(x){\\Delta x}+{\\cal V}\\_{\\cal I}(g'(x){\\Delta x}+{\\Delta \\Gamma})\n \\end{bmatrix}, \\end{aligned}$$ \nwhere ${\\cal V}\\_0$ and ${\\cal V}\\_{\\cal I}\\in \\partial\\_B\\Pi\\_{{\\mathbb{S}}^m\\_+}(A)$ is given by and in Proposition [prop:dif-B-F], respectively. The following proposition addresses the legitimacy of the correction mapping. It essentially shows that under weaker regularity conditions, when *Ẑ* is sufficiently close to $\\overline{Z}$, the corresponding ${\\cal U}\\_{0}$ (or ${\\cal U}\\_{\\cal I}$) in the B-subdifferential of *F* at the corrected point $Z = {\\cal P}\\_{\\delta}(\\widehat Z)$ with properly chosen *δ* is nonsingular and its inverse is uniformly bounded. As one can see later, this will be crucial for the local convergence analysis of the proposed semismooth Newton algorithm.\n[prop:nons] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of the NLSDP with a Lagrange multiplier $(\\overline\\xi,\\overline\\Gamma)\\in M(\\overline{x})$. Let *δ* be any fixed constant in $(0, \\min(\\overline\\lambda\\_\\alpha, |\\overline\\lambda\\_\\gamma|))$.\n* If the W-SOC (Definition [def:W-SOC]) and constraint nondegeneracy (Definition [def:constraint nondegeneracy]) hold at $\\overline{Z}=(\\overline{x},\\overline\\xi,\\overline\\Gamma)\\in\\mathbb{Z}$, then there exists *κ*0 > 0 such that for any $\\widehat{Z}=(\\widehat{x},\\widehat{\\xi},\\widehat\\Gamma)$ sufficient close to $\\overline{Z}$ and $Z={\\cal P}\\_{\\delta}(\\widehat{Z})$, the corresponding ${\\cal U}\\_{0}\\in\\partial\\_B F(Z)$ defined by with respect to *Z* is nonsingular and\n \n$$\\label{eq:Uni-bound-U0}\n\\kappa\\_{0}^{-1}\\|\\Delta Z\\| \\le \\|{\\cal U}\\_{0}(\\Delta Z)\\|\\quad \\forall\\,\\Delta Z\\in\\mathbb{Z}.$$\n* If the S-SOSC (Definition [def:S-SOSC]) and W-SRCQ (Definition [def:W-SRCQ]) hold at $\\overline{Z}=(\\overline{x},\\overline\\xi,\\overline\\Gamma)\\in\\mathbb{Z}$, then there exists *κ*1 > 0 such that for any $\\widehat{Z}=(\\widehat{x},\\widehat{\\xi},\\widehat\\Gamma)$ sufficient close to $\\overline{Z}$ and $Z={\\cal P}\\_{\\delta}(\\widehat{Z})$, the corresponding ${\\cal U}\\_{\\cal I}\\in\\partial\\_B F(Z)$ defined by with respect to *Z* is nonsingular and\n \n$$\\label{eq:Uni-bound-U1}\n\\kappa\\_{1}^{-1}\\|\\Delta Z\\| \\le \\|{\\cal U}\\_{\\cal I}(\\Delta Z)\\|\\quad \\forall\\,\\Delta Z\\in\\mathbb{Z}.$$\nWe focus on the result (i), since (ii) can be obtained similarly.\nIt suffices to show that the inequality holds, as the nonsingularity of ${\\cal U}\\_{0}$ is a direct implication of. Assume, on the contrary, there exist a sequence {*Ẑ**ν*} converging to $ \\overline{Z}$ and a nonzero sequence {Δ*Z**ν* = (Δ*x**ν*, Δ*ξ**ν*, ΔΓ*ν*)} ⊆ Z such that ${\\cal U}^{\\nu}\\_{0}\\in\\partial\\_B F(Z^{\\nu})$ defined by satisfy \n$$\\|{\\cal U}^{\\nu}\\_{0}(\\Delta Z^{\\nu})\\|< 1/\\nu \\|\\Delta Z^{\\nu}\\|.$$ \n Without loss of generality, we may assume that ∥Δ*Z**ν*∥ = 1 for each *ν* and Δ*Z**ν* → Δ*Z*∞ ≠ 0 as *ν* → ∞. For each *ν*, let $\\widehat{A}^{\\nu}=g(\\widehat{x}^{\\nu})+\\widehat\\Gamma^{\\nu}$ enjoy the eigenvalue decomposition with $\\widehat{P}^{\\nu}\\in {\\cal O}^n(\\widehat{A}^{\\nu})$. For each *ν*, denote \n$$\\zeta^{\\nu}:=\\{i\\mid \\widehat\\lambda\\_i^{\\nu}\\ge \\delta\\}, \\quad \\theta^{\\nu}:=\\{i\\mid |\\widehat\\lambda\\_i^{\\nu}|\\le \\delta\\}\\quad \\mbox{and}\\quad \\kappa^{\\nu}:=\\{i\\mid \\widehat\\lambda\\_i^{\\nu}\\le -\\delta\\},$$ \n where *λ̂**ν* : = *λ*(*Â**ν*) is the eigenvalues of *Â**ν*. Since $\\widehat{A}^\\nu\\to \\overline A = g(\\overline{x})+\\overline{\\Gamma}$ as *ν* → ∞, we know from the continuity of eigenvalues that $\\widehat\\lambda^\\nu \\to \\lambda(A)$ as *ν* → ∞. Notice that $\\delta\\in(0, \\min(\\overline\\lambda\\_\\alpha, |\\overline\\lambda\\_\\gamma|))$, for *ν* large enough,\n \n$$\\begin{aligned}\n\\label{eq:zeta}\n\\zeta^{\\nu}\\equiv {\\alpha}, \\quad \\theta^{\\nu}\\equiv {\\beta} \\quad \\mbox{and} \\quad \\kappa^{\\nu}\\equiv {\\gamma},\\end{aligned}$$ \nwhere *α*, *β* and *γ* are the index sets defined by for $\\overline{A}$. Since $Z^{\\nu}={\\cal P}\\_{\\delta}(\\widehat{Z}^{\\nu})$, it holds that\n \n$$\\|Z^{\\nu}-\\widehat{Z}^{\\nu}\\|=\\|{\\cal P}\\_{\\delta}(\\widehat{Z}^{\\nu})-\\widehat{Z}^{\\nu}\\| =\\|\\sum\\_{i\\in \\beta} \\lambda\\_i(\\widehat{A}^{\\nu})\\widehat{P}\\_{i}^{\\nu}(\\widehat{P}\\_i^{\\nu})^T\\| \n \\le \\| \n \\widehat \\lambda\\_\\beta^\\nu \\| \\to 0, \\mbox{ as } \\nu \\to \\infty.$$ \nThus, we know that $Z^{\\nu}\\to \\overline{Z}$ as *ν* → ∞. It also holds that for each *ν*,\n \n$$\\begin{aligned}\n\\label{eq:Anu}\nA^{\\nu}=g(x^{\\nu})+\\Gamma^{\\nu} = \\widehat{P}^{\\nu} \n\\begin{bmatrix}\n \\widehat\\Lambda\\_{ {\\alpha} {\\alpha}}^{\\nu} & 0 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & \\widehat\\Lambda\\_{ {\\gamma} {\\gamma}}^{\\nu}\n\\end{bmatrix}\n(\\widehat{P}^{\\nu})^T \\end{aligned}$$ \nwith $\\widehat{\\Lambda}^{\\nu}={\\rm Diag}(\\widehat\\lambda^{\\nu})$, and $A^\\nu \\to \\overline A$ as *ν* → ∞.\nLet $\\overline{\\cal U}\\_0$ be given by with $\\overline{\\cal V}\\_0$ defined by. By noting that $\\|{\\cal U}\\_{0}^{\\nu}(\\Delta Z^{\\nu})\\|<1/\\nu$, ∥Δ*Z**ν*∥ = 1, *f*, *h* and *g* are twice continous differentiable and $Z^{\\nu}\\to\\overline{Z}$, we have\n \n$$\\begin{aligned}\n && \\| \\overline {\\cal U}\\_{0}(\\Delta Z^{\\infty})\\|\\le \\| \\overline {\\cal U}\\_{0}(\\Delta Z^{\\infty})-{\\cal U}\\_{0}^{\\nu}(\\Delta Z^{\\nu})\\|+\\|{\\cal U}\\_{0}^{\\nu}(\\Delta Z^{\\nu})\\| {{\\nonumber}}\\\\ [3pt]\n &\\le& \\| \\overline {\\cal U}\\_{0}(\\Delta Z^{\\infty})-\\overline{\\cal U}\\_{0}(\\Delta Z^{\\nu})\\| +\\| \\overline {\\cal U}\\_{0}(\\Delta Z^{\\nu})-{\\cal U}\\_{0}^{\\nu}(\\Delta Z^{\\nu})\\|+1/\\nu {{\\nonumber}}\\\\ [3pt]\n &\\le& O(\\| \\Delta Z^{\\infty}-\\Delta Z^{\\nu}\\| )+\\| \\overline{\\cal V}\\_{0}(g'(\\overline{x}){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu}) - {\\cal V}\\_{0}^{\\nu}(g'(x^{\\nu}){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu})\\|+1/\\nu,\\label{eq:U0Uk-diff1}\\end{aligned}$$ \nwhere for each *ν*, ${\\cal V}\\_{0}^{\\nu}$ is defined by with respect to *A**ν*. By Proposition [prop:dis-P] and taking subsequence if necessary, we may further assume that $\\widehat{P}^{\\nu}\\to P^{\\infty}\\in \\mathcal{O}^n(\\overline{A})$. Then, and Proposition [prop:dif-B-F] imply that $\\lim\\_{\\nu\\to\\infty}\\mathcal{V}^\\nu\\_0 = \\overline{\\cal V}\\_0$. Since ${\\cal V}\\_{0}^{\\nu}$ is uniformly bounded, we thus have as *ν* → ∞,\n \n$$\\begin{aligned}\n&&\\| {\\cal V}\\_{0}^{\\nu}(g'(x^{\\nu}){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu})-\\overline{\\cal V}\\_{0}(g'(\\overline x){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu})\\| {{\\nonumber}}\\\\ [3pt]\n&\\le& \\| {\\cal V}\\_{0}^{\\nu}(g'(x^{\\nu}){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu})-{\\cal V}\\_{0}^{\\nu}(g'(\\overline x){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu})\\| {{\\nonumber}}\\\\ [3pt]\n&&+ \\| {\\cal V}\\_{0}^{\\nu}(g'(\\overline x){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu})-\\overline{\\cal V}\\_{0}(g'(\\overline x){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu})\\| {{\\nonumber}}\\\\ [3pt]\n&\\le&O(\\| (g'(x^{\\nu})-g'(\\overline x)){\\Delta x}^{\\nu}\\| )+o(\\| g'(\\overline x){\\Delta x}^{\\nu}+{\\Delta \\Gamma}^{\\nu}\\| ) =o(1).\\label{eq:U0Uk-diff2}\\end{aligned}$$ \nCombining and, we obtain $\\| \\overline {\\cal U}\\_{0}(\\Delta Z^{\\infty})\\| =0$, which implies that $\\overline{\\cal U}\\_{0}$ is singular. This contradicts to Theorem [theorem:bar-N-SDP] (i), since the W-SOC and constraint nondegeneracy are assumed. The proof is then completed. $\\hfill \\Box$\nUsing Proposition [prop:nons], we can further establish a local error bound for the nonsmooth map *F* at the corrected points around $\\overline{Z}$ in the following proposition.\n[prop:error-bound] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of the NLSDP with a Lagrange multiplier $(\\overline\\xi,\\overline\\Gamma)\\in M(\\overline{x})$. Recall the definitions of constants *δ*, *κ*0 and *κ*1 in Proposition [prop:nons]. Then,\n* if the W-SOC and the constraint nondegeneracy hold at $\\overline Z=(\\overline x,\\overline\\xi,\\overline\\Gamma)\\in{\\mathbb{Z}}$, then there exists a neighborhood ${\\cal N}$ of $\\overline{Z}$ such that for any $\\widehat Z\\in {\\cal N}$ and $Z = {\\cal P}\\_\\delta(\\widehat Z)$, \n$$\\| F(Z)\\|\\ge \\frac{1}{2\\kappa\\_{0}}\\| Z-\\overline Z\\|;$$\n* if the S-SOSC and the W-SRCQ hold at $\\overline Z=(\\overline x,\\overline\\xi,\\overline\\Gamma)\\in{\\mathbb{Z}}$, then there exists a neighborhood ${\\cal N}$ of $\\overline{Z}$ such that for any $\\widehat Z\\in {\\cal N}$ and $Z = {\\cal P}\\_\\delta(\\widehat Z)$, \n$$\\| F(Z)\\|\\ge \\frac{1}{2\\kappa\\_{1}}\\| Z-\\overline Z\\|.$$\nWe only show (i), as (ii) can be proved via similar arguments.\nBy the semismoothness of *F* and Proposition [prop:nons] (i), we know that there exists a neighborhood $\\cal N$ of $\\overline{Z}$ such that for any $\\widehat Z \\in {\\cal N}$ and $Z = {\\cal P}\\_\\delta(\\widehat Z)$,\n \n$$\\begin{aligned}\n {{\\nonumber}}\\| F(Z)-{\\cal U}\\_{0}(Z-\\overline Z)\\|\\le \\frac{1}{2\\kappa\\_{0}}\\| Z-\\overline Z\\| \\quad \\mbox{ and } \\quad \\| {\\cal U}\\_{0}\\|\\ge \\frac{1}{\\kappa\\_0},\\end{aligned}$$ \nwhere U0 ∈ ∂*B**F*(*Z*) is given in. Thus,\n \n$$\\begin{aligned}\n{{\\nonumber}}\\|F(Z)\\|\\ge \\|{\\cal U}\\_{0}(Z-\\overline Z)\\|-\\| F(Z)-{\\cal U}\\_{0}(Z-\\overline Z)\\|\\ge \\frac{1}{\\kappa\\_{0}}\\|Z-\\overline Z\\|-\\frac{1}{2\\kappa\\_{0}}\\| Z-\\overline Z\\|=\\frac{1}{2\\kappa\\_{0}}\\| Z-\\overline Z\\|.\\end{aligned}$$ \nThe proof is completed. $\\hfill \\Box$\nNow we are ready to present our inexact semismooth Newton method with correction in Algorithm [algorithm:ASN] for solving the nonsmooth system *F*(*Z*) = 0 with *F* given in. As one can observe, the algorithm is quite similar to the classic inexact semismooth Newton method. The main difference is that in each iteration, an extra correction step, i.e., step 6 in Algorithm [algorithm:ASN], involving the correction mapping ${\\cal P}\\_{\\delta}$ is taken. This correction step, based on Proposition [prop:nons], ensures locally the nonsingularity of ${\\cal U}\\_0^k$ (or ${\\cal U}\\_{\\cal I}^k$), and hence the well-definedness of the proposed algorithm. Meanwhile, we shall emphasize that the additional computational costs brought by the correction step are negligible, as the eigendecompostion of $\\widehat{A}^{k+1} = g(\\widehat x^{k+1}) + \\widehat{\\Gamma}^{k+1}$ in step 6 can be re-used to obtain ${\\cal U}^{k+1} \\in \\partial\\_B F(Z^{k+1})$ in step 3. Moreover, the inexact computation in step 4 indicates that our proposed algorithm has great potential for solving large-scale problems.\n[algorithm:ASN]\n[1]\nAs promised, in the following theorem, we establish the well-definedness and local convergence results of Algorithm [algorithm:ASN] without assuming the subdifferential regularity.\n[theo:alg-convergence] Let $\\overline{x}\\in\\mathbb{X}$ be a stationary point of the NLSDP with a Lagrange multiplier $(\\overline\\xi,\\overline\\Gamma)\\in M(\\overline{x})$. Let $\\delta\\in(0, \\min(\\overline\\lambda\\_\\alpha, |\\overline\\lambda\\_\\gamma|))$. Then,\n* if the W-SOC and the constraint nondegeneracy hold at $\\overline Z=(\\overline x,\\overline\\xi,\\overline\\Gamma)\\in{\\mathbb{Z}}$, then there exists $\\overline\\eta >0$ such that for any $\\widehat Z^0$ sufficiently close to $\\overline Z$ and $\\eta \\le \\overline \\eta$, Algorithm [algorithm:ASN] using ${\\cal U}\\_{0}$ is well defined and the sequence {*Z**k*} so generated converges to $\\overline Z$ superlinearly;\n* if the S-SOSC and W-SRCQ hold at $\\overline Z=(\\overline x,\\overline\\xi,\\overline\\Gamma)\\in{\\mathbb{Z}}$, then there exists $\\overline\\eta >0$ such that for any $\\widehat Z^0$ sufficiently close to $\\overline Z$ and $\\eta \\le \\overline \\eta$, Algorithm [algorithm:ASN] using ${\\cal U}\\_{{\\mathcal{I}}}$ is well-defined and the sequence {*Z**k*} so generated converges to $\\overline Z$ superlinearly.\nMoreover, if *f*, *g* and *h* are twice Lipschitz continuously differentiable, i.e., they have Lipschitz continuous second order differentials, and *τ* = 1, then the corresponding convergence rates obtained in (i) and (ii) are quadratic.\nHere, we focus on the case (i) where the W-SOC and the constraint nondegeneracy hold at $\\overline Z$ and ${\\cal U}\\_0$ is employed. The proof for the case (ii) where the S-SOSC and W-SRCQ hold at $\\overline Z$ can be obtained similarly.\nBy the semismoothness of ΠS+*n*( ⋅ ) on S*n* and the twice continously differentiability of *f* *g* and *h*, we know *F* is semismooth at $\\overline Z$. This, together with Proposition [prop:nons], implies that we can find a positive constant $\\varsigma$, a neighborhood $\\mathbb{B}({\\overline{Z}},\\varsigma)$ of $\\overline{Z}$ and a positve constant *κ*0 such that for any given $ \\widehat Z\\in \\mathbb{B}({\\overline{Z}},\\varsigma)$ and $Z = {\\cal P}\\_{\\delta}(\\widehat Z)$,\n \n$$\\label{eq:semi-neighborhood}\n\\| F(Z) - F(\\overline{Z}) - {\\cal U}\\_0(Z - \\overline{Z}) \\| \\le \\frac{1}{ 4(L\\_g+2)\\kappa\\_0} \\| Z - \\overline{Z} \\| \\quad \\mbox{ and } \\quad \\|({\\cal U}\\_0)^{-1} \\| \\le \\kappa\\_0,$$ \nwhere ${\\cal U}\\_0 \\in \\partial\\_B F(Z)$ is given by and *L**g* > 0 is the Lipschitz constant of *g* over $\\mathbb{B}({\\overline{Z}},\\varsigma)$. Recall the eigenvalue decomposition of $\\overline A = g(\\overline x) + \\overline{\\Gamma"},"data_id":{"kind":"string","value":"arxiv_0181665"}}},{"rowIdx":45394,"cells":{"content":{"kind":"string","value":"Zeta-regularized Lattice Field Theory with Lorentzian background metrics\n========================================================================\nLattice field theory is a very powerful tool to study Feynman’s path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined. On the other hand, a recently developed regularization scheme based on Fourier integral operator *ζ*-functions can treat Feynman’s path integral non-pertubatively in Lorentzian background metrics. In this article, we formally *ζ*-regularize lattice theories with Lorentzian backgrounds and identify conditions for the Fourier integral operator *ζ*-function regularization to be applicable. Furthermore, we show that the classical limit of the *ζ*-regularized theory is independent of the regularization. Finally, we consider the harmonic oscillator as an explicit example. We discuss multiple options for the regularization and analytically show that they all reproduce the correct ground state energy on the lattice and in the continuum limit. Additionally, we solve the harmonic oscillator on the lattice in Minkowski background numerically.\nIntroduction\n============\nLattice Field Theory\n--------------------\nFeynman’s path integral approach to quantum mechanics and quantum field theory has become an indispensable conceptual tool to evaluate physical observables, in particular in high energy physics. One very fascinating aspect of the path integral formulation is that it allows in principle to compute also non-perturbative phenomena and thus reaches out to problems that can not be addressed in perturbation theory.\nThe major obstacle to employ the path integral for studying non-perturbative effects is, however, that it is in general ill defined, if we take it at face value, i.e., using a Lorentzian background metric. An elegant way out is to analytically continue to imaginary – Euclidean – time and discretize the system either on a time lattice in quantum mechanics or on a space-time grid in quantum field theory. The so introduced lattice spacing acts then as an ultraviolet regulator, whereas a finite time or box length serves as an infrared regulator resulting thus in completely regularized version of the path integral, see for introductions to the so obtained *lattice field theories*.\nAnother important advantage of the Euclidean lattice regulated path integral is its resemblance to systems in statistical physics where it corresponds to the partition function. This allows in particular to resort to numerical calculations using Markov Chain Monte Carlo (MCMC) methods to evaluate the path integral. This approach has been employed very successfully in the last years to obtain many non-perturbative results mainly for quantum chromodynamics, the theory of the strong interaction between quarks and gluons. In particular, it became possible to perform an ab initio calculation of the low-lying baryon spectrum using only QCD as the underlying theory. Furthermore, lattice QCD calculations led to a detailed insight into the structure of hadrons and they have provided non-perturbative contributions to electroweak processes and flavor physics, see e.g. for overviews. They were also very successful to determine thermodynamic properties . The remarkable progress in lattice QCD simulations can be seen by the fact that nowadays calculations are performed on large lattices – presently of the order of around 1003 × 200 lattice points – and directly in physical conditions.\nDespite this impressive success, such Euclidean time lattice simulations can not be employed in important and so far open questions such as the matter anti-matter asymmetry, topological theories, the discrepancy of the observed and theoretically predicted amount of charge-parity (CP) violation and non-equilibrium physics. The reason is that in these cases the integrand in the path integral becomes complex such that MCMC can not be used.\nIt would therefore be extremely valuable to have an alternative regularization of the path integral which allows, on one hand, to address non-perturbative physics and, on the other hand, to use a Minkowski background metric or even more general Lorentzian metrics. It is precisely at this point where the proposed *ζ*-regularization of the path integral is filling this so far missing gap. By “gauging” the path integral (cf. ), it allows for a fully non-perturbative regularization for very general metrics. In this way, physical observables can be computed even in cases where the Euclidean lattice techniques such as MCMC simulations fail and even on quantum computers .\nIn this article, we aim to extend the mathematical formulation of *ζ*-regularization within the context of lattice theories and to illustrate its application at the example of the quantum mechanical harmonic oscillator. In particular, we will test different ways of gauging the path integral and demonstrate that all the here employed choices lead to correct physical results. In addition, we will show that the *ζ*-regularization gives the correct classical limit. Our work will therefore not only provide a rather simple example of the *ζ*-regularization on the lattice with Lorentzian background metric, but it also discusses different classes of gauges providing thus suitable choices when applied to a particular physical model of interest.\n*ζ*-regularization\n------------------\nOperator *ζ*-functions are a means of constructing traces on certain operator algebras. Given an algebra A of operators, a trace *τ* that is defined on a sub-algebra A0 ⊆ A, and an operator *A* ∈ A \\ A0, the aim of operator *ζ*-functions is to define *τ*(*A*). To this end, we construct a holomorphic family *φ* : C → A such that *φ* maps an open connected subset of C into A0 and *φ*(0) = *A*. Given this setup and sufficient regularity of the operator algebra and construction of *φ*, we can study the operator *ζ*-function *ζ*(*φ*) which is defined via analytic continuation of *z* ↦ *τ*(*φ*(*z*)) on the domain that is mapped to A0 under *φ*. If this analytic extension exists in a neighborhood of 0 and the value *ζ*(*φ*)(0) is independent of the choices made in the construction of *φ*, then we can use *ζ*(*φ*)(0) as our definition of *τ*(*φ*(0)), i.e., *τ*(*A*).\nFor example, if we consider the absolute value of the differential operator $A:={{\\left\\lvert}{{\\partial}}{\\right\\lvert}}$ on R/2*π*Z, then A is the algebra of classical pseudo-differential operators on R/2*π*Z and *τ* the trace on trace-class operators, i.e., the sum of eigenvalues counting multiplicities. Furthermore, we may construct *φ*(*z*) : = *A**z* + 1. Formally evaluating the trace on *φ*(*z*) thus yields This makes sense for ℜ(*z*) < − 2 and implies that the corresponding operator *ζ*-function is *ζ*(*φ*)(*z*) = 2*ζ**R*( − *z* − 1) where *ζ**R* denotes the Riemann *ζ*-function. In particular, the *ζ*-regularized trace of ${{\\left\\lvert}{{\\partial}}{\\right\\lvert}}$ is given by $\\zeta({\\varphi})(0)=2\\zeta\\_R(-1)=-\\frac{1}{6}$.\nOperator *ζ*-functions were first studied for pseudo-differential operators by Ray and Singer whose work has been extended by many authors including Seeley , Guillemin , Kontsevich and Vishik , Lesch , Paycha and Scott , and Wodzicki . This notion of *ζ*-regularization has been introduced to physics by Hawking in his work on path integrals with a curved space-time background, successfully applied in many physical settings (e.g., the Casimir effect, defining one-loop functional determinants, the stress-energy tensor, conformal field theory, and string theory ), and is related to Hadamard parametrix renormalization . As such the approach plays a foundational role for an effective Lagrangian to be defined , for relatively easy computations of heat kernel coefficients , and for non-trivial extensions of the Chowla-Selberg formula . Furthermore, the residues associated with operator *ζ*-functions give rise to the multiplicative anomaly in perturbation theory and contribute to the energy momentum tensor of a black hole .\nAlthough *ζ*-regularization based on pseudo-differential operators has been very successful, it cannot be applied in general. Radzikowski showed that, in general, Fourier integral operator *ζ*-functions are necessary. The application of Fourier integral operator *ζ*-functions to quantum field theory has recently been developed in the context of the Hamiltonian formulation of quantum field theories . The starting point remains fundamentally the same as with the pseudo-differential application of *ζ*-regularization. Given an algebra of Fourier integral operators A corresponding to a quantum field theory, the time evolution operator *U*, and an observable Ω, we would like to compute the vacuum expectation value\n \n$$\\begin{aligned}\n\\tag{$\\*$}\\label{eq:feynman\\_definition\\_vacuum\\_expectation\\_value}\n \\langle\\Omega\\rangle=\\lim\\_{T\\to\\infty+i0}\\frac{{\\ensuremath{{\\operatorname}{tr}}}(U(0,T)\\Omega)}{{\\ensuremath{{\\operatorname}{tr}}}(U(0,T))}.\\end{aligned}$$ \nBoth *U*(0, *T*) and *U*(0, *T*)Ω are Fourier integral operators in general, *U*(0, *T*) is unitary, and *U*(0, *T*)Ω is usually unbounded. Hence, neither numerator nor denominator are well-defined. This justifies the Fourier integral operator *ζ*-function approach to defining a *ζ*-regularized vacuum expectation value which replaces the traces in by operator *ζ*-functions. To construct these *ζ*-regularized vacuum expectation values , we construct a suitable holomorphic family of operators G called “gauge”. This gauge has to satisfy a number of properties. Most importantly, for ℜ(*z*) ≪ 0 both *U*(0, *T*)G(*z*)Ω and *U*(0, *T*)G(*z*) have to be trace-class operators. This will imply that both *ζ*(*U*(0, *T*)GΩ) and *ζ*(*U*(0, *T*)G) are well-defined meromorphic functions and therefore\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle\\_\\zeta(z)=\\lim\\_{T\\to\\infty+i0}\\frac{\\zeta(U(0,T){\\ensuremath{\\mathfrak{G}}}\\Omega)}{\\zeta(U(0,T){\\ensuremath{\\mathfrak{G}}})}(z)\\end{aligned}$$ \nis a well-defined meromorphic function (provided that *ζ*(*U*(0, *T*)G) is not the constant zero-function). In particular, if ⟨Ω⟩*ζ* is holomorphic in a neighborhood of 0, then we can define the *ζ*-regularized vacuum expectation value as ⟨Ω⟩*ζ*(0). This construction has been shown to be generally possible and to be physically meaningful and accessible using quantum computing .\nFurthermore, the construction is formally applicable to lattice formulations in Lorentzian space-time as well. Applying a lattice discretization to *ζ*(*U*(0, *T*)GΩ)(*z*) for example yields integrals of the form ∫*ω*(Φ)g(*z*)(Φ)*e**i**S*(Φ)D(Φ), i.e., precisely the type of integral we expect from a lattice discretization but with an additional term g(*z*)(Φ). This additional term g(*z*)(Φ) moreover satisfies the asymptotic bound ${{\\left\\lvert}{{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)}{\\right\\lvert}}\\le C (1+{{\\left\\lVert}{\\Phi}{\\right\\lVert}}^2)^{\\delta \\Re(z)}$ for some constants *C*, *δ* > 0 and ${{\\left\\lVert}{\\Phi}{\\right\\lVert}}$ sufficiently large. Since *ω* is polynomially bounded for large ${{\\left\\lVert}{\\Phi}{\\right\\lVert}}$, this implies that Φ ↦ *ω*(Φ)g(*z*)(Φ)*e**i**S*(Φ) is integrable for ℜ(*z*) ≪ 0 and the *ζ*-regularized lattice vacuum expectation can be obtained analytically continuing to *z* = 0.\nIf this general setup can be obtained from lattice discretizing *ζ*-regularized vacuum expectation values starting from the Hamiltonian formulation, we may also take it as a starting point to an a priori *ζ*-regularization of a lattice field theory without needing to construct the Hamiltonian formulation first. However, this raises a number of applicability questions. Most importantly:\n1. What kind of gauge families g can be used to *ζ*-regularize lattice field theories in a Lorentzian background?\n2. Under what conditions is it possible to construct a *ζ*-regularized Hamiltonian theory starting from a “naïvely” *ζ*-regularized lattice field theory?\nThe first question therefore addresses the freedom we have in choosing gauge families most suitable for a given lattice simulation whereas answering the second question allows us to prove equivalence with the Fourier integral operator *ζ*-function approach to *ζ*-regularized vacuum expectation values. This is important because it means that the known applicability and physicality results extend to the thus constructed *ζ*-regularized lattice field theory. In particular, it provides an avenue of using quantum computation to simulate lattice field theories in Lorentzian backgrounds.\nAims of this article\n--------------------\nSince *ζ*-regularized lattice field theories in Lorentzian backgrounds have not been studied from this point of view, we aim to gain insights into questions Q1 and Q2. In we will begin with a general discussion of lattice field theory from a distribution theory point of view. This is important for Q2 because the Hamiltonian theory uses microlocal analysis and therefore is fundamentally a theory of distributions. Formally introducing the gauge family g in the second half of we will show that the *ζ*-regularized lattice field theory and the original lattice field theory coincide in the distributional limit *z* → 0. This is a necessary first step in the construction of gauged transfer matrices and gauged time evolution operators in. Thus, addresses Q1 and addresses Q2. In we will show that *ζ*-regularized lattice field theories have correct classical limits. In particular, we will explicitly compute the classical limit for the harmonic oscillator. We will then continue discussing the harmonic oscillator analytically in and numerically in. For the analytic discussion, we will solve the harmonic oscillator five times, once with Euclidean background and then again with four different gauge families g. Each of these gauges will be representative of a generic property for the gauge that may be useful in different circumstances. The numerical discussion of the harmonic oscillator will focus on classical computing rather than quantum computing and highlight extrapolation and convergence behavior for one of the choices of gauge.\n*ζ*-regularizing lattice vacuum expectation values\n==================================================\nIn this section, we will investigate the connection between lattice vacuum expectation values with Euclidean and Lorentzian backgrounds. In particular, we will highlight how additional complications arise in Lorentzian backgrounds and how a distributional understanding of the lattice integrals and the *ζ*-regularization are used to formally understand the lattice integrals in Lorentzian backgrounds. To this end, approximation on compact subsets will be instrumental. In and, we will provide conditions that ensure the infinite volume limit exists and coincides with the *ζ*-regularized vacuum expectation value considered in .\nIn a Euclidean lattice theory with *T* time slices, we can express the vacuum expectation value ⟨Ω⟩*E* of an observable Ω as\n \n$$\\begin{aligned}\n \\langle \\Omega\\rangle\\_E=\\frac{\\int\\_{X^T}\\Omega(\\Phi)e^{-S(\\Phi)}d\\mu^T(\\Phi)}{\\int\\_{X^T}e^{-S(\\Phi)}d\\mu^T(\\Phi)}\\end{aligned}$$ \nwhere *X* is a locally compact Hausdorff space, *μ* a *σ*-finite Radon measure on *X*, and *μ**T* denotes the *T*-fold product measure *μ* × *μ* × ⋯ × *μ* on *X**T*.\n1. For the topological rotor (i.e., a massive particle on a circle of radius *R*), *X* is the circle of radius *R*.\n2. For quantum mechanics in R*N*, *X* = R*N*.\n3. For a quantum field with values in *V* on *L* spacial lattice points, *X* = *V**L*. In particular, for a gauge field on *L* spacial lattice points, *X* often is *G**L* for some (locally) compact Hausdorff group and *μ* the Haar measure on *X*.\nVacuum expectation values of a lattice theory with Lorentzian background can be expressed as\n \n$$\\begin{aligned}\n \\langle \\Omega\\rangle\\_L=\\frac{\\int\\_{X^T}\\Omega(\\Phi)e^{iS(\\Phi)}d\\mu^T(\\Phi)}{\\int\\_{X^T}e^{iS(\\Phi)}d\\mu^T(\\Phi)}\\end{aligned}$$ \nwhere the integrals ∫*X**T**F*(Φ)*e**i**S*(Φ)*d**μ**T*(Φ) with *F* ∈ {1, Ω} are defined distributionally if *μ*(*X*) = ∞. In terms of the transfer matrix, we can write\n \n$$\\begin{aligned}\n Z=\\int\\_{X^T}e^{iS(\\Phi)}d\\mu^T(\\Phi)=\\int\\_{X^T}\\prod\\_{j=1}^T e^{iS\\_\\Delta(\\Phi\\_{j+1},\\Phi\\_j)}d\\mu^T(\\Phi)\\end{aligned}$$ \nwhere the kernel *κ*Δ of the transfer matrix[1](#fn1) *U*Δ is given by *κ*Δ(Φʹ, Φ) = *e**i**S*Δ(Φʹ, Φ). In this distributional setting, *U*Δ is therefore to be understood as the operator satisfying,\n \n$$\\begin{aligned}\n\\tag{$\\*\\*$}\\label{eq:definition U\\_Delta}\n {\\forall}{\\varphi},\\psi\\in C\\_c^\\infty(X):\\ \\langle{\\varphi},U\\_\\Delta\\psi\\rangle=\\int\\_X\\int\\_X{\\varphi}(\\Phi')^\\*e^{iS\\_\\Delta(\\Phi',\\Phi)}\\psi(\\Phi)d\\mu(\\Phi')d\\mu(\\Phi).\\end{aligned}$$ \nAt this point we can make the connection to one of the necessary assumptions in the construction of *ζ*-regularized vacuum expectation values in the Hamiltonian setting . For the approach to be applicable, we need to ensure that Cauchy surfaces (i.e., time slices) of the spacetime are compact., however, implies that this assumption in the *ζ*-formalism does not restrict generality. Let *K* ⊆ *X* be compact. Then we define *U*Δ∣*K* to be the restriction of *U*Δ to *K* which means that the kernel of *U*Δ∣*K* is given by 1*K* × *K**κ*Δ as a distribution on *K*2 and *U*Δ∣*K* is uniquely determined by\n \n$$\\begin{aligned}\n {\\forall}{\\varphi},\\psi\\in C^\\infty(K):\\ \\langle{\\varphi},U\\_\\Delta|\\_K\\psi\\rangle=\\int\\_K\\int\\_K{\\varphi}(\\Phi')^\\*e^{iS\\_\\Delta(\\Phi',\\Phi)}\\psi(\\Phi)d\\mu|\\_K(\\Phi')d\\mu|\\_K(\\Phi).\\end{aligned}$$ \nThe *ζ*-formalism by construction only works for *U*Δ∣*K*. However, since for every *φ*, *ψ* ∈ *C**c*∞(*X*), there exists a compact *K**φ*, *ψ* ⊆ *X* such that spt*φ* ∪ spt*ψ* ⊆ *K**φ*, *ψ*, where spt*φ* denotes the support of *φ*, we obtain that ⟨*φ*, *U*Δ*ψ*⟩ = ⟨*φ*, *U*Δ∣*K**φ*, *ψ**ψ*⟩. Hence, *U*Δ∣*K* converges to *U*Δ for $K\\nearrow X$ in the sense that the kernel 1*K* × *K**κ*Δ of *U*Δ∣*K* converges to the kernel *κ*Δ of *U*Δ in the space of distributions.\nIn the Euclidean setting, we generally don’t have to discuss this aspect because the integrand *F**e*− *S* is in *L*1(*X*2), and *μ* is a *σ*-finite Radon measure. This implies\n \n$$\\begin{aligned}\n &\\int\\_{X^T}Fe^{-S}d\\mu^T-\\int\\_{K^T}Fe^{-S}d\\mu^T=&\\int\\_{X^T}1\\_{X^T\\setminus K^T}Fe^{-S}d\\mu^T\\to&0\\quad (K\\nearrow X)\\end{aligned}$$ \nand since ∫*X**T**F**e*− *S**d**μ**T* is a well-defined integral, the limit $K\\nearrow X$ usually needs not to be considered.\nIn the *ζ*-regularized setting, we enforce a similar situation by enforcing asymptotic bounds of the form ${{\\left\\lvert}{\\Omega(\\Phi)}{\\right\\lvert}}\\le c{{\\left\\lVert}{\\Phi}{\\right\\lVert}}^d$ for ${{\\left\\lVert}{\\Phi}{\\right\\lVert}}\\to\\infty$ and some constants *c*, *d* ∈ R on the observable Ω, and by introducing a suitably chosen holomorphic family of functions g with g(0) = 1 and ${{\\left\\lvert}{{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)}{\\right\\lvert}}\\le \\alpha{{\\left\\lVert}{\\Phi}{\\right\\lVert}}^{\\beta\\Re(z)}$ for constants *α*, *β* > 0 and ${{\\left\\lVert}{\\Phi}{\\right\\lVert}}\\gg1$. This family g is called a “gauge” in the mathematical literature and ensures that the gauged integrand *F**e**i**S*g(*z*) is in *L*1(*X**T*) for ℜ(*z*) ≪ 0. We therefore obtain\n \n$$\\begin{aligned}\n &\\int\\_{X^T}Fe^{iS}{\\ensuremath{\\mathfrak{g}}}(z)d\\mu^T-\\int\\_{K^T}Fe^{iS}{\\ensuremath{\\mathfrak{g}}}(z)d\\mu^T=&\\int\\_{X^T}1\\_{X^T\\setminus K^T}Fe^{iS}{\\ensuremath{\\mathfrak{g}}}(z)d\\mu^T\\to&0\\end{aligned}$$ \nwhenever ℜ(*z*) ≪ 0. Thus, considering a sequence (*K**n*)*n* ∈ N of compacta in *X* with $K\\_n\\nearrow X\\ (n\\to\\infty)$, we conclude\n \n$$\\begin{aligned}\n \\lim\\_{n\\to\\infty}\\frac{\\int\\_{K\\_n^T}\\Omega e^{iS}{\\ensuremath{\\mathfrak{g}}}(z)d\\mu^T}{\\int\\_{K\\_n^T}e^{iS}{\\ensuremath{\\mathfrak{g}}}(z)d\\mu^T}=\\frac{\\int\\_{X^T}\\Omega e^{iS}{\\ensuremath{\\mathfrak{g}}}(z)d\\mu^T}{\\int\\_{X^T}e^{iS}{\\ensuremath{\\mathfrak{g}}}(z)d\\mu^T}\\end{aligned}$$ \nwhenever ℜ(*z*) ≪ 0. Since the *ζ*-regularized vacuum expectation ⟨Ω⟩*ζ* is defined via analytic continuation of $z\\mapsto\\frac{\\langle \\Omega{\\ensuremath{\\mathfrak{g}}}(z)\\rangle\\_L}{\\langle{\\ensuremath{\\mathfrak{g}}}(z)\\rangle\\_L}$, the above identity implies pointwise convergence on a half-space with ℜ(*z*) ≪ 0.\n[thm:compactificationlimit] Let (*K**n*)*n* ∈ N be a sequence of compacta in *X* with $K\\_n\\nearrow X$ and ${\\ensuremath{\\left}}(\\frac{\\langle 1\\_{K\\_n^T}\\Omega{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}{\\langle 1\\_{K\\_n^T}{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}{\\ensuremath{\\right}})\\_{n\\in{\\ensuremath{\\mathbb{N}}}}$ locally bounded in *C*(*D*) where *D* ⊆ C is open and connected containing a half-space with ℜ(*z*) ≪ 0. Then, ${\\ensuremath{\\left}}(\\frac{\\langle 1\\_{K\\_n^T}\\Omega{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}{\\langle 1\\_{K\\_n^T}{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}{\\ensuremath{\\right}})\\_{n\\in{\\ensuremath{\\mathbb{N}}}}$ converges compactly to the analytic continuation of $\\frac{\\langle\\Omega{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}{\\langle{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}$ on *D*. In particular, if 0 ∈ *D*, we obtain\n \n$$\\begin{aligned}\n \\lim\\_{n\\to\\infty}\\langle 1\\_{K\\_n^T}\\Omega\\rangle\\_\\zeta=\\frac{\\langle\\Omega{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}{\\langle{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}(0)=\\langle \\Omega\\rangle\\_\\zeta.\n \\end{aligned}$$ \nfollows directly from Vitali’s Theorem since the set of pointwise convergence contains a half-space with ℜ(*z*) ≪ 0 and we have asserted local boundedness.\nThe local boundedness assumption in means that for every *z*0 in the domain *D*, there exists a neighborhood *U* ⊆ *D* of *z*0 such that the sequence of functions $\\frac{\\langle 1\\_{K\\_n^T}\\Omega{\\ensuremath{\\mathfrak{g}}}(z)\\rangle\\_L}{\\langle 1\\_{K\\_n^T}{\\ensuremath{\\mathfrak{g}}}(z)\\rangle\\_L}$ with *z* ∈ *U* is uniformly bounded as $K\\_n\\nearrow X$, that is,\n \n$$\\begin{aligned}\n \\sup\\_{n\\in{\\ensuremath{\\mathbb{N}}}}\\ \\sup\\_{z\\in U}\\ {{\\left\\lvert}{\\frac{\\langle 1\\_{K\\_n^T}\\Omega{\\ensuremath{\\mathfrak{g}}}(z)\\rangle\\_L}{\\langle 1\\_{K\\_n^T}{\\ensuremath{\\mathfrak{g}}}(z)\\rangle\\_L}}{\\right\\lvert}}<\\infty.\n \\end{aligned}$$ \nFrom a physical point of view, this means that any finite volume effects arising from the restriction to *K**n* remain bounded under small variation of the gauge parameter *z*. In this sense, the local boundedness assumption requires the physical theory to have locally bounded finite volume effects if the volume is sent to infinity and the observable is subject to holomorphic deformation. In particular, then implies that locally bounded finite volume effects already imply vanishing finite volume effects in the infinite volume limit.\nSince the *ζ*-regularization requires compactification of *X* and ensures that we can reconstruct the non-compact case from compact restrictions, we will from now on only consider the case in which *X* is compact.\nFor quantum mechanics in R*N*, we can generally consider *X* to be the flat torus (R/*L*Z)*N* of side length *L* ∈ R> 0. In this case the limit $K\\nearrow {\\ensuremath{\\mathbb{R}}}^N$ corresponds to $L\\nearrow\\infty$.\nReturning to the transfer matrix *U*Δ on compact *X*, we observe that the kernel\n \n$$\\begin{aligned}\n \\kappa\\_{U\\_\\Delta^T}(\\Phi\\_{T+1},\\Phi\\_1)=\\int\\_{X^{T-1}}\\prod\\_{j=1}^Te^{iS\\_\\Delta(\\Phi\\_{j+1},\\Phi\\_j)}d\\mu^{T-1}(\\Phi\\_2,\\ldots,\\Phi\\_T)\\end{aligned}$$ \nof *U*Δ*T* is integrable along the diagonal which implies that the nuclear trace\n \n$$\\begin{aligned}\n {\\ensuremath{{\\operatorname}{tr}}}(U\\_\\Delta^T)=\\langle\\kappa\\_{U\\_\\Delta^T},\\delta\\_{{\\ensuremath{\\mathrm{diag}}}}\\rangle=\\int\\_X\\kappa\\_{U\\_\\Delta^T}(\\Phi,\\Phi)d\\mu(\\Phi)\\end{aligned}$$ \nof *U*Δ*T* is well-defined. Introducing an observable Ω ∈ *L*∞(*X*) and gauge g, we thus obtain (using *F* ∈ {1, Ω}) that\n \n$$\\begin{aligned}\n \\langle\\kappa\\_{U\\_\\Delta^T}F{\\ensuremath{\\mathfrak{g}}}(z),\\delta\\_{\\ensuremath{\\mathrm{diag}}}\\rangle=&\\int\\_X\\kappa\\_{U\\_\\Delta^T}(\\Phi,\\Phi)F(\\Phi){\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\mu(\\Phi)\\end{aligned}$$ \nis a holomorphic function in *z* which directly implies.\n[thm:analytic continuation] If *X* is compact, then\n \n$$\\begin{aligned}\n \\langle \\Omega\\rangle\\_L=\\frac{\\int\\_{X^T}\\Omega(\\Phi)e^{iS(\\Phi)}d\\mu^T(\\Phi)}{\\int\\_{X^T}e^{iS(\\Phi)}d\\mu^T(\\Phi)}=\\langle\\Omega\\rangle\\_\\zeta.\n \\end{aligned}$$ \nFurthermore, if *X* is not compact, the assumptions of are satisfied, and (*K**n*)*n* ∈ N is as in, then\n \n$$\\begin{aligned}\n \\lim\\_{n\\to\\infty}\\frac{\\int\\_{K\\_n^T}\\Omega(\\Phi)e^{iS(\\Phi)}d\\mu^T(\\Phi)}{\\int\\_{K\\_n^T}e^{iS(\\Phi)}d\\mu^T(\\Phi)}=\\langle \\Omega\\rangle\\_\\zeta.\n \\end{aligned}$$ \nGauging the transfer matrix and time evolution operator\n=======================================================\nIn, we have discussed how the *ζ*-regularization can make sense of the formal integrals arising in a Lorentzian lattice theory through a process of gauging. However, the gauging process as discussed in can be understood as an operation on the observable. More precisely, as was shown in , the entire construction of *ζ*-regularized vacuum expectation values is equivalent to the construction of an analytic continuation of a quotient $\\frac{\\langle O\\_1(z)\\rangle}{\\langle O\\_2(z)\\rangle}$ of expectation values ⟨*O**j*(*z*)⟩ for suitably constructed observables *O**j*(*z*). In order to make the connection to the general *ζ*-regularization theory of vacuum expectation values , on the other hand, we need to consider the gauge as part of the time evolution operator. Furthermore, for the *ζ*-regularization point of view to be as versatile as methods used in Euclidean lattice theories, it would be advantageous to find gauges that can be attributed to the transfer matrix as opposed to the entire time evolution operator only. In this section, we will therefore discuss the relation between the point of view in and the formalism of . As such, we will be able to conclude that all properties proven in also hold for the *ζ*-regularization of lattice field theories. In particular, we will obtain that the *ζ*-regularized lattice vacuum expectation values are independent of the choice of gauge g.\nThe transfer matrix *U*Δ is given by its kernel *κ*Δ(Φʹ, Φ) = *e**i**S*Δ(Φʹ, Φ), i.e.,\n \n$$\\begin{aligned}\n {\\forall}{\\varphi},\\psi\\in C\\_c^\\infty(X):\\ \\langle{\\varphi},U\\_\\Delta\\psi\\rangle=\\int\\_X\\int\\_X{\\varphi}(\\Phi')^\\*e^{iS\\_\\Delta(\\Phi',\\Phi)}\\psi(\\Phi)d\\mu(\\Phi')d\\mu(\\Phi),\\end{aligned}$$ \nand its adjoint *U*Δ\\* has kernel *κ**U*Δ\\*(Φʹ, Φ) = *κ*Δ(Φ, Φʹ)\\* = *e*− *i**S*Δ(Φ, Φʹ). Since the transfer matrix connects neighboring values Φ*j* and Φ*j* + 1 in terms of the global action *e**i**S*(Φ) = ∏*j* = 1*T**e**i**S*Δ(Φ*j* + 1, Φ*j*), it can often be advantageous to mirror this structure in the choice of gauge g. Introducing a gauge g(*z*)(Φ) = ∏*j* = 1*T*gΔ(*z*)(Φ*j* + 1, Φ*j*), we obtain the gauged transfer matrix $\\tilde U\\_\\Delta$ defined as\n \n$$\\begin{aligned}\n {\\ensuremath{\\left}}(\\tilde U\\_\\Delta(z)\\psi{\\ensuremath{\\right}})(\\Phi')=\\int\\_Xe^{iS\\_\\Delta(\\Phi',\\Phi)}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z)(\\Phi',\\Phi)\\psi(\\Phi)d\\mu(\\Phi).\\end{aligned}$$ \nThis directly yields the gauge family GΔ in the operator point of view via\n \n$$\\begin{aligned}\n {\\ensuremath{\\left}}({\\ensuremath{\\mathfrak{G}}}\\_\\Delta(z)\\psi{\\ensuremath{\\right}})(\\Phi'')=&{\\ensuremath{\\left}}(U\\_\\Delta^\\*\\tilde U\\_\\Delta(z)\\psi{\\ensuremath{\\right}})(\\Phi'')\\\\\n =&\\int\\_X\\int\\_Xe^{-iS\\_\\Delta(\\Phi',\\Phi'')}e^{iS\\_\\Delta(\\Phi',\\Phi)}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z)(\\Phi',\\Phi)\\psi(\\Phi)d\\mu(\\Phi)d\\mu(\\Phi')\\end{aligned}$$ \nand thus $\\tilde U\\_\\Delta=U\\_\\Delta{\\ensuremath{\\mathfrak{G}}}\\_\\Delta$. In other words, GΔ(*z*) is an integral operator with kernel\n \n$$\\begin{aligned}\n \\kappa\\_{{\\ensuremath{\\mathfrak{G}}}\\_\\Delta(z)}(\\Phi',\\Phi)=\\int\\_Xe^{-iS\\_\\Delta(y,\\Phi')}e^{iS\\_\\Delta(y,\\Phi)}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z)(y,\\Phi)d\\mu(y).\\end{aligned}$$ \n[thm-gauged-transfer-matrix] Let *U* : = *U*Δ*T* be the time evolution operator defined by a transfer matrix *U*Δ corresponding to an action *S*Δ, g(*z*)(Φ) = ∏*j* = 1*T*gΔ(*z*)(Φ*j* + 1, Φ*j*), GΔ(*z*) the gauge family with the kernel\n \n$$\\begin{aligned}\n \\kappa\\_{{\\ensuremath{\\mathfrak{G}}}\\_\\Delta(z)}(\\Phi',\\Phi)=\\int\\_Xe^{-iS\\_\\Delta(y,\\Phi')}e^{iS\\_\\Delta(y,\\Phi)}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z)(y,\\Phi)d\\mu(y),\n \\end{aligned}$$ \nand *ω* an observable with corresponding operator Ω. Then,\n \n$$\\begin{aligned}\n \\langle\\omega\\rangle(z)=\\frac{\\int\\_{X^T}\\omega(\\Phi)e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\mu^T(\\Phi)}{\\int\\_{X^T}e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\mu^T(\\Phi)}\n \\end{aligned}$$ \nis the *ζ*-regularized lattice vacuum expectation value which, in the operator formulation, corresponds to the *ζ*-regularized vacuum expectation value\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=\\frac{\\zeta{\\ensuremath{\\left}}((U\\_\\Delta{\\ensuremath{\\mathfrak{G}}}\\_\\Delta)^T\\Omega{\\ensuremath{\\right}})(z)}{\\zeta{\\ensuremath{\\left}}((U\\_\\Delta{\\ensuremath{\\mathfrak{G}}}\\_\\Delta)^T{\\ensuremath{\\right}})(z)}.\n \\end{aligned}$$ \nSince (*U*ΔGΔ)*T* is a gauge of *U* (more precisely, (*U*ΔGΔ)*T* = *U*G using the “global” gauge family G = *U*\\*(*U*ΔGΔ)*T*), we obtain gauge independence of ⟨*ω*⟩(0) in the sense of . In particular, if *U*Δ and GΔ commute, then G = (GΔ)*T*.\nWe note that the assumptions of permit gauge families g of the form ${\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)={\\ensuremath{\\left}}(\\prod\\_{j=1}^T{{\\left\\lvert}{\\Phi\\_j}{\\right\\lvert}}{\\ensuremath{\\right}})^{\\frac{z}{T}}$ and ${\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)={\\ensuremath{\\left}}(\\prod\\_{j=1}^T{{\\left\\lvert}{\\Phi\\_{j+1}-\\Phi\\_j}{\\right\\lvert}}{\\ensuremath{\\right}})^{\\frac{z}{T}}$ but not ${\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)={{\\left\\lVert}{\\Phi}{\\right\\lVert}}^z$. The latter is a gauge for *U* = *U*Δ*T* rather than *U*Δ itself. The kernel *κ**U* of *U* is given by\n \n$$\\begin{aligned}\n \\kappa\\_U(\\Phi\\_{T+1},\\Phi\\_1)=\\int\\_{X^{T-1}}e^{iS(\\Phi\\_{T+1},\\ldots,\\Phi\\_1)}d\\mu^{T-1}(\\Phi\\_T,\\ldots,\\Phi\\_2).\\end{aligned}$$ \nHere, we do not identify Φ*T* + 1 with Φ1 yet, since that step is eventually part of taking the trace. To reduce notation, we will introduce $\\hat\\Phi:=(\\Phi\\_T,\\ldots,\\Phi\\_2)$ and $\\hat\\mu:=\\mu^{T-1}$. Then, the action *S* is a function of $(\\Phi\\_{T+1},\\hat\\Phi,\\Phi\\_1)$ due to the open boundary conditions and the gauge family g is a family of functions of $(\\hat\\Phi,\\Phi\\_1)$. Furthermore, *U*\\* has kernel\n \n$$\\begin{aligned}\n \\kappa\\_{U^\\*}(x,y)=\\kappa\\_U(y,x)^\\*=\\int\\_{X^{T-1}}e^{-iS(y,\\hat\\Phi,x)}d\\hat\\mu(\\hat\\Phi).\\end{aligned}$$ \nHence, the gauged time evolution $\\tilde U$ with kernel\n \n$$\\begin{aligned}\n \\kappa\\_{\\tilde U(z)}(\\Phi\\_{T+1},\\Phi\\_1)=\\int\\_{X^{T-1}}e^{iS(\\Phi\\_{T+1},\\hat\\Phi,\\Phi\\_1)}{\\ensuremath{\\mathfrak{g}}}(z)(\\hat\\Phi,\\Phi\\_1)d\\hat\\mu(\\hat\\Phi)\\end{aligned}$$ \ncan be expressed as $\\tilde U=U{\\ensuremath{\\mathfrak{G}}}$ with ${\\ensuremath{\\mathfrak{G}}}=U^\\*\\tilde U$, i.e., G(*z*) has kernel\n \n$$\\begin{aligned}\n \\kappa\\_{{\\ensuremath{\\mathfrak{G}}}(z)}(x,y)=&\\int\\_X\\kappa\\_{U^\\*}(x,w)\\kappa\\_{\\tilde U(z)}(w,y)d\\mu(w)\\\\\n =&\\int\\_X\\int\\_{X^{T-1}}e^{-iS(w,\\hat\\Phi,x)}d\\hat\\mu(\\hat\\Phi)\\int\\_{X^{T-1}}e^{iS(w,\\hat\\Psi,y)}{\\ensuremath{\\mathfrak{g}}}(z)(\\hat\\Psi,y)d\\hat\\mu(\\hat\\Psi)d\\mu(w).\\end{aligned}$$ \nThis directly implies which is the extension of to gauges g that cannot be split into gauges of the transfer matrix.\n[thm-gauged-time-evolution] Let *U* be the time evolution operator corresponding to an action *S*, g(*z*) a family of gauge functions, G(*z*) the gauge family with the kernel\n \n$$\\begin{aligned}\n \\kappa\\_{{\\ensuremath{\\mathfrak{G}}}(z)}(x,y)=\\int\\_X\\int\\_{X^{T-1}}\\int\\_{X^{T-1}}e^{-iS(w,\\hat\\Phi,x)}e^{iS(w,\\hat\\Psi,y)}{\\ensuremath{\\mathfrak{g}}}(z)(\\hat\\Psi,y)d\\hat\\mu(\\hat\\Phi)d\\hat\\mu(\\hat\\Psi)d\\mu(w),\n \\end{aligned}$$ \nand *ω* an observable with corresponding operator Ω. Then,\n \n$$\\begin{aligned}\n \\langle\\omega\\rangle(z)=\\frac{\\int\\_{X^T}\\omega(\\Phi)e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\mu^T(\\Phi)}{\\int\\_{X^T}e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\mu^T(\\Phi)}\n \\end{aligned}$$ \nis the *ζ*-regularized lattice vacuum expectation value which, in the operator formulation, corresponds to the *ζ*-regularized vacuum expectation value\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=\\frac{\\zeta{\\ensuremath{\\left}}(U{\\ensuremath{\\mathfrak{G}}}\\Omega{\\ensuremath{\\right}})(z)}{\\zeta{\\ensuremath{\\left}}(U{\\ensuremath{\\mathfrak{G}}}{\\ensuremath{\\right}})(z)}.\n \\end{aligned}$$ \nIn particular, ⟨*ω*⟩(0) is independent of the gauge g in the sense of .\nThis identifies sufficient conditions on the choice of gauge gΔ. Primarily, we need to ensure that G(*z*) is a Fourier integral operator of order *δ*ℜ(*z*) for some for the construction of *ζ*-regularized vacuum expectation values to make sense in the operator picture . A sufficient condition for this to be true is for g(*z*) to be in the Hörmander class *S**δ*ℜ(*z*) , that is, ${{\\left\\lvert}{{\\partial}\\_\\Phi^\\alpha{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)}{\\right\\lvert}}\\le c\\_\\alpha(1+{{\\left\\lVert}{\\Phi}{\\right\\lVert}})^{\\delta\\Re(z)-{{\\left\\lvert}{\\alpha}{\\right\\lvert}}}$ has to hold for all Φ, *z*, multiindices *α*, and some constant *c**α*. This condition can also be weakened by assuming that the inequality holds asymptotically for ${{\\left\\lVert}{\\Phi}{\\right\\lVert}}\\to\\infty$ . In doing so, we can allow for homogeneous singularities at Φ = 0, such as, ${\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)={{\\left\\lVert}{\\Phi}{\\right\\lVert}}^z$ or even consider g(*z*) to be polyhomogeneous g(*z*) ∼ ∑*ι* ∈ *I**a**ι*(*z*)(Φ) where *a**ι*(*z*) is homogeneous of degree *d**ι* + *z* and the ℜ(*d**ι*) are bounded from above .\nThe classical limit\n===================\nIn the construction of *ζ*-regularized vacuum expectation values, we consider a quotient of *ζ*-functions $\\frac{\\zeta(U{\\ensuremath{\\mathfrak{G}}}\\Omega)}{\\zeta(U{\\ensuremath{\\mathfrak{G}}})}$ which in the lattice setting takes the form $\\frac{\\int e^{iS}{\\ensuremath{\\mathfrak{g}}}\\omega d\\mu}{\\int e^{iS}{\\ensuremath{\\mathfrak{g}}}d\\mu}$. Although the general theory of operator *ζ*-functions would allow for different choices of G (and thus different choices of g) in the numerator and denominator, the definition of *ζ*-regularized vacuum expectation values implicitly requires us to make the same choice. This was introduced to handle the problem that $\\frac{\\zeta(U{\\ensuremath{\\mathfrak{G}}}\\_1\\Omega)}{\\zeta(U{\\ensuremath{\\mathfrak{G}}}\\_2)}$ can depend on the choice of G1 and G2 if *ζ*(*U*G1Ω)(0) = *ζ*(*U*G2)(0) = 0. While enforcing G1 = G2 = G does not obviously solve the problem of $\\frac{\\zeta(U{\\ensuremath{\\mathfrak{G}}}\\Omega)}{\\zeta(U{\\ensuremath{\\mathfrak{G}}})}(0)$ being dependent on G if *ζ*(*U*GΩ)(0) = *ζ*(*U*G)(0) = 0 – in fact, independence of G is a consequence of the physicality proof for *ζ*-regularized vacuum expectation values – there is a physical reason to hope for this to be the correct requirement. In a very loose sense, we can interpret the denominator *ζ*(*U*G)(*z*) to be the partition function of a quantum field theory QFT(*z*) with “time evolution” *U*G(*z*). For ℜ(*z*) ≪ 0 the vacuum expectation ⟨Ω⟩(*z*) of Ω is then well-defined in QFT(*z*) using Feynman’s construction and satisfies $\\langle\\Omega\\rangle(z)=\\frac{\\zeta(U{\\ensuremath{\\mathfrak{G}}}\\Omega)(z)}{\\zeta(U{\\ensuremath{\\mathfrak{G}}})(z)}$. The *ζ*-regularized vacuum expectation value ⟨Ω⟩(0) can therefore be loosely understood as the analytic continuation of well-defined vacuum expectation values in Feynman’s sense with respect to the holomorphic family of quantum field theories QFT.\nOf course, this interpretation requires many philosophical concessions because in the lattice picture, QFT(*z*) corresponds to the “action” *S*(*z*) = *S*(0) − *i*ln(g(*z*)) where *S*(0) is the action of QFT(0), i.e., the action of the QFT we wish to study. However, if we accept this interpretation, then we can ask the question regarding the classical limit. In general, interpreting *S*(*z*) as a classical action is likely to be difficult due to − *i*ln(g(*z*)) being complex. The closest classical action we can extract from *S*(*z*) is the original action *S*(0) which leads to the hypothesis that the classical limit should always be the classical limit of QFT(0) independent of the value of *z*. is precisely this surprising result.\n[thm:classical-limit] Let *S* : R*N* → R be a non-degenerate action with a unique minimum, that is, there is exactly one point *C* ∈ R*N* with *S*ʹ(*C*) = 0 which furthermore satisfies det*S*ʺ(*C*) ≠ 0. Let *ω* : R*N* → R be an observable and g(*z*) a gauge function with ∀*z* ∈ C : g(*z*)(*C*) ≠ 0. Then,\n \n$$\\begin{aligned}\n {\\forall}z\\in{\\ensuremath{\\mathbb{C}}}:\\ \\lim\\_{\\hbar\\searrow0}\\langle\\omega\\rangle(z)=\\lim\\_{\\hbar\\searrow0}\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^N}e^{\\frac{i}{\\hbar}S(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)\\omega(\\Phi)d\\Phi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^N}e^{\\frac{i}{\\hbar}S(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\Phi}=\\omega(C).\n \\end{aligned}$$ \nIn particular, $\\lim\\_{\\hbar\\searrow0}\\langle\\omega\\rangle(0)=\\omega(C)$\nSince we are interested in the classical limit $\\hbar\\searrow0$, we will expand the integrals in\n \n$$\\begin{aligned}\n \\langle\\omega\\rangle(z)=\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^N}e^{\\frac{i}{\\hbar}S(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)\\omega(\\Phi)d\\Phi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^N}e^{\\frac{i}{\\hbar}S(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\Phi}.\n \\end{aligned}$$ \nusing stationary phase approximation and obtain\n \n$$\\begin{aligned}\n \\langle\\omega\\rangle(z)=&\\frac{e^{\\frac{i}{\\hbar}S(C)}{{\\left\\lvert}{\\det S''(C)}{\\right\\lvert}}^{-\\frac12}e^{\\frac{i\\pi}{4}{\\ensuremath{\\mathrm{sgn}}}S''(C)}{\\ensuremath{\\mathfrak{g}}}(z)(C)\\omega(C){\\ensuremath{\\left}}(2\\pi\\hbar{\\ensuremath{\\right}})^{\\frac{N}{2}}+o{\\ensuremath{\\left}}(\\hbar^{\\frac{N}{2}}{\\ensuremath{\\right}})}{e^{\\frac{i}{\\hbar}S(C)}{{\\left\\lvert}{\\det S''(C)}{\\right\\lvert}}^{-\\frac12}e^{\\frac{i\\pi}{4}{\\ensuremath{\\mathrm{sgn}}}S''(C)}{\\ensuremath{\\mathfrak{g}}}(z)(C){\\ensuremath{\\left}}(2\\pi\\hbar{\\ensuremath{\\right}})^{\\frac{N}{2}}+o{\\ensuremath{\\left}}(\\hbar^{\\frac{N}{2}}{\\ensuremath{\\right}})}\\\\\n =&\\frac{\\omega(C)+o(1)}{1+o(1)}.\n \\end{aligned}$$ \nHence, we observe\n \n$$\\begin{aligned}\n {\\forall}z\\in{\\ensuremath{\\mathbb{C}}}:\\ \\lim\\_{\\hbar\\searrow0}\\langle\\omega\\rangle(z)=\\omega(C).\n \\end{aligned}$$ \nConsider the harmonic oscillator on the lattice with action\n \n$$\\begin{aligned}\n S:\\ {\\ensuremath{\\mathbb{R}}}^\\tau\\to{\\ensuremath{\\mathbb{R}}};\\ x\\mapsto a\\sum\\_{j=0}^{\\tau-1}\\frac{m}{2}\\frac{(x\\_{j+1}-x\\_j)^2}{a^2}+\\mu^2x\\_j^2.\n \\end{aligned}$$ \nThen\n \n$$\\begin{aligned}\n {\\partial}\\_kS(x)=&m\\frac{x\\_{k}-x\\_{k-1}}{a}-m\\frac{x\\_{k+1}-x\\_k}{a}+2a\\mu^2x\\_k\\\\\n =&m\\frac{2x\\_{k}-x\\_{k-1}-x\\_{k+1}}{a}+2a\\mu^2x\\_k\\\\\n =&(Mx)\\_k\n \\end{aligned}$$ \nwith $M\\_{j,k}={\\ensuremath{\\left}}(\\frac{2m}{a}+2a\\mu^2{\\ensuremath{\\right}})\\delta\\_{j,k}-\\frac{m}{a}\\delta\\_{j+1,k}-\\frac{m}{a}\\delta\\_{j-1,k}$. Thus, *M* is a circulant matrix and setting Σ0 = (*δ**j* + 1, *k*)*j*, *k* implies $M=\\frac{2m}{a}+2a\\mu^2-\\frac{m}{a}\\Sigma\\_0-\\frac{m}{a}\\Sigma\\_0^{\\tau-1}=p(\\Sigma\\_0)$. Since the eigenvalues of Σ0 are $\\lambda\\_k=e^{\\frac{2\\pi ik}{\\tau}}$, the eigenvalues of *M* are\n \n$$\\begin{aligned}\n p(\\lambda\\_k)=\\frac{2m}{a}+2a\\mu^2-\\frac{m}{a}e^{\\frac{2\\pi ik}{\\tau}}-\\frac{m}{a}e^{-\\frac{2\\pi ik}{\\tau}}=2a\\mu^2+\\frac{2m}{a}{\\ensuremath{\\left}}(1-\\cos{\\ensuremath{\\left}}(\\frac{2\\pi k}{\\tau}{\\ensuremath{\\right}}){\\ensuremath{\\right}}).\n \\end{aligned}$$ \nHence, *S*ʹ(*x*) = 0 fails to have a unique solution (*C* = 0) if and only if ∃*k* : *p*(*λ**k*) = 0. The observation *p*(*λ**k*) ≥ 2*a**μ*2 > 0 therefore implies that *C* = 0 is the unique critical point, and we furthermore observe *S*ʺ(*C*) = *M* ∈ *G**L*(*N*). Thus, the classical limit $\\hbar\\searrow0$ of ⟨*ω*⟩(*z*) is the observable *ω* evaluated on the classical vacuum *C* = 0.\nThe proof of can be extended to the case of multiple critical points *C* with *S*ʹ(*C*) = 0 and det*S*ʺ(*C*) ≠ 0 such as in the case of a double well potential. Let Γ be the set of critical points and *ω* and g as in. Then, stationary phase approximation yields\n \n$$\\begin{aligned}\n \\langle\\omega\\rangle(z)=&\\frac{\\sum\\_{C\\in\\Gamma}e^{\\frac{i}{\\hbar}S(C)}{{\\left\\lvert}{\\det S''(C)}{\\right\\lvert}}^{-\\frac12}e^{\\frac{i\\pi}{4}{\\ensuremath{\\mathrm{sgn}}}S''(C)}{\\ensuremath{\\mathfrak{g}}}(z)(C)\\omega(C)+o{\\ensuremath{\\left}}(1{\\ensuremath{\\right}})}{\\sum\\_{C\\in\\Gamma}e^{\\frac{i}{\\hbar}S(C)}{{\\left\\lvert}{\\det S''(C)}{\\right\\lvert}}^{-\\frac12}e^{\\frac{i\\pi}{4}{\\ensuremath{\\mathrm{sgn}}}S''(C)}{\\ensuremath{\\mathfrak{g}}}(z)(C)+o{\\ensuremath{\\left}}(1{\\ensuremath{\\right}})}\n \\end{aligned}$$ \nand thus\n \n$$\\begin{aligned}\n \\langle\\omega\\rangle(0)=&\\frac{\\sum\\_{C\\in\\Gamma}e^{\\frac{i}{\\hbar}S(C)}{{\\left\\lvert}{\\det S''(C)}{\\right\\lvert}}^{-\\frac12}e^{\\frac{i\\pi}{4}{\\ensuremath{\\mathrm{sgn}}}S''(C)}\\omega(C)+o{\\ensuremath{\\left}}(1{\\ensuremath{\\right}})}{\\sum\\_{C\\in\\Gamma}e^{\\frac{i}{\\hbar}S(C)}{{\\left\\lvert}{\\det S''(C)}{\\right\\lvert}}^{-\\frac12}e^{\\frac{i\\pi}{4}{\\ensuremath{\\mathrm{sgn}}}S''(C)}+o{\\ensuremath{\\left}}(1{\\ensuremath{\\right}})}.\n \\end{aligned}$$ \nIn the case of the double well potential, where we have exactly two critical points *C*1 and *C*2 with *S*(*C*1) = *S*(*C*2) and *S*ʺ(*C*1) = *S*ʺ(*C*2), we can further simplify to\n \n$$\\begin{aligned}\n \\langle\\omega\\rangle(z)=&\\frac{{\\ensuremath{\\mathfrak{g}}}(z)(C\\_1)\\omega(C\\_1)+{\\ensuremath{\\mathfrak{g}}}(z)(C\\_2)\\omega(C\\_2)+o{\\ensuremath{\\left}}(1{\\ensuremath{\\right}})}{{\\ensuremath{\\mathfrak{g}}}(z)(C\\_1)+{\\ensuremath{\\mathfrak{g}}}(z)(C\\_2)+o{\\ensuremath{\\left}}(1{\\ensuremath{\\right}})},\\\\\n \\langle\\omega\\rangle(0)=&\\frac{\\omega(C\\_1)+\\omega(C\\_2)+o{\\ensuremath{\\left}}(1{\\ensuremath{\\right}})}{2+o{\\ensuremath{\\left}}(1{\\ensuremath{\\right}})},\n \\end{aligned}$$ \nand thus\n \n$$\\begin{aligned}\n \\lim\\_{\\hbar\\to0}\\langle\\omega\\rangle(0)=&\\frac{\\omega(C\\_1)+\\omega(C\\_2)}{2}.\n \\end{aligned}$$ \nThe harmonic oscillator: four gauges and a Wick rotation\n========================================================\nIn this section we will consider the harmonic oscillator and analytically compute the ground state energy with five different approaches. First (in ), we will compute it using a Wick rotation; thus reproducing well-known results which will serve as a basis for comparison with the four different choices of gauge g. These four choices of gauge arise from two binary choices. The first choice is whether we wish our gauge to be local in time or global, i.e., whether or not it is possible to split the gauge function g(*z*)(Φ) into a product ∏*j*gΔ(*z*)(Φ*j* + 1, Φ*j*) and thus gauging the transfer matrix *U*Δ. In applications, this choice will likely be a matter of convenience. The more important binary decision is related to the behavior of the gauge near zero. Since the regularization is introduced through the asymptotic behavior of the gauge for large ${{\\left\\lVert}{\\Phi}{\\right\\lVert}}$, there is a considerable degree of freedom in any fixed compact set. In particular, the functional analysis of a *ζ*-lattice field theory is easier if the gauge is of the form ${\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)={{\\left\\lVert}{\\Phi}{\\right\\lVert}}^z$ but this introduces a pole in zero. The corresponding integrals are still well-defined in terms of homogeneous distributions but not absolutely integrable. Thus, for numerical purposes, absolutely integrable gauges like ${\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)=(1+{{\\left\\lVert}{\\Phi}{\\right\\lVert}})^z$ are preferable. In,,, and we will consider each of these cases (cf., ).\n[table:gauges]\n|c|c|c|c| & \n &distributional&abs. integrable \n&&& \n&global in time&& \n&&& \n &&& \n&local in time&& \n&&& \nIn our treatment of the harmonic oscillator, we will absorb ℏ into the action\n \n$$\\begin{aligned}\n S:\\ {\\ensuremath{\\mathbb{R}}}^\\tau\\to{\\ensuremath{\\mathbb{R}}};\\ \\Phi\\mapsto \\frac{a}{\\hbar}\\sum\\_{j=0}^{\\tau-1}\\frac{m}{2}\\frac{(\\Phi\\_{j+1}-\\Phi\\_j)^2}{a^2}+\\frac{m\\omega^2}{2}\\Phi\\_j^2.\\end{aligned}$$ \nGiven an observable Ω and a gauge g, we hence want to solve\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)\\Omega(\\Phi)d\\Phi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\Phi}\\end{aligned}$$ \nin the *ζ*-regularized cases and\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle\\_E=\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{-S(\\Phi)}\\Omega(\\Phi)d\\Phi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{-S(\\Phi)}d\\Phi}\\end{aligned}$$ \nin the Wick rotated case. For the calculations in this section, we will consider observables Ω of the form\n \n$$\\begin{aligned}\n \\Phi^\\alpha=\\frac{1}{\\tau}\\sum\\_{j=0}^{\\tau-1}\\Phi\\_j^\\alpha.\\end{aligned}$$ \nMany interesting observables are related to these Φ*α*. For instance, the ground state energy is given by *i**E*0 = *m**ω*2⟨Φ2⟩(0) or *E*0 = *m**ω*2⟨Φ2⟩*E* where we have lost a factor of *i* due to the Wick rotation.\nWe will compute the continuum limit $E\\_0^{\\mathrm{continuum}}=\\frac{\\hbar\\omega}{2}$ in the Wick rotated case explicitly. For all other cases, we will show that the ground state energy *E*0 as computed with the *ζ*-regularized lattice theory coincides with *E*0 as computed with the Wick rotated lattice theory. To achieve this, we will define the matrix\n \n$$\\begin{aligned}\n \\Sigma:=\\frac{1}{\\hbar}{\\ensuremath{\\left}}({\\ensuremath{\\left}}(\\frac{m}{a}+\\frac{am\\omega^2}{2}{\\ensuremath{\\right}})\\delta\\_{j,k}-\\frac{m}{2a}(\\delta\\_{j,k+1}+\\delta\\_{j+1,k}){\\ensuremath{\\right}})\\_{j,k}\\end{aligned}$$ \nbecause\n \n$$\\begin{aligned}\n S(\\Phi)=\\langle\\Phi,\\Sigma\\Phi\\rangle.\\end{aligned}$$ \nΣ is a circulant matrix and setting Σ0 = (*δ**j* + 1, *k*)*j*, *k* implies\n \n$$\\begin{aligned}\n \\Sigma=\\frac{m}{a\\hbar}+\\frac{am\\omega^2}{2\\hbar}-\\frac{m}{2a\\hbar}\\Sigma\\_0-\\frac{m}{2a\\hbar}\\Sigma\\_0^{\\tau-1}=p(\\Sigma\\_0).\\end{aligned}$$ \nSince the eigenvalues of Σ0 are $\\mu\\_k=e^{\\frac{2\\pi ik}{\\tau}}$, the eigenvalues of Σ are\n \n$$\\begin{aligned}\n p(\\mu\\_k)=\\frac{m}{a\\hbar}+\\frac{am\\omega^2}{2\\hbar}-\\frac{m}{2a\\hbar}e^{\\frac{2\\pi ik}{\\tau}}-\\frac{m}{2a\\hbar}e^{-\\frac{2\\pi ik}{\\tau}}=\\frac{am\\omega^2}{2\\hbar}+\\frac{m}{a\\hbar}{\\ensuremath{\\left}}(1-\\cos{\\ensuremath{\\left}}(\\frac{2\\pi k}{\\tau}{\\ensuremath{\\right}}){\\ensuremath{\\right}}).\\end{aligned}$$ \nHence, Σ is strictly positive and\n \n$$\\begin{aligned}\n S(\\Phi)={{\\left\\lVert}{\\sqrt\\Sigma\\Phi}{\\right\\lVert}}^2.\\end{aligned}$$ \nIn other words, the change of coordinates $\\Psi:=\\sqrt\\Sigma\\Phi$ implies\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{i{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d\\Psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{i{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d\\Psi}\\\\\n =&\\frac{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{ir^2}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(r\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(r\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})r^{\\tau-1}drd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{ir^2}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(r\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})r^{\\tau-1}drd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}\\\\\n =&\\frac{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{is}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})s^{\\frac{\\tau}{2}-1}dsd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{is}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})s^{\\frac{\\tau}{2}-1}dsd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle\\_E=\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{-{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d\\Psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{-{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}d\\Psi}\\end{aligned}$$ \nwhere $\\sqrt\\Sigma^{-1}$ has the eigenvalues\n \n$$\\begin{aligned}\n \\lambda\\_k={\\ensuremath{\\left}}(\\frac{am\\omega^2}{2\\hbar}+\\frac{m}{a\\hbar}{\\ensuremath{\\left}}(1-\\cos{\\ensuremath{\\left}}(\\frac{2\\pi k}{\\tau}{\\ensuremath{\\right}}){\\ensuremath{\\right}}){\\ensuremath{\\right}})^{-\\frac{1}{2}}.\\end{aligned}$$ \nThe Wick rotation\n-----------------\nWe can compute ⟨Ω⟩*E* by choosing an eigenbasis (*e**j*)*j* of $\\sqrt\\Sigma^{-1}$ such that Ψ = ∑*j*Ψ*j**e**j* and $\\sqrt\\Sigma^{-1}e\\_j=\\lambda\\_je\\_j$. Then, we obtain\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle\\_E=&\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{-{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d\\Psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{-{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}d\\Psi}\\\\\n =&\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}\\prod\\_{j=0}^{\\tau-1}e^{-\\Psi\\_j^2}\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}(\\lambda\\_k\\Psi\\_k)^{\\alpha}d\\Psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}\\prod\\_{j=0}^{\\tau-1}e^{-\\Psi\\_j^2}d\\Psi}\\\\\n =&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}}e^{-\\Psi\\_k^2}(\\lambda\\_k\\Psi\\_k)^{\\alpha}d\\Psi\\_k}{\\int\\_{{\\ensuremath{\\mathbb{R}}}}e^{-\\Psi\\_k^2}d\\Psi\\_k}\\\\\n =&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^\\alpha\\frac{{\\ensuremath{\\left}}((-1)^\\alpha+1{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+1}{2}{\\ensuremath{\\right}})}{2\\Gamma{\\ensuremath{\\left}}(\\frac{1}{2}{\\ensuremath{\\right}})}.\\end{aligned}$$ \nFor *α* = 2, we conclude\n \n$$\\begin{aligned}\n \\langle\\Phi^2\\rangle\\_E=&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^2\\frac{\\Gamma{\\ensuremath{\\left}}(\\frac{1}{2}+1{\\ensuremath{\\right}})}{\\Gamma{\\ensuremath{\\left}}(\\frac{1}{2}{\\ensuremath{\\right}})}=\\frac{1}{2\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^2.\\end{aligned}$$ \nTo compute lim*τ* → ∞⟨Φ2⟩*E*, we use $\\lim\\_{\\tau\\to\\infty}\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}f{\\ensuremath{\\left}}(\\frac{2\\pi k}{\\tau}{\\ensuremath{\\right}})=\\frac{1}{2\\pi}\\int\\_0^{2\\pi}f(x)dx$ to obtain[2](#fn2)\n \n$$\\begin{aligned}\n \\lim\\_{\\tau\\to\\infty}\\langle\\Phi^2\\rangle\\_E=&\\frac{1}{4\\pi}\\int\\_0^{2\\pi}{\\ensuremath{\\left}}(\\frac{am\\omega^2}{2\\hbar}+\\frac{m}{a\\hbar}{\\ensuremath{\\left}}(1-\\cos{\\ensuremath{\\left}}(x{\\ensuremath{\\right}}){\\ensuremath{\\right}}){\\ensuremath{\\right}})^{-1}dx\\\\\n =&{\\ensuremath{\\left}}.\\frac{\\arctan{\\ensuremath{\\left}}(\\sqrt{\\frac{2m}{a\\hbar}\\frac{2\\hbar}{am\\omega^2}+1}\\tan{\\ensuremath{\\left}}(\\frac{x}{2}{\\ensuremath{\\right}}){\\ensuremath{\\right}})}{2\\pi\\sqrt{\\frac{am\\omega^2}{2\\hbar}{\\ensuremath{\\left}}(\\frac{am\\omega^2}{2\\hbar}+\\frac{2m}{a\\hbar}{\\ensuremath{\\right}})}}{\\ensuremath{\\right}}|\\_{x=0}^{\\pi}\\\\\n &+{\\ensuremath{\\left}}.\\frac{\\arctan{\\ensuremath{\\left}}(\\sqrt{\\frac{2m}{a\\hbar}\\frac{2\\hbar}{am\\omega^2}+1}\\tan{\\ensuremath{\\left}}(\\frac{x}{2}{\\ensuremath{\\right}}){\\ensuremath{\\right}})}{2\\pi\\sqrt{\\frac{am\\omega^2}{2\\hbar}{\\ensuremath{\\left}}(\\frac{am\\omega^2}{2\\hbar}+\\frac{2m}{a\\hbar}{\\ensuremath{\\right}})}}{\\ensuremath{\\right}}|\\_{x=\\pi}^{2\\pi}\\\\\n =&\\frac{\\hbar}{2m\\omega\\sqrt{\\frac{a^2\\omega^2}{4}+1}}\\end{aligned}$$ \nand thus\n \n$$\\begin{aligned}\n E\\_0^{\\mathrm{continuum}}=\\lim\\_{a\\to0}\\lim\\_{\\tau\\to\\infty}m\\omega^2\\langle\\Phi^2\\rangle\\_E=&\\frac{\\hbar\\omega}{2}.\\end{aligned}$$ \nFor the *ζ*-regularized computations to be consistent and independent of the choice of gauge, it suffices to show $\\langle\\Phi^2\\rangle(0)=\\frac{i}{2\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^2$. If that is the case, we not only obtain the correct continuum limit for the ground state energy of the harmonic oscillator on the lattice, but the ground state energy coincides with the Wick rotated ground state energy result for all choices of lattice spacing *a* and number of lattice sites *τ*.\nThe global distributional gauge\n-------------------------------\nThe first type of gauge will be global in time and distributionally regularizing. These are generally of the form\n \n$$\\begin{aligned}\n {\\forall}r\\in{\\ensuremath{\\mathbb{R}}}\\_{>0}\\ {\\forall}\\Phi\\in{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}:\\ {\\ensuremath{\\mathfrak{g}}}(z)(r,\\Phi)=r^z{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z)(\\Phi)=r^z{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}}).\\end{aligned}$$ \nUsing such a gauge, we can evaluate the integral with respect to *s* in\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{is}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})s^{\\frac{\\tau}{2}-1}dsd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{is}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})s^{\\frac{\\tau}{2}-1}dsd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}\\end{aligned}$$ \nvia analytic continuation of the Laplace transform L and obtain\n \n$$\\begin{aligned}\n I(z):=&\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{is}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})s^{\\frac{\\tau}{2}-1}dsd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)\\\\\n =&\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{is}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})s^{\\frac{\\tau+z+\\alpha}{2}-1}dsd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)\\\\\n =&\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}}){\\ensuremath{\\mathcal{L}}}{\\ensuremath{\\left}}(s\\mapsto s^{\\frac{\\tau+z+\\alpha}{2}-1}{\\ensuremath{\\right}})(-i)d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)\\\\\n =&i^{\\frac{\\tau+z+\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z+\\alpha}{2}{\\ensuremath{\\right}})\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi).\\end{aligned}$$ \nHence,\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{i^{\\frac{\\tau+z+\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z+\\alpha}{2}{\\ensuremath{\\right}})\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{i^{\\frac{\\tau+z}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z}{2}{\\ensuremath{\\right}})\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}\\\\\n =&\\frac{i^{\\frac{\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z+\\alpha}{2}{\\ensuremath{\\right}})\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z}{2}{\\ensuremath{\\right}})\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(0)=&\\frac{i^{\\frac{\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2}{\\ensuremath{\\right}})\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau}{2}{\\ensuremath{\\right}}){\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau})}.\\end{aligned}$$ \nIf we choose the eigenbasis of Σ to parametrize R*τ* again, then\n \n$$\\begin{aligned}\n \\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)=&\\frac1\\tau\\sum\\_{j=0}^{\\tau-1}\\lambda\\_j^\\alpha\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Psi\\_j^\\alpha d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)\\end{aligned}$$ \nneeds to be evaluated. In this case, we can perform the calculation analytically\n \n$$\\begin{aligned}\n \\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Psi\\_j^\\alpha d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)=&\\int\\_0^\\pi\\ldots\\int\\_0^\\pi\\int\\_0^{2\\pi}\\cos({\\varphi}\\_1)^\\alpha\\prod\\_{j=1}^{\\tau-2}\\sin({\\varphi}\\_j)^{\\tau-1-j}d{\\varphi}\\_{\\tau-1}\\ldots d{\\varphi}\\_1\\\\\n =&{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^{\\tau-1}}}{\\ensuremath{\\left}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^{\\tau-1}}{\\ensuremath{\\right}})\\int\\_0^\\pi\\cos({\\varphi})^\\alpha\\sin({\\varphi})^{\\tau-2}d{\\varphi}\\\\\n =&{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^{\\tau-1}}}{\\ensuremath{\\left}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^{\\tau-1}}{\\ensuremath{\\right}})\\frac{{\\ensuremath{\\left}}((-1)^\\alpha+1{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+3}{2}{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau-1}{2}{\\ensuremath{\\right}})}{(\\alpha+1)\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2}{\\ensuremath{\\right}})}\\end{aligned}$$ \nwhich yields\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(0)=&\\frac1\\tau\\sum\\_{j=0}^{\\tau-1}\\lambda\\_j^\\alpha\\frac{i^{\\frac{\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2}{\\ensuremath{\\right}}){\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^{\\tau-1}}}{\\ensuremath{\\left}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^{\\tau-1}}{\\ensuremath{\\right}}){\\ensuremath{\\left}}((-1)^\\alpha+1{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+3}{2}{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau-1}{2}{\\ensuremath{\\right}})}{\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau}{2}{\\ensuremath{\\right}}){\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau})(\\alpha+1)\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2}{\\ensuremath{\\right}})}\\\\\n =&\\frac{i^{\\frac{\\alpha}{2}}}{(\\alpha+1)\\tau}\\sum\\_{j=0}^{\\tau-1}\\lambda\\_j^\\alpha\\frac{{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^{\\tau-1}}}{\\ensuremath{\\left}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^{\\tau-1}}{\\ensuremath{\\right}}){\\ensuremath{\\left}}((-1)^\\alpha+1{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+3}{2}{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau-1}{2}{\\ensuremath{\\right}})}{\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau}{2}{\\ensuremath{\\right}}){\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau})}\\\\\n =&\\frac{i^{\\frac{\\alpha}{2}}{\\ensuremath{\\left}}((-1)^\\alpha+1{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+3}{2}{\\ensuremath{\\right}})}{(\\alpha+1)\\sqrt\\pi\\tau}\\sum\\_{j=0}^{\\tau-1}\\lambda\\_j^\\alpha\\\\\n =&\\frac{i^{\\frac{\\alpha}{2}}{\\ensuremath{\\left}}((-1)^\\alpha+1{\\ensuremath{\\right}})\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+1}{2}{\\ensuremath{\\right}})}{2\\sqrt\\pi\\tau}\\sum\\_{j=0}^{\\tau-1}\\lambda\\_j^\\alpha.\\end{aligned}$$ \nIn particular, for *α* = 2 we obtain\n \n$$\\begin{aligned}\n \\langle\\Phi^2\\rangle(0)=&\\frac{i\\Gamma{\\ensuremath{\\left}}(\\frac{3}{2}{\\ensuremath{\\right}})}{\\sqrt\\pi\\tau}\\sum\\_{j=0}^{\\tau-1}\\lambda\\_j^2=\\frac{i}{2\\tau}\\sum\\_{j=0}^{\\tau-1}\\lambda\\_j^2\\end{aligned}$$ \nand thus the correct ground state energy.\nThe global absolutely integrable gauge\n--------------------------------------\nFor the gauge to yield an absolutely integrable regularization, we need to remove the pole in 0. For instance, a gauge of the form\n \n$$\\begin{aligned}\n {\\forall}r\\in{\\ensuremath{\\mathbb{R}}}\\_{>0}\\ {\\forall}\\Phi\\in{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}:\\ {\\ensuremath{\\mathfrak{g}}}(z)(r,\\Phi)=\\begin{cases}1&,\\ r\\le1\\\\{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z)(\\Phi)r^z&,\\ r>1\\end{cases}\\end{aligned}$$ \nmay be chosen. Then\n \n$$\\begin{aligned}\n I(z):=&\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{is}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt{s}\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})s^{\\frac{\\tau}{2}-1}dsd{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)\\\\\n =&\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)\\int\\_0^1e^{is}s^{\\frac{\\tau+\\alpha}{2}-1}ds\\\\\n &+\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>1}}e^{is}s^{\\frac{\\tau+z+\\alpha}{2}-1}ds\\\\\n =&\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)i^{\\frac{\\tau+\\alpha}{2}}\\gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2},-i{\\ensuremath{\\right}})\\\\\n &+\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)i^{\\frac{\\tau+z+\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z+\\alpha}{2},-i{\\ensuremath{\\right}}).\\end{aligned}$$ \nIn particular, for g∂(*z*) = 1 we obtain\n \n$$\\begin{aligned}\n I(z)=&\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)i^{\\frac{\\tau+\\alpha}{2}}{\\ensuremath{\\left}}(\\gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2},-i{\\ensuremath{\\right}})+\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z+\\alpha}{2},-i{\\ensuremath{\\right}})i^{\\frac{z}{2}}{\\ensuremath{\\right}})\\end{aligned}$$ \nand thus\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=\\frac{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau})}\\frac{i^{\\frac{\\alpha}{2}}{\\ensuremath{\\left}}(\\gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2},-i{\\ensuremath{\\right}})+\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z+\\alpha}{2},-i{\\ensuremath{\\right}})i^{\\frac{z}{2}}{\\ensuremath{\\right}})}{{\\ensuremath{\\left}}(\\gamma{\\ensuremath{\\left}}(\\frac{\\tau}{2},-i{\\ensuremath{\\right}})+\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z}{2},-i{\\ensuremath{\\right}})i^{\\frac{z}{2}}{\\ensuremath{\\right}})}.\\end{aligned}$$ \nAt *z* = 0, we may use\n \n$$\\begin{aligned}\n \\frac{i^{\\frac{\\alpha}{2}}{\\ensuremath{\\left}}(\\gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2},-i{\\ensuremath{\\right}})+\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2},-i{\\ensuremath{\\right}}){\\ensuremath{\\right}})}{{\\ensuremath{\\left}}(\\gamma{\\ensuremath{\\left}}(\\frac{\\tau}{2},-i{\\ensuremath{\\right}})+\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau}{2},-i{\\ensuremath{\\right}}){\\ensuremath{\\right}})}=\\frac{i^{\\frac{\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2}{\\ensuremath{\\right}})}{\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau}{2}{\\ensuremath{\\right}})}\\end{aligned}$$ \nand comparing the remaining integrals to the results of, we observe that the two choices of gauge give the same result (as expected since we have gauge invariance). At the same time this appears a lot more complicated than the computation in. However, there is a major advantage in that the initial R*τ* integrals can be evaluated numerically for ℜ(*z*) ≪ 0 without using the Laplace transform. Hence, we can compute $\\frac{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau})}$, which requires integration over high dimensional spheres, by computing\n \n$$\\begin{aligned}\n \\frac{\\gamma{\\ensuremath{\\left}}(\\frac{\\tau}{2},-i{\\ensuremath{\\right}})+\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z}{2},-i{\\ensuremath{\\right}})i^{\\frac{z}{2}}}{i^{\\frac{\\alpha}{2}}{\\ensuremath{\\left}}(\\gamma{\\ensuremath{\\left}}(\\frac{\\tau+\\alpha}{2},-i{\\ensuremath{\\right}})+\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z+\\alpha}{2},-i{\\ensuremath{\\right}})i^{\\frac{z}{2}}{\\ensuremath{\\right}})}\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{i{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d\\Psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{i{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d\\Psi}\\end{aligned}$$ \nfor values of *z* with ℜ(*z*) ≪ 0. This may be easier to implement as more numerical techniques have been developed for integrating over R*τ* than ∂*B*R*τ*.\nThe local distributional gauge\n------------------------------\nWith the third approach we want to gauge the transfer matrix (as opposed to “only” gauging the entire time evolution operator). This is often useful because it retains locality in time – a property frequently exploited in many lattice methods; e.g., when computing correlation functions. In terms of the following computation, this choice of gauge allows us to use Fubini as in the Wick rotated computation. Let us start with a generic g(*z*)(Φ) = ∏*j* = 0*τ* − 1gΔ(Φ*j*). Then we observe\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{i{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d\\Psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{i{{\\left\\lVert}{\\Psi}{\\right\\lVert}}^2}{\\ensuremath{\\mathfrak{g}}}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d\\Psi}\\\\\n =&\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}\\prod\\_{j=0}^{\\tau-1}e^{i\\Psi\\_j^2}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_j\\Psi\\_j{\\ensuremath{\\right}})\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}(\\lambda\\_k\\Psi\\_k)^{\\alpha}d\\Psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}\\prod\\_{j=0}^{\\tau-1}e^{i\\Psi\\_j^2}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_j\\Psi\\_j{\\ensuremath{\\right}})d\\Psi}\\\\\n =&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}}e^{i\\Psi\\_k^2}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_k\\Psi\\_k{\\ensuremath{\\right}})(\\lambda\\_k\\Psi\\_k)^{\\alpha}d\\Psi\\_k}{\\int\\_{{\\ensuremath{\\mathbb{R}}}}e^{i\\Psi\\_k^2}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_k\\Psi\\_k{\\ensuremath{\\right}})d\\Psi\\_k}\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n \\lambda\\_j={\\ensuremath{\\left}}(\\frac{am\\omega^2}{2}+\\frac{m}{a}{\\ensuremath{\\left}}(1-\\cos{\\ensuremath{\\left}}(\\frac{2\\pi j}{\\tau}{\\ensuremath{\\right}}){\\ensuremath{\\right}}){\\ensuremath{\\right}})^{-\\frac12}.\\end{aligned}$$ \nIf we choose ${\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_k\\Psi\\_k{\\ensuremath{\\right}})={\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_k{{\\left\\lvert}{\\Psi\\_k}{\\right\\lvert}}{\\ensuremath{\\right}})$, we further obtain\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^\\alpha\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{i\\psi}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_k\\sqrt{\\Psi\\_k}{\\ensuremath{\\right}})\\Psi\\_k^{\\frac{\\alpha-1}{2}}d\\Psi\\_k}{\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{i\\psi}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_k\\sqrt{\\Psi\\_k}{\\ensuremath{\\right}})\\Psi\\_k^{-\\frac{1}{2}}d\\Psi\\_k}.\\end{aligned}$$ \nAgain, we have the choice between a distributionally regularizing gauge or an absolutely integrable gauge. In this subsection, we will proceed with a distributionally regularizing gauge ${\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z)(\\psi)={{\\left\\lvert}{\\psi}{\\right\\lvert}}^z$ which implies\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^{\\alpha}\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{i\\psi}\\psi^{\\frac{\\alpha+z-1}{2}}d\\psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{i\\psi}\\psi^{\\frac{z-1}{2}}d\\psi}\n =\\frac{i^{\\frac{\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+z+1}{2}{\\ensuremath{\\right}})}{\\Gamma{\\ensuremath{\\left}}(\\frac{z+1}{2}{\\ensuremath{\\right}})\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^{\\alpha}\\end{aligned}$$ \nand hence the required\n \n$$\\begin{aligned}\n \\langle\\Phi^2\\rangle(0)=&\\lim\\_{z\\to0}\\frac{i\\Gamma{\\ensuremath{\\left}}(\\frac{z+1}{2}+1{\\ensuremath{\\right}})}{\\Gamma{\\ensuremath{\\left}}(\\frac{z+1}{2}{\\ensuremath{\\right}})\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^2\n =\\frac{i}{2\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^2.\\end{aligned}$$ \nThe local absolutely integrable gauge\n-------------------------------------\nThe fourth and final choice of gauge would be an absolutely integrable gauge for the transfer matrix. For example, we may consider\n \n$$\\begin{aligned}\n {\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z)(\\psi)=\\begin{cases}1&,\\ {{\\left\\lvert}{\\psi}{\\right\\lvert}}<1\\\\{{\\left\\lvert}{\\psi}{\\right\\lvert}}^z&,\\ {{\\left\\lvert}{\\psi}{\\right\\lvert}}\\ge1\\end{cases}.\\end{aligned}$$ \nIn this case, we obtain\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^\\alpha\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{i\\psi}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_k\\sqrt\\psi{\\ensuremath{\\right}})\\psi^{\\frac{\\alpha-1}{2}}d\\psi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{i\\psi}{\\ensuremath{\\mathfrak{g}}}\\_\\Delta(z){\\ensuremath{\\left}}(\\lambda\\_k\\sqrt\\psi{\\ensuremath{\\right}})\\psi^{-\\frac{1}{2}}d\\psi}\\\\\n =&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^\\alpha\\frac{\\int\\_0^{\\lambda\\_k^{-1}}e^{i\\psi}\\psi^{\\frac{\\alpha-1}{2}}d\\psi+\\lambda\\_k^z\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>{\\lambda\\_k^{-1}}}}e^{i\\psi}\\psi^{\\frac{\\alpha+z-1}{2}}d\\psi}{\\int\\_0^{\\lambda\\_k^{-1}}e^{i\\psi}\\psi^{-\\frac{1}{2}}d\\psi+\\lambda\\_k^z\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>{\\lambda\\_k^{-1}}}}e^{i\\psi}\\psi^{\\frac{z-1}{2}}d\\psi}\\\\\n =&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^\\alpha\\frac{i^{\\frac{\\alpha+1}{2}}\\gamma{\\ensuremath{\\left}}(\\frac{\\alpha+1}{2},\\lambda\\_k^{-1}{\\ensuremath{\\right}})+\\lambda\\_k^zi^{\\frac{\\alpha+z+1}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+z+1}{2},\\lambda\\_k^{-1}{\\ensuremath{\\right}})}{i^{\\frac{1}{2}}\\gamma{\\ensuremath{\\left}}(\\frac{1}{2},\\lambda\\_k^{-1}{\\ensuremath{\\right}})+\\lambda\\_k^zi^{\\frac{z+1}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{z+1}{2},\\lambda\\_k^{-1}{\\ensuremath{\\right}})}\\end{aligned}$$ \nwhich in the limit *z* → 0 yields\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(0)=&\\frac{1}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^\\alpha\\frac{i^{\\frac{\\alpha}{2}}\\gamma{\\ensuremath{\\left}}(\\frac{\\alpha+1}{2},\\lambda\\_k^{-1}{\\ensuremath{\\right}})+i^{\\frac{\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+1}{2},\\lambda\\_k^{-1}{\\ensuremath{\\right}})}{\\gamma{\\ensuremath{\\left}}(\\frac{1}{2},\\lambda\\_k^{-1}{\\ensuremath{\\right}})+\\Gamma{\\ensuremath{\\left}}(\\frac{1}{2},\\lambda\\_k^{-1}{\\ensuremath{\\right}})}\n =\\frac{i^{\\frac{\\alpha}{2}}}{\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^\\alpha\\frac{\\Gamma{\\ensuremath{\\left}}(\\frac{\\alpha+1}{2}{\\ensuremath{\\right}})}{\\Gamma{\\ensuremath{\\left}}(\\frac{1}{2}{\\ensuremath{\\right}})}\\end{aligned}$$ \nand thus\n \n$$\\begin{aligned}\n \\langle\\Phi^2\\rangle(0)=&\\frac{i}{2\\tau}\\sum\\_{k=0}^{\\tau-1}\\lambda\\_k^2.\\end{aligned}$$ \nThe harmonic oscillator numerically\n===================================\nWhile the harmonic oscillator can be solved analytically, as seen in, more complicated lattice theories will need to be solved numerically. To this end, we may choose between a quantum simulation of the lattice theory or a classical simulation. In this section, we will therefore consider the numerical approach to the *ζ*-regularized harmonic oscillator using classical computation as an example for *ζ*-regularized lattice theories in detail.\nIf we want to treat the *ζ*-regularized harmonic oscillator numerically, we have in principle two options. The first option is to choose a gauge g and compute the dependence on the gauge parameter *z* explicitly. In this case, we may choose a distributionally regularizing gauge g(*z*)(*r*, Φ) = *r**z*g∂(*z*)(Φ) and obtain an expression of the form\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{i^{\\frac{\\alpha}{2}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z+\\alpha}{2}{\\ensuremath{\\right}})\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z}{2}{\\ensuremath{\\right}})\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}{\\ensuremath{\\mathfrak{g}}}\\_{\\partial}(z){\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}.\\end{aligned}$$ \nThis enables us to take the *z* → 0 limit explicitly and we are left with the numerical problem of integrating\n \n$$\\begin{aligned}\n E\\_0 =&\\frac{m\\omega^2\\tau}{2}\\frac{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau})}\\end{aligned}$$ \nwhere Ω is the observable Φ2. Alternatively, if the analytic continuation cannot be computed explicitly, then we can use the fact that *E*0(*z*) = − *i**m**ω*2⟨Ω⟩(*z*) is comprised of integrals that are well-posed for ℜ(*z*) ≪ 0. Hence, *E*0(*z*) can be computed for a sufficiently large number of values *z* with ℜ(*z*) ≪ 0. The *z* = 0 value can then be obtained by fitting the *z*-dependence (which is known up to some parameters) to the thus generated data and extrapolating the fit to *z* = 0. In either case, we need to evaluate high-dimensional spherical integrals.\nSolving the spherical integrals\n-------------------------------\nThe numerical difficulty of computing\n \n$$\\begin{aligned}\n E\\_0=&\\frac{m\\omega^2\\tau}{2}\\frac{\\int\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})d{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}(\\Psi)}{{\\ensuremath{\\mathrm{vol}}}\\_{{\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}}({\\partial}B\\_{{\\ensuremath{\\mathbb{R}}}^\\tau})}\\end{aligned}$$ \nis the quadrature of the high-dimensional spherical integrals. We are using “Sobol’ points on the sphere ∂*B*R*τ*” which can be generated from a “standard” Sobol’ sequence in three steps.\n1. Generate a *τ*-dimensional Sobol’ sequence (uniform distribution on (0, 1)*τ*).\n2. Apply the standard normal (N(0, 1)) percent point function ppf in each coordinate (standard normal distribution on R*τ*).\n3. Normalize each sample (uniform distribution on ∂*B*R*τ*).\nIn order to make full use of the properties of the Sobol’ sequence, we generate 2*n* Sobol’ points for some *n* ∈ N and skip the first point since the first (non-randomized) Sobol’ point is always ${\\ensuremath{\\left}}(\\frac12,\\ldots,\\frac12{\\ensuremath{\\right}})^T$ which gets mapped to the origin under ppf and thus cannot be normalized.\nLet S be the set of generated Sobol’ points. Then we can approximate *E*0 via\n \n$$\\begin{aligned}\n E\\_0\\approx\\frac{m\\omega^2\\tau}{2}\\frac{1}{\\#{\\ensuremath{\\mathcal{S}}}}\\sum\\_{\\Psi\\in{\\ensuremath{\\mathcal{S}}}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}}).\\end{aligned}$$ \nThe convergence in terms of the number of (randomized) Sobol’ samples is shown in. The standard deviations, which were extracted from 100 simulations each, indicate an error scaling proportional to $\\frac{1}{\\sqrt{\\#{\\ensuremath{\\mathcal{S}}}}}$. In other words, the *ζ*-regularized lattice simulation of the harmonic oscillator with Lorentzian background is comparable to a Markov Chain Monte-Carlo simulation in Euclidean background in terms of numerical effort.\n[fig:sobolerror]This figure shows the standard deviation of $\\frac{1}{\\#{\\ensuremath{\\mathcal{S}}}}\\sum\\_{\\Psi\\in{\\ensuremath{\\mathcal{S}}}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}})$ using *N* = #S = 2*n* − 1 sample points. The standard deviation is estimated using 100 simulations with randomized Sobol’ points. Furthermore, a least square error fit *C*(*N* + 1)*α* to the simulation data is shown. The exponent *α* ≈ − 0.575 indicates an error scaling $\\mathrm{error}\\propto\\frac{1}{\\sqrt{\\#{\\ensuremath{\\mathcal{S}}}}}$. The simulation parameters are *m* = *ω* = ℏ = 1, *τ* = 128, and *a* = 0.01.\nHaving a reasonable quadrature to estimate\n \n$$\\begin{aligned}\n E\\_0\\approx\\frac{m\\omega^2\\tau}{2}\\frac{1}{\\#{\\ensuremath{\\mathcal{S}}}}\\sum\\_{\\Psi\\in{\\ensuremath{\\mathcal{S}}}}\\Omega{\\ensuremath{\\left}}(\\sqrt\\Sigma^{-1}\\Psi{\\ensuremath{\\right}}),\\end{aligned}$$ \nwe can now consider the behavior of *E*0(*m*, *ω*, ℏ) as a function of the lattice spacing *a* and the number of lattice points *τ*. The physical volume is then given by *T* = *a**τ*. In order to extract the continuum limit, we are interested in the limit $a\\searrow0$ at constant physical volume *T* and the limit $T\\nearrow\\infty$. (top) shows simulation results of *E*0(*m* = 1, *ω* = 1, ℏ = 1) with lattice spacing *a* ∈ {2− 2, 2− 3, 2− 4, 2− 5} at physical volume *T* = 32. Similarly, (bottom) shows simulation results of with lattice spacing *a* = 2− 5 at physical volume *T* ∈ {4, 8, 16, 32}. The error bars are the standard deviations extracted from 100 simulations using 210 − 1 randomized Sobol’ points and the horizontal lines are the analytic continuum limit $E\\_0(m=1,\\omega=1,\\hbar=1)=\\frac12$. In particular, we note that the results for *a* = 2− 5 and *T* ∈ {16, 32} are already comparable to the continuum limit at the given level of accuracy.\n| |\n| --- |\n| |\n| |\nExtrapolation from ℜ(*z*) ≪ 0\n-----------------------------\nIn more complicated situations, it may not always be possible to compute the dependence on the gauge parameter *z* explicitly, or computing the spherical integrals may be more difficult than computing the R*τ* integrals\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)\\Omega(\\Phi)d\\Phi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\Phi}.\\end{aligned}$$ \nIn such cases, it may be advantageous to compute ⟨Ω⟩(*z*) at ℜ(*z*) ≪ 0 where the integrals are well-defined and numerically well-posed. In order to guide the extrapolation to *z* = 0, we need to account for the unknown *z*-dependence Δ(*z*). However, if the gauge is chosen well, then we can use the fact that the *z*-dependence comes from analytic continuations of Laplace transforms of poly-(log)-homogeneous distributions, for which the general form of these Laplace transforms in terms of the variable *z* are known. Thus, by combining those constituent terms, we obtain a parametric expression Δ(*z*, *p*) such that Δ(*z*, *p*\\*) = ⟨Ω⟩(*z*) is satisfied for some set of parameters *p*\\*. Once this general dependence Δ(*z*, *p*) is identified by analyzing the poly-(log)-homogeneity structure of the integrands, the parameters *p*\\* can be obtained from fitting Δ(*z*, *p*) against the numerical data ⟨Ω⟩(*z*). Finally we obtain ⟨Ω⟩(0) by extrapolation, i.e., by evaluating Δ(*z*, *p*\\*) at *z* = 0.\nConsider an action *S* for which there exists a closed compact manifold *M* in R*τ* such that ∀Φ ∈ R*τ* ∃!*φ* ∈ *M* ∃!*r* ∈ R> 0 : Φ = *r**φ* and *S*(Φ) = *c**r**σ* for some constant *c* ∈ R. Furthermore, let Ω(Φ) = ∑*ι* ∈ *I**r**δ**ι**ω**ι*(*φ*) be a polyhomogeneous observable and choose the gauge g(*z*)(Φ) = (1 + *r*)*z*. Then,\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(z)=&\\frac{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)\\Omega(\\Phi)d\\Phi}{\\int\\_{{\\ensuremath{\\mathbb{R}}}^\\tau}e^{iS(\\Phi)}{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)d\\Phi}\\\\\n =&\\frac{\\sum\\_{\\iota\\in I}\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{icr^\\sigma}(1+r)^zr^{\\delta\\_\\iota+\\tau-1}dr\\int\\_M\\omega\\_\\iota({\\varphi})d{\\ensuremath{\\mathrm{vol}}}\\_M({\\varphi})}{\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{icr^\\sigma}(1+r)^zr^{\\tau-1}dr{\\ensuremath{\\mathrm{vol}}}\\_M(M)}.\n \\end{aligned}$$ \nThus,\n \n$$\\begin{aligned}\n I\\_\\delta(z):=&\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>0}}e^{icr^\\sigma}(1+r)^zr^\\delta dr\\\\\n =&\\int\\_0^1e^{icr^\\sigma}(1+r)^zr^\\delta dr+\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>1}}e^{icr^\\sigma}(1+r)^zr^\\delta dr\\\\\n =&\\sum\\_{k\\in{\\ensuremath{\\mathbb{N}}}\\_0}{z\\choose k}{\\ensuremath{\\left}}(\\int\\_0^1e^{icr^\\sigma}r^{\\delta+k} dr+\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>1}}e^{icr^\\sigma}r^{\\delta+z-k} dr{\\ensuremath{\\right}})\\\\\n =&\\sum\\_{k\\in{\\ensuremath{\\mathbb{N}}}\\_0}{z\\choose k}\\frac{1}{\\sigma}{\\ensuremath{\\left}}(\\int\\_0^1e^{icr}r^{\\frac{\\delta+k+1-\\sigma}{\\sigma}} dr+\\int\\_{{\\ensuremath{\\mathbb{R}}}\\_{>1}}e^{icr}r^{\\frac{\\delta+z-k+1-\\sigma}{\\sigma}} dr{\\ensuremath{\\right}})\\\\\n =&\\sum\\_{k\\in{\\ensuremath{\\mathbb{N}}}\\_0}{z\\choose k}\\frac{1}{\\sigma}{\\ensuremath{\\left}}(\\frac{\\gamma{\\ensuremath{\\left}}(\\frac{\\delta+k+1}{\\sigma},-ic{\\ensuremath{\\right}})}{(-ic)^{\\frac{\\delta+k+1}{\\sigma}}}+\\frac{\\Gamma{\\ensuremath{\\left}}(\\frac{\\delta+z-k+1}{\\sigma},-ic{\\ensuremath{\\right}})}{(-ic)^{\\frac{\\delta+z-k+1}{\\sigma}}}{\\ensuremath{\\right}})\n \\end{aligned}$$ \nimplies\n \n$$\\begin{aligned}\n\\tag{$\\*\\*\\*$}\\label{eq:fit\\_Re(z)<<0}\n \\langle\\Omega\\rangle(z)=&\\sum\\_{\\iota\\in I}C\\_{\\iota}\\frac{\\sum\\_{k\\in{\\ensuremath{\\mathbb{N}}}\\_0}{z\\choose k}\\frac{1}{\\sigma}{\\ensuremath{\\left}}(\\frac{\\gamma{\\ensuremath{\\left}}(\\frac{\\delta\\_\\iota+\\tau+k}{\\sigma},-ic{\\ensuremath{\\right}})}{(-ic)^{\\frac{\\delta\\_\\iota+\\tau+k}{\\sigma}}}+\\frac{\\Gamma{\\ensuremath{\\left}}(\\frac{\\delta\\_\\iota+\\tau+z-k}{\\sigma},-ic{\\ensuremath{\\right}})}{(-ic)^{\\frac{\\delta\\_\\iota+\\tau+z-k}{\\sigma}}}{\\ensuremath{\\right}})}{\\sum\\_{k\\in{\\ensuremath{\\mathbb{N}}}\\_0}{z\\choose k}\\frac{1}{\\sigma}{\\ensuremath{\\left}}(\\frac{\\gamma{\\ensuremath{\\left}}(\\frac{\\tau+k}{\\sigma},-ic{\\ensuremath{\\right}})}{(-ic)^{\\frac{\\tau+k}{\\sigma}}}+\\frac{\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau+z-k}{\\sigma},-ic{\\ensuremath{\\right}})}{(-ic)^{\\frac{\\tau+z-k}{\\sigma}}}{\\ensuremath{\\right}})}.\n \\end{aligned}$$ \nIn this expression, everything is known a priori with the exception of the *C**ι*. In other words, we can fit against simulation data to extract the *C**ι* and obtain\n \n$$\\begin{aligned}\n \\langle\\Omega\\rangle(0)=&\\sum\\_{\\iota\\in I}C\\_{\\iota}\\frac{\\Gamma{\\ensuremath{\\left}}(\\frac{\\delta\\_\\iota+\\tau}{\\sigma}{\\ensuremath{\\right}})}{(-ic)^{\\frac{\\delta\\_\\iota}{\\sigma}}\\Gamma{\\ensuremath{\\left}}(\\frac{\\tau}{\\sigma}{\\ensuremath{\\right}})}.\n \\end{aligned}$$ \n| |\n| --- |\n| |\n| |\n| |\nWith the choice of gauge *g*(*z*)(*r*, *φ*) = *r**z*, the *z*-dependence Δ(*z*, *p*) is much simpler; namely affine linear Δ(*z*, *p*) = *p*0 + *p*1*z*. Using up to 1000 simulations at randomly chosen *z* ∈ [ − 2*τ* − 1, − *τ* − 1] with *τ* = 40, *a* = 0.1, and *m* = *ω* = ℏ = 1, we can fit Δ(*z*, *p*) and obtain the *z* = 0 extrapolation *p*0. (top) shows the least square error fit of Δ(*z*, *p*) to the data of 1000 simulations as well as the *z* = 0 value *E*0(*a*, *τ*). In (middle) the extrapolated value *p*0 extracted from 50, 100, …, 950, 1000 simulations including the standard deviation as error is shown. The comparison to the *z* = 0 value *E*0(*a*, *τ*), which we can compute analytically as in, shows compatibility of the extrapolated value *p*0 with *E*0(*a*, *τ*) already for a relatively small number of simulations. In (bottom), we compare the standard deviation of *p*0 extracted from 50, 100, …, 950, 1000 simulations to a least square error fit of the form *C* ⋅ (#simulations)*α*. The fitted value of *α* ≈ − 0.490 indicates an error scaling proportional to $\\frac{1}{\\sqrt{\\#\\text{simulations}}}$.\nConclusion\n==========\nIn this article, we have studied the *ζ*-regularization of lattice field theories with Lorentzian background metrics. In particular, we have discussed the connection between lattice field theories with Euclidean backgrounds and Lorentzian backgrounds in, in order to understand how the *ζ*-regularization arises as a naïve means of regularizing lattice expectation values when it is not possible to analytically continue the theory to Euclidean time. In, we have then made the connection from the naïvely *ζ*-regularized lattice theory to the Hamiltonian formulation so far discussed in the literature . This allowed us to gain insights into the two main questions that need considering when extending the *ζ*-regularization in the Hamiltonian formulation to the Lagrangian formulation commonly used to describe lattice field theories.\n1. What kind of gauge families g can be used to *ζ*-regularize lattice field theories in a Lorentzian background?\n2. Under what conditions is it possible to construct a *ζ*-regularized Hamiltonian theory starting from a “naïvely” *ζ*-regularized lattice field theory?\nQuestion Q1 is at least partially answered by and. In order to *ζ*-regularize a lattice field theory in a Lorentzian background, g(*z*) must be chosen in such a way, that the corresponding integrals are well-defined for ℜ(*z*) ≪ 0. This is generally satisfied if g(*z*) has an asymptotic behavior of ${{\\left\\lvert}{{\\ensuremath{\\mathfrak{g}}}(z)(\\Phi)}{\\right\\lvert}}\\le c(1+{{\\left\\lVert}{\\Phi}{\\right\\lVert}}^2)^{\\delta\\Re(z)}$ or exhibits even faster convergence to zero as ${{\\left\\lVert}{\\Phi}{\\right\\lVert}}\\to\\infty$. Furthermore, for an increasing, exhaustive sequence of compacta (*K**n*)*n* ∈ N, the compactified vacuum expectation values ${\\ensuremath{\\left}}(\\frac{\\langle 1\\_{K\\_n^T}\\Omega{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}{\\langle 1\\_{K\\_n^T}{\\ensuremath{\\mathfrak{g}}}\\rangle\\_L}{\\ensuremath{\\right}})\\_{n\\in{\\ensuremath{\\mathbb{N}}}}$ need to be locally bounded in *C*(*D*) where *D* ⊆ C is open, connected, and contains 0 as well as a half-space with ℜ(*z*) ≪ 0. This local boundedness condition is likely to be the most intricate assumption to satisfy, as this is the likely point at which renormalization and modeling decisions can have a major impact.\nIt is important to note that the answer to Q1 merely implies the existence of a *ζ*-regularized lattice theory. Without further information, there is no reason to assume that this *ζ*-regularized lattice theory is at all physical. In order to conclude physicality of the expectation values extracted from the *ζ*-regularized lattice theory, we need to connect the Lagrangian formulation of the *ζ*-regularized lattice theory to the Hamiltonian formulation studied to date which includes a proof of physicality. As such, answering Q2 allows us to make physical sense of the *ζ*-regularized lattice vacuum expectation values.\nQuestion Q2 is partially answered by and. In essence, we need to ensure that the resulting gauged transfer matrix or gauged time evolution operator is a suitable family of Fourier integral operators; namely a gauged family in the sense of . This can be ensured by choosing g(*z*) or gΔ(*z*) to be in a Hörmander class *S**δ*ℜ(*z*) or (poly-)homogeneous. While this restricts the class of gauge families g sufficient to construct the *ζ*-regularized lattice theory as per question Q1, computationally this is not a bad class of functions as many analytic results are known and, thus, allowing for computational simplifications in numerical simulations of *ζ*-regularized lattice theories.\nIn, we then considered the classical limit of *ζ*-regularized lattice theories and observed the remarkable fact that the classical limit is independent of the value for *z*. If we consider the impact of the gauge on a classical action, this can be rationalized since the gauge adds an imaginary component to a classical action. Thus, if we expect for only the real-part of the gauged action to be retained in the classical limit, then confirms this point of view.\nFinally, we illustrated the theory of *ζ*-regularized lattice theories developed in this article at the example of the harmonic oscillator. First, we treated it analytically in ; second, we considered it numerically in. Both the analytic and numeric studies of the harmonic oscillator focus on the ground state energy.\nIn, we computed the ground state energy of the harmonic oscillator on the lattice five times. Initially, we computed the ground state energy by using a Wick rotation, i.e., using the canonical approach of analytic continuation to imaginary (Euclidean) time. This served as a comparison for the four choices of gauge families that follow, as well as to show that the lattice theory has the correct continuum limit. The four gauge families that were used to compute the *ζ*-regularized harmonic oscillator on the lattice with Minkowski background highlight the four cases arising from two binary choices we can make. On one hand, we have the choice of gauging local in time vs. global in time, that is, gauging the transfer matrix or the entire time evolution operator. On the other hand, we can choose between absolutely integrable gauge families and distributionally regularizing gauge families, where the former is more directly applicable to numerical simulations but the latter allows for analytic shortcuts via the Laplace transform. Of course, all four gauge families reproduced the correct ground state energy, as is to be expected since we have gauge-independence.\nThe numerical study of the *ζ*-regularized harmonic oscillator on the lattice with Minkowski background in focused on the numerical obstacles arising in classical simulations of *ζ*-regularized lattice theories as opposed to quantum simulations which are a special case of the quantum simulations discussed in . In particular, we noted that distributionally regularizing gauge families allow for numerical shortcuts via Laplace transform and can make for simpler functional dependence on the complex parameter *z*, which needs to be fitted against simulation data for extrapolation if the extrapolation cannot be done analytically to begin with. However, distributionally regularizing gauge families require numerical quadratures over high-dimensional manifolds (a sphere for the harmonic oscillator). Absolutely integrable gauge families, on the other hand, “only” require quadratures in high-dimensional vector spaces R*τ*. As more powerful integration techniques exist for integrating over R*τ* than codimension 1 submanifolds of R*τ*, the absolutely integrable gauge families might be preferable from a numerical point of view. This question may be exacerbated further when considering gauge fields in lattice field theories. Unfortunately, the cost of choosing absolutely integrable gauge families is a more complicated dependence on the complex parameter *z* which may make extrapolation to *z* = 0 more difficult.\n---\n1. It should be noted that in complete generality, the transfer matrix may be “time-dependent.” If that is the case, then we have different *S*Δ, *j*(Φ*j* + 1, Φ*j*), *κ*Δ, *j*, and *U*Δ, *j*. For the purpose of legibility, we will assume ∀*j*, *k* : *S*Δ, *j* = *S*Δ, *k* = *S*Δ throughout this article. This assumption is not necessary for any of the results reported here to be true except that various powers, such as *U*Δ*T*, would need to be replaced with the corresponding products, such as ∏*j* = 1*T**U*Δ, *j*.[↩](#fnref1)\n2. This *τ* → ∞ result coincides with the *N* → ∞ limit of equation (C.29) in .[↩](#fnref2)\n\nZeta-regularized Lattice Field Theory with Lorentzian background metrics\n========================================================================\nLattice field theory is a very powerful tool to study Feynman’s path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined. On the other hand, a recently developed regularization scheme based on Fourier integral operator *ζ*-functions can treat Feynman’s path integral non-pertubatively in Lorentzian background metrics. In this article, we formally *ζ*-regularize lattice theories with Lorentzian backgrounds and identify conditions for the Fourier integral operator *ζ*-function regularization to be applicable. Furthermore, we show that the classical limit of the *ζ*-regularized theory is independent of the regularization. Finally, we consider the harmonic oscillator as an explicit example. We discuss multiple options for the regularization and analytically show that they all reproduce the correct ground state energy on the lattice and in the continuum limit. Additionally, we solve the harmonic oscillator on the lattice in Minkowski background numerically.\nIntroduction\n============\nLattice Field Theory\n--------------------\nFeynman’s path integral approach to quantum mechanics and quantum field theory has become an indispensable conceptual tool to evaluate physical observables, in particular in high energy physics. One very fascinating aspect of the path integral formulation is that it allows in principle to compute also non-perturbative phenomena and thus reaches out to problems that can not be addressed in perturbation theory.\nThe major obstacle to employ the path integral for studying non-perturbative effects is, however, that it is in general ill defined, if we take it at face value, i.e., using a Lorentzian background metric. An elegant way out is to analytically continue to imaginary – Euclidean – time and discretize the system either on a time lattice in quantum mechanics or on a space-time grid in quantum field theory. The so introduced lattice spacing acts then as an ultraviolet regulator, whereas a finite time or box length serves as an infrared regulator resulting thus in completely regularized version of the path integral, see for introductions to the so obtained *lattice field theories*.\nAnother important advantage of the Euclidean lattice regulated path integral is its resemblance to systems in statistical physics where it corresponds to the partition function. This allows in particular to resort to numerical calculations using Markov Chain Monte Carlo (MCMC) methods to evaluate the path integral. This approach has been employed very successfully in the last years to obtain many non-perturbative results mainly for quantum chromodynamics, the theory of the strong interaction between quarks and"},"data_id":{"kind":"string","value":"arxiv_0181671"}}},{"rowIdx":45395,"cells":{"content":{"kind":"string","value":"desc> <conceptsignificance>500</conceptsignificance> </concept> </ccs2012>\nIntroduction\n============\nDataset discovery, or identifying secondary data useful for research, business, or individual purposes, is a vital activity in contemporary life. This is increasingly true for not only data experts, but also for more casual users, who typically do not require large scale, well promoted datasets. Dataset discovery is composed of two parts: searching for data, and assessing the data to ensure it is fit for the intended purpose. This second part is often characterized as sense-making. The predominant methods of discovery currently are searching on a data portal (e.g. data.europa.eu) or via a search engine, including specialist search such as Google Dataset Search.\nMaking data useable requires tools that support individuals’ “data journey”. This includes specifying a goal and finding the data they need; exploring new sources of data; assessing the data’s relevance and credibility, and deriving insights. Additionally, to support non-expert users, such tools need to function effectively in low- to no-code interactions that are suitable to populations with diverse economic, cultural and educational backgrounds.\nCurrent data journeys involve creating ad hoc workflow processes on the fly over a number of tools, many of which are not easy for non-experts to use. A typical set of search and sense-making activities might look as shown in Figure [fig:fivefigures]:\n[fig:fivefigures]\nWhile the user is able to find, explore and use the data, many issues remain. The user is not able to be explicit in their goal in undertaking the search, but only to best estimate the keywords that might locate data that aligns somewhat with the needs. Although the search tool offers summaries and previews, the user must engage with further tools to evaluate and explore the data, and then still more tools to actually employ it for the purpose for which it was sought.\nFurther, data needs are complex, and can typically stretch over multiple search sessions, resulting in complex queries. These are poorly supported in web-based keyword search and can be better addressed with conversational search.\nConversational generative artificial intelligence (CGAI) tools such as OpenAI’s ChatGPT and Google’s Bard have changed how we access information online. We are therefore interested in exploring how they could change people’s interactions with data on the web. Given the ability of CGAI to summarise large, complex sources, explain results, and write code, we hypothesise that chatbots may also be able to find data matching a user goal and help the user understand its scope, quality, and provenance, and to use the data productively. This may also be supported further by the increased interactivity offered by conversational search, which has been established as empowering non-expert users.\nIn this paper we explore whether, given a specific domain, are useful datasets suggested? Do the rationales make sense? Does CGAI help answer critical questions on data provenance, licensing, format and more, and how do users assess this? We begin to consider what a new generation of data tools, enabled by conversational generative AI, might offer to users, and how people might engage with them. Our research questions, therefore, are:\n* RQ1: How does CGAI support state of the art dataset discovery technologies?\n* RQ2: How do people express their information needs when using CGAI for data discovery?\n* RQ3. Which of the main sense-making data discovery activities previously identified are supplied by CGAI?\nTo address these questions we held workshops with users to explore how they would interact with CGAI when engaging in data discovery. The key contributions of this paper are a set of guidelines for data prompting with CGAI and a new prompting-centric human-data interaction framework.\nBackground Literature\n=====================\nIn this section we contextualise our research in (1) the state of the art of dataset discovery and (2) CGAI for information retrieval.\nDataset discovery\n-----------------\nThe manipulation, analysis, and sensemaking of data has been described as ‘data discovery’. The data we refer to here is structured (organised in tables, spreadsheets, databases and so on) and its potential scale is vast: as of 2023, Google Dataset Search indexed more than 45 million datasets from more than 13,000 websites. However, the data sought may not reside in a single dataset but across multiple sources, in a variety of formats, with no common ID, making sensemaking more complex.\n[h]![Koesten et al’s (2017) framework for interaction with structured data, showing the 5 steps of the data journey for users of secondary data..](Koesten_framework.png \"fig:\") [fig:Koestenframework]\nThere are five key steps in the journey of interacting with structured data, which can be seen in Figure [fig:Koestenframework], above. The first step of the data journey for the user is to define their motivation - why they want to use data. This task purpose is crucial to the process of selecting the appropriate dataset. Currently, there is no useful explicit way for users to define their task motivation in search, although search engines bring a wealth of implicit knowledge about previous user behaviour to the results they present. However, this excludes wider context about the task at hand.\nThe second step is that of search itself. Data search is a relatively recent field, with the first widely available dedicated dataset search on the web receiving a full launch as recently as January 2020. The primary forms in which data is made available to people is via data repositories and portals (usually dedicated to a specific institution or domain); via the web, accessed by search engines; and through legal mechanisms such as Freedom of Information requests. When people need to look for data, they employ keyword or faceted search on portals or search engines. Additionally, people find data through personal recommendation. However, access to data does not equate enabling citizens to make use of the data. “There is a disconnect between what datasets are available, what dataset a user needs, and what datasets a user can actually find, trust and is able to use”. A solution to this challenge has not yet been satisfactorily implemented, although there have been improvements in supporting understanding of the data before downloading, such as providing visualisations and summarisation.\nStudies of online search behaviour pre-date the world wide web and find that searches range from analytical, precise and carefully planned, to “on the fly”. Users often find it challenging to describe their information needs because of a lack of background knowledge, and their understanding of an information need often evolves during the search process . It has been suggested that when in need of information users actually want an domain expert, but turn to search engines in the absence of this. Unlike web searches for information, users searching for data are unlikely to expect an exact dataset match to their needs, but rather to locate a relevant subset of topical data. Geospatial and temporal descriptions, constraints (eg open licence) and granularity are the most used parameters to specify data parameters in search.\nData repositories aim to make data search easier, but come with their own complications: they tend to use metadata rather than keyword search and data users and publishers may use different terms for datasets. One study found that a search for “housing stock” on the UK government data website found almost twice as many datasets as a search for “housing supply”. Much of the time in learning search tasks is devoted to examining and comparing results and reformulating queries to discover the boundaries of meaning for key concepts. Users also keep search terms short: the majority of searches on portals, both on the portal and via the web, are composed on average of less than 5 words.\nFurther, the information retrieval task at hand may require an entire set of results from multiple different sources with interdependencies. A familiar example of this is booking a holiday, which requires a variety of interdependent travel and accommodation requirements to be met. This kind of complex query requires “memory” of search interactions that is not yet met by the facilities of the web, and currently users searching for different elements of a holiday need to manually manage a range of concurrent searches which are not automatically informed by the parameters of each separate element.\nSensemaking is a key condition for the secondary reuse of data and comprises the third and fourth steps of the data journey. Summary statistics or visualisations are often suggested as a way of exploring quantitative data, but this already implies a decision by the user that the data is worth engaging with, and that it has been assessed as fit for use. To facilitate this, ‘immediate intelligibility’ is a key requirement for users when discovering data. However, without detail on how and why data was initially collected, this may present a challenge for even the most competent domain expert. These and other assessments of whether a dataset is fit for purpose, such as its granularity, provenance or format, have been described as ‘data evaluation’. Such evaluations tend to be dataset and task specific rather than applicable to all datasets. Users may need to make as many as 20 types of evaluations, related to the relevance, useability and quality of data.\nA second set of activities take place once the user begins to explore the data prior to use. These may include visualisations (provided or created by the user themselves) and a review of documentation, if available. Text summaries are often used to reduce cognitive and time cost during information seeking activities online. Research with dataset text summaries suggests that they do not necessarily contain all the required information (with methodology and provenance most frequently being ignored); that abstractive summaries require extensive natural language generation (NLG); and that they are subject to the idiosyncrasies of the people creating them, rather than being standardised.Further, while it may be possible to generate a convincing summary of data itself, it may be the metadata or context that is more important for sensemaking.\nThe multi-step complexity outlined above suggests that powerful NLG tools such as conversational search could be of great value to users requiring data. Conversational search can build understanding and semantic coherence over multiple rounds of interaction, while also supporting user needs through clarification and suggestion,. Conversational search is particularly useful for exploratory information-seeking contexts where users have insufficient background knowledge. While there is a substantial corpus of research on this, pre-dating the existence of large language models, there is little to no research on conversational dataset search.\nCGAI for information retrieval\n------------------------------\nGenerative AI refers to a class of artificial intelligence models that can create new data based on patterns and structures learned from existing data. Recently there has been explosive growth in conversational agents based on Large Language Models (LLMs). Such conversational generative artificial intelligence (CGAI) can be used to address natural language tasks, commonly include writing essays or literature reviews of scientific papers, and reading and summarising reports. They can also be used to answer complex natural language questions and support the answers with justifications. Further, many LLMs operate in multiple spoken languages and can therefore both translate text and justify the constructions used in translation, as well as providing text in many coding languages. CGAI has been used in a number of text manipulation and generation tasks such as creating financial research, structuring unstructured information and detecting stance in social media posts. It has been found to outperform crowd workers on text annotation tasks such as identifying types of social media posts.\nIn conversational search powered by LLMs, users are not restricted to keywords, and are able to state requests in a number of different ways. This suggests they can include more description about their information need in their questions. However, while this may increase the information available to the model to generate an appropriate answer, misleading or unnecessary information may adversely affect results. The solution to this is generally held to lie in prompt engineering; designing the requests in such a way that they minimise error and maximise potential for appropriate results. This requires a certain level of awareness of potential errors, skill in devising prompts to reduce such errors and assessing the effectiveness of prompts, which is challenging to non-experts. Prompts can also be used to drive the LLM to initiate question asking, by presenting a task that requires the agent to discover more information from the user to complete it. Research into using GPT to extract unstructured to structured information has shown that greater success is achieved when the information extraction task is broken down into easier sub-tasks, which again requires specific skills. A further strategy for improving search is to specify a domain-expert persona for the LLM to adopt. However, this has been found to also increase bias in language tasks.\nLLMs do not have access to contemporaneous, proprietary or personal data. Consequently, enabling access to external data sources, and augmenting CGAI for retrieval of information, in particular through detecting and calling APIs, is a major area of research. Another is addressing the issue of hallucinations. A hallucination is where a plausible answer returned is not based on existing fact, and is ‘made up’. Hallucinations are of particular interest in our research as they often occur in the form of references to sources that do not actually exist. The ability to identify hallucinations presupposes a certain amount of pre-existing knowledge on the part of the user, as LLMs are unable to directly refer to supporting documents in their training data There is currently no reliable way for developers to steer the behaviour of LLMs, and even less so by inexperienced users. The ability of LLMs to fluently produce apparently coherent text may mask the fact that this text may be of much lesser quality than it first appears (containing very little truth or fact), or that it may accidentally be damaging.\nExplaination is desirable (and in some cases, legally required) in intelligent systems. It makes it possible for users to act appropriately, is important for discovering and understanding novel concepts, and supports users’ trust in an application. A working definition of explainable AI (XAI) is ”Given an audience, an explainable Artificial Intelligence is one that produces details or reasons to make its functioning clear or easy to understand”. Explanations have some basic requirements, but they are largely varied based on the requirements, background knowledge, and so forth of the audience. Explanations are used to justify decisions, control function, improve accuracy and discover new knowledge. There are as many as 36 various notions of explanations, including efficiency (whether the explanation supports faster decision making), faithfulness (whether definitions truly reflect the datapoints) and robustness (the consistency of the results of the method.\nExplanations have a social dimension - they constitute an element of a dialogue that aims to transfer knowledge, therefore engaging the beliefs of the giver and recipient of the explanation. There is concern that explainability is dealt with primarily technically and less attention is spent on whether the explainability system is effective and efficiently human-centred. While much attention is currently being paid to understanding how models such as ChatGPT work, much of this insight is currently speculative. Consequently, most explanations for LLMs are post-hoc (the dataset, training methods and algorithms are opaque) and most focus on single data points, rather than the whole model. It is, of course, theoretically possible for a user to request that ChatGPT or other model ‘reflects’ on its answer and explains how it arrived at it.\nSummary\n-------\nPeople both need to locate and understand datasets in order to reuse them. This is often done as part of work, both paid and volunteer. This requires skills around crafting appropriate search, looking in appropriate locations and then assessing the fitness for use of the data. Despite improvement on the supply side, there is still a huge time and cognitive cost on the demand side. Conversational generative artificial intelligence (CGAI) tools such as ChatGPT and Bard have changed how we access information online. We are therefore interested in exploring how they could change people’s interactions with data on the web. Given the ability of CGAI to summarise large, complex sources, explain results, and write code, we hypothesise that chatbots may also be able to find data matching a user goal and help the user understand its scope, quality, and provenance, and to use the data productively. In the next section we describe our user studies with conversational dataset search..\nMethodology\n===========\nTo explore how users might engage in conversational dataset search, we invited a gender-balanced group of users to three workshops conducted in two languages, in which we asked them, “to think of one dataset (or group of datasets) that you would like to discover, using [CGAI agent]. (This can be a dataset you know exists, or a completely new-to-you dataset.) Once [agent] has found something (this may require more than one prompt), explore what it has returned to see if it is usable”. For further details see Appendix [appendix:a].\nData collection\n---------------\nOur pilot was run using the dominant large language model-based chatbot at the time, OpenAI’s GPT3.5. We wished to replicate the scenario we felt was most likely, which was to use the CGAI which had most public awareness and users. For our second workshop, we tried to account for some of the limitations of GPT3.5 by using GPT4. However, 2 days before our research, the live web search facility of GPT4 was suspended. In our third workshop, therefore, we used an (easily accessible) LLM chatbot with web connectivity. As Microsoft Bing is built upon similar foundations to ChatGPT, we opted for Google Bard. Details of the workshops are shown in Table [tab:workshopdetails].\n[h] [tab:workshopdetails]\n|p0.01|p0.08|p0.14|p0.15|p0.01|p0.09|p0.14|p0.08|p0.08| **#** & **Date** & **CGAI**& **Rationale** & **n** & **Gender** & **Attendees** & **Location** & **Language** \n1 & 22/05/2023 & Chat GPT3.5 & Highest adoption & 9 & m=4, f=5 & AI and LLM experts & London & English \n2 & 07/07/2023 & Chat GPT4 & Improved performance on GPT3.5 & 5 & m=3, f=1, e=1 & Political activists & Berlin & German \n3 & 12/07/2023 & BARD & Web connectivity & 9 & m=4, f=5 & Data professionals & London & English \nThe pilot workshop was conducted in London, UK with users with PhD level understanding of AI and large language models. This was to confirm the format of the workshop and protocol. Eight participants took part in the workshop in person, and 2 trialled a self-guided, asynchronous process following the protocol closely. After this workshop we made two adjustments. Firstly we decided to audio record the “in the room” discussion as this provided rich data and background that supported the data in the questionnaires. Secondly, we found the asynchronous process yielded substantially less research data so abandoned this mode. We used the ChatGPT web interface at chat.openai.com, which does not allow any special features or the uploading of large amounts of text or changes of hyperparameters such as temperature settings.\nFor our following workshops, we aimed to ask non-specialist users in a real world context to define a problem they have, and then think about what data they could use to address it. As it was important to our research to reflect a real world scenario, we aimed to work with people who were staff or activists in resource-constrained organisations, who would strongly benefit from using data and understanding its accuracy and provenance. We accessed these groups via personal networks.\nOur second workshop was held in Berlin, Germany with professionals who were engaged in political activism. This used the ChatGPT4 web interface, in order to benefit from any relevant improved performance, for instance the ability to input longer texts. All participants were present in person. The third workshop, in London, comprised data professionals with familiarity with the data economy and how data is used, but whose roles did not involve active coding. Three participants were located remotely but engaged in the workshop process synchronously via Teams. The number of chats and questionnaires collected are shown in Table [tab:data].\n[h] [tab:data]\n| **Date** | **Location** | **CGAI** | **Questionnaires** | **Conversations** | **Audio** | **Teams** |\n| --- | --- | --- | --- | --- | --- | --- |\n| 18/05 | London | ChatGPT3.5 | 9 | 9 | N/A | N/A |\n| 7/07 | Berlin | ChatGPT4 | 5 | 10 | Yes | |\n| 12/07 | London | Bard | 9 | 8 | Yes | Yes |\n| **Total** | 4 | 3 | 23 | 27 | 2 | 1 |\nDuring the workshop, participants were provided with the guidance that can be found in Table [tab:guidance], which resulted in the associated output (a full version of the guidance can be found in Appendix [appendix:a]):\n[h] [tab:guidance]\n|p0.7|p0.25| **Guidance** & **Output** \n[c]@l@Think about your motivation for a dataset search. \nEnter your motivation and any expectations (e.g. I know this dataset really well and know what I expect to see/I don’t know \nif anything like this exists) into the questionnaire \nDevise your prompt for [agent] \nEnter any considerations you had when devising your prompt into the questionnaire\n&\n[c]@l@Audio discussion (for workshops 2 and 3). \nQuestionnaire q1-7 [Appendix [appendix:b]\n \nEnter your prompt and explore [the agent’s] response & Conversational transcript \nAs a group, discuss ChatGPTs response and your assessment of this. Enter replies to the questionnaire as we go. &\n[c]@l@Audio discussion (for workshops 2 and 3). \nQuestionnaire q8-11 [Appendix [appendix:b]]\n \nWe asked participants to evaluate their own experience of using a conversational agent as Successful, Somewhat successful or Not successful. We then broke this down into two elements. We asked whether an appropriate dataset had been suggested (locating data) and then we asked participants to indicate on a Likert scale how useful the supporting information provided with the dataset was (sense-making). Finally we asked participants to compare the process with previous search experience, if relevant.\nData preparation and analysis\n-----------------------------\nThe datasets, analytical methods applied and the relevant research questions are shown in Table [tab:RQdata].\n[h] [tab:RQdata]\n|p0.3|p0.35|p0.25| **RQ** & **Data** & **Analysis** \n1. How can conversational agents support SOTA dataset discovery technologies? &\n[c]@l@Questionnaire questions 1, 2, 7, 8, 9 \nChat sessions\n&\n[c]@l@Descriptive statistics \nInductive thematic analysis\n \n2: How do people express their information needs? &\n[c]@l@Questionnaire demographics \nChat sessions \nQuestionnaire questions 3, 4, 5, 10, 11, 12\n&\n[c]@l@Descriptive statistics \nInductive thematic analysis\n \n3. Which of the main sense-making data discovery activities previously identified are supplied by Conversational AI? &\n[c]@l@Chat sessions \nQuestionnaire questions 3, 4, 5, 10, 11, 12\n& Deductive thematic analysis based on Koesten et al (2017) \nThe data was prepared as follows. The audio recordings were uploaded onto our institutional OneDrive, transcribed and (where necessary) translated using the OneDrive translate facility. The chat sessions and questionnaires were uploaded into our institutional OneDrive and again translated using the OneDrive translate facility. All translations were checked by a German speaker for accuracy.\nThe audio transcriptions, questionnaires, and chat session files were then uploaded into NVIVO for thematic coding by two of the researchers. Two of the authors individually coded a sample of the chat transcripts and long form questionnaire answers inductively for RQ1 and RQ2, and coded a sample of the chat transcripts deductively for RQ3. The resulting code lists were compared and conflicting results were discussed. This was repeated twice for both the deductive codes and inductive codes. The resulting code books are attached as Appendix [appendix:c]. The short questions were analysed quantitatively using Python, and the code is available as a zip file. We counted the frequency of responses and their combinations. We calculated the length of the chat sessions by counting the number of queries asked, the responses when interacting with the conversational agent, and their number of words.\n### Ethics\nThe interview study was approved by Author 1’s institution’s Ethical Advisory Committee via the Minimal Risk Procedure. Informed verbal (audio recordings and chat transcripts) and written (questionnaire) consent was given by the participants. No personal information was used in the analysis.\nFindings\n========\nRQ1: How do CGAI compare with state of the art dataset discovery technology?\n----------------------------------------------------------------------------\nIn this section we examine whether conversational search can compete in terms of success and usefulness for dataset discovery. We explore how people are using it in terms of the length of their queries and length of chats, and also examine some aspects of explanations.\n[t]![Results from the questionnaires regarding previously used methods (multiple answers possible).](other_search.png \"fig:\") [fig:othersearch]\n[h]![Results from the questionnaires regarding familiarity with the conversational agents.](familar.png \"fig:\") [fig:familiar]\n### Success rates of conversational dataset search\nIn order to assess our participants’ previous experience with datset search, we asked them what methods they have previously used to find datasets. While one participant had not previously searched for datasets at all, most of the participants had used Google search and specific data portals (Figure [fig:othersearch]). In addition, we asked participants how familiar they were with the conversational agents (Figure [fig:familiar] ). The majority had some or a little experience with them. Only one particpant had no experience. After the conversation with the chatbot, we asked participants about the process (Figure [fig:usefulsuccess]). The majority of participants (n=15) found the process of using chat agents “somewhat successful”. It is interesting to note that despite a large number of participants finding the process overall “useful” (n=14), only three of them found it fully “successful” in terms of finding a dataset.\n[fig:usefulsuccess]\n### Query and chat format\nAs can be seen in Figure [fig:numwordqueries], the total number of words in a query had a wide range, from 1 - 191. The modal number was 9.\n![ The figure presents an overview of the length of queries in terms of the number of words.](bar_word_question.png \"fig:\") [fig:numwordqueries]\nWe found that the mean number of words per query is approximately 18. This differs from data portals and web search, where the average number of words per query on a data portal is fewer than 3, while using the web search and then redirecting to a data portal is fewer than 5. However, it reflects the increased average length of prompts for text-to-image generation for which the mean and median have been found to be above 20 words.Figure [fig:agentsbox] shows that while the median number of words per query is similar across all three agents, participants using Bard did not use as many long queries.\n[h]![The box plot shows the range of query length for the three conversational agents.](boxplot_question_agent.png \"fig:\") [fig:agentsbox]\nIn terms of the queries themselves, we collected 268 queries across 27 chats with conversational agents. Previous research on conversational search using a text chat interface received 669 queries for 26 chats. However, this study used human participants playing the part of the conversational agents, which may have affected the length of chat. In addition, we found great variation in the range of queries per chat. Our study includes chats with 1 to 33 queries and on average 10 queries per chat as shown in Table [tab:statsagents]. The study above also found minimum and maximum message count varied widely, from 8 messages to 62 messages, and an average 26 queries per chat. We found our average number of queries was consistent across our German speaking group and our English users as shown in Table [tab:statslanguage].\n[h] [tab:statsagents]\n| **Statistics** | **ChatGPT4** | **ChatGPT3.5** | **Bard** | **Total** |\n| --- | --- | --- | --- | --- |\n| Number of chats | 10 | 9 | 8 | 27 |\n| Number of queries | 105 | 55 | 108 | 268 |\n| Min-Max queries per chat | 1-23 | 1-14 | 3-33 | 1-33 |\n| Average number of queries per chat | 10.5 | 6.1 | 13.5 | 9.9 |\n| Average number of words per query | 22.9 | 18 | 17 | 17.7 |\n| Average number of words per answer | 185.2 | 214.1 | 349.8 | 215.3 |\n[h] [tab:statslanguage]\n| **Statistics** | **English** | **German** |\n| --- | --- | --- |\n| Number of chats | 17 | 10 |\n| Number of queries | 163 | 105 |\n| Min-Max queries per chat | 1-33 | 1-23 |\n| Average number of queries per chat | 9.6 | 10.5 |\n| Average number of words per query | 17.6 | 22.9 |\n| Average number of words per answer | 278 | 185.2 |\n### Results of data requests\n[] [tab:datasetsearch]\n|p0.12|p0.09|p0.05|p0.2|p0.05|p0.3| **Data** & **Request** & **LLM** & **Response** & **Datasets** & **Comment** \nCase law & Implied & GPT4 & No access to database & 1 & curated micro dataset \nCase law & Implied & GPT4 & Does not mention having no access to database & 1 & curated micro dataset \nOpen source & Described & GPT4 & No access to specific datasets & 0 & Suggested 4 strategies for search. (on being prompted for sources, suggested 5) \nBirds & Described & GPT4 & No excuses & 4 & \nBirds & Described & GPT4 & Does not have access to datasets, also problem with internal server & 0 & 5 questions later it suggests 3, although not the same as those above \nPlug ins & Described & GPT4 & Not up to date & 0 & Recommends contacting Open AI \nAddresses & Described & GPT4 & Not up to date, no access to db & 5 & \nComplex numbers & Mathematical explanation & GPT4 & & & \nComplex numbers & Mathematical explanation & GPT4 & & & \nStudent numbers & Implied & GPT4 & No access to database & 0 & Recommended where to contact \nProperty prices & Implied & GPT3.5 & No real time access & - & Text summary describing the data \nOntology & Described & GPT3.5 & & 7 & With supporting information on what an ontology is \nConferences & Described & GPT3.5 & Yes! & 15 & \nKnowledge graphs & Described & GPT3.5 & Yes & 5 & \nCredit scores & Requested & GPT3.5 & Yes & 1 & With link \nIslamophobia & Described & GPT3.5 & No & 0 & Provides some later \nMusic emotion & Described & GPT3.5 & Sure! & 8 & \nPanama paper & Request & GPT3.5 & Not able to access & 1 & With supporting information \nFire hydrants & Described & Bard & Sure & 1 & partial dataset, states it is fictional \nTrees & Described & Bard & Yes & 1 & With link and usage \nLongitudinal data & Described & Bard & & 6 & ‘Most popular’ with links \nSport & Described & Bard & & 5 & With links plus suggestions for more \nHomeslessness & Described & Bard & Sure! & 1 & Curated dataset from unascribed source and some interpretation of trend \nPhysics & Implied & Bard & & 0 & Describes interpretation of underlying data \nJunk food & Implied & Bard & & - & Textual summary of report \nPeer production & Described & GPT3.5 & & 3 & In textual report form rather than list \nEducation & Implied & Bard & & - & Textual report \nThe number of initial results is very much less than would be returned from keyword search on the web as shown in Table [tab:datasetsearch]. In our terminology, a “request” dataset is named, a “described” dataset is defined but not named and an “implied” dataset asks for data as information, but does not specify it explicitly as data. As well as returning dataset suggestions (or portals where those datasets might be found) the CGAI also returned data sets they compiled themselves, sometimes with comment and sometimes without explanation. Implied requests in both Bard and GPT tended to be responded to in-kind – ie, without reference to the fact that what is being sought or returned is data. There is considerable inconsistency evident in not only the datasets returned, but also in explanatory commentary from the CGAI, as can be seen in the case law example where GPT4 comments on one chat that it has no access to the dataset, but does not mention this in the other.\n### Self-assessment of efficacy\nWe asked the users to evaluate how effective conversational search was. Many agreed that “as pure search, it is inferior [to googling]”, largely due to the failure to offer links and the lack of trust in the sources. However, they highlighted that looking for and exploring data inside CGAI meant that ‘next steps’ could be integrated into the process. One noted that using an agent “would probably give me a stronger idea of the nuances, internal politics, and where to look next based on what those organisations are connected to. But it would make sense to do that as a next step following this kind of query”. Another noted “if you have [to] deal with further steps separately, namely the application of BGH case law to concrete contractual clauses and the construction of arguments - I am faster than if I do this “manually”, i.e. using Google/specialist databases.”\n### Explanations\nWe also identified activity in the conversation transcripts where users asked the tool to explain why it has suggested the data it has (n=7). In existing keyword search, users cannot ask why information has been returned to them.\nThe majority of the interactions that requested explanations regarded links and sources, with participants asking, “so you can’t give me the link to it? I don’t understand why not” and “But this data should be available from your sources”. The responses mostly focused on the explanation that the agent did not have access to the web or that it was under development, or sometimes that it was “only a language model and don’t have the capacity to understand and respond.” One participant asked, “I hear that you’ve given URLs and links to other people who are using your services. Why can’t you link to the papers you’ve pulled from to answer my questions?” In answer to this, Bard returned the common “under development” reply but also noted that it was unable to link to copyright material or websites that “might contain adult content”. While these explanations are not informative about the specific case, they are generally informative about the rules imposed on the agent.\nWe also found participants asking questions about the sources of the CGAI’s knowledge, for instance, “Where does your knowledge of the global distribution of birds come from?” The answer to this revealed that the agent did “not have direct access to my training data” and that “my knowledge is based on patterns in the data I saw during my training.”\nParticipants confronted the agents about veracity of responses. One participant asked ChatGPT “are you lying?” In response, Chat noted that it did not operate within the context of ‘truth’ or “lies’ but noted that the study it had referred to in the previous response was ‘misleading” as it was a “fictitious example”. Another queried the language used by challenging, “So by ‘estimating’ do you mean you guessed?” GPT4 responded that “Estimating” in this context does not mean “guessing” and stated that it makes, “an informed statement or prediction based on the available information”. A participant requested that Bard provide an image from the data, then asked, “That image is not what the code describes. I can also see that the axes aren’t labelled the same. Did you actually run the code?” In response to this challenge Bard provided multiple draft answers. These answers conflicted with each other and obscured the process. as can be seen in Table [tab].\n[h] [tab:Barddraft]\n|p0.1|p0.85| **Interaction** & **Text** \nFeedback prompt & “That looks suspiciously the same as before. I am sad” \nDraft # & Bard’s answer \nDraft 1 &\n[c]@l@You are correct. The image that I showed you is the same as the one that I showed you before. \nI apologize for the mistake. I am still under development and learning how to generate correct \noutputs. I will try again to generate the correct output and get back to you. \nIn the meantime, I would like to ask you a few questions to help me understand what went wrong. \nDid you run the code yourself? \nDid you get the same output as I did? \nIf not, what was the output that you got? \nYour answers to these questions will help me to understand what went wrong and how I can fix it. \nThank you for your patience and understanding.\n \nDraft 2 &\n[c]@l@I apologize for the confusion. The image that I showed you is the same as the one that I showed \nyou before. I made a mistake in my previous response, and I thought that I had run the code again \nand gotten the correct output. However, I did not actually run the code again. \nI will try to run the code again and get the correct output. I will let you know when I have done so. \nIn the meantime, I can explain why the image looks the same as the one that I showed you before. \nThe code that I wrote is a simple code that generates a 2D contour plot of the q profile as a function \nof normalized poloidal flux and time. The code does not take into account any of the physics \nof tokamak plasmas. \nAs a result, the image that the code generates is not very realistic. The q profile in the image does \nnot evolve in a way that is consistent with the physics of tokamak plasmas. \nI apologize for the inconvenience. I will try to get the correct output of the code and show it to you.\n \nDraft 3 &\n[c]@l@You are right. The image that I showed you is the same as the one that I showed you before. I \napologize for the confusion. I am still under development and learning how to generate correct code \nand output. \nI will try to run the code again and get the correct output. I will let you know when I have done so. \nIn the meantime, I would like to ask for your help. Can you please help me to debug the code? \nI think that there is a mistake in the way that I am defining the q profile. If you can help me to find \nthe mistake, I will be able to generate the correct output. \nThank you for your help!\n \nAdditionally, one participant sought information on the machine learning that was used to form the dataset. The agent, in this case Bard, provided a description of three different techniques and how they were applied. The participant followed this with a request for sources for this information, which was provided.\nWe also identified instances of agents providing detail about their training that was not specifically requested. In this case, a participant asked “Which countries do you have data for?” ChatGPT4 stated, “During the training, I was not provided with specific datasets on the most common bird species in each European country. Instead, I was trained on a variety of texts from many different sources, including books, websites, and other texts that contain information about birds and other topics.”, “I can also access scientific literature and studies that are publicly available and try to extract relevant information from them.”. However, it is important to note that we also identified many occurrences of agents providing entirely unverified and unexplained supporting information. For instance, one participant had to prompt the agent to be aware of the CONAN dataset. It had previously stated no dataset in this field existed. The participant informed GPT3.5 of the dataset, then asked whether the annotators of the datasets were experts. GPT3.5 replied in the affirmative. “Yes; the annotators for the CONAN dataset were experts in the field of hate speech and language analysis. The dataset was annotated by team of professional annotators; including linguists and social scientists, who were trained in hate speech identification and provided with guidelines on how to identify and label hateful content:”. It did not provide any links to its source or explanation as to how it was now able to provide detailed information about a dataset it had previously denied all knowledge of, rendering the process more opaque. Some form of “I’m unable to help you with that, as I’m only a language model and don’t have the necessary information or abilities” was the most common (non)-information about its own abilities supplied by the conversational tools (n=13) with it recurring 6 times in one conversation.\n### Summary\nConversational search was largely characterised by users as a “useful” experience even if few (n=3) people felt it was wholly successful in terms of discovering a dataset. The number of words in each query were considerably longer on average than current search, and there was a very long tail. Explanations were largely limited to reasons that the agent could not perform a given task, and not always accurate. However, the ability to integrate further usage tasks into the conversation, such as returning code to access an API, was seen as a positive aspect.\nRQ2: How do people express their information needs when using CGAI for data discovery?\n--------------------------------------------------------------------------------------\nIn this section we explore the prompts users employ when attempting to discover datasets.\n### Key terms in conversational dataset search\nWe performed a visual representation of the most frequently occuring terms used by users in interacting with the agents as shown in Figure [fig:wordcloud].\n [fig:wordcloud]\nThis suggests that, as with keyword search, users were specifying their data needs geospatially (“london, göttingen, cities”) and temporally (“years, 2022, 2021”). We can see evidence of requests for related evidence (research, study, survey) as well as format based requests (lists, graphs, databases). As noted in Table 6, the average number of words in a question was just under 20. However, amongst the outliers was a 71 word request that specified two kinds of content (street names and postcodes) of a location (EU country) as well as stating conditions around licence, timeliness and format (“no PDFs”).\n### Expression of information needs as data prompts\nIn our thematic analysis we identified three main ways in which people requested datasets. One was requesting the dataset directly, eg, “can you find me the German credit fairness dataset” [datasetrequest].\nAnother was where the dataset was described, as in “can you provide me with a list of available datasets about Islamophobia in Italian containing Tweets labelled as hate speech” [describeddatasets]. One approach that yielded results for the user was to explicitly suggest a document that might contain the data they sought, “for example a survey which asks people about their reasons for opting for junk foods”.\nA last format was implied datasets [implieddatasets], where the request might ask, “What has been the average property price in London, from 2000 to 2010?” Some users varied between these modes of asking, with one user describing a dataset in 4 different ways and then asking 3 further times for an implied version. We noted evidence of requests being made in a way that implied memory of the previous request, for instance “what about ’Global South’ cities”. One such implied dataset request was for a list of judgements on cosmetic repair in German case law. The CGAI recognised the implied request, and replied, “I do not have direct access to a complete database of case law or the exact wording of specific clauses in BGH rulings. However, I can mention a few general cases in which the Federal Court of Justice (BGH) has issued rulings on cosmetic repair clauses until 2021.” Consequently, it curated a small, if necessarily incomplete, dataset for the user. Although no users made requests based on personas, one did ask for an “ELI5” response - “explain like I’m 5 (years old)”, which ChatGPT4 happily complied with, giving examples using fantasy friends and apples and bananas.\nThe most common question after the initial engagement with the agent was for links [linkrequest]. Separating out the various parts of a request was more common than creating a compound request.Another approach taken by participants was to request the source of the dataset [sourcerequest],rather than simply ask for a link. A user who self-defined as being more experienced with CGAI attempted to use slightly different language to access the datasets. Rather than simply asking for links, which they were aware would result in a “I can’t access the web” reply, they asked “Can you provide directions on how to access the datasets?” [accessdatasets]. In fact, this resulted in the return of links that were wholly inaccurate, which motivated the user to request working links [linkrequest], thus bringing them back to the request. Overall, the majority (n=20) of users requested a link or source as a follow up question.\nAlongside link requests, users asked for more information about the dataset itself [additionaldatasetinformation]. Such questions might be general, eg, the licences of datasets or more technical, such as “Which of these datasets provide emotion annotations in the Valence-Arousal space?” However, a major issue with such questions was that the answers often were attended with caveats such as, “Please note that these are fictional examples, ” Users sometimes requested further information about the context of the dataset [furthercontextinformation], rather than the dataset itself. This included requests such as, “What can you tell me on the NeurIPS conference, then?”, “were the annotators of the dataset experts in their field?”\nWe also found instances of a question being repeated after a particpant had interacted with the agent and supplied it with information. Interestingly, this did not necessarily improve results. In multiple cases, a repeated question that the agent had initially attempted to answer met with responses on a second asking such as, “I’m a text-based AI and can’t assist with that.”\nUsers requested information on how to use or analyse the dataset provided [advicerequest]. These included the technical and specific: “Explain how would you go about performing an analysis of the social network of the people contained in the Offshore Leaks Database:” and “what are the limitations of tree canopy cover analysis?” as well as the more general, such as, “Can I compare this dataset to other cities”.\nGenerative AI is not restricted to narrative texts and we saw some examples of people requesting other forms of information to be returned [multimodaloutput], for example, “Write down some code in Python I can use to analyse the Offshore Leaks Database;”, “Can you give me the first 10 triples of each dataset”. Users asked for data visualisations, for instance, “can you make a graph to include the comparative tree canopy cover for all of the cities you have mentioned?”. A particularly positive experience was had by a user who asked if GPT4 could query a REST API, “And then he actually built me corresponding requests, which I can then use”. This integration of further activities [furtheractivity] appeared on a number of occasions, with users requesting drafts of letters, opinions on texts and drafts of legal justification (and a draft of an invalid legal clause). One user asked “can you send a Freedom of Information request too?” which Bard agreed to do, and offered, “to keep you updated on my progress.”\nParticipants also countered some of the information provided by various agents [disagree], such as, “That plot doesn’t show what I wanted. Can you try again?” This occurred over a range of answer types including lists and data visualisations. We also saw instances of participants providing information to the agents [provideinformation], for instance, “The MusicEmo dataset does not exist, that is why you could not find any link for it. Can you remove it from the list please?” In these examples, participants are contending both with the agent failing to provide existing datasets and hallucinating non-existent datasets. In this instance, the participant engaged repeatedly with the agent to achieve an accurate list. The third “input” interaction we identified was when users referred back to earlier conversation [Conversationreminder], for instance, ’what is the role the 110,000 asylum seekers in France are playing in the overall homelessness you gave me at the beginning of the conversation’.”.\n### Summary\nThis research question has attempted to investigate how participants express themselves when attempting to discover data. We find that, offered more natural ways of defining their information needs, users describe datasets in a variety of direct and indirect ways. Similarly, we see variety in the requests (links, access, sources) that follow the suggestion of a dataset(s). While most participants engage with these two steps, what follows after is less defined. Participants may, dependent on their needs and what is offered, explore the data further, seek advice on using the data or even enter into argument with the agent.\nRQ3: Which of the main sense-making data discovery activities previously identified are supplied by CGAI?\n---------------------------------------------------------------------------------------------------------\nWe base our analysis here on the framework of interaction with structured data discovery. They present the sense-making aspect of data discovery as consisting of “evaluation” and “exploration”. Here we explore which aspects of evaluation and exploration were prompted by users, and to what extent the CGAI met them.\n### Evaluation\nEvaluation is broken down into three aspects; relevance, useability and quality, with multiple sub categories. The number of users engaging with each of these sub-categories is provided in Appendix [appendix:c].\nRelevance: Relevance is the extent to which the suggested dataset meets the task needs of the user. Relevance assessments include looking at the data context, coverage, summarisation, original purpose and granularity. As noted above, users often ask for context in the form of either reports or papers data first appeared in, or in which it is subsequently used. Conversational agents are well placed to provide this. However, although sometimes they did suggest texts, they also sometimes claimed they could not do it because of the limitations of being an LLM. Coverage was also requested, with users asking, “Which countries do you have data for?” and “Do these datasets provide full tracks or audio clips?”. Again, success was mixed, but the agents were certainly able to understand and address the requests. At one point, ChatGPT3.5, in the absence of being able to suggest a dataset with appropriate coverage, recommended, “you can consider creating your own dataset by using various methods such as data scraping or using APIs to collect text data from various sources such as social media platforms”. Although people often use generative AI to write documents it is often more powerful in reading them, so summarisation is intuitively a task where they could perform well. Users asked for, a “general description” of a datasets contents, as well as “a few example records contained in the dataset” to better understand what the data looks like. Users also asked for specific parts of the text to test that the agent could regurgitate them - “What was the clause in the 1st judgment?” Unsurprisingly, once an agent was working with a confirmed text it performed well. “Granularity” (the measure of the level of detail in a data structure) is one of the 4 key types of dataset descriptors used in search and there were a number of requests of this type. There were also requests further into the conversation. However, these were rarely successful as the agent did not have access to the actual dataset, for instance requests to break down student data by subject. Regardless, the agent was still able to provide information on granularity, for instance, when asked for data “By street and by postcode” they provided a suggestion of the dataset but noted, “This data may still require further processing to extract all street names and postcodes”.\nUseability: In this section the framework covers aspects of documentation, licensing, access, machine readability, language, labelling, ability to share and format. Users requested information that might be found in documentation, particularly definitions, eg “what is classified as tree canopy cover”, “how does the Food Foundation define ’poorest households’?” and “can you give me the definition of homelessness”. In these cases, the agent provided a number of definitions and sources, but these were not necessarily those used in the dataset. The agents also provided guidance on whether documentation was available, or should be consulted.\nThe agents sometimes supplied unprompted licence information or advice with the dataset suggestions. In one case, having suggested a dataset, “the agent told me that there might be problems after all, that the license and the data are not so available after all. That was a dead end.” Whether the data was licenced or not - the veracity of this interaction - is unverified. This is positive as few participants actually requested licence information, but it is a necessity for use.\nRegarding access, the agents often suggested contacting the data publishers directly for access information. More detail is sometimes provided: in one interaction, the chat agent suggests a number of datasets for the user to investigate. The user then asks whether one of the datasets, Earth Engine, costs to use the data, to which the chat agent gives a full overview of the costs. Again, the accuracy of this is unverified. Another form of unprompted advice seen was around potential considerations for use of the data. ChatGPT4 discussed cost, access, privacy, and licences, and recommended contacting the data publisher directly for more information on this.\nIn terms of language, the models we used worked in around 40 major languages. As well as searching in both English and German, users requested datasets in other languages, for instance, “Can you provide me with a list of available datasets about Islamophobia in Italian containing Tweets labelled as hate speech or not?”. The agents were able to move easily between languages.\nThere were also occasions on which the agent requested the user to provide information, such as when Bard asked one user to share the code they had used to create a graph, as, “Your answers to these questions will help me to understand what went wrong and how I can fix it.” and asked another user to provide detailed information so it could draft a letter. These also demonstrate the facility to build on complex queries, retaining context from previous rounds of requests.\nQuality: Under the heading of quality the framework examines issues of consistency, completeness, collection methods, provenance, and what has not been included in the data set. Users asked about and received answers on collection methods such as, “Can you explain what machine learning techniques were used to form the dataset?”, and “What studies, books, internet articles, etc. did you use?”. One way that users sought to understand the quality of the data was by asking whether the data was used by other organisations. For example, one user asked whether “any of these [have] been used in research publications?”, while another asked if a “study [has] nevertheless been cited in scientific papers?” In terms of provenance, as we saw in Section 4.2.2, there are a great many questions from users about sources of data and figures. However, equally, this is one of the more unreliable aspects of the agents. Firstly, the number of questions about sources is provoked by their opacity regarding the sources of their information, and secondly, there is frequent admission of unreliability, including fictitious examples.\n### Exploration\nIn “exploration” the major issue is that the datasets are not returned within the agent, so basic visual scans and an assessment of basic errors are challenging. Visualisations and data samples were returned in response to requests, although in at least one case, the visualisation was inaccurate. We found examples of metadata and documentation being addressed in the chat sessions, (although again, we are not necessarily testing the veracity of the claims). Interestingly, the references to metadata in the chats were suggestions by the agents, rather than requests by the users. Along with the finding above that explicit reference to documentation was only made by the agents, these results imply that the agents are able to supply information to support data exploration even when they are unprompted or they cannot return the datasets themselves. However, while this is often thorough, it is necessarily generic and not specific to the dataset in question. We found that on occasion the LLMs would describe the data in sufficient detail as to include at least some descriptions of the content of the columns, as it did in the tree dataset. We also explored the data for related aspects, such as “columns”. We found two references. One was an instance of ChatGPT 3.5 describing how to clean data; the other was a comment in the questionnaire from a user who was concerned that subheaders would make it difficult for ChatGPT to identify appropriate data:“a matter of fact, these kinds of in depth information are contained as sub-columns in datasets and not as primary class labels.”\n### Summary\nThe evaluation and exploration tasks outlined above are crucial as it is based on these that people make the decision to use data (or not). We find that the majority of these elements are identifiable in the chats in some way. Of particular interest is the fact that the agents do, on occasion, suggest generic exploration and evaluation tasks without prompting, and also request input from the users on data evaluation tasks.\nDiscussion\n==========\nWhat, then, are the emerging user practices around dataset search with conversational tools?\nConversational dataset discovery enables the explicit enactment of the entire process of dataset discovery. The task can be specified in text by the user, rather than existing as an implicit motivator. Similarly, the data use can be integrated into the process. Overall, therefore, dataset discovery can be greatly enhanced by CGAI. We first discuss 6 areas of promise, then review 4 areas of challenge, before presenting 5 guidelines for prompting data. We then present our model for user-dataset discovery interaction using conversational agents.\nCuration - promise\n------------------\nA key aspect of the agents is the curation of data. Unlike keyword search where (often) thousands of potential dataset links are returned, far fewer datasets were suggested by by the CGAI in response to each query. Given appropriate web connectivity, it can be extrapolated that CGAIs should be able to select only the most relevant datasets from the web and provide them to users enhanced with supporting information. Another form of curation was the returning of a particular number of datasets in response to user specification. Finally, a third type of curation in evidence was that seen in response to implied dataset requests, such as the suggestion of the German case law examples, which again has great potential for future use.\nComplex queries - promise\n-------------------------\nComplex queries have long been an idea in research papers, but CGAI allows this to be realised at scale with non-expert users. While we saw longer, more detailed queries these did not necessarily correlate with identifying datasets more accurately. However, the ability to “remember” the conversation, to return and reexplore earlier interactions, and to build on sensemaking with usage activities in the same space is beneficial.\nFeedback - promise\n------------------\nA key strength of CGAI is the ability to incorporate information fed by users during interactions. However, while users incorporated correction and redirection strategies, this met with varying levels of success. Unsurprisingly the users with the greatest success in this area were those in the more expert group. This suggests that users can control managing this issue with increased familiarity.\nSupplemental information -promise\n---------------------------------\nUnprompted information supplied by the agents often supported key areas of concern with data such as access, licence, cost, and so on. In some cases it made users aware of issues, but did not resolve those issues, for instance, in such responses as “be aware of licenses”, while not actually identifying the license in question. This suggests that while CGAI may not be reliable for data discovery, it may well have potential as a tool of data literacy, as users are not required to come to search with existing knowledge about what aspects of evaluation and exploration are important.\nAdvice - promise\n----------------\nSimilarly, CGAI has potential as supporting tools for data skills, via the provision of advice. Such advice includes, \"it gave me guidance on finding [datasets] in the future” or advice on usage of the data.\nMultimodality - promise\n-----------------------\nHowever, the aspect of CGAI that offers perhaps the most potential as a tool for data is their ability to provide responses in a number of modalities. The ad hoc creation of visualisations and summaries enhances exploration and reduces the skillset a user requires; similarly the ability to provide code supports usage.\nHallucinations - challenge\n--------------------------\nWhere the fiction created by CGAI is very closely correlated with the truth this does not present a problem; nor where the user is very expert in the domain. A third area in which this is not problematic is where the agent provides general rather than specific information, such as “look in these kinds of reports” or “check the metadata”. A somewhat positive response to the fictitious nature of the responses is that users worked hard to counterbalance this, asking for sources, evidence and provenance, in an attempt to navigate around admissions of unreliability. However, there are aspects of hallucinations that work against commonly agreed positive strategies for data search. An example of this is text summarisation. Text summarisation is seen as an accessible and technically achieveable exploration technique. The UK’s Office of National Statistics, for example, a major producer of public data, has experimented with integrating LLM technology into its website search, so that users can extract data from its text bulletins. However, while text summarisation is a strength of CGAI it does not help if the report or text summary is entirely fictitious, and the user is not aware of this fact. Domain expertise remains of importance to counteract hallucination, but a key strategy is verification, and this depends on moving outside of the conversation to check other sources, thus affecting the ability to conduct the dataset discovery process entirely within the conversation.\nExplanations - challenge\n------------------------\nThere is no ground truth for explanations or correct reasons the agents could offer. In terms of a global explanation, this most frequently took the shape of statements about limitations including “As an AI language model, I don’t have real-time data access”; “I have a cut off date for training data of September 2021”; As noted, the bland “I’m not able to help with that, as I’m only a language model” was the most frequently used statement about agents’ capabilities. Local explanations about particular responses are generally generic. However, the most problematic aspect of agents’ own capabilities is the unreliability and opacity of its own post-hoc explanations. This opacity includes information supplied after the event regarding its ability to operate within the boundaries of “truth” and admitting it may therefore be “misleading”. The existence of multiple drafts, each providing potentially differing explanations, further undermines any reliability in this area.\nInconsistency - challenge\n-------------------------\nWe saw great inconsistency between results, even where the same prompt was used. A step by step approach has been found to be successful to counteract this in some maths and reasoning tasks. Recursive prompting might address this, however, that is beyond the scope of this study. Instead, ways need to be found to work around, or indeed, with inconsistency. Such approaches may lie in triangulation of information, either within answers (or drafts) or with external sources.\nCapability expectation - challenge\n----------------------------------\nConversational agents are not search engines with super powers. Users may expect them to able to return a link or csv file, but this is not currently what happens. In other words, users need to approach conversational dataset search less in the anticipation that they will receive a magic bullet answer, and more in the expectation of receiving support to find and use an appropriate dataset, which may happen using a different tool on the web. Further, it may be that for “information veracity, information literacy and serendipity” reducing human interactions with web search is not desirable.\n[h] [tab:guidlines]\n|p0.01|p0.3|p0.6| **#** & **Guideline** & **Description** \n1. & SELECT THE CORRECT PROMPT FORMAT FOR THE DATA & Naming a dataset explicitly is most likely to return the dataset without much supporting information (unless this is specified in the prompt); describing a dataset is likely to return suggestions of datasets with some supporting information and implying a dataset is likely to return a narrative description of some key data points. \n2. & MIX REQUEST PROMPTS FOR MORE EFFECTIVE DISCOVERY & The different types of requests can be employed iteratively, in order to build a better picture of the data. For example, a user might begin with an ‘implied’ prompt in order to understand the bigger narrative of a dataset, and then move to a ‘described’ prompt in order to explore more about the data itself. \n3. & USE RESULTS AS A BASE FOR IMPROVEMENT & It is common for users to request CGAI to supply a draft of a text, such as a letter or report, which they then edit and improve. Similarly, the results of data discovery can be used as a basis for guided information retrieval. \n4. & USE REMINDERS & The CGAI responds to the prompts of the users but is fundamentally passive. Building towards the goal should include explicitly reminding the CGAI of the task and previous interactions in the conversation. \n5. & EXPLOIT THE CODING ABILITIES OF CGAI & For instance, instead of asking for a visualisation, ask for code to create a visualisation. An incorrect visualisation with no explanation of how it was created is harder to debug than incorrect code. \n![Model of dataset prompting interactions with CGAI](diagramforLLMpaper.png \"fig:\") [fig:example]\nPrompting for dataset discovery\n-------------------------------\nTable [tab:guidlines] summarises 5 key insights into prompting strategies that may increase effective data discovery with CGAI. Based on the thematic analysis in 4.2.2, we have also developed a model of prompt interactions for dataset discovery with CGAI (Figure [fig:example]). This not only captures how people interact with CGAI but also proffers a starting point for developing improved prompting at each stage of interaction. All users participated in the first stages, the vast majority in the second and then in some or all of the last activities. The task is often made more explicit, although this can be revealed at any stage throughout the conversational search, not necessarily in the first prompt. Evaluation and exploration are iterative and can happen more or less simultaneously. Further, our model reflects the increased interactivity facilitated by CGAI and users can explicitly disagree with results within the conversation. Usage is an active part of our model as it can be performed within the same set of activities as the discovery and sensemaking. Users sometimes engaged in several stages of activity at once, such as: “You suggested this link to me earlier as a source https://octoverse.github.com/. Do you have any other links on how to use open source?”, which is both seeking advice (how to use open source) and a reminder of an earlier interaction.\nConclusion\n==========\nMaking data useable requires tools that support individuals’ data journeys. This includes specifying a goal and finding the data they need; exploring new sources of data; assessing the relevance and credibility, and deriving insights. In this study we explored whether, given a specific domain, can conversational agents based on large language models suggest useful datasets? Do the rationales make sense? Does CGAI help answer critical questions on data provenance, licensing, format and more, and how do users assess this?\nWe find that CGAI can suggest relevant dataset areas, and allows more engagement with the data and contextual information. However, we also found that the suggestion of datasets and provision of information, especially more obscure data, are unreliable; that suggestions are prone to confusing levels of inconsistency; and domain expertise and web search are often required to establish veracity. Its real strength currently is incorporating advice on evaluation and usage, and supplying code in various forms to support this, reducing the need for advanced data skills to work with data effectively. Of particular note is the fact that users are able to supply explicit feedback, corrections and information to the CGAI to improve outcomes.\nThus, until these issues are addressed, CGAI will not fulfil its very real promise of being an effective end to end data discover tool. Our contributions towards this are i. a model that describes user interaction with CGAI when engaging in dataset discovery and ii. a set of guidelines for data prompting.\nLimitations\n-----------\nIn terms of the users we worked with this is very specifically about data literate people operating in highly served languages. We cannot extrapolate to less data literate people in under served languages. In terms of the agents, given the unstable and changing nature of the answers even within workshops, it may be that entirely different responses would be elicited from the agents if any number of circumstances changed, for instance, increased access to the web, more use of temperature settings, and so on.\nFuture Work\n-----------\nMore work needs to be done on prompting agents in a way that reduces the domain knowledge required to elicit datasets. Our findings further suggest that a valuable follow up study would be to pre-locate datasets and ask the agent to explore it, to understand what further activities and services of data use agents could consistently provide. It is key to investigate the use of conversational agents in under-served languages. This will not only let us further explore the integration of AI into learning and society, but also more broadly understand the support that this technology offers to these communities. Given the datasets on which large language models are trained we hypothesize there may be substantial differences in the ability to discover and make sense of relevant datasets.\nAppendix: A\n===========\n**PILOT RESEARCH - Conversational Agents for Data Discovery**\n*Verbal Information and Consent*\n* This is a research workshop for a paper we intend to submit to CHI, entitled Data Discovery with Conversational Agents.\n* As we are audio recording the session today we may collect personally identifiable information. These will be transcribed and anonymised. Until that happens we will hold your data either securely on KCL equipment used by the researcher, or in the [institutional] secure repository, OneDrive. We will also collect transcripts (screenshots) of CGAI conversations. We do not foresee identifiable information being captured, but if it is, we will anonymise it. We are also using questionnaires, these will be fully anonymous and have their own consent procedure.\n* We will process your personal data (if collected) on a legitimate use basis for university research. We have applied the balance test.\n* We may use quotes in the paper but these will be anonymous.\n* We can’t withdraw your data once it has been anonymised, but you can withdraw from the study at any point.\n* We’d like your consent for future use of your anonymous data collected today by other academics studying conversational agents. If you don’t consent to your data being used in this study or in future you are free to leave now.\n* Any questions?\n*The study*\nIn this research, we would like you to think of one dataset (or group of datasets) that you would like to discover, using ChatGPT. (This can be a dataset you know exists, or a completely new-to-you dataset.) Once ChatGPT has found something (this may require more than one prompt), explore what it has returned to see if it is usable.\n*Timeline* \n[h] [tab:appendA]\n|p0.09|p0.85| **Timing** & **Activity** \n4- 4.05 & Introduction \n4.05 - 4.15 &\n[c]@l@Think about your motivation for a dataset search. \nEnter your motivation and any expectations (eg i know this dataset really well and know what I \nexpect to see/I don’t know if anything like this exists) into the questionnaire \nDevise your prompt for ChatGPT \nEnter any considerations you had when devising your prompt into the questionnaire\n \n4.15 - 4.30 & Enter your prompt and explore ChatGPT’s response \n4.30 - 4.50 & As a group, discuss ChatGPTs response and your assessment of this. Enter replies to the questionnaire as we go. \n4.50 - 5.00 & Wrap up! Please screenshot your ChatGPT conversation and place them [anonymous location]. Please name them according to the kind of dataset you were looking for. \nAppendix: B\n===========\n**Workshop Questionnaire**\nQuestionnaire:\n1. How familiar are you with ChatGPT?\n2. Thinking about either a personal or professional situation, what methods have you previously used to find datasets? Please tick all that apply\n3. What kind of dataset will you ask ChatGPT to find for you today?\n4. Why would you like to find this dataset (or why have you looked for it in the past?)\n5. Do you have an idea of what Chat GPT might suggest for you, either in terms of datasets or explanations? Please describe.\n6. What considerations did you have when devising your prompt (your question to ChatGPT)?\n7. What was the outcome today of discovering data with ChatGPT? Please select one.\n8. Did ChatGPT suggest an appropriate dataset?\n9. How useful was the explanation or supporting information provided by ChatGPT?\n10. How did you find the process of interacting with Chat GPT?\n11. Compared to previous methods of discovering data, how effective did you find the ChatGPT discovery process?\nAppendix: C\n===========\n[h] [tab:inductive]\n| **Code** | **Description** |\n| --- | --- |\n| datasetrequests | Requesting a dataset by name or close approximation |\n| describeddatasets | Describing the contents or function of a dataset or database |\n| implieddatasets | Asking for information based on data |\n| accessdatasets | Directions on how to find the dataset |\n| sourcerequest | Request information on the source of the data |\n| linkrequest | Request a link to the data |\n| advicerequest | Information on how to use or analyse the dataset |\n| furthercontextinformation | Information on the context of the dataset |\n| additionaldatasetinformation | Information on the dataset itself |\n| multimodaloutput | Returning other forms of output eg code |\n| furtheractivity | Integration of usage activity |\n| disagree | Countering information without providing further information |\n| provideinformation | Countering information (hallucinations) by providing further information |\n| conversationreminder | Referring back to earlier interactions |\n| explanationlinks | Querying why links can’t be provided |\n| explanationchallenge | Challenging an explanation provided by the agent |\n| explanationtechnical | Requesting a technical explanation |\n| explanationveracity | Issues with explanation truthfulness |\n[h] [tab:deductive]\n| **Code** | **n=** |\n| --- | --- |\n| data context | 5 |\n| coverage | 3 |\n| summarisation (of data content) | 5 |\n| original purpose | 2 |\n| time frame | 20 |\n| granularity (level of detail) | 4 |\n| documentation | 6 |\n| licensing | 5 |\n| access | 12 |\n| machine readability | 1 |\n| language | 1 |\n| labelling | 5 |\n| ability to share | 0 |\n| format | 6 |\n| consistency | 2 |\n| completeness | 3 |\n| collection methods | 5 |\n| provenance | 20 |\n| what has not been included in the data set | 8 |\n\nPrompting Datasets: Data Discovery with Conversational Agents\n=============================================================\nCan large language models assist in data discovery? Data discovery predominantly happens via search on a data portal or the web, followed by assessment of the dataset to ensure it is fit for the intended purpose. The ability of conversational generative AI (CGAI) to support recommendations with reasoning implies it can suggest datasets to users, explain why it has done so, and provide information akin to documentation regarding the dataset in order to support a use decision. We hold 3 workshops with data users and find that, despite limitations around web capabilities, CGAIs are able to suggest relevant datasets and provide many of the required sensemaking activities, as well as support dataset analysis and manipulation. However, CGAIs may also suggest fictional datasets, and perform inaccurate analysis. We identify emerging practices in data discovery and present a model of these to inform future research directions and data prompt design.\n<ccs2012> <concept> <conceptid>10003120.10003121.10003124.10010870</conceptid> <conceptdesc>Human-centered computing Natural language interfaces</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10003120.10003121.10011748</conceptid> <conceptdesc>Human-centered computing Empirical studies in HCI</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10002951.10003317.10003331.10003333</conceptid> <conceptdesc>Information systems Task models</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10002951.10003317.10003347</conceptid> <conceptdesc>Information systems Retrieval tasks and goals</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10002951.10003317.10003359</conceptid> <conceptdesc>Information systems Evaluation of retrieval results</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10002951.10003317.10003331</conceptid> <conceptdesc>Information systems Users and interactive retrieval</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> </ccs2012>\nIntroduction\n============\nDataset discovery, or identifying secondary data useful for research, business, or individual purposes, is a vital activity in contemporary life. This is increasingly true for not only data experts, but also for more casual users, who typically do not require large scale, well promoted datasets. Dataset discovery is composed of two parts: searching for data, and assessing the data to ensure it is fit for the intended purpose. This second part is often characterized as sense-making. The predominant methods of discovery currently are searching on a data portal (e.g. data.europa.eu) or via a search engine, including specialist search such as Google Dataset Search.\nMaking data useable requires tools that support individuals’ “data journey”. This includes specifying a goal and finding the data they need; exploring new sources of data; assessing the data’s relevance and credibility, and deriving insights. Additionally, to support non-expert users, such tools need to function effectively in low- to no-code interactions that are suitable to populations with diverse economic, cultural and educational backgrounds.\nCurrent data journeys involve creating ad hoc workflow processes on the fly over a number of tools, many of which are not easy for non-experts to use. A typical set of search and sense-making activities might look as shown in Figure [fig:fivefigures]:\n[fig:fivefigures]\nWhile the user is able to find, explore and use the data, many issues remain. The user is not able to be explicit in their goal in undertaking the search, but only to best estimate the keywords that might locate data that aligns somewhat with the needs. Although the search tool offers summaries and previews, the user must engage with further tools to evaluate and explore the data, and then still more tools to actually employ it for the purpose for which it was sought.\nFurther, data needs are complex, and can typically stretch over multiple search sessions, resulting in complex queries. These are poorly supported in web-based keyword search and can be better addressed with conversational search.\nConversational generative artificial intelligence (CGAI) tools such as OpenAI’s ChatGPT and Google’s Bard have changed how we access information online. We are therefore interested in exploring how they could change people’s interactions with data on the web. Given the ability of CGAI to summarise large, complex sources, explain results, and write code, we hypothesise that chatbots may also be able to find data matching a user goal and help the user understand its scope, quality, and provenance, and to use the data productively. This may also be supported further by the increased interactivity offered by conversational search, which has been established as empowering non-expert users.\nIn this paper we explore whether, given a specific domain, are useful datasets suggested? Do the rationales make sense? Does CGAI help answer critical questions on data provenance, licensing, format and more, and how do users assess this? We begin to consider what a new generation of data tools, enabled by conversational generative AI, might offer to users, and how people might engage with them. Our research questions, therefore, are:\n* RQ1: How does CGAI support state of the art dataset discovery technologies?\n* RQ2: How do people express their information needs when using CGAI for data discovery?\n* RQ3. Which of the main sense-making data discovery activities previously identified are supplied by CGAI?\nTo address these questions we held workshops with users to explore how they would interact with CGAI when engaging in data discovery. The key contributions of this paper are a set of guidelines for data prompting with CGAI and a new prompting-centric human-data interaction framework.\nBackground Literature\n=====================\nIn this section we contextualise our research in (1) the state of the art of dataset discovery and (2) CGAI for information retrieval.\nDataset discovery\n-----------------\nThe manipulation, analysis, and sensemaking of data has been described as ‘data discovery’. The data we refer to here is structured (organised in tables, spreadsheets, databases and so on) and its potential scale is vast: as of 2023, Google Dataset Search indexed more than 45 million datasets from more than 13,000 websites. However, the data sought may not reside in a single dataset but across multiple sources, in a variety of formats, with no common ID, making sensemaking more complex.\n[h]![Koesten et al’s (2017) framework for interaction with structured data, showing the 5 steps of the data journey for users of secondary data..](Koesten_framework.png \"fig:\") [fig:Koestenframework]\nThere are five key steps in the journey of interacting with structured data, which can be seen in Figure [fig:Koestenframework], above. The first step of the data journey for the user is to define their motivation - why they want to use data. This task purpose is crucial to the process of selecting the appropriate dataset. Currently, there is no useful explicit way for users to define their task motivation in search, although search engines bring a wealth of implicit knowledge about previous user behaviour to the results they present. However, this excludes wider context about the task at hand.\nThe second step is that of search itself. Data search is a relatively recent field, with the first widely available dedicated dataset search on the web receiving a full launch as recently as January 2020. The primary forms in which data is made available to people is via data repositories and portals (usually dedicated to a specific institution or domain); via the web, accessed by search engines; and through legal mechanisms such as Freedom of Information requests. When people need to look for data, they employ keyword or faceted search on portals or search engines. Additionally, people find data through personal recommendation. However, access to data does not equate enabling citizens to make use of the data. “There is a disconnect between what datasets are available, what dataset a user needs, and what datasets a user can actually find, trust and is able to use”. A solution to this challenge has not yet been satisfactorily implemented, although there have been improvements in supporting understanding of the data before downloading, such as providing visualisations and summarisation.\nStudies of online search behaviour pre-date the world wide web and find that searches range from analytical, precise and carefully planned, to “on the fly”. Users often find it challenging to describe their information needs because of a lack of background knowledge, and their understanding of an information need often evolves during the search process . It has been suggested that when in need of information users actually want an domain expert, but turn to search engines in the absence of this. Unlike web searches for information, users searching for data are unlikely to expect an exact dataset match to their needs, but rather to locate a relevant subset of topical data. Geospatial and temporal descriptions, constraints (eg open licence) and granularity are the most used parameters to specify data parameters in search.\nData repositories aim to make data search easier, but come with their own complications: they tend to use metadata rather than keyword search and data users and publishers may use different terms for datasets. One study found that a search for “housing stock” on the UK government data website found almost twice as many datasets as a search for “housing supply”. Much of the time in learning search tasks is devoted to examining and comparing results and reformulating queries to discover the boundaries of meaning for key concepts. Users also keep search terms short: the majority of searches on portals, both on the portal and via the web, are composed on average of less than 5 words.\nFurther, the information retrieval task at hand may require an entire set of results from multiple different sources with interdependencies. A familiar example of this is booking a holiday, which requires a variety of interdependent travel and accommodation requirements to be met. This kind of complex query requires “memory” of search interactions that is not yet met by the facilities of the web, and currently users searching for different elements of a holiday need to manually manage a range of concurrent searches which are not automatically informed by the parameters of each separate element.\nSensemaking is a key condition for the secondary reuse of data and comprises the third and fourth steps of the data journey. Summary statistics or visualisations are often suggested as a way of exploring quantitative data, but this already implies a decision by the user that the data is worth engaging with, and that it has been assessed as fit for use. To facilitate this, ‘immediate intelligibility’ is a key requirement for users when discovering data. However, without detail on how and why data was initially collected, this may present a challenge for even the most competent domain expert. These and other assessments of whether a dataset is fit for purpose, such as its granularity, provenance or format, have been described as ‘data evaluation’. Such evaluations tend to be dataset and task specific rather than applicable to all datasets. Users may need to make as many as 20 types of evaluations, related to the relevance, useability and quality of data.\nA second set of activities take place once the user begins to explore the data prior to use. These may include visualisations (provided or created by the user themselves) and a review of documentation, if available. Text summaries are often used to reduce cognitive and time cost during information seeking activities online. Research with dataset text summaries suggests that they do not necessarily contain all the required information (with methodology and provenance most frequently being ignored); that abstractive summaries require extensive natural language generation (NLG); and that they are subject to the idiosyncrasies of the people creating them, rather than being standardised.Further, while it may be possible to generate a convincing summary of data itself, it may be the metadata or context that is more important for sensemaking.\nThe multi-step complexity outlined above suggests that powerful NLG tools such as conversational search could be of great value to users requiring data. Conversational search can build understanding and semantic coherence over multiple rounds of interaction, while also supporting user needs through clarification and suggestion,. Conversational search is particularly useful for exploratory information-seeking contexts where users have insufficient background knowledge. While there is a substantial corpus of research on this, pre-dating the existence of large language models, there is little to no research on conversational dataset search.\nCGAI for information retrieval\n------------------------------\nGenerative AI refers to a class of artificial intelligence models that can create new data based on patterns and structures learned from existing data. Recently there has been explosive growth in conversational agents based on Large Language Models (LLMs). Such conversational generative artificial intelligence (CGAI) can be used to address natural language tasks, commonly include writing essays or literature reviews of scientific papers, and reading and summarising reports. They can also be used to answer complex natural language questions and support the answers with justifications. Further, many LLMs operate in multiple spoken languages and can therefore both translate text and justify the constructions used in translation, as well as providing text in many coding languages. CGAI has been used in a number of text manipulation and generation tasks such as creating financial research, structuring unstructured information and detecting stance in social media posts. It has been found to outperform crowd workers on text annotation tasks such as identifying types of social media posts.\nIn conversational search powered by LLMs, users are not restricted to keywords, and are able to state requests in a number of different ways. This suggests they can include more description about their information need in their questions. However, while this may increase the information available to the model to generate an appropriate answer, misleading or unnecessary information may adversely affect results. The solution to this is generally held to lie in prompt engineering; designing the requests in such a way that they minimise error and maximise potential for appropriate results. This requires a certain level of awareness of potential errors, skill in devising prompts to reduce such errors and assessing the effectiveness of prompts, which is challenging to non-experts. Prompts can also be used to drive the LLM to initiate question asking, by presenting a task that requires the agent to discover more information from the user to complete it. Research into using GPT to extract unstructured to structured information has shown that greater success is achieved when the information extraction task is broken down into easier sub-tasks, which again requires specific skills. A further strategy for improving search is to specify a domain-expert persona for the LLM to adopt. However, this has been found to also increase bias in language tasks.\nLLMs do not have access to contemporaneous, proprietary or personal data. Consequently, enabling access to external data sources, and augmenting CGAI for retrieval of information, in particular through detecting and calling APIs, is a major area of research. Another is addressing the issue of hallucinations. A hallucination is where a plausible answer returned is not based on existing fact, and is ‘made up’. Hallucinations are of particular interest in our research as they often occur in the form of references to sources that do not actually exist. The ability to identify hallucinations presupposes a certain amount of pre-existing knowledge on the part of the user, as LLMs are unable to directly refer to supporting documents in their training data There is currently no reliable way for developers to steer the behaviour of LLMs, and even less so by inexperienced users. The ability of LLMs to fluently produce apparently coherent text may mask the fact that this text may be of much lesser quality than it first appears (containing very little truth or fact), or that it may accidentally be damaging.\nExplaination is desirable (and in some cases, legally required) in intelligent systems. It makes it possible for users to act appropriately, is important for discovering and understanding novel concepts, and supports users’ trust in an application. A working definition of explainable AI (XAI) is ”Given an audience, an explainable Artificial Intelligence is one that produces details or reasons to make its functioning clear or easy to understand”. Explanations have some basic requirements, but they are largely varied based on the requirements, background knowledge, and so forth of the audience. Explanations are used to justify decisions, control function, improve accuracy and discover new knowledge. There are as many as 36 various notions of explanations, including efficiency (whether the explanation supports faster decision making), faithfulness (whether definitions truly reflect the datapoints) and robustness (the consistency of the results of the method.\nExplanations have a social dimension - they constitute an element of a dialogue that aims to transfer knowledge, therefore engaging the beliefs of the giver and recipient of the explanation. There is concern that explainability is dealt with primarily technically and less attention is spent on whether the explainability system is effective and efficiently human-centred. While much attention is currently being paid to understanding how models such as ChatGPT work, much of this insight is currently speculative. Consequently, most explanations for LLMs are post-hoc (the dataset, training methods and algorithms are opaque) and most focus on single data points, rather than the whole model. It is, of course, theoretically possible for a user to request that ChatGPT or other model ‘reflects’ on its answer and explains how it arrived at it.\nSummary\n-------\nPeople both need to locate and understand datasets in order to reuse them. This is often done as part of work, both paid and volunteer. This requires skills around crafting appropriate search, looking in appropriate locations and then assessing the fitness for use of the data. Despite improvement on the supply side, there is still a huge time and cognitive cost on the demand side. Conversational generative artificial intelligence (CGAI) tools such as ChatGPT and Bard have changed how we access information online. We are therefore interested in exploring how they could change people’s interactions with data on the web. Given the ability of CGAI to summarise large, complex sources, explain results, and write code, we hypothesise that chatbots may also be able to find data matching a user goal and help the user understand its scope, quality, and provenance, and to use the data productively. In the next section we describe our user studies with conversational dataset search..\nMethodology\n===========\nTo explore how users might engage in conversational dataset search, we invited a gender-balanced group of users to three workshops conducted in two languages, in which we asked them, “to think of one dataset (or group of datasets) that you would like to discover, using [CGAI agent]. (This can be a dataset you know exists, or a completely new-to-you dataset.) Once [agent] has found something (this may require more than one prompt), explore what it has returned to see if it is usable”. For further details see Appendix [appendix:a].\nData collection\n---------------\nOur pilot was run using the dominant large language model-based chatbot at the time, OpenAI’s GPT3.5. We wished to replicate the scenario we felt was most likely, which was to use the CGAI which had most public awareness and users. For our second workshop, we tried to account for some of the limitations of GPT3.5 by using GPT4. However, 2 days before our research, the live web search facility of GPT4 was suspended. In our third workshop, therefore, we used an (easily accessible) LLM chatbot with web connectivity. As Microsoft Bing is built upon similar foundations to ChatGPT, we opted for Google Bard. Details of the workshops are shown in Table [tab:workshopdetails].\n[h] [tab:workshopdetails]\n|p0.01|p0.08|p0.14|p0.15|p0.01|p0.09|p0.14|p0.08|p0.08| **#** & **Date** & **CGAI**& **Rationale** & **n** & **Gender** & **Attendees** & **Location** & **Language** \n1 & 22/05/2023 & Chat GPT3.5 & Highest adoption & 9 & m=4, f=5 & AI and LLM experts & London & English \n2 & 07/07/2023 & Chat GPT4 & Improved performance on GPT3.5 & 5 & m=3, f=1, e=1 & Political activists & Berlin & German \n3 & 12/07/2023 & BARD & Web connectivity & 9 & m=4, f=5 & Data professionals & London & English \nThe pilot workshop was conducted in London, UK with users with PhD level understanding of AI and large language models. This was to confirm the format of the workshop and protocol. Eight participants took part in the workshop in person, and 2 trialled a self-guided, asynchronous process following the protocol closely. After this workshop we made two adjustments. Firstly we decided to audio record the “in the room” discussion as this provided rich data and background that supported the data in the questionnaires. Secondly, we found the asynchronous process yielded substantially less research data so abandoned this mode. We used the ChatGPT web interface at chat.openai.com, which does not allow any special features or the uploading of large amounts of text or changes of hyperparameters such as temperature settings.\nFor our following workshops, we aimed to ask non-specialist users in a real world context to define a problem they have, and then think about what data they could use to address it. As it was important to our research to reflect a real world scenario, we aimed to work with people who were staff or activists in resource-constrained organisations, who would strongly benefit from using data and understanding its accuracy and provenance. We accessed these groups via personal networks.\nOur second workshop was held in Berlin, Germany with professionals who were engaged in political activism. This used the ChatGPT4 web interface, in order to benefit from any relevant improved performance, for instance the ability to input longer texts. All participants were present in person. The third workshop, in London, comprised data professionals with familiarity with the data economy and how data is used, but whose roles did not involve active coding. Three participants were located remotely but engaged in the workshop process synchronously via Teams. The number of chats and questionnaires collected are shown in Table [tab:data].\n[h] [tab:data]\n| **Date** | **Location** | **CGAI** | **Questionnaires** | **Conversations** | **Audio** | **Teams** |\n| --- | --- | --- | --- | --- | --- | --- |\n| 18/05 | London | ChatGPT3.5 | 9 | 9 | N/A | N/A |\n| 7/07 | Berlin | ChatGPT4 | 5 | 10 | Yes | |\n| 12/07 | London | Bard | 9 | 8 | Yes | Yes |\n| **Total** | 4 | 3 | 23 | 27 | 2 | 1 |\nDuring the workshop, participants were provided with the guidance that can be found in Table [tab:guidance], which resulted in the associated output (a full version of the guidance can be found in Appendix [appendix:a]):\n[h] [tab:guidance]\n|p0.7|p0.25| **Guidance** & **Output** \n[c]@l@Think about your motivation for a dataset search. \nEnter your motivation and any expectations (e.g. I know this dataset really well and know what I expect to see/I don’t know \nif anything like this exists) into the questionnaire \nDevise your prompt for [agent] \nEnter any considerations you had when devising your prompt into the questionnaire\n&\n[c]@l@Audio discussion (for workshops 2 and 3). \nQuestionnaire q1-7 [Appendix [appendix:b]\n \nEnter your prompt and explore [the agent’s] response & Conversational transcript \nAs a group, discuss ChatGPTs response and your assessment of this. Enter replies to the questionnaire as we go. &\n[c]@l@Audio discussion (for workshops 2 and 3). \nQuestionnaire q8-11 [Appendix [appendix:b]]\n \nWe asked participants to evaluate their own experience of using a conversational agent as Successful, Somewhat successful or Not successful. We then broke this down into two elements. We asked whether an appropriate dataset had been suggested (locating data) and then we asked participants to indicate on a Likert scale how useful the supporting information provided with the dataset was (sense-making). Finally we asked participants to compare the process with previous search experience, if relevant.\nData preparation and analysis\n-----------------------------\nThe datasets, analytical methods applied and the relevant research questions are shown in Table [tab:RQdata].\n[h] [tab:RQdata]\n|p0.3|p0.35|p0.25| **RQ** & **Data** & **Analysis** \n1. How can conversational agents support SOTA dataset discovery technologies? &\n[c]@l@Questionnaire questions 1, 2, 7, 8, 9 \nChat sessions\n&\n[c]@l@Descriptive statistics \nInductive thematic analysis\n \n2: How do people express their information needs? &\n[c]@l@Questionnaire demographics \nChat sessions \nQuestionnaire questions 3, 4, 5, 10, 11, 12\n&\n[c]@l@Descriptive statistics \nInductive thematic analysis\n \n3. Which of the main sense-making data discovery activities previously identified are supplied by Conversational AI? &\n[c]@l@Chat sessions \nQuestionnaire questions 3, 4, 5, 10, 11, 12\n& Deductive thematic analysis based on Koesten et al (2017) \nThe data was prepared as follows. The audio recordings were uploaded onto our institutional OneDrive, transcribed and (where necessary) translated using the OneDrive translate facility. The chat sessions and questionnaires were uploaded into our institutional OneDrive and again translated using the OneDrive translate facility. All translations were checked by a German speaker for accuracy.\nThe audio transcriptions, questionnaires, and chat session files were then uploaded into NVIVO for thematic coding by two of the researchers. Two of the authors individually coded a sample of the chat transcripts and long form questionnaire answers inductively for RQ1 and RQ2, and coded a sample of the chat transcripts deductively for RQ3. The resulting code lists were compared and conflicting results were discussed. This was repeated twice for both the deductive codes and inductive codes. The resulting code books are attached as Appendix [appendix:c]. The short questions were analysed quantitatively using Python, and the code is available as a zip file. We counted the frequency of responses and their combinations. We calculated the length of the chat sessions by counting the number of queries asked, the responses when interacting with the conversational agent, and their number of words.\n### Ethics\nThe interview study was approved by Author 1’s institution’s Ethical Advisory Committee via the Minimal Risk Procedure. Informed verbal (audio recordings and chat transcripts) and written (questionnaire) consent was given by the participants. No personal information was used in the analysis.\nFindings\n========\nRQ1: How do CGAI compare with state of the art dataset discovery technology?\n----------------------------------------------------------------------------\nIn this section we examine whether conversational search can compete in terms of success and usefulness for dataset discovery. We explore how people are using it in terms of the length of their queries and length of chats, and also examine some aspects of explanations.\n[t]![Results from the questionnaires regarding previously used methods (multiple answers possible).](other_search.png \"fig:\") [fig:othersearch]\n[h]![Results from the questionnaires regarding familiarity with the conversational agents.](familar.png \"fig:\") [fig:familiar]\n### Success rates of conversational dataset search\nIn order to assess our participants’ previous experience with datset search, we asked them what methods they have previously used to find datasets. While one participant had not previously searched for datasets at all, most of the participants had used Google search and specific data portals (Figure [fig:othersearch]). In addition, we asked participants how familiar they were with the conversational agents (Figure [fig:familiar] ). The majority had some or a little experience with them. Only one particpant had no experience. After the conversation with the chatbot, we asked participants about the process (Figure [fig:usefulsuccess]). The majority of participants (n=15) found the process of using chat agents “somewhat successful”. It is interesting to note that despite a large number of participants finding the process overall “useful” (n=14), only three of them found it fully “successful” in terms of finding a dataset.\n[fig:usefulsuccess]\n### Query and chat format\nAs can be seen in Figure [fig:numwordqueries], the total number of words in a query had a wide range, from 1 - 191. The modal number was 9.\n![ The figure presents an overview of the length of queries in terms of the number of words.](bar_word_question.png \"fig:\") [fig:numwordqueries]\nWe found that the mean number of words per query is approximately 18. This differs from data portals and web search, where the average number of words per query on a data portal is fewer than 3, while using the web search and then redirecting to a data portal is fewer than 5. However, it reflects the increased average length of prompts for text-to-image generation for which the mean and median have been found to be above 20 words.Figure [fig:agentsbox] shows that while the median number of words per query is similar across all three agents, participants using Bard did not use as many long queries.\n[h]![The box plot shows the range of query length for the three conversational agents.](boxplot_question_agent.png \"fig:\") [fig:agentsbox]\nIn terms of the queries themselves, we collected 268 queries across 27 chats with conversational agents. Previous research on conversational search using a text chat interface received 669 queries for 26 chats. However, this study used human participants playing the part of the conversational agents, which may have affected the length of chat. In addition, we found great variation in the range of queries per chat. Our study includes chats with 1 to 33 queries and on average 10 queries per chat as shown in Table [tab:statsagents]. The study above also found minimum and maximum message count varied widely, from 8 messages to 62 messages, and an average 26 queries per chat. We found our average number of queries was consistent across our German speaking group and our English users as shown in Table [tab:statslanguage].\n[h] [tab:statsagents]\n| **Statistics** | **ChatGPT4** | **ChatGPT3.5** | **Bard** | **Total** |\n| --- | --- | --- | --- | --- |\n| Number of chats | 10 | 9 | 8 | 27 |\n| Number of queries | 105 | 55 | 108 | 268 |\n| Min-Max queries per chat | 1-23 | 1-14 | 3-33 | 1-33 |\n| Average number of queries per chat | 10.5 | 6.1 | 13.5 | 9.9 |\n| Average number of words per query | 22.9 | 18 | 17 | 17.7 |\n| Average number of words per answer | 185.2 | 214.1 | 349.8 | 215.3 |\n[h] [tab:statslanguage]\n| **Statistics** | **English** | **German** |\n| --- | --- | --- |\n| Number of chats | 17 | 10 |\n| Number of queries | 163 | 105 |\n| Min-Max queries per chat | 1-33 | 1-23 |\n| Average number of queries per chat | 9.6 | 10.5 |\n| Average number of words per query | 17.6 | 22.9 |\n| Average number of words per answer | 278 | 185.2 |\n### Results of data requests\n[] [tab:datasetsearch]\n|p0.12|p0.09|p0.05|p0.2|p0.05|p0.3| **Data** & **Request** & **LLM** & **Response** & **Datasets** & **Comment** \nCase law & Implied & GPT4 & No access to database & 1 & curated micro dataset \nCase law & Implied & GPT4 & Does not mention having no access to database & 1 & curated micro dataset \nOpen source & Described & GPT4 & No access to specific datasets & 0 & Suggested 4 strategies for search. (on being prompted for sources, suggested 5) \nBirds & Described & GPT4 & No excuses & 4 & \nBirds & Described & GPT4 & Does not have access to datasets, also problem with internal server & 0 & 5 questions later it suggests 3, although not the same as those above \nPlug ins & Described & GPT4 & Not up to date & 0 & Recommends contacting Open AI \nAddresses & Described & GPT4 & Not up to date, no access to db & 5 & \nComplex numbers & Mathematical explanation & GPT4 & & & \nComplex numbers & Mathematical explanation & GPT4 & & & \nStudent numbers & Implied & GPT4 & No access to database & 0 & Recommended where to contact \nProperty prices & Implied & GPT3.5 & No real time access & - & Text summary describing the data \nOntology & Described & GPT3.5 & & 7 & With supporting information on what an ontology is \nConferences & Described & GPT3.5 & Yes! & 15 & \nKnowledge graphs & Described & GPT3.5 & Yes & 5 & \nCredit scores & Requested & GPT3.5 & Yes & 1 & With link \nIslamophobia & Described & GPT3.5 & No & 0 & Provides some later \nMusic emotion & Described & GPT3.5 & Sure! & 8 & \nPanama paper & Request & GPT3.5 & Not able to access & 1 & With supporting information \nFire hydrants & Described & Bard & Sure & 1 & partial dataset, states it is fictional \nTrees & Described & Bard & Yes & 1 & With link and usage \nLongitudinal data & Described & Bard & & 6 & ‘Most popular’ with links \nSport & Described & Bard & & 5 & With links plus suggestions for more \nHomeslessness & Described & Bard & Sure! & 1 & Curated dataset from unascribed source and some interpretation of trend \nPhysics & Implied & Bard & & 0 & Describes interpretation of underlying data \nJunk food & Implied & Bard & & - & Textual summary of report \nPeer production & Described & GPT3.5 & & 3 & In textual report form rather than list \nEducation & Implied & Bard & & - & Textual report \nThe number of initial results is very much less than would be returned from keyword search on the web as shown in Table [tab:datasetsearch]. In our terminology, a “request” dataset is named, a “described” dataset is defined but not named and an “implied” dataset asks for data as information, but does not specify it explicitly as data. As well as returning dataset suggestions (or portals where those datasets might be found) the CGAI also returned data sets they compiled themselves, sometimes with comment and sometimes without explanation. Implied requests in both Bard and GPT tended to be responded to in-kind – ie, without reference to the fact that what is being sought or returned is data. There is considerable inconsistency evident in not only the datasets returned, but also in explanatory commentary from the CGAI, as can be seen in the case law example where GPT4 comments on one chat that it has no access to the dataset, but does not mention this in the other.\n### Self-assessment of efficacy\nWe asked the users to evaluate how effective conversational search was. Many agreed that “as pure search, it is inferior [to googling]”, largely due to the failure to offer links and the lack of trust in the sources. However, they highlighted that looking for and exploring data inside CGAI meant that ‘next steps’ could be integrated into the process. One noted that using an agent “would probably give me a stronger idea of the nuances, internal politics, and where to look next based on what those organisations are connected to. But it would make sense to do that as a next step following this kind of query”. Another noted “if you have [to] deal with further steps separately, namely the application of BGH case law to concrete contractual clauses and the construction of arguments - I am faster than if I do this “manually”, i.e. using Google/specialist databases.”\n### Explanations\nWe also identified activity in the conversation transcripts where users asked the tool to explain why it has suggested the data it has (n=7). In existing keyword search, users cannot ask why information has been returned to them.\nThe majority of the interactions that requested explanations regarded links and sources, with participants asking, “so you can’t give me the link to it? I don’t understand why not” and “But this data should be available from your sources”. The responses mostly focused on the explanation that the agent did not have access to the web or that it was under development, or sometimes that it was “only a language model and don’t have the capacity to understand and respond.” One participant asked, “I hear that you’ve given URLs and links to other people who are using your services. Why can’t you link to the papers you’ve pulled from to answer my questions?” In answer to this, Bard returned the common “under development” reply but also noted that it was unable to link to copyright material or websites that “might contain adult content”. While these explanations are not informative about the specific case, they are generally informative about the rules imposed on the agent.\nWe also found participants asking questions about the sources of the CGAI’s knowledge, for instance, “Where does your knowledge of the global distribution of birds come from?” The answer to this revealed that the agent did “not have direct access to my training data” and that “my knowledge is based on patterns in the data I saw during my training.”\nParticipants confronted the agents about veracity of responses. One participant asked ChatGPT “are you lying?” In response, Chat noted that it did not operate within the context of ‘truth’ or “lies’ but noted that the study it had referred to in the previous response was ‘misleading” as it was a “fictitious example”. Another queried the language used by challenging, “So by ‘estimating’ do you mean you guessed?” GPT4 responded that “Estimating” in this context does not mean “guessing” and stated that it makes, “an informed statement or prediction based on the available information”. A participant requested that Bard provide an image from the data, then asked, “That image is not what the code describes. I can also see that the axes aren’t labelled the same. Did you actually run the code?” In response to this challenge Bard provided multiple draft answers. These answers conflicted with each other and obscured the process. as can be seen in Table [tab].\n[h] [tab:Barddraft]\n|p0.1|p0.85| **Interaction** & **Text** \nFeedback prompt & “That looks suspiciously the same as before. I am sad” \nDraft # & Bard’s answer \nDraft 1 &\n[c]@l@You are correct. The image that I showed you is the same as the one that I showed you before. \nI apologize for the mistake. I am still under development and learning how to generate correct \noutputs. I will try again to generate the correct output and get back to you. \nIn the meantime, I would like to ask you a few questions to help me understand what went wrong. \nDid you run the code yourself? \nDid you get the same output as I did? \nIf not, what was the output that you got? \nYour answers to these questions will help me to understand what went wrong and how I can fix it. \nThank you for your patience and understanding.\n \nDraft 2 &\n[c]@l@I apologize for the confusion. The image that I showed you is the same as the one that I showed \nyou before. I made a mistake in my previous response, and I thought that I had run the code again \nand gotten the correct output. However, I did not actually run the code again. \nI will try to run the code again and get the correct output. I will let you know when I have done so. \nIn the meantime, I can explain why the image looks the same as the one that I showed you before. \nThe code that I wrote is a simple code that generates a 2D contour plot of the q profile as a function \nof normalized poloidal flux and time. The code does not take into account any of the physics \nof tokamak plasmas. \nAs a result, the image that the code generates is not very realistic. The q profile in the image does \nnot evolve in a way that is consistent with the physics of tokamak plasmas. \nI apologize for the inconvenience. I will try to get the correct output of the code and show it to you.\n \nDraft 3 &\n[c]@l@You are right. The image that I showed you is the same as the one that I showed you before. I \napologize for the confusion. I am still under development and learning how to generate correct code \nand output. \nI will try to run the code again and get the correct output. I will let you know when I have done so. \nIn the meantime, I would like to ask for your help. Can you please help me to debug the code? \nI think that there is a mistake in the way that I am defining the q profile. If you can help me to find \nthe mistake, I will be able to generate the correct output. \nThank you for your help!\n \nAdditionally, one participant sought information on the machine learning that was used to form the dataset. The agent, in this case Bard, provided a description of three different techniques and how they were applied. The participant followed this with a request for sources for this information, which was provided.\nWe also identified instances of agents providing detail about their training that was not specifically requested. In this case, a participant asked “Which countries do you have data for?” ChatGPT4 stated, “During the training, I was not provided with specific datasets on the most common bird species in each European country. Instead, I was trained on a variety of texts from many different sources, including books, websites, and other texts that contain information about birds and other topics.”, “I can also access scientific literature and studies that are publicly available and try to extract relevant information from them.”. However, it is important to note that we also identified many occurrences of agents providing entirely unverified and unexplained supporting information. For instance, one participant had to prompt the agent to be aware of the CONAN dataset. It had previously stated no dataset in this field existed. The participant informed GPT3.5 of the dataset, then asked whether the annotators of the datasets were experts. GPT3.5 replied in the affirmative. “Yes; the annotators for the CONAN dataset were experts in the field of hate speech and language analysis. The dataset was annotated by team of professional annotators; including linguists and social scientists, who were trained in hate speech identification and provided with guidelines on how to identify and label hateful content:”. It did not provide any links to its source or explanation as to how it was now able to provide detailed information about a dataset it had previously denied all knowledge of, rendering the process more opaque. Some form of “I’m unable to help you with that, as I’m only a language model and don’t have the necessary information or abilities” was the most common (non)-information about its own abilities supplied by the conversational tools (n=13) with it recurring 6 times in one conversation.\n### Summary\nConversational search was largely characterised by users as a “useful” experience even if few (n=3) people felt it was wholly successful in terms of discovering a dataset. The number of words in each query were considerably longer on average than current search, and there was a very long tail. Explanations were largely limited to reasons that the agent could not perform a given task, and not always accurate. However, the ability to integrate further usage tasks into the conversation, such as returning code to access an API, was seen as a positive aspect.\nRQ2: How do people express their information needs when using CGAI for data discovery?\n--------------------------------------------------------------------------------------\nIn this section we explore the prompts users employ when attempting to discover datasets.\n### Key terms in conversational dataset search\nWe performed a visual representation of the most frequently occuring terms used by users in interacting with the agents as shown in Figure [fig:wordcloud].\n [fig:wordcloud]\nThis suggests that, as with keyword search, users were specifying their data needs geospatially (“london, göttingen, cities”) and temporally (“years, 2022, 2021”). We can see evidence of requests for related evidence (research, study, survey) as well as format based requests (lists, graphs, databases). As noted in Table 6, the average number of words in a question was just under 20. However, amongst the outliers was a 71 word request that specified two kinds of content (street names and postcodes) of a location (EU country) as well as stating conditions around licence, timeliness and format (“no PDFs”).\n### Expression of information needs as data prompts\nIn our thematic analysis we identified three main ways in which people requested datasets. One was requesting the dataset directly, eg, “can you find me the German credit fairness dataset” [datasetrequest].\nAnother was where the dataset was described, as in “can you provide me with a list of available datasets about Islamophobia in Italian containing Tweets labelled as hate speech” [describeddatasets]. One approach that yielded results for the user was to explicitly suggest a document that might contain the data they sought, “for example a survey which asks people about their reasons for opting for junk foods”.\nA last format was implied datasets [implieddatasets], where the request might ask, “What has been the average property price in London, from 2000 to 2010?” Some users varied between these modes of asking, with one user describing a dataset in 4 different ways and then asking 3 further times for an implied version. We noted evidence of requests being made in a way that implied memory of the previous request, for instance “what about ’Global South’ cities”. One such implied dataset request was for a list of judgements on cosmetic repair in German case law. The CGAI recognised the implied request, and replied, “I do not have direct access to a complete database of case law or the exact wording of specific clauses in BGH rulings. However, I can mention a few general cases in which the Federal Court of Justice (BGH) has issued rulings on cosmetic repair clauses until 2021.” Consequently, it curated a small, if necessarily incomplete, dataset for the user. Although no users made requests based on personas, one did ask for an “ELI5” response - “explain like I’m 5 (years old)”, which ChatGPT4 happily complied with, giving examples using fantasy friends and apples and bananas.\nThe most common question after the initial engagement with the agent was for links [linkrequest]. Separating out the various parts of a request was more common than creating a compound request.Another approach taken by participants was to request the source of the dataset [sourcerequest],rather than simply ask for a link. A user who self-defined as being more experienced with CGAI attempted to use slightly different language to access the datasets. Rather than simply asking for links, which they were aware would result in a “I can’t access the web” reply, they asked “Can you provide directions on how to access the datasets?” [accessdatasets]. In fact, this resulted in the return of links that were wholly inaccurate, which motivated the user to request working links [linkrequest], thus bringing them back to the request. Overall, the majority (n=20) of users requested a link or source as a follow up question.\nAlongside link requests, users asked for more information about the dataset itself [additionaldatasetinformation]. Such questions might be general, eg, the licences of datasets or more technical, such as “Which of these datasets provide emotion annotations in the Valence-Arousal space?” However, a major issue with such questions was that the answers often were attended with caveats such as, “Please note that these are fictional examples, ” Users sometimes requested further information about the context of the dataset [furthercontextinformation], rather than the dataset itself. This included requests such as, “What can you tell me on the NeurIPS conference, then?”, “were the annotators of the dataset experts in their field?”\nWe also found instances of a question being repeated after a particpant had interacted with the agent and supplied it with information. Interestingly, this did not necessarily improve results. In multiple cases, a repeated question that the agent had initially attempted to answer met with responses on a second asking such as, “I’m a text-based AI and can’t assist with that.”\nUsers requested information on how to use or analyse the dataset provided [advicerequest]. These included the technical and specific: “Explain how would you go about performing an analysis of the social network of the people contained in the Offshore Leaks Database:” and “what are the limitations of tree canopy cover analysis?” as well as the more general, such as, “Can I compare this dataset to other cities”.\nGenerative AI is not restricted to narrative texts and we saw some examples of people requesting other forms of information to be returned [multimodaloutput], for example, “Write down some code in Python I can use to analyse the Offshore Leaks Database;”, “Can you give me the first 10 triples of each dataset”. Users asked for data visualisations, for instance, “can you make a graph to include the comparative tree canopy cover for all of the cities you have mentioned?”. A particularly positive experience was had by a user who asked if GPT4 could query a REST API, “And then he actually built me corresponding requests, which I can then use”. This integration of further activities [furtheractivity] appeared on a number of occasions, with users requesting drafts of letters, opinions on texts and drafts of legal justification (and a draft of an invalid legal clause). One user asked “can you send a Freedom of Information request too?” which Bard agreed to do, and offered, “to keep you updated on my progress.”\nParticipants also countered some of the information provided by various agents [disagree], such as, “That plot doesn’t show what I wanted. Can you try again?” This occurred over a range of answer types including lists and data visualisations. We also saw instances of participants providing information to the agents [provideinformation], for instance, “The MusicEmo dataset does not exist, that is why you could not find any link for it. Can you remove it from the list please?” In these examples, participants are contending both with the agent failing to provide existing datasets and hallucinating non-existent datasets. In this instance, the participant engaged repeatedly with the agent to achieve an accurate list. The third “input” interaction we identified was when users referred back to earlier conversation [Conversationreminder], for instance, ’what is the role the 110,000 asylum seekers in France are playing in the overall homelessness you gave me at the beginning of the conversation’.”.\n### Summary\nThis research question has attempted to investigate how participants express themselves when attempting to discover data. We find that, offered more natural ways of defining their information needs, users describe datasets in a variety of direct and indirect ways. Similarly, we see variety in the requests (links, access, sources) that follow the suggestion of a dataset(s). While most participants engage with these two steps, what follows after is less defined. Participants may, dependent on their needs and what is offered, explore the data further, seek advice on using the data or even enter into argument with the agent.\nRQ3: Which of the main sense-making data discovery activities previously identified are supplied by CGAI?\n---------------------------------------------------------------------------------------------------------\nWe base our analysis here on the framework of interaction with structured data discovery. They present the sense-making aspect of data discovery as consisting of “evaluation” and “exploration”. Here we explore which aspects of evaluation and exploration were prompted by users, and to what extent the CGAI met them.\n### Evaluation\nEvaluation is broken down into three aspects; relevance, useability and quality, with multiple sub categories. The number of users engaging with each of these sub-categories is provided in Appendix [appendix:c].\nRelevance: Relevance is the extent to which the suggested dataset meets the task needs of the user. Relevance assessments include looking at the data context, coverage, summarisation, original purpose and granularity. As noted above, users often ask for context in the form of either reports or papers data first appeared in, or in which it is subsequently used. Conversational agents are well placed to provide this. However, although sometimes they did suggest texts, they also sometimes claimed they could not do it because of the limitations of being an LLM. Coverage was also requested, with users asking, “Which countries do you have data for?” and “Do these datasets provide full tracks or audio clips?”. Again, success was mixed, but the agents were certainly able to understand and address the requests. At one point, ChatGPT3.5, in the absence of being able to suggest a dataset with appropriate coverage, recommended, “you can consider creating your own dataset by using various methods such as data scraping or using APIs to collect text data from various sources such as social media platforms”. Although people often use generative AI to write documents it is often more powerful in reading them, so summarisation is intuitively a task where they could perform well. Users asked for, a “general description” of a datasets contents, as well as “a few example records contained in the dataset” to better understand what the data looks like. Users also asked for specific parts of the text to test that the agent could regurgitate them - “What was the clause in the 1st judgment?” Unsurprisingly, once an agent was working with a confirmed text it performed well. “Granularity” (the measure of the level of detail in a data structure) is one of the 4 key types of dataset descriptors used in search and there were a number of requests of this type. There were also requests further into the conversation. However, these were rarely successful as the agent did not have access to the actual dataset, for instance requests to break down student data by subject. Regardless, the agent was still able to provide information on granularity, for instance, when asked for data “By street and by postcode” they provided a suggestion of the dataset but noted, “This data may still require further processing to extract all street names and postcodes”.\nUseability: In this section the framework covers aspects of documentation, licensing, access, machine readability, language, labelling, ability to share and format. Users requested information that might be found in documentation, particularly definitions, eg “what is classified as tree canopy cover”, “how does the Food Foundation define ’poorest households’?” and “can you give me the definition of homelessness”. In these cases, the agent provided a number of definitions and sources, but these were not necessarily those used in the dataset. The agents also provided guidance on whether documentation was available, or should be consulted.\nThe agents sometimes supplied unprompted licence information or advice with the dataset suggestions. In one case, having suggested a dataset, “the agent told me that there might be problems after all, that the license and the data are not so available after all. That was a dead end.” Whether the data was licenced or not - the veracity of this interaction - is unverified. This is positive as few participants actually requested licence information, but it is a necessity for use.\nRegarding access, the agents often suggested contacting the data publishers directly for access information. More detail is sometimes provided: in one interaction, the chat agent suggests a number of datasets for the user to investigate. The user then asks whether one of the datasets, Earth Engine, costs to use the data, to which the chat agent gives a full overview of the costs. Again, the accuracy of this is unverified. Another form of unprompted advice seen was around potential considerations for use of the data. ChatGPT4 discussed cost, access, privacy, and licences, and recommended contacting the data publisher directly for more information on this.\nIn terms of language, the models we used worked in around 40 major languages. As well as searching in both English and German, users requested datasets in other languages, for instance, “Can you provide me with a list of available datasets about Islamophobia in Italian containing Tweets labelled as hate speech or not?”. The agents were able to move easily between languages.\nThere were also occasions on which the agent requested the user to provide information, such as when Bard asked one user to share the code they had used to create a graph, as, “Your answers to these questions will help me to understand what went wrong and how I can fix it.” and asked another user to provide detailed information so it could draft a letter. These also demonstrate the facility to build on complex queries, retaining context from previous rounds of requests.\nQuality: Under the heading of quality the framework examines issues of consistency, completeness, collection methods, provenance, and what has not been included in the data set. Users asked about and received answers on collection methods such as, “Can you explain what machine learning techniques were used to form the dataset?”, and “What studies, books, internet articles, etc. did you use?”. One way that users sought to understand the quality of the data was by asking whether the data was used by other organisations. For example, one user asked whether “any of these [have] been used in research publications?”, while another asked if a “study [has] nevertheless been cited in scientific papers?” In terms of provenance, as we saw in Section 4.2.2, there are a great many questions from users about sources of data and figures. However, equally, this is one of the more unreliable aspects of the agents. Firstly, the number of questions about sources is provoked by their opacity regarding the sources of their information, and secondly, there is frequent admission of unreliability, including fictitious examples.\n### Exploration\nIn “exploration” the major issue is that the datasets are not returned within the agent, so basic visual scans and an assessment of basic errors are challenging. Visualisations and data samples were returned in response to requests, although in at least one case, the visualisation was inaccurate. We found examples of metadata and documentation being addressed in the chat sessions, (although again, we are not necessarily testing the veracity of the claims). Interestingly, the references to metadata in the chats were suggestions by the agents, rather than requests by the users. Along with the finding above that explicit reference to documentation was only made by the agents, these results imply that the agents are able to supply information to support data exploration even when they are unprompted or they cannot return the datasets themselves. However, while this is often thorough, it is necessarily generic and not specific to the dataset in question. We found that on occasion the LLMs would describe the data in sufficient detail as to include at least some descriptions of the content of the columns, as it did in the tree dataset. We also explored the data for related aspects, such as “columns”. We found two references. One was an instance of ChatGPT 3.5 describing how to clean data; the other was a comment in the questionnaire from a user who was concerned that subheaders would make it difficult for ChatGPT to identify appropriate data:“a matter of fact, these kinds of in depth information are contained as sub-columns in datasets and not as primary class labels.”\n### Summary\nThe evaluation and exploration tasks outlined above are crucial as it is based on these that people make the decision to use data (or not). We find that the majority of these elements are identifiable in the chats in some way. Of particular interest is the fact that the agents do, on occasion, suggest generic exploration and evaluation tasks without prompting, and also request input from the users on data evaluation tasks.\nDiscussion\n==========\nWhat, then, are the emerging user practices around dataset search with conversational tools?\nConversational dataset discovery enables the explicit enactment of the entire process of dataset discovery. The task can be specified in text by the user, rather than existing as an implicit motivator. Similarly, the data use can be integrated into the process. Overall, therefore, dataset discovery can be greatly enhanced by CGAI. We first discuss 6 areas of promise, then review 4 areas of challenge, before presenting 5 guidelines for prompting data. We then present our model for user-dataset discovery interaction using conversational agents.\nCuration - promise\n------------------\nA key aspect of the agents is the curation of data. Unlike keyword search where (often) thousands of potential dataset links are returned, far fewer datasets were suggested by by the CGAI in response to each query. Given appropriate web connectivity, it can be extrapolated that CGAIs should be able to select only the most relevant datasets from the web and provide them to users enhanced with supporting information. Another form of curation was the returning of a particular number of datasets in response to user specification. Finally, a third type of curation in evidence was that seen in response to implied dataset requests, such as the suggestion of the German case law examples, which again has great potential for future use.\nComplex queries - promise\n-------------------------\nComplex queries have long been an idea in research papers, but CGAI allows this to be realised at scale with non-expert users. While we saw longer, more detailed queries these did not necessarily correlate with identifying datasets more accurately. However, the ability to “remember” the conversation, to return and reexplore earlier interactions, and to build on sensemaking with usage activities in the same space is beneficial.\nFeedback - promise\n------------------\nA key strength of CGAI is the ability to incorporate information fed by users during interactions. However, while users incorporated correction and redirection strategies, this met with varying levels of success. Unsurprisingly the users with the greatest success in this area were those in the more expert group. This suggests that users can control managing this issue with increased familiarity.\nSupplemental information -promise\n---------------------------------\nUnprompted information supplied by the agents often supported key areas of concern with data such as access, licence, cost, and so on. In some cases it made users aware of issues, but did not resolve those issues, for instance, in such responses as “be aware of licenses”, while not actually identifying the license in question. This suggests that while CGAI may not be reliable for data discovery, it may well have potential as a tool of data literacy, as users are not required to come to search with existing knowledge about what aspects of evaluation and exploration are important.\nAdvice - promise\n----------------\nSimilarly, CGAI has potential as supporting tools for data skills, via the provision of advice. Such advice includes, \"it gave me guidance on finding [datasets] in the future” or advice on usage of the data.\nMultimodality - promise\n-----------------------\nHowever, the aspect of CGAI that offers perhaps the most potential as a tool for data is their ability to provide responses in a number of modalities. The ad hoc creation of visualisations and summaries enhances exploration and reduces the skillset a user requires; similarly the ability to provide code supports usage.\nHallucinations - challenge\n--------------------------\nWhere the fiction created by CGAI is very closely correlated with the truth this does not present a problem; nor where the user is very expert in the domain. A third area in which this is not problematic is where the agent provides general rather than specific information, such as “look in these kinds of reports” or “check the metadata”. A somewhat positive response to the fictitious nature of the responses is that users worked hard to counterbalance this, asking for sources, evidence and provenance, in an attempt to navigate around admissions of unreliability. However, there are aspects of hallucinations that work against commonly agreed positive strategies for data search. An example of this is text summarisation. Text summarisation is seen as an accessible and technically achieveable exploration technique. The UK’s Office of National Statistics, for example, a major producer of public data, has experimented with integrating LLM technology into its website search, so that users can extract data from its text bulletins. However, while text summarisation is a strength of CGAI it does not help if the report or text summary is entirely fictitious, and the user is not aware of this fact. Domain expertise remains of importance to counteract hallucination, but a key strategy is verification, and this depends on moving outside of the conversation to check other sources, thus affecting the ability to conduct the dataset discovery process entirely within the conversation.\nExplanations - challenge\n------------------------\nThere is no ground truth for explanations or correct reasons the agents could offer. In terms of a global explanation, this most frequently took the shape of statements about limitations including “As an AI language model, I don’t have real-time data access”; “I have a cut off date for training data of September 2021”; As noted, the bland “I’m not able to help with that, as I’m only a language model” was the most frequently used statement about agents’ capabilities. Local explanations about particular responses are generally generic. However, the most problematic aspect of agents’ own capabilities is the unreliability and opacity of its own post-hoc explanations. This opacity includes information supplied after the event regarding its ability to operate within the boundaries of “truth” and admitting it may therefore be “misleading”. The existence of multiple drafts, each providing potentially differing explanations, further undermines any reliability in this area.\nInconsistency - challenge\n-------------------------\nWe saw great inconsistency between results, even where the same prompt was used. A step by step approach has been found to be successful to counteract this in some maths and reasoning tasks. Recursive prompting might address this, however, that is beyond the scope of this study. Instead, ways need to be found to work around, or indeed, with inconsistency. Such approaches may lie in triangulation of information, either within answers (or drafts) or with external sources.\nCapability expectation - challenge\n----------------------------------\nConversational agents are not search engines with super powers. Users may expect them to able to return a link or csv file, but this is not currently what happens. In other words, users need to approach conversational dataset search less in the anticipation that they will receive a magic bullet answer, and more in the expectation of receiving support to find and use an appropriate dataset, which may happen using a different tool on the web. Further, it may be that for “information veracity, information literacy and serendipity” reducing human interactions with web search is not desirable.\n[h] [tab:guidlines]\n|p0.01|p0.3|p0.6| **#** & **Guideline** & **Description** \n1. & SELECT THE CORRECT PROMPT FORMAT FOR THE DATA & Naming a dataset explicitly is most likely to return the dataset without much supporting information (unless this is specified in the prompt); describing a dataset is likely to return suggestions of datasets with some supporting information and implying a dataset is likely to return a narrative description of some key data points. \n2. & MIX REQUEST PROMPTS FOR MORE EFFECTIVE DISCOVERY & The different types of requests can be employed iteratively, in order to build a better picture of the data. For example, a user might begin with an ‘implied’ prompt in order to understand the bigger narrative of a dataset, and then move to a ‘described’ prompt in order to explore more about the data itself. \n3. & USE RESULTS AS A BASE FOR IMPROVEMENT & It is common for users to request CGAI to supply a draft of a text, such as a letter or report, which they then edit and improve. Similarly, the results of data discovery can be used as a basis for guided information retrieval. \n4. & USE REMINDERS & The CGAI responds to the prompts of the users but is fundamentally passive. Building towards the goal should include explicitly reminding the CGAI of the task and previous interactions in the conversation. \n5. & EXPLOIT THE CODING ABILITIES OF CGAI & For instance, instead of asking for a visualisation, ask for code to create a visualisation. An incorrect visualisation with no explanation of how it was created is harder to debug than incorrect code. \n![Model of dataset prompting interactions with CGAI](diagramforLLMpaper.png \"fig:\") [fig:example]\nPrompting for dataset discovery\n-------------------------------\nTable [tab:guidlines] summarises 5 key insights into prompting strategies that may increase effective data discovery with CGAI. Based on the thematic analysis in 4.2.2, we have also developed a model of prompt interactions for dataset discovery with CGAI (Figure [fig:example]). This not only captures how people interact with CGAI but also proffers a starting point for developing improved prompting at each stage of interaction. All users participated in the first stages, the vast majority in the second and then in some or all of the last activities. The task is often made more explicit, although this can be revealed at any stage throughout the conversational search, not necessarily in the first prompt. Evaluation and exploration are iterative and can happen more or less simultaneously. Further, our model reflects the increased interactivity facilitated by CGAI and users can explicitly disagree with results within the conversation. Usage is an active part of our model as it can be performed within the same set of activities as the discovery and sensemaking. Users sometimes engaged in several stages of activity at once, such as: “You suggested this link to me earlier as a source https://octoverse.github.com/. Do you have any other links on how to use open source?”, which is both seeking advice (how to use open source) and a reminder of an earlier interaction.\nConclusion\n==========\nMaking data useable requires tools that support individuals’ data journeys. This includes specifying a goal and finding the data they need; exploring new sources of data; assessing the relevance and credibility, and deriving insights. In this study we explored whether, given a specific domain, can conversational agents based on large language models suggest useful datasets? Do the rationales make sense? Does CGAI help answer critical questions on data provenance, licensing, format and more, and how do users assess this?\nWe find that CGAI can suggest relevant dataset areas, and allows more engagement with the data and contextual information. However, we also found that the suggestion of datasets and provision of information, especially more obscure data, are unreliable; that suggestions are prone to confusing levels of inconsistency; and domain expertise and web search are often required to establish veracity. Its real strength currently is incorporating advice on evaluation and usage, and supplying code in various forms to support this, reducing the need for advanced data skills to work with data effectively. Of particular note is the fact that users are able to supply explicit feedback, corrections and information to the CGAI to improve outcomes.\nThus, until these issues are addressed, CGAI will not fulfil its very real promise of being an effective end to end data discover tool. Our contributions towards this are i. a model that describes user interaction with CGAI when engaging in dataset discovery and ii. a set of guidelines for data prompting.\nLimitations\n-----------\nIn terms of the users we worked with this is very specifically about data literate people operating in highly served languages. We cannot extrapolate to less data literate people in under served languages. In terms of the agents, given the unstable and changing nature of the answers even within workshops, it may be that entirely different responses would be elicited from the agents if any number of circumstances changed, for instance, increased access to the web, more use of temperature settings, and so on.\nFuture Work\n-----------\nMore work needs to be done on prompting agents in a way that reduces the domain knowledge required to elicit datasets. Our findings further suggest that a valuable follow up study would be to pre-locate datasets and ask the agent to explore it, to understand what further activities and services of data use agents could consistently provide. It is key to investigate the use of conversational agents in under-served languages. This will not only let us further explore the integration of AI into learning and society, but also more broadly understand the support that this technology offers to these communities. Given the datasets on which large language models are trained we hypothesize there may be substantial differences in the ability to discover and make sense of relevant datasets.\nAppendix: A\n===========\n**PILOT RESEARCH - Conversational Agents for Data Discovery**\n*Verbal Information and Consent*\n* This is a research workshop for a paper we intend to submit to CHI, entitled Data Discovery with Conversational Agents.\n* As we are audio recording the session today we may collect personally identifiable information. These will be transcribed and anonymised. Until that happens we will hold your data either securely on KCL equipment used by the researcher, or in the [institutional] secure repository, OneDrive. We will also collect transcripts (screenshots) of CGAI conversations. We do not foresee identifiable information being captured, but if it is, we will anonymise it. We are also using questionnaires, these will be fully anonymous and have their own consent procedure.\n* We will process your personal data (if collected) on a legitimate use basis for university research. We have applied the balance test.\n* We may use quotes in the paper but these will be anonymous.\n* We can’t withdraw your data once it has been anonymised, but you can withdraw from the study at any point.\n* We’d like your consent for future use of your anonymous data collected today by other academics studying conversational agents. If you don’t consent to your data being used in this study or in future you are free to leave now.\n* Any questions?\n*The study*\nIn this research, we would like you to think of one dataset (or group of datasets) that you would like to discover, using ChatGPT. (This can be a dataset you know exists, or a completely new-to-you dataset.) Once ChatGPT has found something (this may require more than one prompt), explore what it has returned to see if it is usable.\n*Timeline* \n[h] [tab:appendA]\n|p0.09|p0.85| **Timing** & **Activity** \n4- 4.05 & Introduction \n4.05 - 4.15 &\n[c]@l@Think about your motivation for a dataset search. \nEnter your motivation and any expectations (eg i know this dataset really well and know what I \nexpect to see/I don’t know if anything like this exists) into the questionnaire \nDevise your prompt for ChatGPT \nEnter any considerations you had when devising your prompt into the questionnaire\n \n4.15 - 4.30 & Enter your prompt and explore ChatGPT’s response \n4.30 - 4.50 & As a group, discuss ChatGPTs response and your assessment of this. Enter replies to the questionnaire as we go. \n4.50 - 5.00 & Wrap up! Please screenshot your ChatGPT conversation and place them [anonymous location]. Please name them according to the kind of dataset you were looking for. \nAppendix: B\n===========\n**Workshop Questionnaire**\nQuestionnaire:\n1. How familiar are you with ChatGPT?\n2. Thinking about either a personal or professional situation, what methods have you previously used to find datasets? Please tick all that apply\n3. What kind of dataset will you ask ChatGPT to find for you today?\n4. Why would you like to find this dataset (or why have you looked for it in the past?)\n5. Do you have an idea of what Chat GPT might suggest for you, either in terms of datasets or explanations? Please describe.\n6. What considerations did you have when devising your prompt (your question to ChatGPT)?\n7. What was the outcome today of discovering data with ChatGPT? Please select one.\n8. Did ChatGPT suggest an appropriate dataset?\n9. How useful was the explanation or supporting information provided by ChatGPT?\n10. How did you find the process of interacting with Chat GPT?\n11. Compared to previous methods of discovering data, how effective did you find the ChatGPT discovery process?\nAppendix: C\n===========\n[h] [tab:inductive]\n| **Code** | **Description** |\n| --- | --- |\n| datasetrequests | Requesting a dataset by name or close approximation |\n| describeddatasets | Describing the contents or function of a dataset or database |\n| implieddatasets | Asking for information based on data |\n| accessdatasets | Directions on how to find the dataset |\n| sourcerequest | Request information on the source of the data |\n| linkrequest | Request a link to the data |\n| advicerequest | Information on how to use or analyse the dataset |\n| furthercontextinformation | Information on the context of the dataset |\n| additionaldatasetinformation | Information on the dataset itself |\n| multimodaloutput | Returning other forms of output eg code |\n| furtheractivity | Integration of usage activity |\n| disagree | Countering information without providing further information |\n| provideinformation | Countering information (hallucinations) by providing further information |\n| conversationreminder | Referring back to earlier interactions |\n| explanationlinks | Querying why links can’t be provided |\n| explanationchallenge | Challenging an explanation provided by the agent |\n| explanationtechnical | Requesting a technical explanation |\n| explanationveracity | Issues with explanation truthfulness |\n[h] [tab:deductive]\n| **Code** | **n=** |\n| --- | --- |\n| data context | 5 |\n| coverage | 3 |\n| summarisation (of data content) | 5 |\n| original purpose | 2 |\n| time frame | 20 |\n| granularity (level of detail) | 4 |\n| documentation | 6 |\n| licensing | 5 |\n| access | 12 |\n| machine readability | 1 |\n| language | 1 |\n| labelling | 5 |\n| ability to share | 0 |\n| format | 6 |\n| consistency | 2 |\n| completeness | 3 |\n| collection methods | 5 |\n| provenance | 20 |\n| what has not been included in the data set | 8 |\n"},"data_id":{"kind":"string","value":"arxiv_0181675"}}},{"rowIdx":45396,"cells":{"content":{"kind":"string","value":"On Gauduchon connections with Kähler-like curvature\n===================================================\nWe study Hermitian metrics with a Gauduchon connection being ``Kähler-like\", namely, satisfying the same symmetries for curvature as the Levi-Civita and Chern connections. In particular, we investigate 6-dimensional solvmanifolds with invariant complex structures with trivial canonical bundle and with invariant Hermitian metrics. The results for this case give evidence for two conjectures that are expected to hold in more generality: first, if the Strominger-Bismut connection is Kähler-like, then the metric is pluriclosed; second, if another Gauduchon connection, different from Chern or Strominger-Bismut, is Kähler-like, then the metric is Kähler. As a further motivation, we show that the Kähler-like condition for the Levi-Civita connection assures that the Ricci flow preserves the Hermitian condition along analytic solutions.\nIntroduction\n============\nWe study Hermitian manifolds whose curvature tensor, with respect to a ``canonical\" connection, satisfies further symmetries, as initiated by A. Gray, and by B. Yang and F. Zheng. Symmetries clearly include the case of flat curvature. This is ultimately related to Yau’s Problem 87 concerning compact Hermitian manifolds with holonomy reduced to subgroups of ${\\mathrm U}(n)$.\nThe setting is the following. Let *X* be a complex manifold endowed with a Hermitian metric. One may wonder which connection is more ``natural\" to investigate the complex geometry of *X*.\nOn the one side, the *Levi-Civita connection* ∇*L**C* is the only torsion-free metric connection. In general, it does not preserve the complex structure, this condition forcing the metric to be Kähler. The absence of torsion yields that its curvature tensor *R**L**C* satisfies the symmetry called *first Bianchi identity*: \n∑*σ* ∈ G*R**L**C*(*σ**x*, *σ**y*)*σ**z* = 0. \nOn the other side, the *Chern connection* ∇*C**h* is another tool to investigate the differential complex geometry of *X*. It is the only Hermitian connection (namely, that preserves both the metric and the complex structure,) being compatible with the Cauchy-Riemann operator on the holomorphic tangent bundle. This yields that its curvature tensor *R**C**h* satisfies the *type condition* symmetry: \n*R**C**h* ∈ ∧ 1, 1(*X*; End(*T*1, 0*X*)). \nThe condition *R**C**h* = *R**L**C* forces the metric to be Kähler, as proven in. In the same spirit, in, the authors prove that if the total scalar curvatures *s**C**h* = *s**L**C*, then the metric is balanced. Notwithstanding, there are plenty of non-Kähler Hermitian metrics such that either *R**L**C* or *R**C**h* satisfies both the symmetries of the Levi-Civita curvature and the symmetries of the Chern curvature. Such metrics are called *G-Kähler-like*, respectively *Kähler-like* by B. Yang and F. Zheng.\nNotice that no locally homogeneous examples of compact complex Hermitian surfaces with odd first Betti number exist such that the Levi-Civita connection is Kähler-like, thanks to a result by Muškarov that prevents *J*-invariant Ricci tensor.\nWe point out three remarks that motivate the (G-)Kähler-like conditions. First, they imply the cohomological property of the metric being *balanced* in the sense of Michelsohn, as proven in. This property plays a role in the Strominger system, and this is a first motivation in. Second, symmetries are clearly satisfied by zero curvature: classification results in case of certain specific connections are investigated in. This is related to Yau’s Problem 87 in on compact Hermitian manifolds with holonomy reduced to a subgroup of ${\\mathrm U}(n)$ (here the connection needs not to be the Levi-Civita connection). Third, we prove that the Kähler-like condition for the Levi-Civita connection assures that the Ricci flow preserves the Hermitian condition along (analytic, compare ) solutions:\nLet *g*0 be a Hermitian metric on a compact complex manifold, and consider an analytic solution (*g*(*t*))*t* ∈ ( − *ɛ*, *ɛ*) for *ɛ* > 0 of the Ricci flow\n \n$$\\label{eq:ricci-flow}\\tag{RF}\n\\frac{d}{dt}g(t) = -\\mathrm{Ric}(g(t)), \\qquad g(0)=g\\_0.$$ \nIf the Levi-Civita connection of *g*0 is Kähler-like, then *g*(*t*) is Hermitian for any *t*.\nIn fact, there are infinitely-many Hermitian connections, namely, preserving both the metric and the complex structure, but possibly having torsion. Prescribing components of the torsion selects a special family of Hermitian connections ∇*ɛ* called *canonical connections in the sense of Gauduchon*, varying $\\varepsilon\\in\\mathbb R$. Such family includes and interpolates the Chern connection ∇*C**h* = ∇0 and the *Strominger-Bismut connection* $\\nabla^{+}=\\nabla^{\\sfrac{1}{2}}$, both of which have a role in complex geometry and heterotic string theory. In this note, we extend the notion of *Kähler-like* (Definition [def:KL]) to the family $\\{\\nabla^{\\varepsilon}\\}\\_{\\varepsilon\\in\\mathbb R}$ of Gauduchon connections.\nWhereas the balanced condition is related to the Bott-Chern cohomology and to the Chern connection, the Strominger-Bismut connection is related to the Aeppli cohomology and to the notion of *pluriclosed* metrics (also known as SKT), see *e.g.*. Therefore, as a Strominger-Bismut counterpart of (see Theorem [thm:yang-zheng-balanced]), the following conjecture is suggested:\n[conj:bismut-skt] Consider a compact complex manifold endowed with a Hermitian metric. If the Strominger-Bismut connection is Kähler-like, then the metric is pluriclosed.\nNotice that the particular case of Strominger-Bismut-flat metrics is studied in, where it is proven that a compact Hermitian manifold whose Strominger-Bismut connection is flat admits, as finite unbranched cover, a local Samelson space, given by the product of a compact semisimple Lie group and a torus,.\nMoreover, they prove in that a balanced Strominger-Bismut-flat metric is actually Kähler. It is known that Hermitian metrics that are both pluriclosed and balanced are in fact Kähler, see *e.g.* or. In fact, it is conjectured that no compact non-Kähler complex manifold can admit at the same time a balanced metric and a pluriclosed metric, see. This conjecture has proven to be true, for example, for the twistor spaces of compact anti-self-dual Riemannian manifolds, for compact complex manifolds in the Fujiki class $\\mathcal C$, for $\\sharp\\_k \\mathbb S^3 \\times \\mathbb S^3$ for *k* ≥ 2 endowed with the complex structures constructed from the conifold transition, for nilmanifolds with invariant complex structures. So this last theorem of Wang, Yang, and Zheng is explained by the fact that the flatness of the Strominger-Bismut connection implies that the metric is pluriclosed, which gives a partial evidence for the above Conjecture [conj:bismut-skt] (see Section [relations] for details).\nThe canonical connections by Gauduchon allow to tie the Chern and Strominger-Bismut connections. So, one may claim that a generic canonical connection shares properties with both of them. Again since no non-Kähler metric can be both balanced and pluriclosed, we conjecture the following, that extends on Gauduchon-flat connections.\n[conj:gauduchon-kahler-like] Consider a compact complex manifold endowed with a Hermitian metric. Consider a canonical connection in the Gauduchon family, different from the Strominger-Bismut connection and the Chern connection. If it is Kähler-like, then the metric is Kähler.\nWe test these conjectures on a class of compact complex manifolds of complex dimension three given by quotients of Lie groups[1](#fn1). More precisely, we restrict to *Calabi-Yau solvmanifolds*, that is, compact quotients of solvable Lie groups by discrete subgroups, endowed with an invariant complex structure having a non-zero invariant closed (3,0)-form, which implies that the canonical bundle is holomorphically-trivial. We choose this class because of their role *e.g.* in constructing explicit invariant solutions to the Strominger system with respect to a Gauduchon connection. They will be equipped with an invariant Hermitian metric. When the group is nilpotent, we get *nilmanifolds* and they always satisfy the latter condition on triviality of the canonical bundle by. Invariant complex structures on 6-dimensional nilmanifolds are ultimately classified by, see also the references therein, and they never admit Kähler metrics, unless the nilmanifold be a torus, by topological obstructions. Invariant complex structures on 6-dimensional solvmanifolds with trivial canonical bundle are classified by.\nThe following theorem is a consequence of direct inspection performed with the help of the symbolic computation softwares, Sage and Mathematica.\n[thm:solvmanifolds] Consider the class of compact 6-dimensional solvmanifolds endowed with an invariant complex structure with holomorphically-trivial canonical bundle and with an invariant Hermitian metric.\n* If the Strominger-Bismut connection is Kähler-like, then the metric is pluriclosed.\n* If a canonical connection in the Gauduchon family, different from the Strominger-Bismut connection and the Chern connection, is Kähler-like, then the metric is Kähler.\nTherefore, Conjectures [conj:bismut-skt] and [conj:gauduchon-kahler-like] are satisfied for this class of manifolds.\nA more detailed statement can be found in Theorem [thm:solv-precise], and Proposition [prop:lc] finalizes the study showing that, on six-dimensional solvmanifolds with holomorphically-trivial canonical bundle endowed with an invariant Hermitian non-Kähler metric, the Levi-Civita connection is never Kähler-like.\nIn particular, we observe here some evidence following from the performed study.\nHolomorphically-parallelizable complex structures are Chern-flat and then their Chern connection is clearly Kähler-like. Conversely, by, compact Hermitian Chern-flat manifolds are given by quotients of complex Hermitian Lie groups. We have examples of non-Strominger-Bismut-flat metric whose Strominger-Bismut connection is Kähler-like.\nThe examples here confirm, stating that compact Hermitian manifolds with a flat Gauduchon connection (different from Chern or Strominger-Bismut) must be Kähler. The analogous question for invariant Hermitian structures on Lie groups is asked and investigated in, and our result gives further evidence for a positive answer, since our computations are also valid on Lie groups endowed with left-invariant structures (see Remark [rmk:non-cpt]).\nOn a compact complex manifold, being balanced is a necessary condition for either the Levi-Civita or the Chern connections being Kähler-like, thanks to, but the converse does not hold true, as many examples below also show. It is expected that compact complex manifolds satisfying the $\\partial\\overline\\partial$-Lemma admit balanced metrics. We ask *whether the $\\partial\\overline\\partial$-Lemma property has any relation with the Kähler-like property*.\nAccording to, no compact complex non-Kähler manifold is expected to bear both a balanced metric and a pluriclosed metric. Then, according to and Conjecture [conj:bismut-skt], we expect that *a compact complex manifold admitting both a Hermitian metric with Kähler-like Chern connection and a (possibly different) Hermitian metric with Kähler-like Strominger-Bismut connection admits a Kähler metric*. By Theorem [thm:solv], this is obviously confirmed in the class of Calabi-Yau solvmanifolds of complex dimension 3. More in general, we ask *whether it is possible to admit two different connections (with respect to possibly different metrics) being both Kähler-like*.\nThe Iwasawa manifold and its small deformations show that the property of being Kähler-like with respect to the Chern connection is not open under deformations of the complex structures. It is not even closed, *i.e.* the central limit of a holomorphic family of compact Hermitian manifolds with the Kähler-like property may not admit any Hermitian metric satisfying the Kähler-like property with respect to the Chern connection, see Corollary [cor:kl-non-closed]. We ask *whether the Kähler-like property with respect to the Strominger-Bismut connection (or, more generally, any other Gauduchon connection,) is open and/or closed by holomorphic deformations of the complex structure*.\n This work has been written during several stays of the first-named author at the Departamento de Matemáticas de la Universidad de Zaragoza, which he thanks for the warm hospitality. He thanks also Jose Fernando for his warm hospitality at the Facultad de Matemáticas of the Universidad Complutense de Madrid. The authors would like to thank Francesco Pediconi, Fabio Podestà, Cristiano Spotti, and Luigi Vezzoni for several interesting discussions. Many thanks also to the anonymous Referees for their useful comments and suggestions. \nKähler-like connections\n=======================\nCurvatures\n----------\nLet *X* be a manifold endowed with a complex structure *J* and a Hermitian structure $h=g-\\sqrt{-1}\\omega$.\nLet ∇ be any linear *metric* connection on *X*, that is, preserving the Riemannian metric *g*, namely, ∇*g* = 0. (Denote with the same symbol its $\\C$-bi-linear extension to $TX\\otimes\\_\\R\\C$.) Consider its curvature operator: \n*R*∇(*x*, *y*) : = ∇2(*x*, *y*) = [∇*x*, ∇*y*] − ∇[*x*, *y*], \n and define also the (4, 0)-tensor \n*R*∇(*x*, *y*, *z*, *w*) : = *g*(*R*∇(*x*, *y*)*z*, *w*). \n Notice that, by the very definition, respectively by the condition ∇*g* = 0, \n*R*∇(\\_, \\_) ∈ ∧ 2(*X*; End(*T**X*)) and *R*∇(\\_, \\_, \\_, \\_) ∈ ∧ 2*X* ⊗ ∧ 2*X*. \nThe *first Ricci curvature* and the *second Ricci curvature* are, respectively, the traces \nRic(*i*)(*x*, *y*) = tr *R*∇(*x*, *y*) ∈ ∧ 2*X*, \n \nRic(*i**i*)(*z*, *w*) = tr*g**R*∇(\\_, \\_, *z*, *w*) ∈ ∧ 2*X* ⊂ End(*T**X*). \n The *scalar curvature* is \n$$\\mathrm{Scal} = \\mathrm{tr}\\_g \\mathrm{Ric}^{(i)} = \\mathrm{tr}\\, \\mathrm{Ric}^{(ii)} \\in\\mathcal{C}^\\infty(X;\\R).$$ \nGauduchon connections\n---------------------\nWe focus on the *canonical connections* in the Gauduchon family as defined in : for $t\\in\\mathbb R$, the Hermitian connection $\\nabla^{G\\_t}=:\\nabla^{\\frac{1-t}{4}, \\frac{1+t}{4}}$ associated to (*J*, *g*) is defined by \n$$g(\\nabla^{G\\_t}\\_xy, z) = g(\\nabla^{LC}\\_xy, z) + \\frac{1-t}{4}\\, T(x,y,z) + \\frac{1+t}{4}\\, C(x,y,z),$$ \n where \n*T* : = *J**d**ω* : = − *d**ω*(*J*\\_, *J*\\_, *J*\\_), *C* : = *d**ω*(*J*\\_, \\_, \\_), \n and ∇*L**C* denotes the Levi-Civita connection, that we can compute by\n \n$$\\begin{aligned}\ng(\\nabla^{LC}\\_xy, z)\n&=& \\frac{1}{2}\\,\\left(\nx\\,g(y,z) + y\\,g(x,z) - z\\,g(x,y) \\right. \\\\[5pt]\n&& \\left. + g([x,y],z) - g([y,z],x) - g([x,z],y) \\right).\\end{aligned}$$ \nNote that, in the space of metric connections ∇*ɛ*, *ρ* as defined in, Gauduchon connections sit on the line $\\varepsilon+\\rho=\\frac{1}{2}$.\nOther than \n∇*L**C* = ∇0, 0, \n special values are \n$$\\nabla^{Ch} = \\nabla^{G\\_1} = \\nabla^{0,\\frac{1}{2}},\n\\qquad\n\\nabla^+ = \\nabla^{G\\_{-1}} = \\nabla^{\\frac{1}{2},0},\n\\qquad\n\\nabla^- = \\nabla^{-\\frac{1}{2},0}$$ \n corresponding to the Chern, Strominger-Bismut, and anti-Strominger-Bismut connections, respectively. Moreover, \n$$\\nabla^{G\\_0}=\\nabla^{\\frac{1}{4},\\frac{1}{4}}=:\\nabla^{lv}$$ \n is the *first canonical connection*, also called *associated connection*, namely the projection onto the holomorphic tangent bundle of the Levi-Civita connection. Another important connection is given by \n$$\\nabla^{G\\_{\\sfrac{1}{3}}},$$ \n called the *minimal Gauduchon connection* because it is distinguished by the property that it has the smallest total torsion among Gauduchon connections.\nSymmetries of the curvature\n---------------------------\nIf we focus on the *Levi-Civita connection* ∇*L**C*, (namely, the unique metric torsion-free connection on *X*,) respectively on the *Chern connection* ∇*C**h*, (namely, the unique Hermitian connection being compatible with the Cauchy-Riemann operator $\\overline\\partial$ on the holomorphic tangent bundle,) we have further symmetries.\nThe Levi-Civita connection ∇ : = ∇*L**C* satisfies the *first Bianchi identity*:\n \n$$\\label{eq:bianchi1}\\tag{1Bnc}\n\\sum\\_{\\sigma\\in\\mathfrak{G}} R^{\\nabla}(\\sigma x,\\sigma y)\\sigma z = 0.$$ \n(This follows by the Levi-Civita connection being torsion-free, where the torsion is defined as \n*T*∇(*x*, *y*) : = ∇*x**y* − ∇*y**x* − [*x*, *y*], \n and by the Jacobi identity for the Lie bracket [\\_, \\_].) More in general, for a metric connection ∇ with possibly non-zero torsion *T*∇, we have (compare *e.g.* ): \n∑*σ* ∈ G*R*∇(*σ**x*, *σ**y*)*σ**z* = *d*∇*T*(*x*, *y*, *z*). \nThe Chern connection ∇ : = ∇*C**h* satisfies the *type condition*: \n*R*∇ ∈ ∧ 1, 1(*X*; End(*T*1, 0*X*)), \n namely,\n \n$$\\label{eq:type}\\tag{Cplx}\nR^{\\nabla}(x,y,z,w) \\;=\\; R^{\\nabla}(x,y,Jz,Jw) \\;=\\; R^{\\nabla}(Jx,Jy,z,w)\\;.$$ \n(The *J*-invariance in the third and fourth arguments, namely the property *R*∇ ∈ ∧ 2(*X*; End(*T*1, 0*X*)), follows from ∇*J* = 0 and *g*(*J*\\_, *J*\\_) = *g*(\\_, \\_). The conclusion follows from $\\nabla^{0,1}=\\overline\\partial$ yielding (∇0, 1)2 = 0 and by ∇ being real.)\nCondition for a metric connection implies\n \n$$\\label{eq:symm}\\tag{Symm}\nR^\\nabla \\in S^2\\wedge^2X,$$ \nnamely, *R*∇(*x*, *y*, *z*, *w*) = *R*∇(*z*, *w*, *x*, *y*). In this case, the first Bianchi identity holds for the (4, 0)-tensor *R*∇ for permutations of any triple of indices. If moreover *R*∇ ∈ ∧ 2(*X*; End(*T*1, 0*X*)), then actually $R^\\nabla \\in \\wedge^{1,1}(X;\\mathrm{End(T^{1,0}X))}$.\nKähler-like symmetries\n----------------------\nOn a Kähler manifold, we have \n∇*L**C* = ∇*C**h* , \n since ∇*L**C**J* = 0. In particular, ∇ : = ∇*L**C* satisfies both and.\n[def:KL] Let *X* be a complex manifold endowed with a Hermitian structure. Let ∇ be a metric connection on it. We say that ∇ is *Kähler-like* if it satisfies and.\n[comparison with Yang and Zheng’s ] We claim that the above definition specializes to Yang and Zheng’s definitions in in case of Chern connection and Levi-Civita connection. Recall that, in, B. Yang and F. Zheng call a Hermitian structure:\n* *Kähler-like (in the sense of )* if, for any (1, 0)-tangent vector *X*, *Y*, *Z*, and *W*, it holds \n$$R^{Ch}(X, \\bar Y, Z, \\bar W) \\;=\\; R^{Ch}(Z, \\bar Y, X, \\bar W) \\;;$$\n* *G-Kähler-like (in the sense of )*, in honour of Gray, if, for any (1, 0)-tangent vector *X*, *Y*, *Z*, and *W*, it holds \n$$R^{LC}(X, Y, \\bar Z, \\bar W) \\;=\\; R^{LC}(X, Y, Z, \\bar W) \\;=\\; 0 \\;.$$\nThen:\n1. a Hermitian structure is Kähler-like in the sense of if and only if its Chern connection is Kähler-like in the sense of Definition [def:KL];\n2. a Hermitian structure is G-Kähler-like in the sense of if and only if its Levi-Civita connection is Kähler-like in the sense of Definition [def:KL].\n*(1)* Assume that the Chern connection ∇*C**h* is Kähler-like in the sense of Definition [def:KL]. In particular, by, we get:\n \n$$\\begin{aligned}\nR^{Ch}(X, \\bar Y, Z, \\bar W) &=& -R^{Ch}(\\bar Y, Z, X, \\bar W) - R^{Ch}(Z, X, \\bar Y, \\bar W) \\\\[5pt]\n&=& R^{Ch}(Z, \\bar Y, X, \\bar W) + g(R^{Ch}(X, Z)\\bar Y, \\bar W) \\\\[5pt]\n&=& R^{Ch}(Z, \\bar Y, X, \\bar W),\\end{aligned}$$ \nsince $g(R^{Ch}(X, Z)\\bar Y, \\bar W)=0$.\nConversely, assume that the Hermitian structure is Kähler-like in the sense of. We already noticed that the Chern connection satisfies. We prove. It suffices to prove it for \n$$(x,y,z,w) \\in \\{ (\\bar X, Y, Z, \\bar W),\\, (X, \\bar Y, Z, \\bar W),\\, (X, Y, \\bar Z, \\bar W) \\},$$ \n where *X*, *Y*, *Z*, and *W* are (1, 0)-tangent vector fields. For example, in the first case, we have:\n \n$$\\begin{aligned}\n\\lefteqn{ R^\\nabla(\\bar X, Y, Z, \\bar W)+R^\\nabla(Y, Z, \\bar X, \\bar W)+R^\\nabla(Z, \\bar X, Y, \\bar W) } \\\\[5pt]\n&=& -R^\\nabla(Y, \\bar X, Z, \\bar W)+R^\\nabla(Z, \\bar X, Y, \\bar W) = 0,\\end{aligned}$$ \nwhere we used the skew-symmetries of *R*∇, the type properties of *R*∇, and the condition of Kähler-like in the sense of.\n*(2)* Assume that the Levi-Civita connection ∇*L**C* is Kähler-like in the sense of Definition [def:KL]. In particular, by, we get: for any (1, 0)-tangent vector fields *X*, *Y*, *Z*, and *W*, \n$$R^{LC}(X, Y, \\bar Z, \\bar W) = R^{LC}(X, Y, J\\bar Z, J\\bar W) = -R^{LC}(X, Y, \\bar Z, \\bar W),$$ \n whence \n$$R^{LC}(X, Y, \\bar Z, \\bar W) = 0 \\;;$$ \n moreover, by using, we get:\n \n$$\\begin{aligned}\nR^{LC}(X, Y, Z, \\bar W) &=& R^{LC}(Z, \\bar W, X, Y) = R^{LC}(Z, \\bar W, JX, JY) \\\\[5pt]\n&=& -R^{LC}(Z, \\bar W, X, Y) = - R^{LC}(X, Y, Z, \\bar W),\\end{aligned}$$ \nwhence \n$$R^{LC}(X, Y, Z, \\bar W) = 0.$$ \nConversely, assume that the Hermitian structure is G-Kähler-like in the sense of. By, we have also: for any (1, 0)-tangent vector fields *X*, *Y*, *Z*, and *W*,\n \n$$\\label{eq:gray}\\tag{Gray}\nR^{LC}(X,Y,Z,W) = 0.$$ \nWe already noticed that the Levi-Civita connection satisfies. We prove. By, it suffices to show *R*(*x*, *y*, *z*, *w*) = *R*(*z*, *y*, *J**z*, *J**w*). By the definition of G-Kähler-like and by, and up to conjugation and (skew-)symmetries, we are reduced to the terms of curvatures:\n \n$$\\begin{aligned}\nR(\\bar X, Y, Z, \\bar W) = -R(Y, \\bar X, Z, \\bar W), & \\qquad & R(X, \\bar Y, Z, \\bar W), \\\\[5pt]\nR(X, Y, Z, \\bar W) = 0, & \\qquad & R(X, Y, \\bar Z, \\bar W) = 0, \\\\[5pt]\nR(\\bar X, Y, \\bar Z, \\bar W) = 0, & \\qquad & R(X, \\bar Y, \\bar Z, \\bar W) = 0, \\\\[5pt]\nR(\\bar X, \\bar Y, Z, \\bar W) = 0, & \\qquad & R(\\bar X, \\bar Y, \\bar Z, \\bar W) = 0,\\end{aligned}$$ \nwhence to the case: $(x,y,z,w)=(X, \\bar Y, Z, \\bar W)$. Of course, in this case holds.\nWe will use also the following characterization of the Kähler-like condition.\n[caracterizacionKL] A metric linear connection ∇ on a manifold endowed with a Hermitian structure is Kähler-like in the sense of Definition [def:KL] if and only if, for any (1, 0)-tangent vector fields *X*, *Y*, *Z*, and *W*, for any tangent vector fields *x*, *y*, *z*, and *w*,\n \n$$\\label{eq:bianchi1-v2}\\tag{1Bnc'}\nB(X, \\bar Y, Z, \\bar W) := R^\\nabla(X, \\bar Y, Z, \\bar W) - R^\\nabla(Z, \\bar Y, X, \\bar W) = 0,$$ \n \n$$\\label{eq:type-v2}\\tag{Cplx'}\nR^\\nabla(x, y, Z, W) = R^\\nabla(X, Y, z, w) = 0.$$ \nAssume that ∇ is Kähler-like in the sense of Definition [def:KL]. Moreover, we get also. Then:\n \n$$\\begin{aligned}\nR^\\nabla(x, y, Z, W) &=& R^\\nabla(x, y, JZ, JW) \\;=\\; -R^\\nabla(x, y, Z, W) \\;, \\\\[5pt]\nR^\\nabla(X, Y, z, w) &=& R^\\nabla(z, w, X, Y) \\;=\\; R^\\nabla(z, w, JX, JY) \\\\[5pt]\n&=& -R^\\nabla(z, w, X, Y) \\;=\\; -R^\\nabla(X, Y, z, w) \\;,\\end{aligned}$$ \nwhence we get. Now, we have:\n \n$$\\begin{aligned}\nR^\\nabla(X, \\bar Y, Z, \\bar W) &=& -R^\\nabla(\\bar Y, Z, X, \\bar W)-R^\\nabla(Z, X, \\bar Y, \\bar W) \\\\[5pt]\n&=& R^\\nabla(Z, \\bar Y, X, \\bar W) \\;,\\end{aligned}$$ \nthat is,.\nOn the other hand, up to anti-symmetries and conjugation, the only non-zero terms of the curvature are \n$$R^\\nabla(X, \\bar Y, Z, \\bar W) \\;.$$ \n Therefore, is satisfied. Also, it follows that yields.\nWe notice that the possible several traces become equal as soon as we assume further symmetries of the curvature tensor. In this direction, compare the results by Liu and Yang in, comparing scalar curvatures. In particular, state conditions concerning equalities between total scalar curvatures that force the metric to be Kähler, respectively balanced.\nRelations to special Hermitian metrics\n--------------------------------------\nA Hermitian metric *ω* on a complex manifold is called *balanced* if *d**ω*\\**ω* = 0, where *d**ω*\\* denotes the adjoint operator of *d* with respect to the *L*2-pairing induced by the metric associated to *ω*; in other words, *d**ω**n* − 1 = 0, where *n* is the complex dimension of the manifold.\nWe recall the following obstruction to being Kähler-like, respectively G-Kähler-like in the sense of.\n[][thm:yang-zheng-balanced] If a Hermitian metric on a compact complex manifold has either Levi-Civita or Chern connection being Kähler-like, then it is balanced in the sense of Michelsohn.\nA Hermitian metric *ω* on a complex manifold is called *pluriclosed* (also known as *SKT*) if $\\partial\\overline\\partial\\omega=0$. Using a result in, and inspired by the argument in, we show the following result for Strominger-Bismut-flat Hermitian manifolds, which can be seen as an evidence for Conjecture [conj:bismut-skt] in the special case of flat connection.\n[thm:bismut-flat-skt] Let *X* be a complex manifold of complex dimension *n* endowed with a Hermitian metric. If the Strominger-Bismut connection is flat, then the metric is pluriclosed.\nDenote by *ω* the Strominger-Bismut-flat Hermitian metric on the manifold *X*. The pluriclosed condition $\\partial\\overline\\partial\\omega=0$ is a local property. The Strominger-Bismut connection ∇+ being flat, we can choose an unitary parallel frame $(e\\_j, \\bar e\\_j)\\_j$ in a neighbourhood of any point *p* of *X*, that is, $\\nabla^+e\\_j=\\nabla^+\\bar e\\_j=0$. Furthermore, by, it has (local) structure constants \n$$[e\\_i,e\\_j]=c\\_{ij}^k e\\_k,\n\\qquad\n[e\\_i, \\bar e\\_j] = \\overline{c\\_{jk}^i} e\\_k - c\\_{ik}^j \\bar e\\_k,$$ \n where $c\\_{ij}^k\\in\\mathbb C$ are constants satisfying the required identities, and *i*, *j*, *k* ∈ {1, …, *n*}. In addition, the metric satisfies \n⟨[*X*, *Y*], *Z*⟩ = − ⟨[*X*, *Z*], *Y*⟩, \n for any *X*, *Y*, *Z* in $(e\\_j, \\bar e\\_j)\\_{1\\leq j\\leq n}$. Hereafter, we shorten *e.g.* $e^{i\\bar j}:=e^i\\wedge \\bar e^j$, where $(e^j,\\bar e^j)\\_j$ is the dual coframe of $(e\\_j,\\bar e\\_j)\\_j$, and we use Einstein notation on summation over repeated indices. In this notation, we have that with respect to the unitary parallel frame $(e\\_j, \\bar e\\_j)\\_j$ the metric *ω* is expressed locally as \n$$\\omega = \\frac{\\sqrt{-1}}{2} e^{k \\bar k}.$$ \nIn other words, we are locally reduced to the Lie algebra case where the complex structure and the metric are both left invariant, and moreover, the metric is actually bi-invariant. In these conditions one has $\\partial\\overline\\partial\\omega=0$ (see also ), so the metric *ω* is pluriclosed.\nHermitian geometry on six-dimensional Calabi-Yau solvmanifolds\n==============================================================\nWe consider 6-dimensional *solvmanifolds* $X = \\left. \\Gamma \\middle\\backslash G \\right.$, namely, compact quotients of connected simply-connected solvable Lie groups by co-compact discrete subgroups. In particular, we consider solvmanifolds endowed with invariant Hermitian structures (*J*, *g*), that is, structures whose lift to the universal cover is invariant under left-translations. Such structures are encoded in the associated Lie algebra g. As a matter of notation, we will denote Lie algebras in the concise form in : *e.g.* rh3 = (0, 0, − 12, 0) means that the four-dimensional Lie algebra rh3 admits a basis (*e*1, *e*2, *e*3, *e*4) such that [*e*1, *e*2] = *e*3, the other brackets being trivial. The notation actually refers to the dual rh3\\*: the dual basis (*e*1, *e*2, *e*3, *e*4) satisfies *d**e*1 = *d**e*2 = *d**e*4 = 0 and *d**e*3 = − *e*1 ∧ *e*2, where we will shorten *e*12 : = *e*1 ∧ *e*2. Here, we use the following formula to relate the differential *d*: g∨ → ∧ 2g∨ and the bracket [\\_, \\_]: ∧ 2g → g: \n*d**α*(*x*, *y*) = − *α*([*x*, *y*]). \nMetric formulas\n---------------\nLet \n(*φ*1, *φ*2, *φ*3) \n be an invariant co-frame of (1, 0)-forms with respect to *J*. With respect to the frame $\\left( \\varphi\\_1, \\varphi\\_2, \\varphi\\_3, \\bar\\varphi\\_1, \\bar\\varphi\\_2, \\bar\\varphi\\_3 \\right)$ dual to $\\left( \\varphi^1, \\varphi^2, \\varphi^3, \\bar\\varphi^1, \\bar\\varphi^2, \\bar\\varphi^3 \\right)$, we first set the structure constants \n[*φ**I*, *φ**H*] = : *c**I**H**K**φ**K*. \n (Capital letters here vary in $\\{1,2,3,\\bar1,\\bar2,\\bar3\\}$ and refer to the corresponding component; the Einstein summation is assumed.)\nThe generic invariant Hermitian structure *ω* = *g*(\\_, *J*\\_) is given by\n \n$$\\begin{aligned}\n\\label{eq:metric}\n2\\,\\omega &=& \\sqrt{-1} r^2\\, \\varphi^{1\\bar1} + \\sqrt{-1} s^2\\,\\varphi^{2\\bar2} + \\sqrt{-1} t^2\\, \\varphi^{3\\bar3} \\\\[5pt]\n&& + u\\, \\varphi^{1\\bar2} - \\bar u\\, \\varphi^{2\\bar1} + v\\, \\varphi^{2\\bar3} - \\bar v\\, \\varphi^{3\\bar2} + z\\, \\varphi^{1\\bar3} - \\bar z\\, \\varphi^{3\\bar1} \\nonumber\\end{aligned}$$ \nwhere the coefficients satisfy \n*r*2 > 0, *s*2 > 0, *t*2 > 0, \n \n*r*2*s*2 > ∣*u*∣2, *r*2*t*2 > ∣*z*∣2, *s*2*t*2 > ∣*v*∣2, \n \n$$r^2s^2t^2 + 2 {\\mathsf{Re}\\,}(\\sqrt{-1}\\,\\bar u z \\bar v) > t^2|u|^2+r^2|v|^2+s^2|z|^2.$$ \n (Hereafter, we shorten *e.g.* $\\varphi^{3\\bar1}:=\\varphi^3\\wedge\\bar\\varphi^1$.) That is to say, the matrix (*g**K**L*)*K*, *L* is positive-definite. Its inverse is denoted by \n(*g**K**L*)*K*, *L* : = (*g**K**L*)*K*, *L*− 1. \nLet Ω be the matrix associated to the Hermitian metric, i.e. \n$$\\Omega=\\left(\\!\\!\\! \\begin{array}{ccc}\n\\sqrt{-1}\\, \\frac{r^2}{2} & \\frac{u}{2} & \\frac{z}{2} \\\\\n-\\frac{\\overline{u}}{2} & \\sqrt{-1}\\, \\frac{s^2}{2} & \\frac{v}{2} \\\\\n-\\frac{\\overline{z}}{2} & -\\frac{\\overline{v}}{2} & \\sqrt{-1}\\, \\frac{t^2}{2}\n\\end{array} \\!\\right)\\!$$ \n Then,\n \n$$\\label{det}\n8\\sqrt{-1}\\, \\det\\Omega = r^2s^2t^2-r^2|v|^2-s^2|z|^2-t^2|u|^2 + 2\\,{\\mathsf{Re}\\,}(\\sqrt{-1}\\,\\bar u\\bar v z) > 0.$$ \nWe can express the Christoffel symbols of ∇*L**C* as\n \n$$\\begin{aligned}\n(\\Gamma^{LC})\\_{IH}^{K} &=& \\frac{1}{2}\\, g^{KL}\\, \\left( g([\\varphi\\_I,\\varphi\\_H], \\varphi\\_L) - g([\\varphi\\_H,\\varphi\\_L], \\varphi\\_I) - g([\\varphi\\_I,\\varphi\\_L], \\varphi\\_H) \\right) \\\\[5pt]\n&=& \\frac{1}{2}\\, c\\_{IH}^{K} - \\frac{1}{2}\\, g^{KA}g\\_{BI}c\\_{HA}^{B} - \\frac{1}{2}\\, g^{KA}g\\_{BH}c\\_{IA}^{B}.\\end{aligned}$$ \nTherefore, for (*ɛ*, *ρ*), we compute the Christoffel symbols of the ∇*ɛ*, *ρ* connection: \n(Γ*ɛ*, *ρ*)*I**H**K* = (Γ*L**C*)*I**H**K* + *ɛ**g**K**L**T**I**H**L* + *ρ**g**K**L**C**I**H**L*, \n where \n*T**I**H**L* = − *d**ω*(*J**φ**I*, *J**φ**H*, *J**φ**L*), *C**I**H**L* = *d**ω*(*J**φ**I*, *φ**H*, *φ**L*). \n We can then express the (4, 0)-Riemannian curvature of ∇*ɛ*, *ρ* as\n \n$$\\begin{aligned}\n(R^{\\varepsilon,\\rho})\\_{IHKL} &=& g\\_{AL}(\\Gamma^{\\varepsilon,\\rho})\\_{HK}^{B}(\\Gamma^{\\varepsilon,\\rho})\\_{IB}^{A} - g\\_{AL} (\\Gamma^{\\varepsilon,\\rho})\\_{IK}^{B} (\\Gamma^{\\varepsilon,\\rho})\\_{HB}^{A} \\\\[5pt]\n&& - g\\_{AL} c\\_{IH}^{B} (\\Gamma^{\\varepsilon,\\rho})\\_{BK}^{A}.\\end{aligned}$$ \nComplex structures\n------------------\nWe consider 6-dimensional *Calabi-Yau solvmanifolds*, meaning that they are endowed with an invariant complex structure having a non-zero invariant closed (3,0)-form, thus their canonical bundle is holomorphically-trivial. This includes nilmanifolds and the solvmanifolds in. We recall here their classification.\nSix-dimensional nilpotent Lie algebras have been classified by Morozov in 34 different classes, 18 of which admit invariant complex structures by Salamon : they are h1, …, h16, h19−, h26+. The complex structures on them are classified into four families in. We recall the complex structure equations and the underlying Lie algebras in Table [table:nil-cplx].\n[h!]\n[table:nil-cplx]\nWe also consider solvmanifolds other than nilmanifolds. Classification of invariant complex structures in dimension 6 such that the canonical bundle is holomorphically-trivial is obtained in. We recall the complex structure equations and the underlying Lie algebras in Table [table:solv-cplx].\n[h!]\n[table:solv-cplx]\nIn particular, notice that the family (Np) and the family (Siv1) consist of *holomorphically-parallelizable* structures, namely, such that the holomorphic tangent bundle is holomorphically-trivial. The case (Np) with *ρ* = 0 corresponds to the complex torus, the case (Np) with *ρ* = 1 corresponds to the holomorphically-parallelizable Iwasawa manifold, and the case (Siv1) corresponds to the holomorphically-parallelizable Nakamura manifold.\nWe recall that, by, holomorphically-parallelizable manifolds can be regarded, up to a holomorphic homeomorphism, as quotients of a connected simply-connected complex Lie group by a discrete subgroup. Clearly, any invariant metric on a holomorphically-parallelizable manifold is Chern-flat. (On the other side, by, a compact complex Hermitian manifold is Chern-flat if and only if its universal cover is holomorphically isometric to a complex Lie group endowed with an invariant Hermitian metric.) In, the authors prove that, on a compact complex manifold of complex dimension *n* ≥ 3, a metric such that both its Levi-Civita and its Chern connection are Kähler-like is actually Kähler. Recall that, by, holomorphically-parallelizable compact Kähler manifolds are complex torus. Then, on holomorphically-parallelizable nilmanifolds and solvmanifolds different than tori, the Chern connection associated to an invariant Hermitian structure is always Kähler-like, while the Levi-Civita connection is never.\nNote that this case also includes the case of the *special Lie algebra $\\mathfrak{sl}(2;\\mathbb C)$*, with structure equations\n \n$$\\tag{sl2C}\\label{eq:sl2C}\nd\\varphi^1=\\varphi^{23}, \\quad d\\varphi^2=-\\varphi^{13}, \\quad d\\varphi^3=\\varphi^{12},$$ \nother than the already mentioned *Iwasawa manifold* (Np) with *ρ* = 1, and *Nakamura manifold* (Siv1).\nSpecial Hermitian metrics\n-------------------------\nIn view of and Conjectures [conj:bismut-skt] and [conj:gauduchon-kahler-like], we are particularly interested on nilmanifolds and solvmanifolds admitting invariant complex structures with holomorphically-trivial canonical bundle and (invariant) balanced or pluriclosed metrics.\nAs for balanced, according to, and, they are:\n* either h1 with a holomorphically-parallelizable complex structure in Family (Np) with *ρ* = 0; any Hermitian metric is in fact Kähler;\n* or h2, …, h6 with a nilpotent complex structure in Family (Ni); a generic Hermitian metric as in is balanced if and only if *r*2 = 1, *v* = *z* = 0, and $s^2 + D = \\sqrt{-1}\\, \\bar u\\, \\lambda$;\n* or h5 with a holomorphically-parallelizable complex structure in Family (Np) with *ρ* = 1, corresponding to the Iwasawa manifold; in this case, any invariant Hermitian metric is balanced;\n* or h19− with a non-nilpotent complex structure in Family (Niii); balanced metrics in are characterized by *u* = *z* = 0, and either *t*2 = 1 and *v* = 0, or *t*2 > 0 and *v* = 1;\n* or $\\mathfrak g\\_1$ or $\\mathfrak g\\_2^{\\alpha}$ with a splitting-type complex structure in Family (Si); a generic metric as in is balanced if and only if *v* = *z* = 0; they are non-Kähler except for $\\mathfrak g\\_2^{0}$ with *u* = *v* = *z* = 0;\n* or g3 (Sii) or g5 (Siii2) or g7 (Siii4); balanced metrics in are characterized by *v* = *z* = 0, and moreover in case g5 take $u\\in\\mathbb R$, respectively in case $\\mathfrak g\\_7$ take *r*2 = *s*2;\n* or $\\mathfrak g\\_8$ with a holomorphically-parallelizable splitting-type complex structure in Family (Siv1): any metric is balanced; or with a splitting-type complex structure in Family (Siv3): balanced metrics as in are characterized by *v* = *z* = 0.\nAs for pluriclosed metrics, they exist only on:\n* h1 with a holomorphically-parallelizable complex structure in Family (Np) with *ρ* = 0; any Hermitian metric is in fact Kähler;\n* $\\mathfrak h\\_2$, $\\mathfrak h\\_4$, $\\mathfrak h\\_5$, $\\mathfrak h\\_8$ in Family (Ni), and in this case any Hermitian metric is pluriclosed, see ;\n* $\\mathfrak g\\_2^0$ in Family (Si) and $\\mathfrak g\\_4$ (Siii1); pluriclosed metrics are given by *u* = 0 in the form ; see.\nFinally, we review the existence of Kähler metrics. By and, more in general,, Kähler metrics do not exist on non-tori nilmanifolds (with invariant or non-invariant complex structures). On the other side, Kähler metrics exist on:\n* $\\mathfrak h\\_1$, namely, the complex torus;\n* $\\mathfrak g\\_2^0$ in Family (Si), and the Kähler metrics are given by the diagonal ones, *u* = *v* = *z* = 0, see.\nObserve that the Lie algebras $\\frh\\_7,\\ldots,\\frh\\_{16},\\frh\\_{26}^+$ and g9 do not admit neither balanced nor pluriclosed metrics.\nKähler-like Gauduchon connections on six-dimensional Calabi-Yau solvmanifolds\n=============================================================================\nIn this section we study the existence of Kähler-like connections on the class of 6-dimensional Calabi-Yau solvmanifolds.\nLet us denote the Gauduchon connections by ∇*ɛ* : = ∇*G*1 − 4*ɛ* and by *R**ɛ* their corresponding curvature. We recall that with respect to this notation, ∇*ɛ* = 0 = ∇*C**h* and $\\nabla^{\\varepsilon = \\sfrac{1}{2}} = \\nabla^+$.\n[thm:solv] Let *X* = Γ ∖ *G* be a six-dimensional solvmanifold endowed with an invariant complex structure *J* such that the canonical bundle is holomorphically-trivial. Denote by g the Lie algebra associated to *G* and let *ω* be the (1, 1)-form associated to any invariant *J*-Hermitian metric on *X*. We have the following.\n* If the Chern connection ∇*C**h* is Kähler-like, then g is isomorphic to h1, h5, g1, g2*α* ≥ 0, or g8, and the Hermitian metric *ω* is balanced.\n* If the Strominger-Bismut connection ∇+ is Kähler-like, then g is isomorphic to h1, h2, h8, g20, or g4, and the Hermitian metric *ω* is pluriclosed.\n* If a Gauduchon connection ∇*ɛ* is Kähler-like for some $\\varepsilon \\in \\mathbb{R}\\setminus\\{0,\\frac12\\}$, then g is isomorphic to h1 or g20, and the Hermitian metric *ω* is Kähler.\nNotice that Theorem [thm:solv] yields that Conjectures [conj:bismut-skt] and [conj:gauduchon-kahler-like] are satisfied for such class of six-dimensional Calabi-Yau solvmanifolds. It also shows that the property of being Kähler-like for the Chern connection is neither open nor closed under holomorphic deformations of the complex structure: as for non-closedness, it follows by\n[cor:kl-non-closed] The Lie algebra $\\mathfrak g\\_8$ admits Chern-flat Hermitian metrics for complex structures in Family (Siv3), which admit limits in Family (Siv1) which are not even balanced.\nThe following result makes Theorem [thm:solv] more precise, in specifying the complex structures *J* and the *J*-Hermitian metrics *ω* in each case when the connections are Kähler-like.\n[thm:solv-precise] In the conditions of Theorem [thm:solv], we have the following.\n* The Chern connection ∇*C**h* is Kähler-like precisely in the following cases:\n\t+ h1, with *J* in Family (Np) with *ρ* = 0, with any *ω* given by, which is in fact Kähler and Chern-flat;\n\t+ h5, with *J* in the holomorphically-parallelizable Family (Np) with *ρ* = 1, with any *ω* given by, which is in fact Chern-flat;\n\t+ g1, with *J* in Family (Si) with *θ* = 0, with *ω* given by with *u* = *v* = *z* = 0, which is in fact Chern-flat;\n\t+ g20, with *J* in Family (Si) with $\\theta=\\frac{\\pi}{2}$, with *ω* given by with *u* = *v* = *z* = 0, which is in fact Kähler and Chern-flat;\n\t+ g2*α* with *α* > 0, with *J* in Family (Si) with $\\theta\\not\\in\\{0,\\frac{\\pi}{2}\\}$, with *ω* given by with *u* = *v* = *z* = 0, which is in fact Chern-flat;\n\t+ g8, with *J* in the holomorphically-parallelizable Family (Siv1), with any *ω* given by, which is in fact Chern-flat;\n\t+ g8, with *J* in Family (Siv3), with *ω* given by with *u* = *v* = *z* = 0, which is in fact Chern-flat.In any case, the metric is balanced by.\n* The Strominger-Bismut connection ∇+ is Kähler-like precisely in the previous Kähler cases (see $\\mathfrak h\\_1$ and $\\mathfrak g\\_2^0$) and in the following cases:\n\t+ h2, with *J* in Family (Ni) with *ρ* = *λ* = 0 and $D=\\sqrt{-1}$, with *ω* given by with *r*2 = 1, *u* = *v* = *z* = 0 (not Strominger-Bismut-flat);\n\t+ h8, with *J* in Family (Ni) with *ρ* = *λ* = *D* = 0, with any *ω* given by (not Strominger-Bismut-flat);\n\t+ g4, with *J* in Family (Siii1), with *ω* given by with *u* = *v* = *z* = 0 (not Strominger-Bismut-flat).In any case, the metric is pluriclosed.\n* The Gauduchon connection ∇*ɛ* for some $\\varepsilon \\in \\mathbb{R}\\setminus\\{0,\\frac12\\}$ is Kähler-like precisely in the previous Kähler cases (see $\\mathfrak h\\_1$ and $\\mathfrak g\\_2^0$).\nThe rest of the section is devoted to the proofs of Theorems [thm:solv] and [thm:solv-precise]. Recall that by Remark [caracterizacionKL] we know that ∇*ɛ* is Kähler-like if and only if the identities and hold; more concretely, \n$$R^\\varepsilon\\_{ij\\bullet\\bullet}=R^\\varepsilon\\_{\\bullet\\bullet k\\ell}=0,\\text{ and }\nB^\\varepsilon\\_{i\\bar jk \\bar\\ell}:= R^\\varepsilon\\_{i\\bar j k\\bar \\ell}- R^\\varepsilon\\_{k\\bar j i\\bar \\ell}=0,$$ \n for any *i*, *j*, *k*, ℓ ∈ {1, 2, 3}, where for instance $B^\\varepsilon\\_{1\\bar 2 3\\bar 1}$ denotes $B^\\varepsilon(X\\_1,\\bar X\\_2,X\\_3,\\bar X\\_1)$. Throughout this section, and also Section [LCKL], we make use of this notation.\n[rmk:non-cpt] We remark that we are doing calculations at the level of the Lie algebra, equivalently, on invariant objects on the Lie group. In particular, this means that our results hold true also at the level of the *non-compact* Lie group. Compare also, where no compactness assumptions is supposed, but the extra assumption on invariant parallel frame, to study flat Gauduchon connections on Lie groups.\nNilmanifolds\n------------\n### Holomorphically-parallelizable nilmanifolds in Family (Np)\nConsider the complex structure equations \n*d**φ*1 = *d**φ*2 = 0, *d**φ*3 = *φ*12, \n and a generic (balanced) Hermitian metric given by. Let {*X*1, *X*2, *X*3} be the (1,0)-basis of vectors dual to {*φ*1, *φ*2, *φ*3}, that is, [*X*1, *X*2] = − *X*3.\nThe Gauduchon connection ∇*ɛ* is never Kähler-like unless *ɛ* = 0.\nA direct calculation shows that *R**i**j* • •*ɛ* = *R*• • *k*ℓ*ɛ* = 0 for any *i*, *j*, *k*, ℓ ∈ {1, 2, 3}. On the other hand, what symmetries of Bianchi-type concerns, we can observe that: \n$$B^{\\varepsilon}\\_{1\\bar{1}3\\bar{3}} = \\frac{2\\varepsilon^2t^4(r^2t^2-|z|^2)}{8\\sqrt{-1}\\det\\Omega}= 0\\Longleftrightarrow \\varepsilon = 0.$$ \nMoreover, *B*0 ≡ 0, *i.e.* ∇*C**h* is Kähler-like; in fact, this is true for any holomorphically-parallelizable manifold.\n### Nilmanifolds in Family (Ni)\nAccording to, any Hermitian structure can be expressed as:\n \n$$\\begin{gathered}\n\\label{F-I}\nd\\varphi^1=d\\varphi^2=0,\\quad d\\varphi^3 = \\rho\\,\\varphi^{12} + \\varphi^{1\\bar 1} + \\lambda\\,\\varphi^{1\\bar2} + D\\,\\varphi^{2\\bar2}, \\\\\n2\\omega = \\sqrt{-1}(\\varphi^{1\\bar 1} + s^2\\,\\varphi^{2\\bar2} + t^2\\,\\varphi^{3\\bar3}) + u\\,\\varphi^{1\\bar2} -\\bar u\\,\\varphi^{2\\bar1},\\nonumber\\end{gathered}$$ \nwhere *ρ* ∈ {0, 1}, *λ* ≥ 0, Im *D* ≥ 0, and *s*2 > ∣*u*∣2, *t*2 > 0; *i.e.* we can take *v* = *z* = 0 and *r*2 = 1 in the generic expression.\nIn the notation as above, if ∇*ɛ* is Kähler-like, then $\\varepsilon=\\frac12$, *ρ* = 0 and *ω* is pluriclosed.\nLet us take the element $R^{\\varepsilon}\\_{231\\bar{3}}=\\frac{2 \\varepsilon^2 \\rho s^2 t^6}{8\\sqrt{-1}\\det\\Omega}$. It vanishes if and only if *ɛ**ρ* = 0. If *ɛ* = 0, then $B^{0}\\_{1\\bar{1}3\\bar{3}} = -\\frac{s^2 t^4}{2(s^2-|u|^2)}\\neq0$, hence *ρ* = 0.\nNow, substituting *ρ* = 0 we find: \n$$B^{\\varepsilon}\\_{1\\bar{1}2\\bar{1}}=\\frac{-\\lambda (1-2\\varepsilon)^2t^2}{2}=0 \\Longleftrightarrow \\varepsilon=\\frac12\\text{ or } \\lambda = 0.$$ \n Finally, if *λ* = 0, then $B^{\\varepsilon}\\_{1\\bar{3}3\\bar{1}} = \\frac{s^2 t^4\\varepsilon (2\\varepsilon-1) }{s^2-|u|^2}\\neq 0$, hence $\\varepsilon=\\frac12$.\nFor the last statement, according to, the Hermitian metric *ω* is pluriclosed if and only if the parameters in the complex structure (Ni) satisfy \n$$\\rho + \\lambda^2 -(D+\\bar D) = 0.$$ \n Now, in the case *ρ* = 0, we have that: \n$$B^{\\frac12}\\_{1\\bar{1}2\\bar{2}} = \\frac{-t^2}{2} (\\lambda^2 - (D+\\bar D)),$$ \n whence the statement.\nIn the notation as above, ∇*ɛ* is Kähler-like if and only if $\\varepsilon=\\frac12$ and\n* either the Lie algebra is $\\frh\\_2$ and the Hermitian structure is given by $(\\rho, \\lambda, D) = (0,0,\\sqrt{-1})$ and *u* = 0,\n* or the Lie algebra is $\\frh\\_8$ and the Hermitian structure is anyone defined on the complex structure (*ρ*, *λ*, *D*) = (0, 0, 0).\nIn, it is shown that the only Lie algebras underlying equations with *ρ* = 0 admitting a Hermitian pluriclosed metric are $\\frh\\_2$ and $\\frh\\_8$. Moreover, ${\\mathsf{Re}\\,}D = \\frac{\\lambda^2}{2}$. Taking into account the classification of complex structures up to equivalence, *λ* can take only the value 0, so we are forced to:\n* $(\\rho, \\lambda, D) = (0,0,\\sqrt{-1})$, namely, the Lie algebra $\\frh\\_2$;\n* (*ρ*, *λ*, *D*) = (0, 0, 0), namely, the Lie algebra $\\frh\\_8$.\nWe are going to study the two cases above. First, the computation of the curvature elements for *ρ* = 0 and $\\varepsilon = \\sfrac{1}{2}$ yields that, for any *i*, *j*, *k*, ℓ ∈ {1, 2, 3}, \n$$R^{\\frac12}\\_{ij\\bullet\\bullet}=R^{\\frac12}\\_{\\bullet\\bullet k\\ell}=0.$$ \n With respect to the relations coming from the Bianchi identity, we summarize the results of the computations in Appendix [app:BC-Ni]. In particular, $\\varepsilon=\\frac12$, *ρ* = 0, *λ* = 0, and Re *D* = 0 yield that\n \n$$\\label{SKT}\n B^{\\frac12}\\equiv 0\\Longleftrightarrow B^{\\frac12}\\_{1\\bar{3}3\\bar{2}}=\\frac{ t^4 u (D-\\bar D)}{16\\det\\Omega}=0 \\Longleftrightarrow u\\, {\\mathsf{Im}\\,}D=0.$$ \nIn the cases above, the Strominger-Bismut connection is not flat. In fact, $R^+\\_{1\\bar{1}1\\bar{1}} = R^+\\_{2\\bar{2}2\\bar{2}} = t^2\\neq 0$ for $\\frak h\\_2$, and $R^+\\_{1\\bar{1}1\\bar{1}} = t^2\\neq 0$ for $\\frak h\\_8$, the other components of the curvature being zero.\n### Nilmanifolds in Family (Nii)\nConsider the complex structure equations \n$$d\\varphi^1=0,\\quad d\\varphi^2=\\varphi^{1\\bar1},\\quad d\\varphi^3 = \\rho\\,\\varphi^{12} +B \\varphi^{1\\bar 2} +c \\,\\varphi^{2\\bar1},$$ \n where $\\rho\\in\\{0,1\\},\\, c\\geq 0,\\, B\\in \\mathbb C$ satisfying (*ρ*, *B*, *c*) ≠ (0, 0, 0).\nFor nilmanifolds in Family (Nii), the Gauduchon connection ∇*ɛ* is never Kähler-like.\nFor the Chern connection, i.e. *ɛ* = 0, the result follows directly from the fact that *ω* is never balanced and by.\nAs a consequence, we consider *ɛ* ≠ 0 and the following elements: \n$$\\begin{array}{lcl}\n B^{\\varepsilon}\\_{3\\bar{2}2\\bar{3}} &=& -\\frac{t^4 (s^2t^2-|v|^2)}{4\\sqrt{-1}\\,\\det\\Omega} \\left(\\varepsilon^2 \\rho^2 - 2|B|^2 \\left(\\varepsilon-\\frac12\\right)\\left(\\varepsilon-\\frac14\\right)\\right),\\\\[1em]\n B^{\\varepsilon}\\_{3\\bar{3}2\\bar{2}} &=& \\frac{\\varepsilon\\, t^4 (s^2t^2-|v|^2)}{4\\sqrt{-1}\\,\\det\\Omega} \\left(\\varepsilon c^2 -2 |B|^2 \\left(\\varepsilon-\\frac14\\right)\\right),\n\\end{array}$$ \n where recall the expression for detΩ as in.\nObserve first that $R^{\\varepsilon}\\_{232\\bar{3}}=\\frac{2 \\varepsilon^2 \\rho \\bar B (s^2 t^2-|v|^2)t^4}{8\\sqrt{-1}\\det\\Omega}=0$ if and only if *ρ**B* = 0. If *B* = 0, then \n$$(B^{\\varepsilon}\\_{3\\bar{2}2\\bar{3}}, B^{\\varepsilon}\\_{3\\bar{3}2\\bar{2}}) = (0,0)\\Longleftrightarrow (\\rho, c)=(0,0)$$ \n which is a contradiction. Therefore, we may assume *B* ≠ 0 and *ρ* = 0. Now, \n$$B^{\\varepsilon}\\_{3\\bar{2}2\\bar{3}}=0 \\Longleftrightarrow \\varepsilon = \\frac12 \\text{ or } \\varepsilon = \\frac14.$$ \nIf $\\varepsilon = \\frac14$, then \n$$B^{\\frac14}\\_{3\\bar{3}2\\bar{2}}=0 \\Longleftrightarrow c=0$$ \n but in this case one can check that $B\\_{2\\bar{1}1\\bar{2}}^{\\frac14} = \\frac{|B|^2 t^2}{4}\\neq 0$.\nIf $\\varepsilon = \\frac12$, then $R^{\\frac12}\\_{231\\bar{2}}=\\frac{c (s^2 t^2 - |v|^2)^2}{16\\sqrt{-1}\\det\\Omega}$ and $R^{\\frac12}\\_{232\\bar{1}}=\\frac{-\\bar B (s^2 t^2 - |v|^2)^2}{16\\sqrt{-1}\\det\\Omega}$, therefore \n$$(R^{\\frac12}\\_{231\\bar{2}}, R^{\\frac12}\\_{232\\bar{1}}) = (0,0)\\Longleftrightarrow (B, c)=(0,0),$$ \n providing a contradiction.\n### Nilmanifolds in Family (Niii)\nConsider the complex structure equations \n$$d\\varphi^1=0,\\quad d\\varphi^2=\\varphi^{13}+\\varphi^{1\\bar3},\\quad d\\varphi^3 =\\sqrt{-1}\\nu\\,\\varphi^{1\\bar 1}+ \\sqrt{-1}\\,\\delta\\, (\\varphi^{1\\bar 2} -\\varphi^{2\\bar1}),$$ \n where *ν* = {0, 1} and *δ* = ± 1.\nFor nilmanifolds in Family (Niii), the Gauduchon connection ∇*ɛ* is never Kähler-like.\nConsider the Bianchi identity relation $B^{\\varepsilon}\\_{3\\bar{3}2\\bar{3}}$. Now,\n \n$$\\begin{aligned}\nB^{\\varepsilon}\\_{3\\bar{3}2\\bar{3}}&=&\\frac{(4 \\varepsilon^2+4 \\varepsilon-1)(-\\sqrt{-1} s^2+\\delta t^2) (s^2 t^2-|v|^2)v}{16\\sqrt{-1}\\det\\Omega}=0 \\\\\n&\\Longleftrightarrow& \\begin{cases}\nv=0,\\\\\n\\varepsilon = \\frac{-1\\pm \\sqrt 2}{2}.\n\\end{cases}\\end{aligned}$$ \nWe start supposing *v* = 0. Now, $B^{\\varepsilon}\\_{3\\bar{2}1\\bar{2}} = \\frac{2 \\varepsilon^2 \\bar z (\\sqrt{-1} s^2 +\\delta t^2)s^4 }{8\\sqrt{-1}\\det\\Omega}=0$ if and only if *z**ɛ* = 0. In the case *ɛ* = 0, one can check that $B^0\\_{3\\bar{3}2\\bar{2}}=\\frac{-s^4 t^2 (s^2 +\\delta \\sqrt{-1} t^2)}{16\\sqrt{-1}\\det\\Omega}$ is always a non-zero element. On the other hand, if *z* = 0, just take \n$$B^{\\varepsilon}\\_{2\\bar{2}3\\bar{3}}=\\frac{s^2t^2(s^2+ \\sqrt{-1} \\delta t^2)\n\\left(2\\varepsilon s^2+\\sqrt{-1}(1-2\\varepsilon)\\delta t^2\\right)(4 \\varepsilon-1)}{16\\sqrt{-1}\\det\\Omega}.$$ \nOne has $B^{\\varepsilon}\\_{2\\bar{2}3\\bar{3}}=0 \\Longleftrightarrow \\varepsilon = \\frac14$. But in this case, $B^{\\frac14}\\_{1\\bar{1}3\\bar{3}}=\\frac{-s^2}{2}\\neq 0$.\nFinally, if $\\varepsilon\\_0=\\frac{-1\\pm \\sqrt 2}{2}$ is a root of the polynomial 4*ɛ*2 + 4*ɛ* − 1, then:\n \n$$\\begin{aligned}\n\\lefteqn{ B^{\\varepsilon\\_0}\\_{3\\bar{3}2\\bar{2}}=0}\\\\\n&\\Longleftrightarrow& (s^2t^2-|v|^2)\\left((s^2 + \\sqrt{-1}\\,\\delta t^2) \\left[(\\mp 7 +5\\sqrt 2) s^2 \\pm \\sqrt{-1} t^2 (4\\pm 3\\sqrt2)\\right] \\right. \\\\\n&& \\left. -\\sqrt{-1}\\,\\delta |v|^2 (\\mp 3+2\\sqrt 2)\\right)=0,\\end{aligned}$$ \nbut the last expression is always non-zero.\nSolvmanifolds\n-------------\n### Solvmanifolds in Family (Si)\nBy Section [subsec:complex-st], for this family of complex structures the underlying Lie algebras are $\\frak g\\_1$ or $\\frak g\\_2^{\\alpha}$ with *α* ≥ 0.\nThe Chern connection ∇*C**h* is Kähler-like if and only *ω* given by satisfies *u* = *v* = *z* = 0. Moreover, in these cases, the Chern curvature vanishes identically.\nUsing, a necessary condition for ∇*C**h* to be Kähler-like is that *ω* must be a balanced metric. According to, *ω* is balanced if and only if *v* = *z* = 0. Now, with this restriction, it is immediate to see that *B*0 ≡ 0 if and only if *u* = 0 (see Appendix [sec:app:g2a-Chern]). The last statement follows by direct inspection.\n[rmk:g12] The previous result gives an example of a non-Kähler metric with Kähler-like Chern connection on a solvmanifold endowed with a left-invariant complex structure which does not admit any basis of (left-invariant) holomorphic vector fields.\nNote indeed that the above diagonal metrics are Kähler if and only if the parameter satisfies $A=\\sqrt{-1}$, corresponding to the Lie algebra $\\mathfrak g^0\\_2$. Clearly, by, the complex solvmanifold is in fact biholomorphic to a holomorphically-parallelizable manifold. But it is presented in a different way as quotient of a Lie group, and as such it is not parallelizable by left-invariant holomorphic vector fields.\nFor *ɛ* ≠ 0, the Gauduchon connection ∇*ɛ* is Kähler-like if and only if $\\frak g= \\frak g\\_2^0$ and *ω* is Kähler.\nWe prove firstly that the underlying Lie algebra must be $\\frak g\\_2^0$, corresponding to $A=\\sqrt{-1}$ in family (Si), as: \n$$B^{\\varepsilon}\\_{1\\bar{1}2\\bar{2}} = \\frac{8\\,\\varepsilon^2(r^2s^2-|u|^2)}{8\\sqrt{-1}\\, \\det\\Omega} \\left[({\\mathsf{Re}\\,}A)^2 (r^2s^2-|u|^2) + |A|^2|u|^2\\right]=0,$$ \n implies $A=\\sqrt{-1}$ and *u* = 0.\nNow, as it is shown in Appendix [sec:app:g2a], $B^{\\varepsilon}\\_{1\\bar{1}3\\bar{3}} = B^{\\varepsilon}\\_{2\\bar{2}3\\bar{3}} = 0$ if and only if (2*ɛ* − 1)*v* = (2*ɛ* − 1)*z* = 0.\nIf *v* = *z* = 0, in particular the metric *ω* is Kähler and one has that *B**ɛ* ≡ 0 and *R**ɛ* ≡ 0.\nOn the other hand, consider now the Strominger-Bismut connection, *i.e.* $\\varepsilon = \\frac12$. Then, $B^{\\frac12}\\equiv 0$ if and only if $(B^{\\frac12}\\_{1\\bar{3}3\\bar{3}}, B^{\\frac12}\\_{2\\bar{3}3\\bar{3}}) = (\\frac{\\sqrt{-1}z}{2}, \\frac{\\sqrt{-1}v}{2}) = (0,0)$, which implies *v* = *z* = 0.\n### Solvmanifolds in Family (Sii)\nBy Section [subsec:complex-st], the Lie algebra underlying this family of complex structures is $\\frak g\\_3$.\nFor any solvmanifold in Family (Sii), the Gauduchon connection ∇*ɛ* is never Kähler-like.\nFirst let us suppose *ɛ* = 0. Using, a necessary condition for ∇*C**h* to be Kähler-like is that *ω* must be a balanced metric. According to, *ω* is balanced if and only if *u* = *z* = 0. Now, with this restriction, \n$$B^0\\_{1\\bar{2}2\\bar{1}} = \\frac{-(t^2 - 2 \\sqrt{-1} s^2 x)(1+4x^2)}{32 x^2}\\neq 0,$$ \n whence the statement.\nIn what follows, we will suppose that *ɛ* ≠ 0 and a generic Hermitian metric *ω*. Consider the elements $R^{\\varepsilon}\\_{232\\bar{3}}$ and $R^{\\varepsilon}\\_{233\\bar{3}}$: \n$$\\begin{cases}\nR^{\\varepsilon}\\_{232\\bar{3}}=\\frac{\\varepsilon (\\sqrt{-1} + 2 x) (s^2 t^2 - |v|^2) (-v^2 + 2 \\sqrt{-1} s^2 t^2 x -\n 4 s^4 x^2 + 2 \\varepsilon (t^4 + 4 s^4 x^2) - 2 \\sqrt{-1} |v|^2 x)}{64x\\sqrt{-1}\\det\\Omega},\\\\[5pt]\nR^{\\varepsilon}\\_{233\\bar{3}}=\\frac{\\varepsilon (\\sqrt{-1} + 2 x) (s^2 t^2 -\n |v|^2) (v (\\sqrt{-1} (-1 + 2 \\varepsilon) t^2 + 4 \\varepsilon s^2 x) +\n 4 \\sqrt{-1} x (\\sqrt{-1} \\varepsilon t^2 - s^2 x + 2 \\varepsilon s^2 x) \\bar v)}{64x\\sqrt{-1}\\det\\Omega}.\n\\end{cases}$$ \nNow: \n$$\\begin{cases}\nR^{\\varepsilon}\\_{232\\bar{3}}=0\\Longleftrightarrow 2\\varepsilon t^4 - v^2 + 2\\sqrt{-1}x (s^2t^2 - |v|^2) + 4s^4x^2(2\\varepsilon-1)=0,\\\\[5pt]\nR^{\\varepsilon}\\_{233\\bar{3}}=0\\Longleftrightarrow v\\left(4\\sqrt{-1}\\,\\varepsilon s^2 x - (2\\varepsilon-1) t^2\\right) - 4x\\bar v\\left(\\sqrt{-1}\\,\\varepsilon t^2 + (2\\varepsilon - 1)s^2 x\\right)=0.\n\\end{cases}$$ \n Adding and subtracting the two expressions above, we obtain the equivalent system: \n$$\\begin{cases}\n(2\\sqrt{-1}\\,s^2x + 2\\varepsilon (t^2-2\\sqrt{-1}\\,s^2 x) - v)(t^2+2\\sqrt{-1}\\,s^2 x + v + 2\\sqrt{-1}\\,x\\bar v)=0,\\\\[5pt]\n(2\\sqrt{-1}\\,s^2x + 2\\varepsilon (t^2-2\\sqrt{-1}\\,s^2 x) + v)(t^2+2\\sqrt{-1}\\,s^2 x - v - 2\\sqrt{-1}\\,x\\bar v)=0.\n\\end{cases}$$ \n As a matter of notation, let us write the first equation as *A**B* = 0 and the second as *C**D* = 0.\nLet us focus our attention in the first equation and consider the case *A* = 0. We can express *v* as: \n$$v = 2\\sqrt{-1}\\,s^2x + 2\\varepsilon (t^2-2\\sqrt{-1}\\,s^2 x).$$ \n Substituting this value in the second equation we obtain that: \n$$\\begin{cases}\nC=4(\\varepsilon t^2 - \\sqrt{-1}s^2 x (2\\varepsilon -1))\\neq 0,\\\\[5pt]\nD = (1-2\\varepsilon) (t^2-4s^2x^2) + 4\\sqrt{-1}\\,\\varepsilon x (s^2-t^2)=0\\Longleftrightarrow \\begin{cases}\n{\\mathsf{Re}\\,}D=0,\\\\ {\\mathsf{Im}\\,}D=0,\n\\end{cases}\n\\end{cases}$$ \n Now, \n$$\\begin{cases}\n{\\mathsf{Re}\\,}D=0,\\\\ {\\mathsf{Im}\\,}D=0,\n\\end{cases}\\Longleftrightarrow \\begin{cases}\n(1-2\\varepsilon) (t^2-4s^2x^2)=0,\\\\ s^2-t^2=0,\n\\end{cases} \\Longleftrightarrow \\begin{cases}\n(1-2\\varepsilon) (1-4x^2)=0,\\\\ s^2-t^2=0,\n\\end{cases}$$ \n so, we obtain two different possibilities: \n$$s^2 = t^2,\\quad \\varepsilon = \\frac12,\\quad \\text{or} \\quad s^2 = t^2,\\quad x = \\frac12.$$ \nIf we consider *A* ≠ 0, then *B* = 0 and therefore $v + 2\\sqrt{-1}\\,x\\bar v = - t^2-2\\sqrt{-1}\\,s^2 x$. Substituting this expression in *D* we get that $D=2(t^2+2\\sqrt{-1}x^2x)\\neq0$, which implies that *C* = 0. Observe that system *B* = *C* = 0 is the same as *A* = *D* = 0 just changing the sign of *v*. One get the same solutions as before.\nTo finish the proof, we study first the case *s*2 = *t*2 and $\\varepsilon = \\frac12$. Directly, \n$$R^{\\frac12}\\_{121\\bar{2}} = \\frac{t^2(2x-\\sqrt{-1})}{16x}\\neq 0.$$ \n On the other hand, if *s*2 = *t*2 and $x = \\frac12$, $B^{\\varepsilon}\\_{2\\bar{2}3\\bar{1}} =\\frac{\\sqrt{-1}\\varepsilon(2\\varepsilon-1)t^4}{\\bar u+\\sqrt{-1}\\bar z}\\neq 0$.\n### Solvmanifolds in Families (Siii1), (Siii3), (Siii4)\nRecall that from Section [subsec:complex-st], the Lie algebras underlying (Siii1), (Siii3), and (Siii4) are, respectively, $\\frak g\\_4$, $\\frak g\\_6$, and $\\frak g\\_7$. In order to give an unified argument, we will gather the complex equations as follows: \n$$d\\varphi^1=\\sqrt{-1}\\,(\\varphi^{13} + \\varphi^{1\\bar3}),\\quad d\\varphi^2 = -\\sqrt{-1}\\,(\\varphi^{23} + \\varphi^{2\\bar3}),$$ \n \n$$d\\varphi^3 = x\\,\\varphi^{1\\bar1} +y\\,\\varphi^{2\\bar 2},$$ \n where (*x*, *y*) = ( ± 1, 0) for $\\frak g\\_4$, (*x*, *y*) = (1, 1) for $\\frak g\\_6$ and (*x*, *y* = − *x*) = ( ± 1, ∓ 1) for $\\frak g\\_7$. In particular, *x* ≠ 0.\nThe Chern connection ∇*C**h* is never Kähler-like.\nUsing, a necessary condition for ∇*C**h* to be Kähler-like is that *ω* must be a balanced metric. A direct computation shows that *ω* is balanced if and only if *v* = *z* = 0 and *s*2*x* + *r*2*y* = 0. Observe that this last equation has no solutions on $\\frak g\\_4$ and $\\frak g\\_6$, which implies that ∇*C**h* is never Kähler-like for these algebras. On $\\frak g\\_7$, since *x* = − *y*, the only option is *r*2 = *s*2. Now, if we compute the Bianchi identity-symmetries *B*0, we get that $B^0\\_{1\\bar{1}2\\bar{2}} = \\frac{t^2}{2}\\neq 0$.\n[prop-g467] In the notation as above, consider *ɛ* ≠ 0. Then, ∇*ɛ* is Kähler-like if and only if $\\frak g$ is isomorphic to $\\frak g\\_4$, $\\varepsilon=\\frac12$, and *u* = *v* = *z* = 0 in. In this case, the Strominger-Bismut connection is non-flat.\nFrom now on, consider *ɛ* ≠ 0. Let us focus on Chern-symmetries $R^{\\varepsilon}\\_{121\\bar{3}}$ and $R^{\\varepsilon}\\_{122\\bar{3}}$:\n \n$$\\begin{aligned}\nR^{\\varepsilon}\\_{121\\bar{3}} &=& \\frac{-\\varepsilon\\, z}{8\\sqrt{-1}\\, \\det\\Omega}\\, \\left(v\\,\\left[\\sqrt{-1}\\,(2\\varepsilon + 1) uv x + z \\left(s^2 x (2\\varepsilon + 1) - r^2y (2\\varepsilon - 1)\\right)\\right] \\right. \\\\\n&& \\left. + \\sqrt{-1}\\,(2\\varepsilon - 1) y \\bar u z^2\\right), \\\\\nR^{\\varepsilon}\\_{122\\bar{3}} &=& \\frac{\\varepsilon\\, v}{8\\sqrt{-1}\\, \\det\\Omega}\\, \\left(v\\,\\left[\\sqrt{-1}\\,(2\\varepsilon - 1) uv x + z \\left(s^2 x (2\\varepsilon - 1) - r^2y (2\\varepsilon + 1)\\right)\\right] \\right. \\\\\n&& \\left. + \\sqrt{-1}\\,(2\\varepsilon + 1) y \\bar u z^2\\right).\\end{aligned}$$ \nTrivially, this two elements vanishes if *v* = *z* = 0. Suppose that *v**z* ≠ 0. Then, the system $R^{\\varepsilon}\\_{121\\bar{3}} = R^{\\varepsilon}\\_{122\\bar{3}}=0$ is equivalent to \n$$\\left\\{\\begin{array}{rcl}\nA &:=& v\\,\\left[\\sqrt{-1}\\,(2\\varepsilon + 1) uv x + z \\left(s^2 x (2\\varepsilon + 1) - r^2y (2\\varepsilon - 1)\\right)\\right] \\\\\n&& + \\sqrt{-1}\\,(2\\varepsilon - 1) y \\bar u z^2 =0,\\\\[5pt]\nB &:=& v\\,\\left[\\sqrt{-1}\\,(2\\varepsilon - 1) uv x + z \\left(s^2 x (2\\varepsilon - 1) - r^2y (2\\varepsilon + 1)\\right)\\right] \\\\\n&& + \\sqrt{-1}\\,(2\\varepsilon + 1) y \\bar u z^2=0.\n\\end{array}\\right.$$ \n Consider the following homogeneous system, whose solutions are precisely *A* = *B* = 0: \n$$\\begin{cases}\nA(1+2\\varepsilon) + B(1-2\\varepsilon)=0,\\\\[5pt]\nA(1-2\\varepsilon) + B(1+2\\varepsilon)=0.\n\\end{cases}$$ \n This last system reduces to\n \n$$\\begin{aligned}\n\\begin{cases}\n8\\,\\varepsilon\\,v\\,x\\,(\\sqrt{-1}uv + s^2z)=0,\\\\[5pt]\n8\\,\\varepsilon\\,\\sqrt{-1} \\,z\\,y \\,(\\sqrt{-1}r^2v + \\bar u z)=0.\n\\end{cases} &\\Longleftrightarrow&\n\\begin{cases}\n\\sqrt{-1}uv + s^2z=0,\\\\[5pt]\ny \\,(\\sqrt{-1}r^2v + \\bar u z)=0.\n\\end{cases} \\\\\n&\\Longleftrightarrow&\n\\begin{cases}\nu = \\dfrac{\\sqrt{-1}\\,s^2\\,z}{v},\\\\[5pt]\n\\dfrac{\\sqrt{-1}\\,y}{\\bar v} \\,(r^2|v|^2 - s^2|z|^2)=0.\n\\end{cases}\\end{aligned}$$ \nMoreover, substituting $u = \\dfrac{\\sqrt{-1}\\,s^2\\,z}{v}$ in, we get: \n$$8\\sqrt{-1}\\, \\det\\Omega = \\frac{(s^2t^2-|v|^2)(r^2|v|^2 - s^2|z|^2)}{|v|^2}\\neq 0.$$ \n So, we are forced to consider *y* = 0. However, in this particular case, $R^{\\varepsilon}\\_{123\\bar{2}} = \\frac{4\\sqrt{-1}\\,\\varepsilon^2 s^4 z}{s^2t^2-|v|^2}\\neq 0$.\nLet us study now the case when *v**z* = 0. Suppose first *z* = 0 and *v* ≠ 0. Now, $R^{\\varepsilon}\\_{121\\bar{2}}=\\frac{-\\varepsilon^2u^2v^2x}{2\\det\\Omega}=0$ if and only if *u* = 0. But imposing this condition, we get that $R^{\\varepsilon}\\_{231\\bar{1}}=-\\varepsilon v x\\neq 0$.\nOn the other hand, if *v* = 0 and *z* ≠ 0, symmetry $R^{\\varepsilon}\\_{121\\bar{3}}=\\frac{-\\varepsilon (2 \\varepsilon-1 ) y z^3 \\bar u}{8\\det\\Omega}=0$ is equivalent to (2*ɛ* − 1)*u**y* = 0. If $\\varepsilon = \\frac12$ (corresponding to the Strominger-Bismut connection), then \n$$\\begin{cases}\nR^{\\frac12}\\_{132\\bar{2}}=\\frac{yz}{2}=0,\\\\[5pt]\nR^{\\frac12}\\_{121\\bar{1}}=\\frac{s^2z^2 x \\bar u}{8\\sqrt{-1}\\det \\Omega}=0,\n\\end{cases} \\Longleftrightarrow \\quad u=y=0.$$ \n But now, $R^{\\frac12}\\_{133\\bar{3}}=\\frac{\\sqrt{-1}z}{2}\\neq 0$.\nSuppose now $\\varepsilon\\neq \\frac12$. If *u* = 0, directly from $R^{\\varepsilon}\\_{132\\bar{2}}=\\varepsilon yz$ one gets *y* = 0. Analogously, if we suppose *y* = 0, then $R^{\\varepsilon}\\_{121\\bar{1}}=\\frac{-\\sqrt{-1} \\varepsilon s^2 z^2 \\bar u x}{4\\det\\Omega}$ implies *u* = 0. So, if $\\varepsilon\\neq \\frac12$ we may assume *u* = *y* = 0. At this point, observe that $R^{\\varepsilon}\\_{131\\bar{3}}=\\frac{\\varepsilon(2\\varepsilon-1)(r^2-\\sqrt{-1}xt^2)z^2}{8\\sqrt{-1}\\det \\Omega}=0$ if and only if $\\varepsilon=\\frac12$.\nFinally, if *v* = *z* = 0, then \n$$R^{\\varepsilon}\\_{133\\bar{1}}=\\frac{\\varepsilon ((-1 + 2 \\varepsilon) s^2 t^4 x^2 + (8 \\varepsilon r^2 u + 2 \\sqrt{-1} t^2 u x) \\bar u)}{8it^2\\det\\Omega}=0$$ \n if and only if the complex number \n$$(2\\varepsilon-1) s^2 t^4 x^2 + 8\\varepsilon r^2 |u|^2 + \\sqrt{-1} (2t^2 x |u|^2)=0.$$ \n Observe that the imaginary part vanishes if and only if *u* = 0. Furthermore, one can check that if *u* = 0, then all the Chern-symmetries are zero if and only if $\\varepsilon = \\frac12$. As for Bianchi identity-symmetries, in the case *u* = *v* = *z* = 0 and $\\varepsilon = \\frac12$, we have $R^{\\frac12}\\_{ij\\bullet\\bullet}=R^{\\frac12}\\_{\\bullet\\bullet k\\ell}=0$ for any *i*, *j*, *k*, ℓ ∈ {1, 2, 3} if and only if *y* = 0, so the Lie algebra is isomorphic to $\\frak g\\_4$.\nFinally, note that $ R^\\frac12\\_{1\\bar11\\bar1} = t^2 $, whence the last statement.\nWe notice that any diagonal metric on $\\frak g\\_4$ gives examples of a non-flat Kähler-like Strominger-Bismut connection on a non-Kähler manifold. (For the existence of lattices, see.)\n### Solvmanifolds in Family (Siii2)\nBy Section [subsec:complex-st], the underlying Lie algebra is $\\frak g\\_5$.\nThe connection ∇*ɛ* is never Kähler-like.\nLet us suppose *ɛ* = 0. Using, a necessary condition for ∇*C**h* to be Kähler-like is that *ω* must be a balanced metric. According to, *ω* is balanced if and only if *v* = *z* = 0 and $u\\in\\mathbb R$. Now, with these restrictions, \n$$B^0\\_{1\\bar{1}2\\bar{1}} = -r^2\\neq 0,$$ \n whence the statement.\nFrom now on, consider *ɛ* ≠ 0 and a generic Hermitian metric *ω*. Let us focus on Chern-symmetries $R^{\\varepsilon}\\_{121\\bar{3}}$ and $R^{\\varepsilon}\\_{122\\bar{3}}$ and develop a similar argument as that in Proposition [prop-g467]:\n \n$$\\begin{aligned}\nR^{\\varepsilon}\\_{121\\bar{3}} &=& \\frac{\\varepsilon\\, z}{8\\sqrt{-1}\\, \\det\\Omega}\\, \\left(\\sqrt{-1}\\,\\left[(2\\varepsilon + 1) r^2 v^2-z (2\\varepsilon - 1)(\\sqrt{-1}uv + s^2z)\\right] \\right. \\\\\n&& \\left. + (2\\varepsilon + 1) \\bar u v z\\right),\\\\\nR^{\\varepsilon}\\_{122\\bar{3}} &=& \\frac{-\\varepsilon\\, v}{8\\sqrt{-1}\\, \\det\\Omega}\\, \\left(\\sqrt{-1}\\,\\left[(2\\varepsilon - 1) r^2 v^2-z (2\\varepsilon + 1)(\\sqrt{-1}uv + s^2z)\\right] \\right. \\\\\n&& \\left. + (2\\varepsilon - 1) \\bar u v z\\right).\\end{aligned}$$ \nClearly, this two elements vanishes if *v* = *z* = 0. Suppose that *v**z* ≠ 0. Then, the system $R^{\\varepsilon}\\_{121\\bar{3}} = R^{\\varepsilon}\\_{122\\bar{3}}=0$ is equivalent to \n$$\\begin{cases}\nA := \\sqrt{-1}\\,\\left[(2\\varepsilon + 1) r^2 v^2-z (2\\varepsilon - 1)(\\sqrt{-1}uv + s^2z)\\right] + (2\\varepsilon + 1) \\bar u v z =0,\\\\[5pt]\nB := \\sqrt{-1}\\,\\left[(2\\varepsilon - 1) r^2 v^2-z (2\\varepsilon + 1)(\\sqrt{-1}uv + s^2z)\\right] + (2\\varepsilon - 1) \\bar u v z=0.\n\\end{cases}$$ \n Consider the following homogeneous system, whose solutions are precisely *A* = *B* = 0: \n$$\\begin{cases}\nA(1+2\\varepsilon) + B(1-2\\varepsilon)=0,\\\\[5pt]\nA(1-2\\varepsilon) + B(1+2\\varepsilon)=0.\n\\end{cases}$$ \n This last system is equivalent to \n$$\\begin{cases}\n\\sqrt{-1}uv + s^2z=0,\\\\[5pt]\n\\sqrt{-1}r^2v + \\bar u z=0.\n\\end{cases} \\Longleftrightarrow \\quad\nv=z=0,$$ \n which contradicts our initial hypothesis.\nWe can suppose now that *v**z* = 0. In fact, in this case *v* = *z* = 0 (use $R^{\\varepsilon}\\_{122\\bar{2}}=\\frac{\\sqrt{-1}\\varepsilon^2 s^4 z^2}{2\\det\\Omega}=0$ when *v* = 0 and $R^{\\varepsilon}\\_{121\\bar{1}}=\\frac{\\sqrt{-1}\\varepsilon^2 r^4v^2}{2\\det\\Omega}$ if *z* = 0). Now, $B^{\\varepsilon}\\_{1\\bar{1}2\\bar{1}}=(4\\varepsilon-1) r^2=0$ if and only if $\\varepsilon=\\frac14$ but now $B^{\\frac14}\\_{1\\bar{1}3\\bar{3}}=-r^2\\neq 0$.\n### Solvmanifolds in Families (Siv1), (Siv2), and (Siv3)\nRecall that by Section [subsec:complex-st], the underlying Lie algebra to these complex structures is $\\frak g\\_8$.\nFor the holomorphically-parallelizable structure in Family (Siv1), we have that the Chern connection ∇*C**h* is flat and ∇*ɛ* ≠ 0 is never Kähler-like.\nThe result about Chern connection comes from the fact that the structure is complex-parallelizable. When *ɛ* ≠ 0, one can see that *R**i**j* • •*ɛ* = *R*• • *k*ℓ*ɛ* ≡ 0 for any *i*, *j*, *k*, ℓ ∈ {1, 2, 3} if and only if $\\varepsilon=\\frac12$ (see for example $R^\\varepsilon\\_{133\\bar1}=-(2\\varepsilon-1)\\varepsilon r^2$). But now, $B^{\\frac12}\\_{1\\bar{1}2\\bar{2}} = \\frac{r^4 s^4 - |u|^4}{16\\sqrt{-1}\\det\\Omega}$ if and only if *r*4*s*4 − ∣*u*∣4 = 0 which is a contradiction.\nFor the complex structures in Family (Siv2), ∇*ɛ* is never Kähler-like.\nThe result for the Chern connection (namely *ɛ* = 0) follows directly observing that complex structures in Family (Siv2) do not admit balanced metrics and by.\nIf *ɛ* ≠ 0, then \n$$R^{\\varepsilon}\\_{123\\bar{1}}=-\\frac{\\varepsilon^2(r^2s^2-|u|^2)(x r^2s^2 + x|u|^2 + 2\\sqrt{-1} r^2 \\bar u)}{2 \\det \\Omega},$$ \n and \n$$R^{\\varepsilon}\\_{123\\bar{2}}=-\\frac{\\varepsilon^2(r^2s^2-|u|^2)(r^2s^2 - 2\\sqrt{-1} x s^2 u + |u|^2)}{2 \\det \\Omega}.$$ \n Hence, $R^{\\varepsilon}\\_{123\\bar{1}}=R^{\\varepsilon}\\_{123\\bar{2}}=0$ if and only if \n$$xr^2s^2+x|u|^2 + 2\\sqrt{-1} r^2 \\bar u=0,\n\\quad\nr^2s^2 + |u|^2 - 2\\sqrt{-1} x s^2 u =0.$$ \n If *x* = 1, these equations imply Re *u* = 0, Im *u* = − *r*2, and *r*2 = *s*2, which is a contradiction to the positive definiteness of the metric. Hence, *x* = 0 and $R^{\\varepsilon}\\_{123\\bar{2}}$ is always different from zero.\nFor the complex structures in Family (Siv3), ∇*ɛ* is Kähler-like if and only if *ɛ* = 0 and *ω* is diagonal. Moreover, in these cases, ∇*C**h* is flat.\nLet us start with the case *ɛ* = 0 and imposing *ω* to be balanced. In this situation, *v* = *z* = 0. Moreover, *B*0 ≡ 0 if and only if *u* = 0 (see for example $B^0\\_{1\\bar33\\bar1}=\\frac{2r^2|u|^2}{r^2s^2-|u|^2}$). Observe that we do not get restriction on the parameter *A*.\nFinally, if *ɛ* ≠ 0, we focus on Bianchi identity-symmetry $B^{\\varepsilon}\\_{1\\bar{1}2\\bar{2}}$. Observe that \n$$B^{\\varepsilon}\\_{1\\bar{1}2\\bar{2}} = \\frac{2\\varepsilon^2(r^2s^2-|u|^2) (r^2s^2|A-1|^2 + |u|^2 |A+1|^2)}{8\\sqrt{-1}\\, \\det\\Omega}\\neq 0,$$ \n whence the statement.\n### Solvmanifolds in Family (Sv)\nBy Section [subsec:complex-st], the underlying Lie algebra is $\\frak g\\_9$.\nThe connection ∇*ɛ* is never Kähler-like.\nIf *ɛ* = 0, then the result follows directly by observing that $\\frak g\\_9$ does not admit balanced metrics and by.\nFrom now on, consider *ɛ* ≠ 0. Let us focus on the Chern-symmetry $R^{\\varepsilon}\\_{121\\bar{2}}$: \n$$R^{\\varepsilon}\\_{121\\bar{2}} = \\frac{\\bar v\\left[(2r^2s^2-|u|^2)v - (\\sqrt{-1}s^2\\bar u)z\\right]}{32\\, \\sqrt{-1} \\det \\Omega}.$$ \n Hence, \n$$R^{\\varepsilon}\\_{121\\bar{2}} = 0 \\quad \\Longleftrightarrow \\quad \\bar v\\left[(2r^2s^2-|u|^2)v - (\\sqrt{-1}s^2\\bar u)z\\right]=0.$$ \n Suppose first that *v* ≠ 0. Then, we can express \n$$v=\\frac{\\sqrt{-1}s^2\\bar u z}{2r^2s^2-|u|^2}.$$ \n Observe that, since *v* ≠ 0, also *u**z* ≠ 0. Using this particular value for *v*, $R^{\\varepsilon}\\_{122\\bar{1}}=\\frac{\\varepsilon(1-2\\varepsilon)s^2\\bar u^2}{4(r²s^2-|u|^2)}=0$ if and only if $\\varepsilon = \\frac12$, but now $R^{\\frac12}\\_{131\\bar{2}}=\\frac{s^2u\\bar z}{4(r^2s^2-|u|^2)}\\neq0$.\nOn the other hand, if *v* = 0, then $R^{\\varepsilon}\\_{122\\bar{3}}=\\frac{\\varepsilon(1-2\\varepsilon)s^2z\\bar u^2}{16\\sqrt{-1}\\det\\Omega}=0$ holds if and only if (2*ɛ* − 1)*u**z* = 0. None of the three cases leads to Kähler-like connections: if $\\varepsilon = \\frac12$, then $R^{\\frac12}\\_{233\\bar{3}}=\\frac{s^4|z|^2}{32\\sqrt{-1}\\det\\Omega}=0$ to conclude that *z* = 0 but then $R^{\\frac12}\\_{123\\bar{1}}=\\frac{-(r^2s^2-|u|^2)}{4t^2}\\neq 0$; if *z* = 0, then $R^{\\varepsilon}\\_{131\\bar{3}}=\\frac{\\varepsilon(2\\varepsilon-1)r^4}{t^2}$ vanishes if and only if $\\varepsilon = \\frac12$, that has already been studied; finally, if *u* = 0, then take $R^{\\varepsilon}\\_{121\\bar{1}}=\\frac{-\\varepsilon r^2s^2z}{4(r^2s^2-|z|^2)}$. This element is zero if and only if *z* = 0, which has already been analysed.\nKähler-like Levi-Civita connection\n==================================\nHermitian condition along the Ricci-flow\n----------------------------------------\nIn this section, we prove that the Kähler-like condition for the Levi-Civita connection assures that the Ricci flow preserves the Hermitian condition of the initial metric along analytic solutions. We wonder whether, as an application, one may prove that non-toral nilmanifolds do not admit invariant Hermitian metrics whose Levi-Civita connection is Kähler-like, thanks to the results by Lauret on the Ricci flow for nilpotent Lie groups. In fact, we will prove that this is true in dimension six, see Proposition [prop:lc].\n[thm:ricci-flow-hermitian] Let *g*0 be a Hermitian metric on a compact complex manifold, and consider an analytic solution (*g*(*t*))*t* ∈ ( − *ɛ*, *ɛ*) for *ɛ* > 0 of the Ricci flow\n \n$$\\label{eq:ricci-flow}\\tag{RF}\n\\frac{d}{dt}g(t) = -\\mathrm{Ric}(g(t)), \\qquad g(0)=g\\_0.$$ \nIf the Levi-Civita connection of *g*0 is Kähler-like, then *g*(*t*) is Hermitian for any *t*.\nThe proof follows by the evolution equation for the curvatures along the Ricci flow in :\n \n$$\\begin{aligned}\n\\frac{d}{dt} \\mathrm{Ric}(t)\\_{JK} &=& \\Delta\\_{t} \\mathrm{Ric}(t)\\_{JK}+2g(t)^{PQ}g(t)^{RS}R(t)\\_{PJKR}\\mathrm{Ric}(t)\\_{QS} \\\\\n&& -2g(t)^{PQ}\\mathrm{Ric}(t)\\_{JP}\\mathrm{Ric}(t)\\_{QK}, \\\\\n\\frac{d}{dt} R(t)\\_{IJKL} &=& \\Delta\\_{t} R(t)\\_{IJKL} \\\\\n&& + 2 \\left( g(t)^{PR} g(t)^{QS} R(t)\\_{IPJQ} R(t)\\_{KRLS} \\right. \\\\\n&& \\left. - g(t)^{PR} g(t)^{QS} R(t)\\_{IPJQ} R(t)\\_{LRKS} \\right. \\\\\n&& \\left. - g(t)^{PR} g(t)^{QS} R(t)\\_{IPLQ} R(t)\\_{JRKS} \\right. \\\\\n&& \\left. + g(t)^{PR} g(t)^{QS} R(t)\\_{IPKQ} R(t)\\_{JRLS} \\right) \\\\\n&& -g(t)^{PQ} \\left( R(t)\\_{PJKL} \\mathrm{Ric}(t)\\_{QI} + R(t)\\_{IPKL} \\mathrm{Ric}(t)\\_{QJ} \\right. \\\\\n&& \\left. + R(t)\\_{IJPL} \\mathrm{Ric}(t)\\_{QK} + R(t)\\_{IJKP} \\mathrm{Ric}(t)\\_{QL} \\right).\\end{aligned}$$ \nHere, capital letters *I*, *J*, …vary in $\\{1, \\ldots, n, \\bar1, \\ldots, \\bar n\\}$ while small letters *i*, *j*, …vary in {1, …, *n*}, and for simplicity we just write the dependence in *t* meaning to refer to the quantities for *g*(*t*). The dot denotes the derivative at zero, *e.g.* $\\dot{\\mathrm{Ric}}\\_{jk} = \\left.\\frac{d}{dt}\\right\\lfloor\\_{t=0} \\mathrm{Ric}\\_{jk}(t)$.\nThe Kähler-like condition assures that \n*R**i**j**K**L*(0) = 0, Ric*i**j*(0) = 0. \n This, together with the formula for the evolution of the Ricci curvature tensor, whence \n$$\\dot{\\mathrm{Ric}}\\_{jk} = 0,$$ \n yields that \n$$\\dot g\\_{jk} = \\ddot g\\_{jk} = 0.$$ \nMoreover, notice that the Kähler-like assumption forces *R**I**J**K**L*(0) to be zero whenever three or more of the indices in (*I*, *J*, *K*, *L*) have the same type. The same holds for the tensor *g*(0)*P**R**g*(0)*Q**S**R*(0)*P**I**Q**J**R*(0)*R**K**S**L*, that appears in the right-hand-side for $\\dot R$. From the evolution equation of the Riemann curvature tensor, we get the same property also for $\\dot{R}\\_{IJKL}$. Then we get \n$$\\dddot{g}\\_{jk}=0$$ \n by straightforward computations.\nFurther derivatives behave similarly. Indeed, the only remark to be noticed is that $\\dot{R}\\_{IJKL}$ involves the tensors *R*• • • •(0), $\\dot{R}\\_{\\bullet\\bullet\\bullet\\bullet}$, Ric• •(0), $\\dot{\\mathrm{Ric}}\\_{\\bullet\\bullet}$, other than the Hermitian metric *g*• •(0) and its time-derivative $\\dot{g}\\_{\\bullet\\bullet}$, whence it still satisfies the property to vanish whenever three out of (*I*, *J*, *K*, *L*) have the same type.\nSo we get that the solution *g*(*t*)*j**k* has vanishing time-derivatives of any order at 0. Being analytic, it is constant, so *g*(*t*)*j**k* = *g*(0)*j**k* = 0, that is, *g*(*t*) is still Hermitian.\nKähler-like Levi-Civita connection on six-dimensional Calabi-Yau solvmanifolds\n------------------------------------------------------------------------------\nIn this section, we prove the following:\n[prop:lc] Let (*X*, *J*, *g*) be a complex six-dimensional solvmanifold with holomorphically-trivial canonical bundle endowed with an invariant Hermitian metric. The Levi-Civita connection is Kähler-like if and only if the metric is Kähler.\nFirst, consider the nilpotent case. We provide explicit computations. Without loss of generality, we can reduce to the case when the Hermitian metric is balanced, thanks to, and we can exclude the holomorphically-parallelizable case (Np) thanks to the fact that holomorphically-parallelizable manifolds are Chern-flat and to. So we only remain with Family (Ni) with the exception of Lie algebra $\\mathfrak h\\_8$, and the Family (Niii) in fact only for $\\mathfrak h\\_{19}^-$. We distinguish complex structures of nilpotent, respectively non-nilpotent type, see.\nFor nilpotent complex structures on nilpotent Lie algebras, argue as follows. First of all, the generic Hermitian metric as in is balanced if and only if it is equivalent to one with parameters *r*2 = 1, *v* = *z* = 0, and also $s^2+D=\\sqrt{-1}\\bar{u}\\lambda$, see. We compute, for example, $R^{LC}\\_{1\\bar112}=-\\frac{3}{8}\\,\\rho\\, t^2$. So the Levi-Civita connection of a balanced metric on a nilmanifold with complex structure in Family (Ni) with *ρ* = 1 is never Kähler-like. This forces *ρ* = 0, and the Lie algebra to be either h3 (with *λ* = 0 and *D* = − 1) or h5 (with *λ* = 1 and $D\\in[0,\\frac{1}{4})$). Moreover, since $R^{LC}\\_{3\\bar213}=\\frac{1}{8}\\,\\frac{\\sqrt{-1}\\, t^4\\, u\\, \\bar{D}}{s^2 - |u|^2}$, then either *u* = 0 (case h3) or *D* = 0 (case h5).\nIn the first case, that is, when *ρ* = *λ* = 0 and *D* = − 1, we compute $B^{LC}\\_{1\\bar13\\bar3} = \\frac{{\\left(\\lambda^{2} + \\rho^{2} + s^{2} + \\sqrt{-1} \\, \\lambda u - \\sqrt{-1} \\, \\lambda \\overline{u}\\right)} t^{4}}{8 \\, {\\left(s^{2} - u \\overline{u}\\right)}}$ that reduces to $B^{LC}\\_{1\\bar13\\bar3} = \\frac{{s^{2}} t^{4}}{8 \\, {\\left(s^{2} - u \\overline{u}\\right)}}\\neq0$, in fact, $B^{LC}\\_{1\\bar13\\bar3} = \\frac{t^{4}}{8}\\neq0$, by using *s*2 = 1 and *u* = 0. Therefore, in this case, the Levi-Civita connection is not Kähler-like.\nIn the second case, that is, when *ρ* = *D* = 0 and *λ* = 1, we compute $B^{LC}\\_{1\\bar12\\bar2} = -\\frac{1}{8} \\, {\\left(\\lambda^{2} + 2 \\, \\rho^{2} - D - \\overline{D}\\right)} t^{2}$ that reduces to $B^{LC}\\_{1\\bar12\\bar2} = -\\frac{1}{8} \\, t^{2}\\neq 0$. Therefore, also in this second case, the Levi-Civita connection is not Kähler-like.\nFinally, for non-nilpotent complex structure on nilpotent Lie algebras, argue as follows. The generic Hermitian balanced metric as in forces *ν* = 0 (case $\\mathfrak h\\_{19}^-$) and can be reduced to parameters *u* = *z* = 0, (and moreover either *v* = 1, or *v* = 0 and *t*2 = 1,) see. We compute, for example, $R^{LC}\\_{3\\bar22\\bar3}=-\\frac{s^4+t^4}{8r^2}\\neq 0$ and $R^{LC}\\_{2\\bar23\\bar3}=0$. Therefore the Levi-Civita connection is not Kähler-like.\nWe restrict now our attention to the solvable non-nilpotent case.\nWe remind that by , the Levi-Civita connection be Kähler-like implies that the metric is balanced. Hence and by we restrict our attention to the Lie algebras $\\frak g\\_1$, $\\frak g\\_2^{\\alpha}$, $\\frak g\\_3$, $\\frak g\\_5$, $\\frak g\\_7$, and $\\frak g\\_8$.\nIf the Lie algebra is isomorphic to $\\frak g\\_1$ or $\\frak g\\_2^{\\alpha}$ (case (Si)) then:\n \n$$\\begin{aligned}\n\\lefteqn{ R^{\\text{LC}}\\_{12\\bar{1}\\bar{2}}=\\frac{(A^2+\\bar A^2)(r^2s^2-|u|^2)+2|A|^2(r^2s^2+|u|^2)}{4t^2}=0 } \\\\\n&\\Leftrightarrow& ({\\mathsf{Re}\\,}A)^2r^2s^2+({\\mathsf{Im}\\,}A)^2|u|^2=0,\\\\\n&\\Leftrightarrow& {\\mathsf{Re}\\,}A=0,\\, u=0, \\\\\n&\\Leftrightarrow&A=\\sqrt{-1},\\, u=0.\\end{aligned}$$ \nThat is, the Lie algebra is $\\frak g\\_2^0$ and *ω* is Kähler.\nIf the Lie algebra is isomorphic to $\\frak g\\_3$ (case (Sii)) then: \n$$R^{\\text{LC}}\\_{23\\bar{2}\\bar{3}}=\\frac{(1+4x^2)(t^4+4s^4x^2)}{128r^2x^2}\\neq0.$$ \nIf the Lie algebra is isomorphic to $\\frak g\\_5$ (case (Siii2)) then: \n$$\\begin{array}{lclcl}\nR^{\\text{LC}}\\_{13\\bar{2}\\bar{3}}=\\frac{-\\sqrt{-1}(4r^2s^2+t^4)u}{8(r^2s^2-u^2)}=0 &\\Leftrightarrow& \\sqrt{-1}(4r^2s^2+t^4)u=0&\\Leftrightarrow&u=0.\n\\end{array}$$ \n But *u* = 0 implies $R^{\\text{LC}}\\_{233\\bar{1}}=\\frac{-t^2}{2}\\neq0$.\nIf the Lie algebra is isomorphic to $\\frak g\\_7$ (case (Siii4)) then: \n$$R^{\\text{LC}}\\_{23\\bar{2}\\bar{3}}=\\frac{s^2(t^4+4|u|^2)}{8(s^4-|u|^2)}\\neq0.$$ \n If the Lie algebra is isomorphic to $\\frak g\\_8$, then the complex structure belongs to Families (Siv1) or (Siv3). In the first case we directly find that $R^{\\text{LC}}\\_{133\\bar{1}}=\\frac{r^2}{8}\\neq 0$. In the second case the Kähler-like condition implies the following system: \n$$\\begin{array}{lcl}\nR^{\\text{LC}}\\_{133\\bar{1}}&=&\\frac{r^2\\left[(A-1)^2r^2s^2-(A^2-6A+1)|u|^2\\right]}{8(r^2s^2-|u|^2)}=0,\\\\\nR^{\\text{LC}}\\_{133\\bar{2}}&=&\\frac{-\\sqrt{-1}u\\left[(A^2+6A+1)r^2s^2-(A+1)^2|u|^2\\right]}{8(r^2s^2-|u|^2)}=0.\n\\end{array}$$ \n If *u* = 0 then the first equation implies (*A* − 1)2*r*2*s*2 = 0 which is not possible. On the other hand if *u* ≠ 0 then the vanishing of the brackets corresponds to a homogeneous linear system in *r*2*s*2 and ∣*u*∣2. The equations are linear dependent if and only if *A* = 0, but in this case the second equation reduces to *r*2*s*2 − ∣*u*∣2 = 0.\nBianchi and Chern symmetries for the Gauduchon connections on nilmanifolds in Family (Ni)\n=========================================================================================\n \n$$\\begin{aligned}\nR^{\\varepsilon}\\_{121\\bar{1}}& =&2\\,t^2\\,\\varepsilon (1-\\varepsilon) \\rho,\\\\[5pt]\nR^{\\varepsilon}\\_{122\\bar{1}}& =&\\lambda\\,R^{\\varepsilon}\\_{121\\bar{1}},\\\\[5pt]\nR^{\\varepsilon}\\_{122\\bar{2}}& =&\\bar D\\,R^{\\varepsilon}\\_{121\\bar{1}},\\\\[5pt]\nR^{\\varepsilon}\\_{123\\bar{3}}& =&\\frac{t^4\\,\\varepsilon}{s^2-|u|^2}\\rho(2\\varepsilon-1) (s^2+\\sqrt{-1}\\lambda u + \\bar D),\\\\[5pt]\nR^{\\varepsilon}\\_{131\\bar{3}} & =& \\frac{-2\\sqrt{-1}\\,t^4\\,\\varepsilon^2}{s^2-|u|^2} \\rho\\, u,\\\\[5pt]\nR^{\\varepsilon}\\_{132\\bar{3}} & =& \\frac{-2\\,t^4\\,\\varepsilon^2}{s^2-|u|^2} \\rho\\, (\\sqrt{-1}\\lambda u + \\bar D),\\\\[5pt]\nR^{\\varepsilon}\\_{133\\bar{1}} & =& \\frac{-t^4\\,\\varepsilon}{s^2-|u|^2} (2\\varepsilon-1) (s^2+\\sqrt{-1}\\lambda u),\\\\[5pt]\nR^{\\varepsilon}\\_{133\\bar{2}} & =& \\frac{-\\sqrt{-1}\\,t^4\\,\\varepsilon}{s^2-|u|^2} (2\\varepsilon-1)\\,u\\, \\bar D,\\\\[5pt]\nR^{\\varepsilon}\\_{231\\bar{3}} & =& \\frac{2\\,s^2\\,t^4\\,\\varepsilon^2}{s^2-|u|^2} \\rho,\\\\[5pt]\nR^{\\varepsilon}\\_{232\\bar{3}} & =& \\frac{2\\,t^4\\,\\varepsilon^2}{s^2-|u|^2} \\rho\\, (\\lambda s^2-\\sqrt{-1}\\bar u \\bar D),\\\\[5pt]\nR^{\\varepsilon}\\_{233\\bar{1}} & =& \\frac{-t^4\\,\\varepsilon}{s^2-|u|^2} (2\\varepsilon-1) (\\lambda (s^2+\\sqrt{-1}\\lambda u + \\bar D)-\\sqrt{-1}\\bar u \\bar D),\\\\[5pt]\nR^{\\varepsilon}\\_{233\\bar{2}} & =& \\frac{-t^4\\,\\varepsilon}{s^2-|u|^2} (2\\varepsilon-1)\\bar D (\\sqrt{-1}\\lambda u + \\bar D).\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{1}2\\bar{1}}& =& \\frac{-t^2}{2} (2\\varepsilon-1)^2\\,\\lambda,\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{1}2\\bar{2}}& =& \\frac{-t^2}{2} \\left[2\\varepsilon (\\lambda^2 - D + 4\\rho \\varepsilon) + \\bar D (4\\varepsilon^2 - 6 \\varepsilon + 1)\\right],\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{1}3\\bar{3}}& =&\\frac{t^4}{2 (s^2-|u|^2)} \\left[4\\,\\rho\\varepsilon^2 -s^2 (2\\varepsilon-1)(4\\varepsilon - 1)\\right],\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{2}2\\bar{1}}& =& \\frac{t^2}{2} \\left[2\\varepsilon (4\\rho \\varepsilon-\\bar D) + (D-\\lambda^2) (4\\varepsilon^2 - 6 \\varepsilon + 1)\\right],\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{2}2\\bar{2}}& =& \\bar D B^{\\varepsilon}\\_{1\\bar{1}2\\bar{1}},\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{2}3\\bar{3}}& =& \\frac{-t^4}{2 (s^2-|u|^2)} \\left[4\\,\\sqrt{-1}\\,\\rho u \\varepsilon^2 +(2\\varepsilon-1)(4\\varepsilon - 1) (\\lambda s^2 + \\sqrt{-1} u D)\\right],\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{3}2\\bar{3}}& =& \\frac{2\\,t^4\\,\\varepsilon^2}{s^2-|u|^2} \\rho (D+ s^2 - \\sqrt{-1} \\lambda \\bar u),\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{3}3\\bar{1}}& =&\\frac{-t^4\\,\\varepsilon}{s^2-|u|^2} \\left[s^2 + 2\\varepsilon (\\lambda^2- s^2 - 2 \\lambda {\\mathsf{Im}\\,}u)\\right],\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{3}3\\bar{2}}& =&\\frac{t^4\\,\\varepsilon}{s^2-|u|^2} \\left[(4\\varepsilon-1) (\\lambda s^2 +\\sqrt{-1} u D) - 2\\varepsilon \\bar D (\\lambda + \\sqrt{-1} u)\\right],\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{1}3\\bar{3}}& =&\\overline{B^{\\varepsilon}\\_{1\\bar{2}3\\bar{3}}},\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{2}3\\bar{3}}& =&\\frac{-t^4}{2(s^2-|u|^2)} \\left[(2\\varepsilon-1)(4\\varepsilon-1) (\\lambda^2 s^2 -2\\lambda {\\mathsf{Im}\\,}(uD) + |D|^2) - 4s^2\\rho\\varepsilon^2\\right],\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{3}3\\bar{1}}& =&\\overline{B^{\\varepsilon}\\_{1\\bar{3}3\\bar{2}}},\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{3}3\\bar{2}}& =&\\frac{t^4\\,\\varepsilon}{s^2-|u|^2} \\left[(2\\varepsilon-1)|D|^2 + (4\\varepsilon-1)\\lambda (\\lambda s^2 -2 {\\mathsf{Im}\\,}(uD))\\right].\\end{aligned}$$ \nBianchi symmetries for the Levi-Civita connection on solvmanifolds $\\frak g\\_1$ and $\\frak g\\_2^{\\alpha}$ in Family (Si)\n========================================================================================================================\n \n$$\\begin{aligned}\nB^{0}\\_{1\\bar{3}3\\bar{1}}& =&\\frac{2\\,r^2 |u|^2}{r^2s^2-|u|^2},\\qquad\nB^{0}\\_{1\\bar{3}3\\bar{2}}=\\frac{-2\\,\\sqrt{-1}\\,r^2 s^2 u}{r^2s^2-|u|^2},\\\\[5pt]\nB^{0}\\_{2\\bar{3}3\\bar{1}}& =&\\overline{B^{\\varepsilon}\\_{1\\bar{3}3\\bar{2}}},\\qquad \\qquad\nB^{0}\\_{2\\bar{3}3\\bar{2}}=\\frac{2\\,s^2 |u|^2}{r^2s^2-|u|^2}.\\end{aligned}$$ \nBianchi and Chern symmetries for the Gauduchon connections on solvmanifolds $\\frak g\\_2^0$ in Family (Si)\n=========================================================================================================\n \n$$\\begin{aligned}\nR^{\\varepsilon}\\_{131\\bar{3}}& =&\\frac{\\varepsilon (2\\varepsilon-1) r^2 s^2 z^2}{8\\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nR^{\\varepsilon}\\_{132\\bar{3}}& =&\\frac{-\\varepsilon (2\\varepsilon-1) r^2 s^2 v z}{8\\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nR^{\\varepsilon}\\_{133\\bar{3}}& =&\\frac{\\varepsilon\\,z}{8 \\det\\Omega} \\left[r^2s^2t^2 - 2\\varepsilon r^2|v|^2 + 2(\\varepsilon-1)s^2|z|^2\\right],\\\\[5pt]\nR^{\\varepsilon}\\_{231\\bar{3}}& =&-R^{\\varepsilon}\\_{132\\bar{3}},\\\\[5pt]\nR^{\\varepsilon}\\_{232\\bar{3}}& =&\\frac{\\varepsilon (2\\varepsilon-1) r^2 s^2 v^2}{8\\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nR^{\\varepsilon}\\_{233\\bar{3}}& =&\\frac{\\varepsilon\\,v}{8 \\det\\Omega} \\left[r^2s^2t^2 + 2(\\varepsilon-1) r^2|v|^2 - 2 \\varepsilon s^2|z|^2\\right].\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{1}3\\bar{3}}& =&\\frac{-\\varepsilon (2\\varepsilon-1) r^2 s^2 |z|^2}{8\\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{2}3\\bar{3}}& =&\\frac{\\varepsilon (2\\varepsilon-1) r^2 s^2 \\bar v z}{8\\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{3}3\\bar{1}}& =&\\frac{(2\\varepsilon-1)(4\\varepsilon-1) r^2 s^2 |z|^2}{16 \\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{3}3\\bar{2}}& =&\\frac{-(2\\varepsilon-1)(4\\varepsilon-1) r^2 s^2 \\bar v z}{16 \\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nB^{\\varepsilon}\\_{1\\bar{3}3\\bar{3}}& =&\\frac{z}{16 \\det\\Omega}\\left[(4\\varepsilon-1)r^2s^2t^2 + 2 (2\\varepsilon^2-4\\varepsilon+1) r^2 |v|^2 - 4\\varepsilon^2 s^2 |z|^2\\right],\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{1}3\\bar{3}}& =&\\overline{B^{\\varepsilon}\\_{1\\bar{2}3\\bar{3}}},\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{2}3\\bar{3}}& =&\\frac{-\\varepsilon (2\\varepsilon-1) r^2 s^2 |v|^2}{8\\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{3}3\\bar{1}}& =&\\overline{B^{\\varepsilon}\\_{1\\bar{3}3\\bar{2}}},\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{3}3\\bar{2}}& =&\\frac{(2\\varepsilon-1)(4\\varepsilon-1) r^2 s^2 |v|^2}{16 \\sqrt{-1}\\, \\det\\Omega},\\\\[5pt]\nB^{\\varepsilon}\\_{2\\bar{3}3\\bar{3}}& =&\\frac{v}{16 \\det\\Omega}\\left[(4\\varepsilon-1)r^2s^2t^2 - 4\\varepsilon^2 r^2 |v|^2 + 2 (2\\varepsilon^2-4\\varepsilon+1) s^2 |z|^2\\right].\\end{aligned}$$ \n99\nB. Alexandrov, S. Ivanov, Vanishing theorems on Hermitian manifolds, *Differential Geom. Appl.* **14** (2001), no. 3, 251–265.\nA. Andrada, M. L. Barberis, I. G. Dotti, Classification of abelian complex structures on 6-dimensional Lie algebras, *J."},"data_id":{"kind":"string","value":"arxiv_0181679"}}},{"rowIdx":45397,"cells":{"content":{"kind":"string","value":"Critical phenomena in gravitational collapse\n============================================\nAs first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term “critical phenomena”. They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. Critical phenomena give a natural route from smooth initial data to arbitrarily large curvatures visible from infinity, and are therefore likely to be relevant for cosmic censorship, quantum gravity, astrophysics, and our general understanding of the dynamics of general relativity.\nIntroduction\n============\nOverview of the subject\n-----------------------\nTake generic initial data in general relativity, adjust any one parameter *p* of the initial data to the threshold of black hole formation, and compare the resulting spacetimes as a function of *p*. In many situations, the following *critical phenomena* are then observed:\n* Near the threshold, black holes with arbitrarily small masses can be created, and the black hole mass scales as\n \n*M* ∝ (*p* − *p*\\*)*γ*, \nwhere *p* parameterises the initial data and black holes form for *p* > *p*\\*.\n* The *critical exponent* *γ* is universal with respect to initial data, that is, independent of the particular 1-parameter family, although it depends on the type of collapsing matter.\n* In the region of large curvature before black hole formation, the spacetime approaches a self-similar solution which is also universal with respect to initial data, the *critical solution*.\nCritical phenomena were discovered by Choptuik in numerical simulations of a spherical scalar field. They have been found in numerous other numerical and analytical studies in spherical symmetry, and a few in axisymmetry; in particular critical phenomena have been seen in the collapse of axisymmetric gravitational waves in vacuum. Scaling laws similar to the one for black hole mass have been discovered for black hole charge and conjectured for black hole angular momentum.\nIt is still unclear how universal critical phenomena in collapse are with respect to matter types and beyond spherical symmetry, in particular for vacuum collapse. Progress is likely to be made over the next few years with better numerical simulations.\nCritical phenomena can usefully be described in dynamical systems terms. A critical solution is then characterised as an attracting fixed point within a surface that divides two basins of attraction, a *critical surface* in phase space. Such a fixed point can be either a stationary spacetime, or one that is scale-invariant and self-similar. The latter is relevant for the (“type II”) critical phenomena sketched above. We shall also see how the dynamical systems approach establishes a connection with critical phenomena in statistical mechanics.\nTherefore, we could define the field of critical phenomena in gravitational collapse as the study of the boundaries among the basins of attraction of different end states of self-gravitating systems, such as black hole formation or dispersion. In our view, the main physical motivation for this study is that those critical solutions which are self-similar provide a way of achieving arbitrarily large spacetime curvature outside a black hole, and in the limit a naked singularity, by fine-tuning generic initial data for generic matter to the black hole threshold. Those solutions are therefore likely to be important for quantum gravity and cosmic censorship.\nPlan of this review\n-------------------\nFaced with more material to review, we have attempted to make this update shorter and more systematic than the original 1999 version. We begin with the abstract theory in Sec. [section:theory]. Since 1999, progress on the theory side has mainly been made on the global spacetime structure of the critical solution and cosmic censorship in the spherical scalar field model. We have included this new material in an enlarged Sec. [section:scalarfield] on the spherical scalar field, although we hope it will turn out to be sufficiently generic to merit inclusion in Sec. [section:theory]. Nonspherical perturbations of the spherical scalar field are also discussed in Sec. [section:scalarfield].\nIn Sec. [section:spherical] we review the rich phenomenology that has been found in many other systems restricted to spherical symmetry. Numerical work in spherical symmetry has proliferated since 1999, but we have tried to keep this section as short as possible. There has been less progress in going beyond spherical symmetry than we anticipated in 1999, even though we continue to believe that important results await there. What is known today is summarised in Sec. [section:nonspherical].\nThe reader unfamiliar with the topic is advised to begin with either Secs. [section:universality]-[section:scaling], which give the key theory of universality, self-similarity and scaling, or Secs. [section:scalarfieldequations]-[section:threshold], which describe the classic example, the massless scalar field.\nThis review is limited to numerical and theoretical work on phenomena at the threshold of black hole formation in 3+1-dimensional general relativity. We report only briefly on work in higher and lower spacetime dimensions and non-gravity systems that may be relevant as toy models for general relativity. We exclude other work on self-similarity in general relativity and work on critical phenomena in other areas of physics.\nOther reviews on the subject are,,,,,,,,. The 2002 review by Gundlach gives more detailed explanations on some of the basic aspects of the theory.\nTheory\n======\nIn this Section we describe the basic theory underlying critical collapse of the type that forms arbitrarily small black holes (later called type II, see also Sec. [section:typeI]). We begin with the mathematical origin of its three main characteristics, which were already summarised in the introduction:\n* universality with respect to initial data;\n* scale-invariance of the critical solution;\n* black hole mass scaling.\nUniversality\n------------\nConsider GR as an infinite-dimensional continuous dynamical system. Points in the phase space are initial data sets (3-metric, extrinsic curvature and suitable matter variables, which together obey the Einstein constraints). We evolve with the Einstein equations in a suitable gauge (see Sec. [section:coordinates]). Solution curves of the dynamical system are spacetimes obeying the Einstein-matter equations, sliced by specific Cauchy surfaces of constant time *t*.\nAn isolated system in GR can end up in qualitatively different stable end states. Two possibilities are the formation of a single black hole in collapse, or complete dispersion of the mass-energy to infinity. For a massless scalar field in spherical symmetry, these are the only possible end states (see Sec. [section:scalarfield]). Any point in phase space can be classified as ending up in one or the other type of end state. The entire phase space therefore splits into two halves, separated by a “critical surface”.\nA phase space trajectory that starts on a critical surface by definition never leaves it. A critical surface is therefore a dynamical system in its own right, with one dimension fewer than the full system. If it has an attracting fixed point, such a point is called a critical point. It is an attractor of codimension one in the full system, and the critical surface is its attracting manifold. The fact that the critical solution is an attractor of codimension one is visible in its linear perturbations: it has an infinite number of decaying perturbation modes spanning the tangent plane to the critical surface, and a single growing mode not tangential to it.\nAs illustrated in Figs. [fig:dynsim] and [fig:phasespace3d], any trajectory beginning near the critical surface, but not necessarily near the critical point, moves almost parallel to the critical surface towards the critical point. Near the critical point the evolution slows down, and eventually moves away from the critical point in the direction of the growing mode. This is the origin of universality. All details of the initial data have been forgotten, except for the distance from the black hole threshold. The closer the initial phase point is to the critical surface, the more the solution curve approaches the critical point, and the longer it will remain close to it. We should stress that this phase picture is extremely simplified. Some of the problems associated with this simplification are discussed in Sec. [section:coordinates].\nSelf-similarity\n---------------\nFixed points of dynamical systems often have additional symmetries. In the case of type II critical phenomena, the critical point is a spacetime that is self-similar, or scale-invariant. These symmetries can be discrete or continuous. The critical solution of a spherically symmetric perfect fluid (see Sec. [section:perfectfluid]), has continuous self-similarity (CSS). A CSS spacetime is one that admits a homothetic vector field *ξ*, defined by :\n \n$$\\label{homothetic\\_metric}\n {\\cal L}\\_\\xi g\\_{ab} = 2 g\\_{ab}.$$ \nIn coordinates *x**μ* = (*τ*, *x**i*) adapted to the symmetry, so that\n \n$$\\label{xi\\_in\\_coordinates}\n \\xi = - {\\partial\\over\\partial\\tau},$$ \nthe metric coefficients are of the form\n \n$$\\label{CSS\\_coordinates}\n g\\_{\\mu\\nu}(\\tau, x^i) = e^{-2\\tau} \\tilde g\\_{\\mu\\nu}(x^i),$$ \nwhere the coordinate *τ* is the negative logarithm of a spacetime scale, and the remaining three coordinates *x**i* can be thought of angles around the singular spacetime point *τ* = ∞ (see Sec. [section:globalstructure]).\nThe critical solution of other systems, in particular the spherical scalar field (see Sec. [section:scalarfield]) and axisymmetric gravitational waves (see Sec. [section:brillwaves]), show discrete self-similarity (DSS). The simplest way of defining DSS is in adapted coordinates, where\n \n$$\\label{DSS\\_coordinates}\n g\\_{\\mu\\nu}(\\tau, x^i) = e^{-2\\tau} \\tilde g\\_{\\mu\\nu}(\\tau,x^i),$$ \nsuch that *g**μ**ν*(*τ*, *x**i*) is periodic in *τ* with period Δ. More formally, DSS can be defined as a discrete conformal isometry .\nUsing the gauge freedom of general relativity, the lapse and shift in the ADM formalism can be chosen (non-uniquely) so that the coordinates become adapted coordinates if and when the solution becomes self-similar (see Sec. [section:coordinates]). *τ* is then both a time coordinate (in the usual sense that surfaces of constant time are Cauchy surfaces), and the logarithm of overall scale at constant *x**i*. The minus sign in ([xiincoordinates]) and hence ([CSScoordinates]) and ([DSScoordinates]), is a convention assuming that smaller scales are in the future. The time parameter used in Fig. [fig:phasespace3d] is of this type.\nMass scaling\n------------\nLet *Z* stand for a set of scale-invariant variables of the problem, such as $\\tilde g\\_{\\mu\\nu}$ and suitably rescaled matter variables. If the dynamics is scale-invariant (this is the case exactly for example for the scalar field, and approximately for other systems, see Sec. [section:universalityclasses]), then *Z*(*x*) is an element of the phase space factored by overall scale, and *Z*(*x*, *τ*) a solution. Note that *Z*(*x*) is an initial data set for GR only up to scale. The overall scale is supplied by *τ*.\nFor simplicity, assume that the critical solution is CSS. It can then be written as *Z*(*x*, *τ*) = *Z*\\*(*x*). Its linear perturbations can depend on *τ* only exponentially. To linear order, the solution near the critical point must be of the form\n \n*Z*(*x*, *τ*) ≃ *Z*\\*(*x*) + ∑*i* = 1∞*C**i*(*p*) *e**λ**i**τ**Z**i*(*x*). \nThe perturbation amplitudes *C**i* depend on the initial data, and hence on *p*. As *Z*\\* is a critical solution, by definition there is exactly one *λ**i* with positive real part (in fact it is purely real), say *λ*0. As *τ* → ∞, all other perturbations vanish. In the following we consider this limit, and retain only the one growing perturbation.\nFrom our phase space picture, the evolution ends at the critical solution for *p* = *p*\\*, so we must have *C*0(*p*\\*) = 0. Linearising in *p* around *p*\\*, we obtain\n \n$$\\label{echoing\\_region}\n \\lim\\_{\\tau\\to\\infty} Z(x,\\tau) \\simeq Z\\_\\*(x) + {dC\\_0\\over dp} (p-p\\_\\*)\n e^{\\lambda\\_0\\tau} Z\\_0(x).$$ \nFor *p* ≠ *p*\\*, but close to it, the solution has the approximate form ([echoingregion]) over a range of *τ*. Now we extract Cauchy data at one particular *p*-dependent value of *τ* within that range, namely *τ*\\* defined by [taustardef]\n \n$${dC\\_0\\over dp} (p-p\\_\\*) e^{-\\lambda\\_0\\tau\\_\\*} \\equiv \\epsilon,$$ \nwhere *ε* is some constant ≪ 1 such that at this *τ* the linear approximation is still valid. At sufficiently large *τ*, the linear perturbation has grown so much that the linear approximation breaks down, and for *C*0 > 0 a black hole forms while for *C*0 < 0 the solution disperses. The crucial point is that we need not follow this evolution in detail, nor does the precise value of *ε* matter. It is sufficient to note that the Cauchy data at *τ* = *τ*\\* are\n \n*Z*(*x*, *τ*\\*) ≃ *Z*\\*(*x*) + *ε**Z*0(*x*). \nDue to the funnelling effect of the critical solution, the data at *τ*\\* is always the same, except for an overall scale, which is given by *e*− *τ*\\*. For example, the physical spacetime metric, with dimension $(\\hbox{length})^2$ is given by $g\\_{\\mu\\nu}=e^{-2\\tau}\n\\tilde g\\_{\\mu\\nu}$, and similar scalings hold for the matter variables according to their dimension. In particular, as *e*− *τ*\\* is the only scale in the initial data ([pdata]), the mass of the final black hole must be proportional to that scale. Therefore\n \n*M* ∝ *e*− *τ*\\* ∝ (*p* − *p*\\*)1/*λ*0, \nand, comparing with Eq. ([eq1]), we have found the critical exponent *γ* = 1/*λ*0.\nWhen the critical solution is DSS, a periodic or fine structure of small amplitude is superimposed on this basic power law :\n \nln*M* = *γ*ln(*p* − *p*\\*) + *c* + *f*(*γ*ln(*p* − *p*\\*) + *c*), \nwhere *f*(*z*) has period Δ and is universal, and only *c* depends on the initial data. As the critical solution is periodic in *τ* with period Δ the number *N* of scaling “echos” is approximated by\n \n$$N \\simeq \\Delta^{-1} \\gamma \\ln|p-p\\_\\*| + {\\rm const.}$$ \nNote that this holds for both supercritical and subcritical solutions.\nType I\n------\nIn type I critical phenomena, the same phase space picture as in Sec. [section:universality] applies, but the critical solution is now stationary or time-periodic instead of self-similar or scale-periodic. It also has a finite mass and can be thought of as a metastable star. (Type I and II were so named after first and second order phase transitions in statistical mechanics, in which the order parameter is discontinuous and continuous, respectively.) Universality in this context implies that the black hole mass near the threshold is independent of the initial data, namely a certain fraction of the mass of the stationary critical solution. The dimensionful quantity that scales is not the black hole mass, but the lifetime *t**p* of the intermediate state where the solution is approximated by the critical solution. This is clearly\n \n$$\\label{typeIscaling}\nt\\_p = - {1\\over \\lambda\\_0} \\ln|p-p\\_\\*| + {\\rm const.}$$ \nType I critical phenomena occur when a mass scale in the field equations becomes dynamically relevant. (This scale does not necessarily set the mass of the critical solution absolutely: there could be a family of critical solutions selected by the initial conditions.) Conversely, as the type II power law is scale-invariant, type II phenomena occur in situations where either the field equations do not contain a scale, or this scale is dynamically irrelevant. Many systems, such as the massive scalar field, show both type I and type II critical phenomena, in different regions of the space of initial data.\nCoordinate choices for the dynamical systems picture\n----------------------------------------------------\nThe time evolution of Cauchy data in GR can only be considered as a dynamical system if the ADM evolution equations are complemented by a prescription for the lapse and shift. To realise the phase space picture of Sec. [section:universality], the critical solution must be a fixed point or limit cycle. We have seen how coordinates adapted to the self-similarity can be constructed, but is there a prescription of the lapse and shift for *arbitrary* initial data, such that, given initial data for the critical solution, the resulting time evolution actively drives the metric to a form ([DSScoordinates]) that explicitly displays the self-similarity?\nGarfinkle and Gundlach have suggested several combinations of lapse and shift conditions that leave CSS spacetimes invariant and turn the Choptuik DSS spacetime into a limit cycle. (See for partial successes.) Among these, the combination of maximal slicing with minimal strain shift has been suggested in a different context but for related reasons. Maximal slicing requires the initial data slice to be maximal (*K**a**a* = 0), but other prescriptions, such as freezing the trace of *K* together with minimal distortion, allow for an arbitrary initial slice with arbitrary spatial coordinates.\nAll these coordinate conditions are elliptic equations that require boundary conditions, and will turn CSS spacetimes into fixed points (or DSS into limit cycles) only given correct boundary conditions. Roughly speaking, these boundary conditions require a guess of how far the slice is from the accumulation point *t* = *t*\\*, and answers to this problem only exist in spherical symmetry. Appropriate boundary conditions are also needed if the dynamical system is extended to include the lapse and shift as evolved variables, turning the elliptic equations for the lapse and shift into hyperbolic or parabolic equations.\nTurning a CSS or stationary spacetime into a fixed point of the dynamical system also requires an appropriate choice of the phase space variables *Z*(*x**i*). To capture CSS (or DSS) solutions, one needs scale-invariant variables. Essentially, these can be constructed by dimensional analysis. The coordinates *x**i* and *τ* are dimensionless, *l**e*− *τ* has dimension length, and *g**μ**ν* has dimension *l*2. The scaling for the ADM and any matter variables follows.\nEven with a prescription for the lapse and shift in place, a given spacetime does not correspond to a unique trajectory in phase space. Rather, for each initial slice through the same spacetime one obtains a different slicing of the entire spacetime. A possibility for avoiding this ambiguity would be to restrict the phase space further, for example by restricting possible data sets to maximal or constant extrinsic curvature slices.\nAnother open problem is that in order to talk about attractors and repellers on the phase space we need to define a norm on a suitable function space which includes both asymptotically flat data and data for the exact critical solution. The norm itself must favour the central region and ignore what is further out and asymptotically flat if all black holes of the same mass are to be considered as the same endstate.\nApproximate self-similarity and universality classes\n----------------------------------------------------\nThe field equations for the massless scalar field coupled to the Einstein equations are scale-free. Realistic matter models introduce length scales, and the field equations then do not allow for exactly self-similar solutions. They may however admit solutions which are CSS or DSS asymptotically on small spacetime scales as the dimensionful parameters become irrelevant, including type II critical solutions. This can be explored by a formal expansion in powers of the small parameter *L**e*− *τ*, where *L* is a parameter with dimensions length in the evolution equations. The zeroth order of the expansion is the self-similar critical solution of the system with *L* = 0. A similar ansatz can be made for the linear perturbations of the resulting background. The zeroth order of the background expansion determines Δ exactly and independently of *L*, and the zeroth order term of the linear perturbation expansion determines the critical exponent 1/*λ*0 exactly, so that there is no need in practice to calculate any higher orders in *L* to make predictions for type II critical phenomena where they occur. (With *L* ≠ 0, the basin of attraction of the type II critical solution will depend on *L*, and type I critical phenomena may also occur, see Sec. [section:typeI]. A priori, there could also be more than one type II critical solution for *L* = 0, although this has not been observed.)\nThis procedure has been carried out for the Einstein-Yang-Mills system and for massless scalar electrodynamics. Both systems have a single length scale 1/*e* (in geometric units *c* = *G* = 1), where *e* is the gauge coupling constant. All values of *e* can be said to form one universality class of field equations represented by *e* = 0. This notion of universality classes is fundamentally the same as in statistical mechanics. Other examples include modifications to the perfect fluid equation of state that do not affect the limit of high density. A simple example is that any scalar field potential *V*(*ϕ*) becomes dynamically irrelevant compared to the kinetic energy ∣∇*ϕ*∣2 in a self-similar solution, so that all scalar fields with potentials are in the universality class of the free massless scalar field. Surprisingly, even two different models like the *S**U*(2) Yang-Mills and *S**U*(2) Skyrme models in spherical symmetry are members of the same universality class .\nIf there are several scales *L*0, *L*1, *L*2 etc. present in the problem, a possible approach is to set the arbitrary scale in ([xtau]) equal to one of them, say *L*0, and define the dimensionless constants *l**i* = *L**i*/*L*0 from the others. The scope of the universality classes depends on where the *l**i* appear in the field equations. If a particular *L**i* appears in the field equations only in positive integer powers, the corresponding *l**i* appears only multiplied by *e*− *τ*, and will be irrelevant in the scaling limit. All values of this *l**i* therefore belong to the same universality class. From the example above, adding a quartic self-interaction *λ**ϕ*4 to the massive scalar field gives rise to the dimensionless number *λ*/*m*2 but its value is an irrelevant (in the language of renormalisation group theory) parameter.\nContrary to the statement in , we conjecture that massive scalar electrodynamics, for any values of *e* and *m*, is in the universality class of the massless uncharged scalar field in a region of phase space where type II critical phenomena occur. Examples of dimensionless parameters which do change the universality class are the *k* of the perfect fluid, the *κ* of the 2-dimensional sigma model or, probably, a conformal coupling of the scalar field (the numerical evidence is weak but a dependence should be expected).\nThe analogy with critical phase transitions\n-------------------------------------------\nSome basic aspects of critical phenomena in gravitational collapse, such as fine-tuning, universality, scale-invariant physics, and critical exponents for dimensionful quantities, can also be identified in critical phase transitions in statistical mechanics (see for an introductory textbook).\nFrom an abstract point of view, the objective of statistical mechanics is to derive relations between macroscopic observables *A* of the system and macroscopic external forces *f* acting on it, by considering ensembles of microscopic states of the system. The expectation values ⟨*A*⟩ can be generated as partial derivatives of the partition function\n \n$$Z(\\mu, f)=\\sum\\_{\\rm microstates} \\ e^{- H({\\rm\nmicrostate}, \\mu, f)}$$ \nHere the *μ* are parameters of the Hamiltonian such as the strength of intermolecular forces, and *f* are macroscopic quantities which are being controlled, such as the temperature or magnetic field.\nPhase transitions in thermodynamics are thresholds in the space of external forces *f* at which the macroscopic observables *A*, or one of their derivatives, change discontinuously. We consider two examples: the liquid-gas transition in a fluid, and the ferromagnetic phase transition.\nThe liquid-gas phase transition in a fluid occurs at the boiling curve *p* = *p**b*(*T*). In crossing this curve, the fluid density changes discontinuously. However, with increasing temperature, the difference between the liquid and gas density on the boiling curve decreases, and at the the critical point (*p*\\* = *p**b*(*T*\\*), *T*\\*) it vanishes as a non-integer power:\n \n$$\\rho\\_{\\rm liquid}-\\rho\\_{\\rm gas}\\sim (T\\_\\*-T)^\\gamma.$$ \nAt the critical point an otherwise clear fluid becomes opaque, due to density fluctuations appearing on all scales up to scales much larger than the underlying atomic scale, and including the wavelength of light. This indicates that the fluid near its critical point is approximately scale-invariant (for some range of scales between the size of molecules and the size of the container).\nIn a ferromagnetic material at high temperatures, the magnetisation ${\\bf m}$ of the material (alignment of atomic spins) is determined by the external magnetic field ${\\bf B}$. At low temperatures, the material shows a spontaneous magnetisation even at zero external field. In the absence of an external field this breaks rotational symmetry: the system makes a random choice of direction. With increasing temperature, the spontaneous magnetisation ${\\bf m}$ decreases and vanishes at the Curie temperature *T*\\* as\n \n$$|{\\bf m}|\\sim (T\\_\\*-T)^\\gamma.$$ \nAgain, the correlation length, or length scale of a typical fluctuation, diverges at the critical point, indicating scale-invariant physics.\nQuantities such as $|\\bf m|$ or $\\rho\\_{\\rm liquid}-\\rho\\_{\\rm gas}$ are called order parameters. In statistical mechanics, one distinguishes between first-order phase transitions, where the order parameter changes discontinuously, and second-order, or critical, ones, where it goes to zero continuously. One should think of a critical phase transition as the critical point where a line of first-order phase transitions ends as the order parameter vanishes. This is already clear in the fluid example. In the ferromagnet example, at first one seems to have only the one parameter *T* to adjust. But in the presence of a very weak external field, the spontaneous magnetisation aligns itself with the external field ${\\bf B}$, while its strength is to leading order independent of ${\\bf B}$. The function ${\\bf m}({\\bf\nB},T)$ therefore changes discontinuously at ${\\bf B}=0$. The line ${\\bf B}=0$ for *T* < *T*\\* is therefore a line of first order phase transitions between directions (if we consider one spatial dimension only, between ${\\bf m}$ up and ${\\bf m}$ down). This line ends at the critical point $({\\bf B}=0,T=T\\_\\*)$ where the order parameter $|{\\bf\nm}|$ vanishes. The critical value ${\\bf B}=0$ of ${\\bf B}$ is determined by symmetry; by contrast *p*\\* depends on microscopic properties of the material.\nWe have already stated that a critical phase transition involves scale-invariant physics. In particular, the atomic scale, and any dimensionful parameters associated with that scale, must become irrelevant at the critical point. This is taken as the starting point for obtaining properties of the system at the critical point.\nOne first defines a semi-group acting on micro-states: the renormalisation group. Its action is to group together a small number of adjacent particles as a single particle of a fictitious new system by using some averaging procedure. This can also be done in a more abstract way in Fourier space. One then defines a dual action of the renormalisation group on the space of Hamiltonians by demanding that the partition function is invariant under the renormalisation group action:\n \n$$\\sum\\_{\\rm microstates} e^{-H}=\\sum\\_{\\rm microstates'} e^{-H'}.$$ \nThe renormalised Hamiltonian is in general more complicated than the original one, but it can be approximated by the same Hamiltonian with new values of the parameters *μ* and external forces *f*. (At this stage it is common to drop the distinction between *μ* and *f*, as the new *μ*ʹ and *f*ʹ depend on both *μ* and *f*.) Fixed points of the renormalisation group correspond to Hamiltonians with the parameters at their critical values. The critical values of many of these parameters will be 0 or ∞, meaning that the dimensionful parameters they were originally associated with are irrelevant. Because a fixed point of the renormalisation group can not have a preferred length scale, the only parameters that can have nontrivial values are dimensionless.\nThe behaviour of thermodynamical quantities at the critical point is in general not trivial to calculate. But the action of the renormalisation group on length scales is given by its definition. The blowup of the correlation length *ξ* at the critical point is therefore the easiest critical exponent to calculate. The same is true for the black hole mass, which is just a length scale. We can immediately reinterpret the mathematics of Section [section:scaling] as a calculation of the critical exponent for *ξ*, by substituting the correlation length *ξ* for the black hole mass *M*, *T*\\* − *T* for *p* − *p*\\*, and taking into account that the *τ*-evolution in critical collapse is towards smaller scales, while the renormalisation group flow goes towards larger scales: *ξ* therefore diverges at the critical point, while *M* vanishes.\nIn type II critical phenomena in gravitational collapse, we should think of the black hole mass as being controlled by the functions *P* and *Q* on phase space defined by Eq. ([PQ]). Clearly, *P* is the equivalent of the reduced temperature *T* − *T*\\*. Gundlach has suggested that the angular momentum of the initial data can play the role of ${\\bf B}$, and the final black hole angular momentum the role of ${\\bf m}$. Like the magnetic field, angular momentum is a vector, with a critical value that must be zero because all other values break rotational symmetry.\nThe scalar field\n================\nCritical phenomena in gravitational collapse were first discovered by Choptuik in the model of a spherically symmetric, massless scalar field *ϕ* minimally coupled to general relativity. The scalar field matter is both simple, and acts as a toy model in spherical symmetry for the effects of gravitational radiation. Given that it is still the best-studied model in spherical symmetry, we review it here as a case study. For other numerical work on this model, see. Important analytical studies of gravitational collapse in this model have been carried out by Christodoulou.\nWe first review the field equations and Choptuik’s observations at the black hole threshold, mainly as a concrete example for the general ideas discussed above. We then summarise more recent work on the global structure of Choptuik’s critical solution, which throws an interesting light on cosmic censorship. In particular, the exact critical solution contains a curvature singularity that is locally and globally naked, and any critical solution obtained in the limit of perfect fine-tuning of asymptotically flat initial data is at least locally naked. By perturbing around spherical symmetry, the stability of the Choptuik solution in the full phase space can be investigated, and the scaling of black hole angular momentum can be predicted. By embedding the real scalar field in scalar electrodynamics and perturbing around the Choptuik solution, the scaling of black hole charge can be predicted.\nField equations in spherical symmetry\n-------------------------------------\nThe Einstein equations are\n \n$$\\label{scalar\\_stress\\_energy}\n G\\_{ab} = 8 \\pi \\left(\\nabla\\_a \\phi \\nabla\\_b \\phi - {1\\over 2} g\\_{ab}\n \\nabla\\_c \\phi \\nabla^c \\phi\\right),$$ \nand the matter equation is\n \n∇*a*∇*a**ϕ* = 0. \nNote that the matter equation of motion is contained within the contracted Bianchi identities. Choptuik chose Schwarzschild-like coordinates\n \n*d**s*2 = − *α*2(*r*, *t*) *d**t*2 + *a*2(*r*, *t*) *d**r*2 + *r*2 *d*Ω2, \nwhere *d*Ω2 = *d**θ*2 + sin2*θ* *d**φ*2 is the metric on the unit 2-sphere. This choice of coordinates is defined by the radius *r* giving the surface area of 2-spheres as 4*π**r*2, and by *t* being orthogonal to *r* (polar-radial coordinates). One more condition is required to fix the coordinate completely. Choptuik chose *α* = 1 at *r* = 0, so that *t* is the proper time of the central observer.\nIn the auxiliary variables\n \n$$\\Phi = \\phi\\_{, r}, \\qquad \\Pi={a\\over\\alpha} \\phi\\_{, t},$$ \nthe wave equation becomes a first-order system,\n \n$$\\label{wave}\n \\begin{array}{l}\n \\displaystyle \\Phi\\_{, t} = \\left({\\alpha\\over a}\\Pi\\right)\\_{, r},\n \\\\ \\\\\n \\displaystyle \\Pi\\_{, t} = {1\\over r^2} \\left(r^2{\\alpha\\over a}\n \\Phi\\right)\\_{, r}.\n \\end{array}$$ \nIn spherical symmetry there are four algebraically independent components of the Einstein equations. Of these, one is a linear combination of derivatives of the other and can be disregarded. The other three contain only first derivatives of the metric, namely *a*, *t*, *a*, *r* and *α*, *r*, and are\n \n$$\\begin{aligned}\n \\label{da\\_dr}\n {a\\_{, r}\\over a} + {a^2 -1 \\over 2r} \n &=& 2\\pi r (\\Pi^2 + \\Phi^2), \\\\\n \\label{dalpha\\_dr}\n {\\alpha\\_{, r}\\over \\alpha} - {a\\_{, r}\\over a} - {a^2 -1 \\over r}\n &=& 0, \\\\\n \\label{da\\_dt}\n {a\\_{, t}\\over \\alpha} &=& 4\\pi r \\Phi \\Pi.\\end{aligned}$$ \nBecause of spherical symmetry, the only dynamics is in the scalar field equations ([wave]). The metric can be found by integrating the ODEs ([dadr]) and ([dalphadr]) for *a* and *α* at any fixed *t*, given *ϕ* and Π. Eq. ([dadt]) can be ignored in this “fully constrained” evolution scheme.\nThe black hole threshold\n------------------------\nThe free data for the system are the two functions Π(*r*, 0) and Φ(*r*, 0). Choptuik investigated several one-parameter families of such data by evolving the data for many different values of the parameter. Simple examples of such families are Π(*r*, 0) = 0 and a Gaussian for Φ(*r*, 0), with the parameter *p* taken to be either the amplitude of the Gaussian, with the width and centre fixed, or the width, with position and amplitude fixed, or the position, with width and amplitude fixed. For sufficiently small amplitude (or the peak sufficiently wide), the scalar field will disperse, and for sufficiently large amplitude it will form a black hole.\nGeneric one-parameter families behave in this way, but this is difficult to prove in generality. Christodoulou showed for the spherically symmetric scalar field system that data sufficiently weak in a well-defined way evolve to a Minkowski-like spacetime , and that a class of sufficiently strong data forms a black hole .\nChoptuik found that in all 1-parameter families of initial data he investigated he could make arbitrarily small black holes by fine-tuning the parameter *p* close to the black hole threshold. An important fact is that there is nothing visibly special to the black hole threshold. One cannot tell that one given data set will form a black hole and another one infinitesimally close will not, short of evolving both for a sufficiently long time.\nAs *p* → *p*\\* along the family, the spacetime varies on ever smaller scales. Choptuik developed numerical techniques that recursively refine the numerical grid in spacetime regions where details arise on scales too small to be resolved properly. In the end, he could determine *p*\\* up to a relative precision of 10− 15, and make black holes as small as 10− 6 times the ADM mass of the spacetime. The power-law scaling ([massscaling]) was obeyed from those smallest masses up to black hole masses of, for some families, 0.9 of the ADM mass, that is, over six orders of magnitude . There were no families of initial data which did not show the universal critical solution and critical exponent. Choptuik therefore conjectured that *γ* is the same for all one-parameter families of smooth, asymptotically flat initial data that depend smoothly on the parameter, and that the approximate scaling law holds ever better for arbitrarily small *p* − *p*\\*.\nIt is an empirical fact that typical one-parameter families cross the threshold only once, so that there is every indication that it is a smooth submanifold, as we assumed in the phase space picture. Taking into account the discussion of mass scaling above, we can formally write the black hole mass as a functional of the initial data *z* = (*ϕ*(*r*, 0), Π(*r*, 0)) *exactly* as\n \n*M*[*z*] = *Q*[*z*]*H*(*P*[*z*]) (*P*[*z*])*γ*, \nwhere *P* and *Q* are *smooth* functions on phase space and *H* is the Heaviside function. (*Q* could be absorbed into *P*).\nIn hindsight, polar-radial gauge is well-adapted to self-similarity. In this gauge, discrete self-similarity corresponds to\n \n*Z*(*r*, *t*) = *Z*(*e**n*Δ*r*, *e**n*Δ*t*) \nfor any integer *n*, where *Z* stands for any one of the dimensionless quantities *a*, *α* or *ϕ* (and therefore also for *r*Π or *r*Φ). With\n \n$$\\label{x\\_tau}\n x = -{r \\over t-t\\_\\*}, \\quad \\tau = - \\ln\\left(-{t-t\\_\\*\\over\n L}\\right), \\quad t<t\\_\\*,$$ \ndiscrete self-similarity is\n \n*Z*(*x*, *τ* + Δ) = *Z*(*x*, *τ*). \nThe dimensionful constants *t*\\* and *L* depend on the particular one-parameter family of solutions, but the dimensionless critical fields *a*\\*, *α*\\* and *ϕ*\\*, and in particular their dimensionless period Δ, are universal. Empirically, Δ ≃ 3.44 for the scalar field in numerical time evolutions, and Δ = 3.445452402(3) from a numerical construction of the critical solution based on exact self-similarity and analyticity.\nGlobal structure of the critical solution\n-----------------------------------------\nIn adapted coordinates, the metric of the critical spacetime is of the form *e*− 2*τ* times a regular metric. From this general form alone, one can conclude that *τ* = ∞ is a curvature singularity, where Riemann and Ricci invariants blow up like *e*4*τ* (unless the spacetime is flat), and which is at finite proper time from regular points in its past. The Weyl tensor with index position *C**a**b**c**d* is conformally invariant, so that components with this index position remain finite as *τ* → ∞. This type of singularity is called “conformally compactifiable” or “isotropic” . For a classification of all possible *global* structures of spherically symmetric self-similar spacetimes see.\nThe global structure of the scalar field critical solution was determined accurately in by assuming analyticity at the centre of spherical symmetry and at the past light cone of the singularity (the self-similarity horizon, or SSH). The critical solution is then analytic up to the future lightcone of the singularity (the Cauchy horizon, or CH). Global adapted coordinates *x* and *τ* can be chosen so that the regular centre *r* = 0, the SSH and the CH are all lines of constant *x*, and surfaces of constant *τ* are never tangent to *x* lines. (A global *τ* is no longer a global time coordinate). This is illustrated in Fig. [figure:redshiftmatched].\nApproaching the Cauchy horizon, the scalar field oscillates an infinite number of times but with the amplitude of the oscillations decaying to zero. The scalar field in regular adapted coordinates (*x*, *τ*) is of the form\n \n$$\\label{phicritcont}\n\\phi(x,\\tau)\\simeq F\\_\\mathrm{reg}(\\tau)\n+|x|^\\epsilon F\\_\\mathrm{sing}\\Bigl[\n\\tau+H(\\tau)+K\\ln|x|\\Bigr],$$ \nwhere *F*reg(*τ*), *F*sing(*τ*) and *H*(*τ*) are periodic with period Δ, and the SSH is at *x* = 0. These functions have been computed numerically to high accuracy, together with the constants *K* and *ε*. The scalar field itself is smooth with respect to *τ*, and as *ε* > 0, it is continuous but not differentiable with respect to *x* on the CH itself. The same is true for the metric and the curvature. Surprisingly, the ratio *m*/*r* of the Hawking mass over the area radius on the Cauchy horizon is of order 10− 6 but not zero (the value is known to eight significant figures).\nAs the CH itself is regular with smooth null data except for the singular point at its base, it is not intuitively clear why the continuation is not unique. A partial explanation is given in, where all *DSS* continuations are considered. Within a DSS ansatz, the solution just to the future of the CH has the same form as ([phicritcont]). *F*reg(*τ*) is the same on both sides, but *F*sing(*τ*) can be chosen freely on the future side of the CH. Within the restriction to DSS this function can be taken to parameterise the information that comes out of the naked singularity.\nThere is precisely one choice of *F*sing(*τ*) on the future side that gives a regular centre to the future of the CH, with the exception of the naked singularity itself, which is then a point. This continuation was calculated numerically, and is almost but not quite Minkowski in the sense that *m*/*r* remains small everywhere to the future of the SSH.\nAll other DSS continuations have a naked, timelike central curvature singularity with negative mass. More exotic continuations including further Cauchy horizons would be allowed kinematically but are not achieved dynamically if we assume that the continuation is DSS. The spacetime diagram of the generic DSS continued solution is given in Fig. [figure:critcont].\nNear-critical spacetimes and naked singularities\n------------------------------------------------\nChoptuik’s results have an obvious bearing on the issue of cosmic censorship. (See for a general review of cosmic censorship.) Roughly speaking, fine-tuning to the black hole threshold provides a set of data which is codimension one in the space of generic, smooth, asymptotically flat initial data, and whose time evolution contains at least the point singularity of the critical solution. The cosmic censorship hypothesis must therefore be formulated as “*generic* smooth initial data for reasonable matter do not form naked singularities”. Here we look at the relation between fine-tuning and naked singularities in more detail.\nChristodoulou proves rigorously that naked singularity formation is not generic, but in a rather larger function space, functions of bounded variation, than one would naturally consider. In particular, the instability of the naked singularity found by Christodoulou is not differentiable on the past light cone. This is unnatural in the context of critical collapse, where the naked singularity can arise from generic (up to fine-tuning) smooth initial data, and the intersection of the past light cone of the singularity with the initial data surface is as smooth as the initial data elsewhere. It is therefore not clear how this theorem relates to the numerical and analytical results strongly indicating that naked singularities are codimension-1 generic within the space of smooth initial data.\nFirst, consider the exact critical solution. The lapse *α* defined by ([trmetric]) is bounded above and below in the critical solution. Therefore the redshift measured between constant *r* observers located at any two points on an outgoing radial null geodesic in the critical spacetime to the past of the CH is bounded above and below. Within the exact critical solution, a point with arbitrarily high curvature can therefore be observed from a point with arbitrarily low curvature. Next consider a spacetime where the critical solution in a central region is smoothly matched to an asymptotically flat outer region such that the resulting asymptotically flat spacetime contains a part of the critical solution that includes the singularity and a part of the CH. In this spacetime, a point of arbitrarily large curvature can be seen from with finite redshift. This is illustrated in Fig. [figure:redshiftmatched].\nNow consider the evolution of asymptotically flat initial data that have been fine-tuned to the black hole threshold. The global structure of such spacetimes has been investigated numerically in. Empirically, these spacetimes can be approximated near the singularity by ([perturbations]). Almost all perturbations decay as the singularity is approached and the approximation becomes better, until the one growing perturbation (which by the assumption of fine-tuning starts out small) becomes significant. A maximal value of the curvature is then reached which is still visible from and which scales as \n \n*R*max ∝ (*p* − *p*\\*)2*γ*. \nFinally, consider the limit of perfect fine-tuning. The growing mode is then absent, and all other modes decay as the naked singularity is approached. Note that ([perturbations]) suggests this is true regardless of the direction (future, past, or spacelike, depending on the value of *x*) in which the singularity is approached. This seems to be in conflict with causality: the issue is sensitively connected to the completeness of the modes *Z**i* and to the stability of the Cauchy horizon and requires more investigation.\nA complication in the supercritical case has been pointed out in. In the literature on supercritical evolutions, what is quoted as the black hole mass is in fact the mass of the first apparent horizon (AH) that appears in the time slicing used by the code (spacelike or null). The black hole mass can and generically will be larger than the AH mass when it is first measured because of matter falling in later, and the region of maximal curvature may well be inside the event horizon, and hidden from observers at. (See Fig. [figure:smallBHlargeBH] for an illustration). The true black hole mass can only be measured at, where it is defined to be the limit of the Bondi mass *m*B as the Bondi time *u*B → ∞. This was implemented in. Only one family of initial data was investigated, but in this family it was found that *m*B converges to 10− 4 of the initial Bondi mass in the fine tuning limit. More numerical evidence would be helpful, but the result is plausible. As the underlying physics is perfectly scale-invariant in the massless scalar field model, the minimum mass must be determined by the family of initial data through the infall of matter into the black hole. Simulations of critical collapse of a perfect fluid in a cosmological context show a similar lower bound due to matter falling back after shock formation, but this may not be true for all initial data.\nElectric charge\n---------------\nGiven the scaling power law for the black hole mass in critical collapse, one would like to know what happens if one takes a generic 1-parameter family of initial data with both electric charge and angular momentum (for suitable matter), and fine-tunes the parameter *p* to the black hole threshold. A simple model for charged matter is a complex scalar field coupled to electromagnetism with the substitution ∇*a* → ∇*a* + *i**e**A**a*, or scalar electrodynamics. (Note that in geometric units, black hole charge *Q* has dimension of length, but the charge parameter *e* has dimension 1/length.)\nGundlach and Martín-García have studied scalar massless electrodynamics in spherical symmetry perturbatively. Clearly, the real scalar field critical solution of Choptuik is a solution of this system too. In fact, it remains a critical solution within massless scalar electrodynamics in the sense that it still has only one growing perturbation mode within the enlarged solution space. Some of its perturbations carry electric charge, but as they are all decaying, electric charge is a subdominant effect. The charge of the black hole in the critical limit is dominated by the most slowly decaying of the charged modes. From this analysis, a universal power-law scaling of the black hole charge\n \n*Q* ∼ (*p* − *p*\\*)*δ* \nwas predicted. The predicted value *δ* ≃ 0.88 of the critical exponent (in scalar electrodynamics) was subsequently verified in collapse simulations by Hod and Piran and later again by Petryk. (The mass scales with *γ* ≃ 0.37 as for the uncharged scalar field.) No other type of criticality can be found in the phase space of this system as dispersion and black holes are the only possible end states, though black holes with $|Q|\\lesssim M $ can be formed.\nGeneral considerations similar to those in Sec. [section:universalityclasses] led Gundlach and Martín-García to the general prediction that the two critical exponents are always related, for any matter model, by the inequality\n \n*δ* ≥ 2*γ* \n(with the equality holding if the critical solution is charged), so that black hole charge can always be treated perturbatively at the black hole threshold. This has not yet been verified in any other matter model.\nSelf-interaction potential\n--------------------------\nAn example of the richer phenomenology in the presence of a scale in the field equations is the spherical *massive* scalar field with a potential *m*2*ϕ*2 coupled to gravity: In one region of phase space, with characteristic scales smaller than 1/*m*, the black hole threshold is dominated by the Choptuik solution and type II critical phenomena occur. In another it is dominated by metastable oscillating boson stars (whose mass is of order 1/*m* in geometric units) and type I critical phenomena occur. (For the real scalar field, the type I critical solution is an (unstable) oscillating boson star while for the complex scalar field it can be a static (unstable) boson star.)\nWhen the scalar field with a potential is coupled to electromagnetism, type II criticality is still controlled by a solution which asymptotically resembles the uncharged Choptuik spacetime, but type I criticality is now controlled by charged boson stars. There are indications that subcritical type I evolutions lead to slow, large amplitude oscillations of stable boson stars and not to dispersion to infinity, as had been conjectured in. Another interesting extension is the study of the dynamics of a real scalar field with a symmetric double-well potential, in which the system displays type I criticality between the two possible vacua.\nNonspherical perturbations: stability and angular momentum\n----------------------------------------------------------\nCritical collapse is really relevant for cosmic censorship only if it is not restricted to spherical symmetry. Martín-García and Gundlach have analysed all nonspherical perturbations of the scalar field critical solution by solving a linear eigenvalue problem with an ansatz of regularity at the centre and the SSH. They find that the only growing mode is the known spherical one, while all other spherical modes and all non-spherical modes decay. This strongly suggests that the critical solution is an attractor of codimension one not only in the space of spherically symmetric data but (modulo linearisation stability) of all data in a finite neighbourhood of spherical symmetry.\nMore recently, Choptuik and collaborators have carried out axisymmetric time evolutions for the massless scalar field using adaptive mesh refinement. They find that in the limit of fine-tuning generic axisymmetric initial data the spherically symmetric critical solution is approached at first but then deviates from spherical symmetry and eventually develops two centres, each of which approaches the critical solution and bifurcates again in a universal way. This suggests that the critical solution has non-spherical growing perturbation modes, possibly a single *l* = 2 even parity mode (in axisymmetry, only *m* = 0 is allowed). There appears to be a conflict between the time evolution results and the perturbative results, which needs to be resolved by more work (see Sec. [section:brillwaves]).\nPerturbing the scalar field around spherical symmetry, angular momentum comes in to second order in perturbation theory. All angular momentum perturbations were found to decay, and a critical exponent *μ* ≃ 0.76 for the angular momentum was derived for the massless scalar field in. This prediction has not yet been tested in nonlinear collapse simulations.\nMore spherical symmetry\n=======================\nThe pioneering work of Choptuik on the spherical massless scalar field has been followed by a plethora of further investigations. These could be organised under many different criteria. We have chosen the following rough categories:\n* Systems in which the field equations, *when reduced to spherical symmetry*, form a single wave-like equation, typically with explicit *r*-dependence in its coefficients. This includes Yang-Mills fields, sigma models, vector and spinor fields, scalar fields in 2+1 or in 4+1 and more spacetime dimensions, and scalar fields in a semi-classical approximation to quantum gravity.\n* Perfect fluid matter, either in an asymptotically flat or a cosmological context. The linearised Euler equations are in fact wave-like, but the full non-linear equations admit shock heating and are therefore not even time-reversal symmetric.\n* Collisionless matter described by the Vlasov equation is a partial differential equation on particle phase space as well as spacetime. Therefore even in spherical symmetry, the matter equation is a partial differential equation in four dimensions (rather than two). Intuitively speaking, there are infinitely more matter degrees of freedom than in the scalar field or in non-spherical vacuum gravity.\n* Spherically symmetric nonlinear wave equations on (3+1) Minkowski spacetime, and other nonlinear partial differential equations which show a transition between singularity formation and dispersal.\nSome of these examples were constructed because they may have intrinsic physical relevance (semiclassical gravity, primordial black holes), others as toy models for 3+1-dimensional gravity, and others mostly out of a purely mathematical interest. Table [table:mattermodels] gives an overview of these models.\nMatter obeying wave equations\n-----------------------------\n### 2d nonlinear *σ* model\nWe have already discussed in subsection [section:self-interaction] the effects of adding a potential to the evolution of the scalar field. An alternative generalization is a modification of the kinetic term in the Lagrangian, with the general form\n \n$$\\frac{1}{2} g^{\\mu\\nu} G\\_{IJ}(\\phi^K)\\nabla\\_\\mu\\phi^I \\nabla\\_\\nu \\phi^J$$ \nfor an N-dimensional vector field *ϕ**I*(*x*) with *I* = 1..*N*, and where *G**I**J*(*ϕ**K*) is a fixed nonlinear function acting as a metric on the so-called *target space* of the fields *ϕ**I*. Such a system is called a nonlinear *σ* model, harmonic map or wave map. The fields *ϕ**I* and metric *G**I**J* are dimensionless, and this allows the introduction of dimensionless parameters in the system, which cannot be asymptotically neglected using the arguments of subsection [section:universalityclasses]. (Compare with the potential *V*(*ϕ**I*), which has dimensions (length)− 2, and hence requires dimensionful parameters).\nThe case *N* = 1 gives nothing new, and so Hirschmann and Eardly (HE from now on) studied the *N* = 2 case with a target manifold with constant curvature, proportional to a real dimensionless constant *κ*. Using a single complex coordinate *ϕ* the action of the system can be written as\n \n$$\\int d^4x \\sqrt{g} \\left[ \\frac{R}{2} \n - \\frac{|\\nabla\\phi|^2}{(1-\\kappa|\\phi|^2)^2} \\right].$$ \nFor *κ* ≥ 0 this system is equivalent to the problem of a real massless scalar field coupled to Brans-Dicke (BD) gravity (with the BD coupling constant given by 8*ω**B**D* = − 12 + *κ*− 1). Liebling and Choptuik have shown that there is a smooth transition in the BD system from DSS criticality for low *κ* (the flat target space case *κ* = 0 is equivalent to a self-gravitating complex massless scalar field, whose critical solution is the original DSS spacetime found by Choptuik) to CSS criticality for larger *κ* (the case *κ* = 1 is equivalent to the axion-dilaton system, and has been shown to display CSS criticality in ).\nGeneralizing their previous results for *κ* = 0, HE constructed for each *κ* a CSS solution based on the ansatz *ϕ*(*τ*, *x*) = *e**i**ω**τ**ϕ*(*x*) for the critical scalar field. Studying its perturbations HE concluded that this solution is critical for *κ* > 0.0754, but has three unstable modes for *κ* < 0.0754 and even more for *κ* < − 0.28. Below 0.0754 a DSS solution takes over, as shown in the simulations of Liebling and Choptuik, and HE conjectured that the transition is a Hopf bifurcation, such that the DSS cycle smoothly shrinks with growing *κ*, collapsing onto the CSS solution at the transition and then disappearing with a finite value of the echoing period Δ.\nThe close relation between the CSS and DSS critical solutions is also manifest in the construction of their global structures. In particular, the results of and for the CSS *κ* = 0 and *κ* = 1 solutions respectively show that the Cauchy horizon of the singularity is almost but not quite flat, exactly as was the case with the Choptuik DSS spacetime (see subsection [section:globalstructure]).\n### Spherical Einstein-*S**U*(2) sigma model\nThis is an *N* = 3 sigma model, and it also displays a transition between CSS and DSS criticality, but this is a totally different type of transition, in particular showing a divergence in the echoing period Δ.\nIn a reduction to spherical symmetry, the effective action is\n \n$$\\int d^4x \\sqrt{g} \\left[\\frac{R}{2}- \\eta\n\\left( |\\nabla\\phi|^2 + \\frac{2\\sin^2\\phi}{r^2} \\right)\\right],$$ \nwhere *r* is the area radius, and the coupling constant *η* is dimensionless. It has been shown (numerically and then analytically ) that for 0 ≤ *η* < 1/2, there is an infinite sequence of CSS solutions *ϕ**n* labelled by a nodal number *n*, and having *n* growing modes. (The case *η* = 0, in which the sigma field decouples from gravity, will be revisited below.) The *n*-th solution is always regular in the past light cone of the singularity, but is regular up to the future light cone only for *η* < *η**n* where *η*0 ≃ 0.0688, *η*1 ≃ 0.1518 and *η**n* < *η**n* + 1 < 1/2. For larger couplings an apparent horizon develops and the solution cannot be smoothly continued. These results suggest that *ϕ*0 is a stable naked singularity for *η* < *η*0, and *ϕ*1 acts as a critical solution between naked singularity formation and dispersal for *η* < *η*0 and between black hole formation and dispersal for *η*0 < *η* < *η*1. The numerical experiments agree with this scenario in the range 0 ≤ *η* < 0.14. Other CSS solutions of this system are investigated in, and the possibility of chaos in.\nAichelburg and collaborators have shown that for $\\eta\\gtrsim 0.2$ there is clear DSS type II criticality at the black hole threshold. The period Δ depends on *η*, monotonically decreasing towards an asymptotic value for *η* → ∞. Interesting new behavior occurs in the intermediate range 0.14 < *η* < 0.2 that lies between clear CSS and clear DSS. With decreasing *η* the overall DSS includes episodes of approximate CSS, of increasing length (measured in the log-scale time *τ*). As *η* → *η**c* ≃ 0.170 from above the duration of the CSS epochs, and hence the overall DSS period Δ diverges. For 0.14 < *η* < 0.17 time evolutions of initial data near the black hole threshold no longer show overal DSS, but they still show CSS episodes. Black hole mass scaling is unclear in this regime.\nIt has been conjectured that this transition from CSS to DSS can be interpreted, in the language of the theory of dynamical systems, as the infinite-dimensional analogue of a three-dimensional Shil’nikov bifurcation. High-precision numerics in further supports this picture: for *η* > *η**c* a codimension-1 CSS solution coexists in phase space with a codimension-1 DSS attractor such that the (one-dimensional) unstable manifold of the DSS solution lies on the stable manifold of the CSS solution. For *η* close to *η**c* the two solutions are close and the orbits around the DSS solution become slower because they spend more time in the neighbourhood of the CSS attractor. A linear stability analysis predicts a law $\\Delta\\simeq -\\frac{2}{\\lambda}\n\\log(\\eta-\\eta\\_c)+b$ for some constant *b*, where *λ* is the Lyapunov exponent of the CSS solution. For *η* = *η**c* both solutions touch and the DSS cycle dissapears.\n### Einstein-Yang-Mills\nChoptuik, Chmaj and Bizoń have found both type I and type II critical collapse in the spherical Einstein–Yang-Mills system with SU(2) gauge potential, restricting to the purely magnetic case, in which the matter is described by a single real scalar field. The situation is very similar to that of the massive scalar field, and now the critical solutions are the well-known static *n* = 1 Bartnik-Mckinnon solution for type I and a DSS solution (later constructed in ) for type II. In both cases the black holes produced in the supercritical regime are Schwarzschild black holes with zero Yang-Mills field strength, but the final states (and the dynamics leading to them) can be distinguished by the value of the Yang-Mills final gauge potential at infinity, which can take two values, corresponding to two distinct vacuum states.\nChoptuik, Hirschmann and Marsa have investigated the boundary in phase space between formation of those two types of black holes, using a code that can follow the time evolutions for long after the black hole has formed. This is a new “type III” phase transition whose critical solution is an unstable static black hole with Yang-Mills hair, which collapses to a hairless Schwarzschild black hole with either vacuum state of the Yang-Mills field, depending on the sign of its one growing perturbation mode. This “colored” black hole is actually a member of a 1-parameter family parameterized by its apparent horizon radius and outside the horizon it approaches the corresponding BM solution. When the horizon radius approaches zero the three critical solutions meet at a “triple point”. What happens there deserves further investigation.\nMillward and Hirschmann have further coupled a Higgs field to the Einstein-Yang-Mills system. New possible end states appear: regular static solutions, and stable hairy black holes (different from the colored black holes referred to above). Again there are type I or type II critical phenomena depending on the initial conditions.\nIt is known that the spherical critical solutions within the magnetic ansatz become more unstable when other components of the gauge field are taken into account, and so they will not be critical in the general case.\n### Vacuum 4+1\nBizoń, Chmaj and Schmidt have found a way of constructing asymptotically flat vacuum spacetimes in 4+1 dimensions which are spherically symmetric while containing gravitational waves (Birkhoff’s theorem does not hold in more than 3+1 dimensions). Recall that in (3+1 dimensional) Bianchi IX cosmology the manifold is *M*1 × *S*3 where the *S*3 is equipped with an *S**U*(2) invariant (homogeneous but anisotropic) metric\n \n*d**s*2 = *L*12*σ*12 + *L*22*σ*32 + *L*32*σ*32, \nwhere the *σ**i* are the *S**U*(2) left-invariant 1-forms, and the *L**i* are functions of time only. Similarly, the spacetimes of Bizoń, Chmaj and Schmidt are of the product form *M*2 × *S*3 where the *L**i* now depend only on *r* and *t*. This gives rise to nontrivial dynamics, including a threshold between dispersion and black hole formation. With the additional *U*(1) symmetry *L*1 = *L*2 there is only one dynamical degree of freedom. At the black hole threshold, type II critical phenomena are seen with Δ ≃ 0.49 and *γ* ≃ 0.3289. In evolutions with the general ansatz where all *L**i* are different, the *U*(1) symmetry is recovered dynamically. Furthermore, the fact that each *U*(1)-symmetric solution exists in 3 copies gives rise to additional critical surfaces, and this lead to the prediction of 2-mode-unstable solutions, which was verified numerically, and the conjecture that there is a countable family of DSS solutions with *n* unstable modes. A similar ansatz can be made in other odd spacetime dimensions, and in 8+1 dimensions type II critical behaviour is again observed.\n### Scalar field collapse in 2+1\nSpacetime in 2+1 dimensions is flat everywhere where there is no matter, so that gravity is not acting at a distance in the usual way. There are no gravitational waves, and black holes can only be formed in the presence of a negative cosmological constant. (See for a review.)\nScalar field collapse in circular symmetry was investigated numerically by Pretorius and Choptuik, and Husain and Olivier. In a regime where the cosmological constant is small compared to spacetime curvature they find type II critical phenomena with a universal CSS critical solution, and *γ* = 1.20 ± 0.05. The value *γ* ≃ 0.81 appears to be less accurate.\nLooking for the critical solution in closed form, Garfinkle found a countable family of exact spherically symmetric CSS solutions for a massless scalar field with Λ = 0, but his results remain inconclusive. The *q* = 4 solution appears to match the numerical evolutions inside the past light cone, but its past light cone is also an apparent horizon. The *q* = 4 solution has three growing modes although the top one would give *γ* = 8/7 ≃ 1.14 if only the other two could be ruled out. An attempt at this seems unmotivated. At the same time, it is possible to embed the Λ = 0 solutions into a family of Λ < 0 ones, which can be constructed along the lines of Sec. [section:universalityclasses], so that Garfinkle’s solution could be the leading term in an expansion in *e*− *τ*( − Λ)− 1/2.\n### Scalar field collapse in higher dimensions\nCritical collapse of a massless scalar field in spherical symmetry in 5+1 spacetime dimensions was investigated in. Results are similar to 3+1 dimensions, with a DSS critical solution and mass scaling with *γ* ≃ 0.424. Birukou et al have developed a code for arbitrary spacetime dimension. They confirm known results in 3+1 (*γ* ≃ 0.36) and 5+1 (*γ* ≃ 0.44) dimensions, and investigate 4+1 dimensions. Without a cosmological constant they find mass scaling with *γ* ≃ 0.41 for one family of initial data and *γ* ≃ 0.52 for another. They see wiggles in the ln*M* versus ln(*p* − *p* \\* ) plot that indicate a DSS critical solution, but have not investigated the critical solution directly. With a negative cosmological constant and the second family, they find *γ* = 0.49. has made a more precise determination: *γ* = 0.4131 ± 0.0001. This was motivated by an attempt to explain this exponent using an holographic duality between the strong coupling regime of 4+1 gravity and the weak coupling regime of 3+1 QCD, which had predicted *γ* = 0.409552.\nKol relates a solution that is related to the Choptuik solution to a variant of the critical solution in the black-string black hole transition, and claims to obtain analytic estimates for *γ* and Δ. This has motivated a numerical determination of *γ* and Δ for the spherical massless scalar field in noninteger dimension up to 14.\n### Other systems obeying wave equations\nChoptuik, Hirschmann and Liebling have presented perturbative indications that the static solutions found by van Putten in the vacuum Brans-Dicke system are critical solutions. They have also performed full numerical simulations, but only starting from small deviations with respect to those solutions.\nVentrella and Choptuik have performed numerical simulations of collapse of a massless Dirac field in a special state: an incoherent sum of two independent left-handed zero-spin fields having opposite orbital angular momentum. This is prepared so that the total distribution of energy-momentum is spherically symmetric. The freedom in the system is then contained in a single complex scalar field obeying a modified linear wave equation in spherical symmetry. There are clear signs of CSS criticality in the metric variables, and the critical complex field exhibits a phase of the form *e**i**ω**τ* for a definite *ω* (the Hirschmann and Eardly ansatz for the complex scalar field critical solutions), which can be considered as a trivial form of DSS.\nGarfinkle, Mann and Vuille have found coexistence of types I and II criticality in the spherical collapse of a massive vector field (the Proca system), the scenario being almost identical to that of a massive scalar field. In the self-similar phase the collapse amplifies the longitudinal mode of the Proca field with respect to its transverse modes, which become negligible, and the critical solution is simply the gradient of the Choptuik DSS spacetime.\nSarbach and Lehner find type I critical behaviour in *q* + 3-dimensional spacetimes with *U*(1) × *S**O*(*q* + 1) symmetry in Einstein-Maxwell theory at the threshold between dispersion and formation of a black string.\n| l | c | c | c | c | c |\nMatter & Type & Collapse & Critical & Perturbations \n& & simulations & solution & of crit. soln. \nPerfect fluid *p* = *k**ρ* & II & & CSS & \nVlasov & I? & & & \nReal scalar field: &&&& \n– massless, min. coupled & II & &\nDSS & \n– massive & I & & oscillating & \n& II & & DSS & \n– conformally coupled & II & & DSS & \n– 4+1 & II & & & \n– 5+1 & II & & & \nMassive complex scalar field & I, II & & & \nMassless scalar electrodynamics & II & & DSS & \nMassive vector field & II & & DSS & \nMassless Dirac & II & & CSS & \nVacuum Brans-Dicke & I & & static & \n2-d sigma model &&&& \n– complex scalar (*κ* = 0) & II & & DSS & \n– axion-dilaton (*κ* = 1) & II & & CSS & \n– scalar-Brans-Dicke (*κ* > 0) & II & & CSS, DSS & \n– general *κ* including *κ* < 0 & II & & CSS, DSS & \n*S**U*(2) Yang-Mills & I & & static & \n& II & & DSS & \n& “III” & & colored BH & \n*S**U*(2) Yang-Mills-Higgs & (idem) & & (idem) & \n*S**U*(2) Skyrme model & I & & static & \n& II & \n& DSS & \n*S**O*(3) Mexican hat & II & & DSS & \nPerfect fluid matter\n--------------------\n### Spherical symmetry\nEvans and Coleman performed the first simulations of critical collapse with a perfect fluid with equation of state *p* = *k**ρ* (where *ρ* is the energy density and *p* the pressure) for *k* = 1/3 (radiation), and found a CSS critical solution with a mass-scaling critical exponent *γ* ≈ 0.36. Koike, Hara and Adachi constructed that critical solution and its linear perturbations from a CSS ansatz as an eigenvalue problem, computing the critical exponent to high precision. Independently, Maison constructed the regular CSS solutions and their linear perturbations for a large number of values of *k*, showing for the first time that the critical exponents were model-dependent. As Ori and Piran before, he claimed that there are no regular CSS solutions for *k* > 0.89, but Neilsen and Choptuik have found CSS critical solutions for all values of *k* right up to 1, both in collapse simulations and by making a CSS ansatz. The difficulty comes from a change in character of the sonic point, which becomes a nodal point for *k* > 0.89, rather than a focal point, making the ODE problem associated with the CSS ansatz much more difficult to solve. Harada has also found that the critical solution becomes unstable to a “kink” (discontinuous at the sonic point of the background solution) mode for *k* > 0.89, but because it is not smooth it does not seem to have any influence on the numerical simulations of collapse. On the other hand, the limit *k* → 1 leading to the stiff equation of state *p* = *ρ* is singular in that during evolution the fluid 4-velocity can become spacelike and the density *ρ* negative. The stiff fluid equations of motion are in fact equivalent to the massless-scalar field, but the critical solutions can differ, dependending on how one deals with the issue of negative density. Summarizing, it is possible to construct the Evans-Coleman CSS critical (codimension 1) solution for all values 0 < *k* < 1. This solution can be identified in the general classification of CSS perfect-fluid solutions as the unique spacetime that is analytic at the center and at the sound cone, is ingoing near the center, and outgoing everywhere else. There is even a Newtonian counterpart of the critical solution: the Hunter(a) solution.\n*p* = *k**ρ* is the only equation of state compatible with exact CSS (homothetic) solutions for perfect fluid collapse and therefore we might think that other equations of state would not display critical phenomena, at least of type II. Neilsen and Choptuik have given evidence that for the ideal gas equation of state (from now on EOS) *p* = *k**ρ*0*ε* (where *ρ*0 is the rest mass density and *ε* is the internal energy per rest mass unit) the black hole threshold also contains a CSS attractor, and that it coincides with the CSS exact critical solution of the ultrarelativistic case with the same *k*. This is interpreted a posteriori as a sign that the critical CSS solution is highly ultrarelativistic *ρ* = (1 + *ε*)*ρ*0 ≃ *ε**ρ*0 ≫ *ρ*0, and hence rest mass is irrelevant. Novak has also shown in the case *k* = 1, or even with a more general tabulated EOS, that type II critical phenomena can be found by velocity-induced perturbations of static TOV solutions. A thorough and much more precise analysis by Noble and Choptuik of the possible collapse scenarios of the stiff *k* = 1 ideal gas has confirmed this surprising result, and again the critical solution (and hence the critical exponent) is that of the ultrarelativistic limit problem. Parametrizing, as usual, the TOV solutions by the central density *ρ**c*, they find that for low-density initial stars it is not possible to form a black hole by velocity-induced collapse; for intermediate initial values of *ρ**c*, it is possible to induce type II criticality for large enough velocity perturbations; for large initial central densities they always get type I criticality, as we might have anticipated.\nNoble and Choptuik have also investigated the evolution of a perfect fluid interacting with a massless scalar field indirectly through gravity. By tuning of the amplitude of the pulse it is possible to drive a fluid star to collapse. For massive stars type I criticality is found, in which the critical solution oscillates around a member of the unstable TOV branch. For less massive stars a large scalar amplitude is required to induce collapse, and the black hole threshold is always dominated by the scalar field DSS critical solution, with the fluid evolving passively.\n### Nonspherical perturbations\nNon-spherically symmetric perturbations around the spherical critical solution for the perfect fluid can be used to study angular momentum perturbatively. All nonspherical perturbations of the perfect fluid critical solution decay for equations of state *p* = *k**ρ* with *k* in the range 1/9 < *k* < 0.49, and so the spherically symmetric critical solution is stable under small deviations from spherical symmetry. Infinitesimal angular momentum is carried by the axial parity perturbations with angular dependence *l* = 1. From these two facts one can derive the angular momentum scaling law at the black hole threshold \n \n*L* ∼ (*p* − *p*\\*)*μ*, \nwhich should be valid in the range 1/9 < *k* < 0.49. The angular momentum exponent *μ*(*k*) is related to the mass exponent *γ*(*k*) by\n \n$$\\label{muofk}\n\\mu(k) = \\Big(2+\\lambda\\_1(k)\\Big)\\gamma(k).$$ \nwhere\n \n$$\\label{lambda1}\n\\lambda\\_1(k)={9k-1\\over 3k+3}$$ \nis the growth or decay rate of the dominant *l* = 1 axial perturbation mode. In particular for the value *k* = 1/3, where *γ* ≃ 0.3558, *μ* = (5/2)*γ* ≃ 0.8895.\nOri and Piran have pointed out that there exists a CSS perfect fluid solution for 0 < *k* < 0.036 generalizing the Larson-Penston solution of Newtonian fluid collapse, and which has a naked singularity for 0 < *k* < 0.0105. Harada and Maeda have shown that this solution has no growing perturbative modes in spherical symmetry and hence a naked singularity becomes a global attractor of the evolution for the latter range of *k*. This is also true in the limit *k* = 0, which can be considered as the Newtonian limit. Their result has been confirmed with very high precision numerics by Snajdr. This seems to violate cosmic censorship, as generic spherical initial data would create a naked singularity. However, the exact result ([lambda1]) holds for any regular CSS spherical perfect fluid solution, and so all such solutions with *k* < 1/9 have at least one unstable nonspherical perturbation. Therefore the naked singularity is unstable to infinitesimal perturbations with angular momentum when one lifts the restriction to spherical symmetry.\nIn the early universe, quantum fluctuations of the metric and matter can be important, for example providing the seeds of galaxy formation. Large enough fluctuations will collapse to form primordial black holes. As large quantum fluctuations are exponentially more unlikely than small ones, *P*(*δ*) ∼ *e*− *δ*2, where *δ* is the density contrast of the fluctuation, one would expect the spectrum of primordial black holes to be sharply peaked at the minimal *δ* that leads to black hole formation, giving rise to critical phenomena. See also.\nAn approximation to primordial black hole formation is a spherically symmetric distribution of a radiation gas (*p* = *ρ*/3) with cosmological rather than asymptotically flat boundary conditions. In type II critical phenomena were found, which would imply that the mass of primordial black holes formed are much smaller than the naively expected value of the mass contained within the Hubble horizon at the time of collapse. The boundary conditions and initial data were refined in, and a minimum black hole mass of ∼ 10− 4 of the horizon mass was found, due to matter accreting onto the black hole after strong shock formation.\nCollisionless matter\n--------------------\nA cloud of collisionless particles can be described by the Vlasov equation, i.e., the Boltzmann equation without collision term. This matter model differs from field theories by having a much larger number of matter degrees of freedom: The matter content is described by a statistical distribution *f*(*x**μ*, *p**ν*) on the point particle phase space, instead of a finite number of fields *ϕ*(*x**μ*). When restricted to spherical symmetry, individual particles move tangentially as well as radially, and so individually have angular momentum, but the stress-energy tensor averages out to a spherically symmetric one, with zero total angular momentum. The distribution *f* is then a function *f*(*r*, *t*, *p**r*, *L*2) of radius, time, radial momentum and (conserved) angular momentum.\nSeveral numerical simulations of critical collapse of collisionless matter in spherical symmetry have been published to date, and remarkably no type II scaling phenomena has been discovered. Indications of type I scaling have been found, but these do not quite fit the standard picture of critical collapse. Rein et al. find that black hole formation turns on with a mass gap that is a large part of the ADM mass of the initial data, and this gap depends on the initial matter condition. No critical behavior of either type I or type II was observed. Olabarrieta and Choptuik find evidence of a metastable static solution at the black hole threshold, with type I scaling of its life time as in Eq. ([typeIscaling]). However, the critical exponent depends weakly on the family of initial data, ranging from 5.0 to 5.9, with a quoted uncertainty of 0.2. Furthermore, the matter distribution does not appear to be universal, while the metric seems to be universal up to an overall rescaling, so that there appears to be no universal critical solution. More precise computations by Stevenson and Choptuik, using finite volume HRSC methods, have confirmed the existence of static intermediate solutions and non-universal scaling with exponents ranging now from 5.27 to 11.65.\nMartín-García and Gundlach have constructed a family of CSS spherically symmetric solutions for massless particles that is generic by function counting. There are infinitely many solutions with different matter configurations but the same stress-energy tensor and spacetime metric, due to the existence of an exact symmetry: two massless particles with energy-momentum *p**μ* in the solution can be replaced by one particle with 2*p**μ*. A similar result holds for the perturbations. As the growth exponent *λ* of a perturbation mode can be determined from the metric alone, this means that there are infinitely many perturbation modes with the same *λ*. If there is one growing perturbative mode, there are infinitely many. Therefore a candidate critical solution (either static or CSS) cannot be isolated or have only one growing mode. This argument rules out the existence of both type I and type II critical phenomena (in their standard form, i.e., including universality) for massless particles in the complete system, but some partial form of criticality could still be found by restricting to sections of phase space in which that symmetry is broken, for example by prescribing a fixed form for the dependence of the distribution function *f* on angular momentum *L*, as those numerical simulations have done.\nA recent investigation of Andréasson and Rein with massive particles has confirmed again the existence of a mass gap and the existence of metastable static solutions at the black hole threshold, though there is no estimation of the scaling of their life-times. More interestingly, they show that the subcritical regime can lead to either dispersion or an oscillating steady state depending on the binding energy of the system. They also conclude, based on perturbative arguments, that there cannot be an isolated universal critical solution.\nMore numerical work is still required, but current evidence suggests that there are no type II critical phenomena, and that there is a continuum of critical solutions in type I critical phenomena and hence only limited universality.\nCriticality in singularity formation without gravitational collapse\n-------------------------------------------------------------------\nIt is well known that the Yang-Mills field does not form singularities from smooth initial conditions in 3+1 dimensions, but Bizoń and Tabor have shown singularity formation in 4+1 (the critical dimension for this system from the point of view of energy scaling arguments) and 5+1 (the first supercritical dimension). In 5+1 there is countable family *W**n* of CSS solutions with *n* unstable modes, such that *W*1 acts as a critical solution separating singularity (*W*0) formation from dispersal to infinity. In 4+1 there are no self-similar solutions and the formation of singularities seems to proceed through adiabatic shrinking of a static solution.\nCompletely parallel results can be found for wave maps, for which the critical dimension is 2+1. For the wave map from 3+1 Minkowski to the 3-sphere Bizoń has shown that there is a countable family of regular (before the Cauchy horizon) CSS solutions labeled by a nodal number *n* ≥ 0, such that each solution has *n* unstable modes. Simulations of collapse in spherical symmetry and in 3D show that *n* = 0 is a global attractor and the *n* = 1 solution is the critical solution. See also for computations of the largest perturbation-eigenvalues of *W*0 and *W*1. Again, for the wave map from 2+1 Minkowski to the 2-sphere generic singularity formation proceeds through adiabatic shrinking of a static solution.\nThese results have led to the suggestion in that criticality (in the sense of the existence of a codimension-1 solution separating evolution towards qualitatively different end states) could be a generic and robust feature of evolutionary PDE systems in supercritical dimensions, and not an effect particular of gravity.\nGarfinkle and Isenberg examine the threshold between the round endstate and pinching off in Ricci flow for a familiy of spherically symmetric geometries on *S*3. They have found intermediate approach to a special “javelin” geometry, but have not investigated whether this is universal.\nA scaling of the shape of the event horizon at the moment of merger in binary black hole mergers is noted in, but is really a kinematic effect.\nAnalytic studies and toy models\n-------------------------------\n### Exact solutions of Einstein–Klein-Gordon\nA number of authors have explored the possibility of finding critical phenomena with CSS (rather than DSS) massless scalar critical solutions. The Roberts 1-parameter family of 3+1 solutions has been analyzed along this line in. This family contains black holes whose masses (with a suitable matching to an asymptotically flat solution) scale as (*p* − 1)1/2 for $p\\gtrsim 1$, but such a special family of solutions has no direct relevance for collapse from generic data. Its generalization to other dimensions has been considered in. A fully analytic construction of all (spherical and nonspherical) linear perturbations of the Roberts solution by Frolov has shown that there is a continuum of unstable spherical modes filling a sector of the complex plane with *R**e**λ* ≥ 1, so that it cannot be a critical solution. Interestingly, all nonspherical perturbations decay.\nFrolov has also suggested that the critical (*p* = 1) Roberts solution, which has an outgoing null singularity, plus its most rapidly growing (spherical) perturbation mode would evolve into the Choptuik solution, which would inherit the oscillation in *τ* with a period 4.44 of that mode.\nA similar transition within a single 1-parameter family of solutions has been pointed out in for the Wyman solution.\nHayward and Clement and Fabbri have also proposed critical solutions with a null singularity, and have attempted to construct black hole solutions from their linear perturbations. This is probably irrelevant to critical collapse, as the critical spacetime does not have an outgoing null singularity. Rather, the singularity is naked but first appears in a point. The future light cone of that point is not a null singularity but a Cauchy horizon with finite curvature.\nOther authors have attempted analytic approximations to the Choptuik solution. Pullin has suggested describing critical collapse approximately as a perturbation of the Schwarzschild spacetime. Price and Pullin have approximated the Choptuik solution by two flat space solutions of the scalar wave equation that are matched at a “transition edge” at constant self-similarity coordinate *x*. The nonlinearity of the gravitational field comes in through the matching procedure, and its details are claimed to provide an estimate of the echoing period Δ.\n### Toy models\nBirmingham and Sen considered the formation of a black hole from the collision of two point particles of equal mass in 2+1 gravity. Peleg and Steif have investigated the collapse of a dust ring. In both cases the mass of the black holes is a known function of the parameters of the initial condition, giving a “critical exponent” 1/2, but no underlying self-similar solution is involved.\nFrolov and Frolov, Larsen and Christensen consider a stationary 2+1-dimensional Nambu-Goto membrane held fixed at infinity in a stationary 3+1-dimensional black hole background spacetime. The induced 2+1 metric on the membrane can have wormhole, black hole, or Minkowski topology. The critical solution between Minkowski and black hole topology has 2+1 CSS. The mass of the apparent horizon of induced black hole metrics scales with *γ* = 2/3, superimposed with a wiggle of period $3\\pi/\\sqrt{7}$ in ln*p*. The mass scaling is universal with respect to different background black hole metrics, as they can be approximated by Rindler space in the mass scaling limit.\nHorowitz and Hubeny and Birmingham have attempted to calculate the critical exponent in toy models from the adS-CFT correspondence.\nBurko considers the transition between existence and non-existence of a null branch of the singularity inside a spherically symmetric charged black hole with massless scalar field matter thrown in.\nWang has constructed homothetic cylindrically symmetric solutions of 3+1 Einstein-Klein-Gordon and studied their cylindrically symmetric perturbations. It is not clear how these are related to a critical surface in phase space.\nQuantum effects\n---------------\nType II critical phenomena provide a relatively natural way of producing arbitrarily high curvatures, where quantum gravity effects should become important, from generic initial data. Approaching the Planck scale from above, one would expect to be able to write down a critical solution that is the classical critical solution asymptotically at large scales, as an expansion in inverse powers of the Planck length, see Sec.[section:universalityclasses].\nBlack hole evolution in semiclassical gravity has been investigated in 1+1 dimensional models which serve as toy models for spherical symmetry (see for a review). The black hole threshold in such models has been investigated in,,,,,. In some of these models, the critical exponent is 1/2 for kinematical reasons.\nIn a 3+1-dimensional but perturbative approach is taken. The quantum effects then give an additional unstable mode with *λ* = 2. If this is larger than the positive Lyapunov exponent *λ*0, it will become the dominant perturbation for sufficiently good fine-tuning, and therefore sufficiently good fine-tuning will reveal a mass gap.\nBeyond spherical symmetry\n=========================\nNumerical studies of critical collapse should go beyond spherical symmetry (and in the first instance to axisymmetry) for three reasons:\n* Weak gravitational waves in vacuum general relativity can focus and collapse. The black hole threshold in this process shows what in critical phenomena in gravitational collapse is intrinsic to gravity rather than the matter model.\n* Black holes are characterised by charge and angular momentum as well as mass. Angular momentum is the more interesting of the two because it is again independent of matter, but cannot be studied in spherical symmetry.\n* Angular momentum resists collapse, but angular momentum in the initial data is needed to make a black hole with angular momentum. Therefore it is an interesting question to ask what happens to the dimensionless ratio *J*/*M*2 at the black hole threshold.\nIn the following we review what has been done so far.\nPerturbative approach to angular momentum\n-----------------------------------------\nWe have already mentioned that when angular momentum is small, a critical exponent for ∣*J*∣ can be derived in perturbation theory. This has been done for the perfect fluid (see Sec. [section:perfectfluid]) in first-order perturbation theory and for the massless scalar field (see Sec. [section:scalarnonspherical]), where second-order perturbation theory in the scalar field is necessary to obtain an angular momentum perturbation in the stress-energy tensor. However, neither of the predicted angular momentum scaling laws has been verified in numerical evolutions.\nFor a perfect fluid with equation of state *p* = *k**ρ* with 0 < *k* < 1/9, precisely one mode that carries angular momentum is unstable, and this mode and the known spherical mode are the only two unstable modes of the spherical critical solution. (Note by comparison that from dimensional analysis one would not expect an uncharged critical solution to have a growing perturbation mode carrying charge.) The presence of two growing modes of the critical solution is expected to give rise to interesting phenomena. Near the critical solution, the two growing modes compete. *J* and *M* of the final black hole are expected to depend on the distance to the black hole threshold and the angular momentum of the initial data through universal functions of *one* variable that are similar to “universal scaling functions” in statistical mechanics (see also the end of Sec. [section:analogy]). While they have not yet been computed, these functions can in principle be determined from time evolutions of a single 2-parameter family of initial data, and then determine *J* and *M* for all initial data near the black hole threshold and with small angular momentum. They would extend the simple power-law scalings of *J* and *M* into a region of initial data space with larger angular momentum.\nAxisymmetric vacuum gravity\n---------------------------\nAbrahams and Evans have numerically investigated black hole formation in axisymmetric vacuum gravity. They write the metric as\n \n*d**s*2 = − *α*2 *d**t*2 + *ϕ*4[*e*2*η*/3(*d**r* + *β**r* *d**t*)2 + *r*2*e*2*η*/3(*d**θ* + *β**θ* *d**t*)2 + *e*− 4*η*/3*r*2sin2*θ* *d**φ*2], \nwhere the lapse *α*, shift components *β**r* and *β**θ*, and 3-metric coefficients *ϕ* and *η* are functions of *r*, *t* and *θ*. Axisymmetry limits gravitational waves to one polarisation out of two, so that there are as many physical degrees of freedom as in a single wave equation. On the initial slice, *η* and *K**θ**r* are given as free data, and *ϕ*, *K**r**r*, and *K**φ**φ* are determined by solving the Hamiltonian constraint and the two independent components of the momentum constraint. Afterwards, *η*, *K**θ**r*, *K**r**r*, and *K**φ**φ* are evolved, and only *ϕ* is obtained by solving the Hamiltonian constraint. *α* is obtained by solving the maximal slicing condition *K**i**i* = 0 for *α*, and *β**r* and *β**θ* are obtained from the time derivatives quasi-isotropic spatial gauge conditions *g**θ**θ* = *r*2*g**r**r* and *g**r**θ* = 0.\nIn order to keep their numerical grid as small as possible, Abrahams and Evans chose their initial data to be mostly ingoing. The two free functions *η* and *K**θ**r* in the initial data were chosen to have the same functional form they would have in a linearised gravitational wave with pure *l* = 2, *m* = 0 angular dependence. This ansatz reduced the freedom in the initial data to one free function of advanced time. A specific peaked function was chosen, and only the overall amplitude was varied.\nLimited numerical resolution allowed Abrahams and Evans to find black holes with masses only down to 0.2 of the ADM mass. Even this far from criticality, they found power-law scaling of the black hole mass, with a critical exponent *γ* ≃ 0.36. The black hole mass was determined from the apparent horizon surface area, and the frequencies of the lowest quasi-normal modes of the black hole. There was tentative evidence for scale echoing in the time evolution, with Δ ≃ 0.6, with about three echos seen. Here *η* has the echoing property *η*(*e*Δ*r*, *e*Δ*t*) = *η*(*r*, *t*), and the same echoing property is expected to hold also for *α*, *ϕ*, *β**r* and *r*− 1*β**θ*. In a subsequent paper, some evidence for universality of the critical solution, echoing period and critical exponent was given in the evolution of a second family of initial data, one in which *η* = 0 at the initial time. In this family, black hole masses down to 0.06 of the ADM mass were achieved.\nIt is striking that at 14 years later, these results have not yet been independently verified. An attempt with a 3D numerical relativity code (but axisymmetric initial data), using free evolution in the BSSN formulation with maximal slicing and zero shift, to repeat the results of Abrahams and Evans was not successful. The reason could be a combination of rather lower resolution than that of Abrahams and Evans, growing constraint violations, and an inappropriate choice of gauge. (It is now known that BSSN with maximal slicing and zero shift is an ill-posed system.)\nScalar field\n------------\nIn 2003, Choptuik, Hirschmann, Liebling and Pretorius reported on numerical evolutions at the black hole threshold of an axisymmetric massless scalar field. In axisymmetry with scalar field matter there is no angular momentum and only one polarisation of gravitational waves. The slicing condition is maximal slicing and the spatial gauge, in cylindrical coordinates, is *g**z**z* = *g**ρ**ρ*, *g**z**ρ* = 0, similar to the gauge used by Abrahams and Evans. The Hamiltonian constraint is solved at every time step, and the time derivatives of the spatial gauge conditions are substituted into the momentum constraints to obtain second-order elliptic equations for the two shift components. Thus the evolution is partially constrained. Adaptive mesh-refinement was used in the numerical time evolution.\nThe initial data were either time-symmetric or approximately ingoing, with the scalar field either symmetric or antisymmetric in *z*. In the symmetric case, even strongly non-spherical data were attracted to the known spherical critical solution for the massless scalar field. Scaling with the known *γ* was observed in the Ricci scalar. However, with sufficiently good fine-tuning to the black hole threshold, the approximately spherical region that approaches the critical solution suffers an *l* = 2 (and by ansatz *m* = 0) instability and splits into two new spherical regions which again approach the critical solution. The spatial separation of the two new centres is related to the smallest length scale that developed prior to the branching. There is evidence that with increasing fine-tuning each of these centres splits again. The antisymmetric initial data cannot approach a single spherical critical solution but the solution splits initially into two approximately spherical regions where the critical solution is approached (up to an overall sign in the scalar field). The separation of these initial two centres is determined by the initial data, but there is evidence that they in turn split.\nAll this is consistent with the assumption that the spherical critical solution has, besides the known one spherical unstable mode, precisely one further *l* = 2 unstable mode. (Without the restriction to axisymmetry, if such a mode exists, it would be 5-fold degenerate with *m* = − 2, …, 2.) This contradicts the calculation of the perturbation spectrum in. Choptuik and co-workers do not state with certainty that the mode they see in numerical evolutions is a continuum mode, although they have no indication that it is a numerical artifact. The growth rate of the putative mode is measured to be *λ* ≃ 0.1 − 0.4, which should be compared with the growth rate *λ* ≃ 2.7 of the spherical mode and the relatively small decay rate of *λ* ≃ − 0.02 claimed in for the least damped mode, which is also an *l* = 2 mode. We observe that the range of *τ* in Figs. 6 and 7 of is about 10, and over this range the plot of the amplitude of the *l* = 2 perturbation against the log-scale coordinate *τ* seems equally consistent with linear growth in *τ* as with exponential growth. (In the notation of, *τ* denotes proper time and *τ*\\* the accumulation point of echos, so that the log-scale coordinate *τ* used in this review corresponds to − ln(*τ* − *τ*\\*) in the notation of.)\nAn interesting extension was made in by considering a complex scalar field giving rise to an axisymmetric spacetime with angular momentum. The stress-energy tensor of a complex scalar field Ψ is\n \n$$T\\_{ab}={1\\over2}\\left(\\Psi\\_{,a}\\bar\\Psi\\_{,b}+\\bar\\Psi\\_{,a}\\Psi\\_{,b}\n- g\\_{ab}\\bar\\Psi\\_{,c}\\Psi^{,c}\\right)$$ \nAn axisymmetric spacetime with azimuthal Killing vector *ξ* admits a conserved vector field *T**a**b**ξ**b*, so that the total angular momentum\n \n*J* = ∫Σ*T**a**b**ξ**a**n**b* *d**V* \nis independent of the Cauchy surface Σ. With the ansatz\n \nΨ(*ρ*, *z*, *t*, *φ*) = Φ(*ρ*, *z*, *t*)*e**i**m**φ*, \nin adapted coordinates where *ξ* = ∂/∂*φ* and with *m* an integer, the stress-energy tensor becomes axisymmetric and hence compatible with the Einstein equations for an axisymmetric spacetime.\nFor any Σ tangent to *ξ*, in particular a hypersurface $t={\\rm const.}$, the angular momentum density measured by a normal observer becomes\n \n$$j=T\\_{ab}\\xi^a n^b = -{im\\over 2}\\left(\\Pi\\bar\\Phi-\\bar\\Pi\\Phi\\right)\n=m A^2 n^a \\delta\\_{,a},$$ \nwhere Φ = *A**e**i**δ* with *δ* and *A* real and Π = *n**a*Φ, *a*. By comparison the energy density measured by a normal observer is\n \n$$\\rho=T\\_{ab}n^an^b=\\Pi\\bar\\Pi+D\\_a\\Phi D^a\\bar\\Phi,$$ \nwhere *D**a* is the derivative operator projected into Σ. This means that the ratio of energy density to angular momentum density can be adjusted arbitrarily in the initial data, including zero angular momentum for a Φ that is real (up to a constant phase). On the other hand, even in the absence of angular momentum a purely real Φ obeys a wave equation with an explicit *m*2/*ρ*2 centrifugal term. For the same reason, regular solutions must have Φ ∼ *ρ**m* on the axis, and there are no spherically symmetric solutions. Intuitively speaking, the centrifugal force resisting collapse appears unrelated to the angular momentum component of the stress-energy tensor in a way that differs from what one would expect in rotating fluid collapse or rotating (non-axisymmetric) vacuum collapse.\nFor all initial data in numerical evolutions, a critical solution is approached that is discretely self-similar with log-scale period Δ ≃ 0.42. (By ansatz this solution is axisymmetric but spherical symmetry is ruled out and so the critical solution cannot be the Choptuik solution.) The same critical solution is approached in particular for initial data with Π = 0 and hence no angular momentum, and initial data where $\\Pi=\\bar\\Phi$ and hence with large angular momentum. A scaling exponent of *γ* ≃ 0.11 is observed in the Ricci scalar in subcritical evolutions. The critical solution is purely real (up to an initial data-dependent constant phase) and hence has no angular momentum. Only *m* = 1 was investigated, but it is plausible that a different critical solution exists for each integer *m*.\nFar from the black hole threshold, *J* ∼ *M*2 in the final black hole, but nearer the black hole threshold, *J* ∼ *M*6, where *J* and *M* are measured on the apparent horizon when it first forms. *J*/*M*2 → 0 is compatible with a non-rotating critical solution.\nIn the absence of angular momentum, the wave equations for the real and imaginary part of Φ decouple. Assuming that the background critical solution is purely real with *δ* = 0 and *A* ∼ 1, and angular momentum is provided by a perturbation with *δ* ∼ *e**λ**τ*, one would expect *ρ* ∼ ∣∇*A*∣2 ∼ *e*2*τ* and *j* ∼ *A*∣∇*δ*∣ ∼ *e*(1 + *λ*)*τ*. Integrating over a region of size *e*− 3*τ* when the black hole forms, we find *M* ∼ *e*− *τ*\\* and *J* ∼ *e*( − 2 + *λ*)*τ*\\*. Then *J* ∼ *M*6 would imply *λ* = − 4.\nLai has studied type I critical phenomena for boson (massive complex scalar field) stars in axisymmetry, the first study of type I in axisymmetry. He finds that the subcritical endstate is a boson star with a large amplitude fundamental mode oscillation.\nNeutron star collision in axisymmetry\n-------------------------------------\nA first investigation of type I critical collapse in an astrophysically motivated scenario was carried out in. The matter is a perfect fluid with “Gamma law” equation of state *P* = (*γ* − 1)*ρ**ε*, where Γ ≃ 2 is a constant, *P* is the pressure, *ρ* the rest mass density and *ε* the internal energy per rest mass (so that *ρ*(1 + *ε*) is the total energy density). The initial data are constructed with *P* = *k**ρ*Γ for a constant *k*, which corresponds to the “cold” (constant entropy) limit of the Gamma law equation of state. The initial data correspond to two identical stars which have fallen from infinity. (The evolution starts at finite distance, with an initial velocity calculated in the first post-Newtonian approximation). The entire solution is axisymmetric with an additional reflection symmetry that maps one star to the other.\nThe parameter of the initial data that is varied is the mass of the two stars. Supercritical data form a single black hole, while subcritical data form a single star. The diagnostics given are plots against time of the lapse and the Ricci scalar at the symmetry centre of the spacetime, and of outgoing *l* = 2 gravitational waves. Collapse of the lapse and blowup of the Ricci scalar are taken as indications of black hole formation. The limited numerical evidence is compatible with type I critical phenomena, with the putative critical solution showing oscillations with the same period in the lapse and Ricci scalar. For the critical solution to be exactly time-periodic, it would have to be spherical in order to not lose energy through gravitational waves, and there is some evidence that indeed it does not radiate gravitational waves.\nOther one-parameter families of initial data were obtained by fixing the mass of the stars in the first family very near its critical value and then varying one of the following: the initial separation, the initial speed, and Γ. For all approximately the same scaling law for the survival time of the critical solution was found. However, as all these initial data are very close together, this only confirms the validity of the general perturbation theory explanation of critical phenomena, but does not provide evidence of universality.\nA key question that has not been answered is how shock heating affects the standard critical collapse scenario. A priori the existence of a universal spherical critical solution is unlikely in the presence of shock heating as there is no dynamical mechanism to arrive at a universal spherical temperature profile, in contrast to non-dissipative matter models or vacuum gravity where all equations are wave-like.\nBlack hole collisions\n---------------------\nInteresting numerical evidence for critical phenomena in the glancing collision of two black holes has been found by Pretorius. He evolved initial data for two equal mass nonrotating black holes, with a reflection symmetry through the orbital plane, parameterised by their relative boost at constant impact parameter. The threshold in initial data space is between data which merge immmediately and those which do not (although they will merge later for initial data which are bound). The critical solution is a circular orbit that loses 1 − 1.5% of total energy per orbit through radiation. On both sides of the threshold, the number of orbits scales as\n \n*n* ≃ − *γ*ln∣*p* − *p*\\*∣, \nfor *γ* ≃ 0.31 − 0.38. The simulations are currently limited by numerical accuracy to *n* < 5 and Δ*E*/*E* < 0.08 but Pretorius conjectures that the total energy loss and hence the number of orbits is limited only by the irreducible mass of the initial data, a much larger number. In particular, he speculates that for highly boosted initial data such that the total energy of the initial data is dominated by kinetic energy, almost all the energy can be converted into gravitational radiation.\nThe standard dynamical systems picture of critical collapse, with the critical solution an attractor in the threshold hypersurface, appears to be consistent with these observation. Pretorius compares his data with the unstable circular geodesics in the spacetime of the hypothetical rotating black hole that would result if merger occurred promptly. These give orbital periods, and their linear perturbations give a critical exponent, in rough agreement with the numerical values for the full black hole collision.\nPretorius does not comment on the nature of the critical solution, but because of the mass loss through gravitational radiation it cannot be strictly stationary. The mass loss would be compatible with self-similarity, with a helical homothetic vector field, but exact self-similarity would be compatible with the presence of black holes only in the infinite boost limit. Therefore the critical solution is likely to be more complicated, and can perhaps be written as an expansion with an exactly stationary or homothetic spacetime as the leading term.\nPretorius also speculates that these phenomena generalise to generic initial data with unequal masses and black hole spins which are not aligned. This seems uncertain, given the claim by Levin that the threshold of immediate merger is fractal if the spins are not aligned, and that the system is therefore chaotic. However, Levin’s analysis is based on a 2nd order post-Newtonian approximation to general relativity, in which there is no radiation reaction, while the rapid energy loss observed here may suppress chaos. Nevertheless, the phase space is much bigger when the orbit is not confined to an orbital plane, and so the critical solution observed here may not be an attractor in the full critical surface.\nThanks\n------\nWe would like to thank David Garfinkle for a critical reading of the manuscript and Louisiana State University for hospitality while this work was begun. JMM was supported by the I3P framework of CSIC and the European Social Fund, and by the Spanish MEC Project FIS2005-05736-C03-02.\n\nCritical phenomena in gravitational collapse\n============================================\nAs first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term “critical phenomena”. They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. Critical phenomena give a natural route from smooth initial data to arbitrarily large curvatures visible from infinity, and are therefore likely to be relevant for cosmic censorship, quantum gravity, astrophysics, and our general understanding of the dynamics of general relativity.\nIntroduction\n============\nOverview of the subject\n-----------------------\nTake generic initial data in general relativity, adjust any one parameter *p* of the initial data to the threshold of black hole formation, and compare the resulting spacetimes as a function of *p*. In many situations, the following *critical phenomena* are then observed:\n* Near the threshold, black holes with arbitrarily small masses can be created, and the black hole mass scales as\n \n*M* ∝ (*p* − *p*\\*)*γ*, \nwhere *p* parameterises the initial data and black holes form for *p* > *p*\\*.\n* The *critical exponent* *γ* is universal with respect to initial data, that is, independent of the particular 1-parameter family, although it depends on the type of collapsing matter.\n* In the region of large curvature before black hole formation, the spacetime approaches a self-similar solution which is also universal with respect to initial data, the *critical solution*.\nCritical phenomena were discovered by Choptuik in numerical simulations of a spherical scalar field. They have been found in numerous other numerical and analytical studies in spherical symmetry, and a few in axisymmetry; in particular critical phenomena have been seen in the collapse of axisymmetric gravitational waves in vacuum. Scaling laws similar to the one for black hole mass have been discovered for black hole charge and conjectured for black hole angular momentum.\nIt is still unclear how universal critical phenomena in collapse are with respect to matter types and beyond spherical symmetry, in particular for vacuum collapse. Progress is likely to be made over the next few years with better numerical simulations.\nCritical phenomena can usefully be described in dynamical systems terms. A critical solution is then characterised as an attracting fixed point within a surface that divides two basins of attraction, a *critical surface* in phase space. Such a fixed point can be either a stationary spacetime, or one that is scale-invariant and self-similar. The latter is relevant for the (“type II”) critical phenomena sketched above. We shall also see how the dynamical systems approach establishes a connection with critical phenomena in statistical mechanics.\nTherefore, we could define the field of critical phenomena in gravitational collapse as the study of the boundaries among the basins of attraction of different end states of self-gravitating systems, such as black hole formation or dispersion. In our view, the main physical motivation for this study is that those critical solutions which are self-similar provide a way of achieving arbitrarily large spacetime curvature outside a black hole, and in the limit a naked singularity, by fine-tuning generic initial data for generic matter to the black hole threshold. Those solutions are therefore likely to be important for quantum gravity and cosmic censorship.\nPlan of this review\n-------------------\nFaced with more material to review, we have attempted to make this update shorter and more systematic than the original 1999 version. We begin with the abstract theory in Sec. [section:theory]. Since 1999, progress on the theory side has mainly been made on the global spacetime structure of the critical solution and cosmic censorship in the spherical scalar field model. We have included this new material in an enlarged Sec. [section:scalarfield] on the spherical scalar field, although we hope it will turn out to be sufficiently generic to merit inclusion in Sec. [section:theory]. Nonspherical perturbations of the spherical scalar field are also discussed in Sec. [section:scalarfield].\nIn Sec. [section:spherical] we review the rich phenomenology that has been found in many other systems restricted to spherical symmetry. Numerical work in spherical symmetry has proliferated since 1999, but we have tried to keep this section as short as possible. There has been less progress in going beyond spherical symmetry than we anticipated in 1999, even though we continue to believe that important results await there. What is known today is summarised in Sec. [section:nonspherical].\nThe reader unfamiliar with the topic is advised to begin with either Secs. [section:universality]-[section:scaling], which give the key theory of universality, self-similarity and scaling, or Secs. [section:scalarfieldequations]-[section:threshold], which describe the classic example, the massless scalar field.\nThis review is limited to numerical and theoretical work on phenomena at the threshold of black hole formation in 3+1-dimensional general relativity. We report only briefly on work in higher and lower spacetime dimensions and non-gravity systems that may be relevant as toy models for general relativity. We exclude other work on self-similarity in general relativity and work on critical phenomena in other areas of physics.\nOther reviews on the subject are,,,,,,,,. The 2002 review by Gundlach gives more detailed explanations on some of the basic aspects of the theory.\nTheory\n======\nIn this Section we describe the basic theory underlying critical collapse of the type that forms arbitrarily small black holes (later called type II, see also Sec. [section:typeI]). We begin with the mathematical origin of its three main characteristics, which were already summarised in the introduction:\n* universality with respect to initial data;\n* scale-invariance of the critical solution;\n* black hole mass scaling.\nUniversality\n------------\nConsider GR as an infinite-dimensional continuous dynamical system. Points in the phase space are initial data sets (3-metric, extrinsic curvature and suitable matter variables, which together obey the Einstein constraints). We evolve with the Einstein equations in a suitable gauge (see Sec. [section:coordinates]). Solution curves of the dynamical system are spacetimes obeying the Einstein-matter equations, sliced by specific Cauchy surfaces of constant time *t*.\nAn isolated system in GR can end up in qualitatively different stable end states. Two possibilities are the formation of a single black hole in collapse, or complete dispersion of the mass-energy to infinity. For a massless scalar field in spherical symmetry, these are the only possible end states (see Sec. [section:scalarfield]). Any point in phase space can be classified as ending up in one or the other type of end state. The entire phase space therefore splits into two halves, separated by a “critical surface”.\nA phase space trajectory that starts on a critical surface by definition never leaves it. A critical surface is therefore a dynamical system in its own right, with one dimension fewer than the full system. If it has an attracting fixed point, such a point is called a critical point. It is an attractor of codimension one in the full system, and the critical surface is its attracting manifold. The fact that the critical solution is an attractor of codimension one is visible in its linear perturbations: it has an infinite number of decaying perturbation modes spanning the tangent plane to the critical surface, and a single growing mode not tangential to it.\nAs illustrated in Figs. [fig:dynsim] and [fig:phasespace3d], any trajectory beginning near the critical surface, but not necessarily near the critical point, moves almost parallel to the critical surface towards the critical point. Near the critical point the evolution slows down, and eventually moves away from the critical point in the direction of the growing mode. This is the origin of universality. All details of the initial data have been forgotten, except for the distance from the black hole threshold. The closer the initial phase point is to the critical surface, the more the solution curve approaches the critical point, and the longer it will remain close to it. We should stress that this phase picture is extremely simplified. Some of the problems associated with this simplification are discussed in Sec. [section:coordinates].\nSelf-similarity\n---------------\nFixed points of dynamical systems often have additional symmetries. In the case of type II critical phenomena, the critical point is a spacetime that is self-similar, or scale-invariant. These symmetries can be discrete or continuous. The critical solution of a spherically symmetric perfect fluid (see Sec. [section:perfectfluid]), has continuous self-similarity (CSS). A CSS spacetime is one that admits a homothetic vector field *ξ*, defined by :\n \n$$\\label{homothetic\\_metric}\n {\\cal L}\\_\\xi g\\_{ab} = 2 g\\_{ab}.$$ \nIn coordinates *x**μ* = (*τ*, *x**i*) adapted to the symmetry, so that\n \n$$\\label{xi\\_in\\_coordinates}\n \\xi = - {\\partial\\over\\partial\\tau},$$ \nthe metric coefficients are of the form\n \n$$\\label{CSS\\_coordinates}\n g\\_{\\mu\\nu}(\\tau, x^i) = e^{-2\\tau} \\tilde g\\_{\\mu\\nu}(x^i),$$ \nwhere the coordinate *τ* is the negative logarithm of a spacetime scale, and the remaining three coordinates *x**i* can be thought of angles around the singular spacetime point *τ* = ∞ (see Sec. [section:globalstructure]).\nThe critical solution of other systems, in particular the spherical scalar field (see Sec. [section:scalarfield]) and axisymmetric gravitational waves (see Sec. [section:brillwaves]), show discrete self-similarity (DSS). The simplest way of defining DSS is in adapted coordinates, where\n \n$$\\label{DSS\\_coordinates}\n g\\_{\\mu\\nu}(\\tau, x^i) = e^{-2\\tau} \\tilde g\\_{\\mu\\nu}(\\tau,x^i),$$ \nsuch that *g**μ**ν*(*τ*, *x**i*) is periodic in *τ* with period Δ. More formally, DSS can be defined as a discrete conformal isometry .\nUsing the gauge freedom of general relativity, the lapse and shift in the ADM formalism can be chosen (non-uniquely) so that the coordinates become adapted coordinates if and when the solution becomes self-similar (see Sec. [section:coordinates]). *τ* is then both a time coordinate (in the usual sense that surfaces of constant time are Cauchy surfaces), and the logarithm of overall scale at constant *x**i*. The minus sign in ([xiincoordinates]) and hence ([CSScoordinates]) and ([DSScoordinates]), is a convention assuming that smaller scales are in the future. The time parameter used in Fig. [fig:phasespace3d] is of this type.\nMass scaling\n------------\nLet *Z* stand for a set of scale-invariant variables of the problem, such as $\\tilde g\\_{\\mu\\nu}$ and suitably rescaled matter variables. If the dynamics is scale-invariant (this is the case exactly for example for the scalar field, and approximately for other systems, see Sec. [section:universalityclasses]), then *Z*(*x*) is an element of the phase space factored by overall scale, and *Z*(*x*, *τ*) a solution. Note that *Z*(*x*) is an initial data set for GR only up to scale. The overall scale is supplied by *τ*.\nFor simplicity, assume that the critical solution is CSS. It can then be written as *Z*(*x*, *τ*) = *Z*\\*(*x*). Its linear perturbations can depend on *τ* only exponentially. To linear order, the solution near the critical point must be of the form\n \n*Z*(*x*, *τ*) ≃ *Z*\\*(*x*) + ∑*i* = 1∞*C**i*(*p*) *e**λ**i**τ**Z**i*(*x*). \nThe perturbation amplitudes *C**i* depend on the initial data, and hence on *p*. As *Z*\\* is a critical solution, by definition there is exactly one *λ**i* with positive real part (in fact it is purely real), say *λ*0. As *τ* → ∞, all other perturbations vanish. In the following we consider this limit, and retain only the one growing perturbation.\nFrom our phase space picture, the evolution ends at the critical solution for *p* = *p*\\*, so we must have *C*0(*p*\\*) = 0. Linearising in *p* around *p*\\*, we obtain\n \n$$\\label{echoing\\_region}\n \\lim\\_{\\tau\\to\\infty} Z(x,\\tau) \\simeq Z\\_\\*(x) + {dC\\_0\\over dp} (p-p\\_\\*)\n e^{\\lambda\\_0\\tau} Z\\_0(x).$$ \nFor *p* ≠ *p*\\*, but close to it, the solution has the approximate form ([echoingregion]) over a range of *τ*. Now we extract Cauchy data at one particular *p*-dependent value of *τ* within that range, namely *τ*\\* defined by [taustardef]\n \n$${dC\\_0\\over dp} (p-p\\_\\*) e^{-\\lambda\\_0\\tau\\_\\*} \\equiv \\epsilon,$$ \nwhere *ε* is some constant ≪ 1 such that at this *τ* the linear approximation is still valid. At sufficiently large *τ*, the linear perturbation has grown so much that the linear approximation breaks down, and for *C*0 > 0 a black hole forms while for *C*0 < 0 the solution disperses. The crucial point is that we need not follow this evolution in detail, nor does the precise value of *ε* matter. It is sufficient to note that the Cauchy data at *τ* = *τ*\\* are\n \n*Z*(*x*, *τ*\\*) ≃ *Z*\\*(*x*) + *ε**Z*0(*x*). \nDue to the funnelling effect of the critical solution, the data at *τ*\\* is always the same, except for an overall scale, which is given by *e*− *τ*\\*. For example, the physical spacetime metric, with dimension $(\\hbox{length})^2$ is given by $g\\_{\\mu\\nu}=e^{-2\\tau}\n\\tilde g\\_{\\mu\\nu}$, and similar scalings hold for the matter variables according to their dimension. In particular, as *e*− *τ*\\* is the only scale in the initial data ([pdata]), the mass of the final black hole must be proportional to that scale. Therefore\n \n*M* ∝ *e*− *τ*\\* ∝ (*p* − *p*\\*)1/*λ*0, \nand, comparing with Eq. ([eq1]), we have found the critical exponent *γ* = 1/*λ*0.\nWhen the critical solution is DSS, a periodic or fine structure of small amplitude is superimposed on this basic power law :\n \nln*M* = *γ*ln(*p* − *p*\\*) + *c* + *f*(*γ*ln(*p* − *p*\\*) + *c*), \nwhere *f*(*z*) has period Δ and is universal, and only *c* depends on the initial data. As the critical solution is periodic in *τ* with period Δ the number *N* of scaling “echos” is approximated by\n \n$$N \\simeq \\Delta^{-1} \\gamma \\ln|p-p\\_\\*| + {\\rm const.}$$ \nNote that this holds for both supercritical and subcritical solutions.\nType I\n------\nIn type I critical phenomena, the same phase space picture as in Sec. [section:universality] applies, but the critical solution is now stationary or time-periodic instead of self-similar or scale-periodic. It also has a finite mass and can be thought of as a metastable star. (Type I and II were so named after first and second order phase transitions in statistical mechanics, in which the order parameter is discontinuous and continuous, respectively.) Universality in this context implies that the black hole mass near the threshold is independent of the initial data, namely a certain fraction of the mass of the stationary critical solution. The dimensionful quantity that scales is not the black hole mass, but the lifetime *t**p* of the intermediate state where the solution is approximated by the critical solution. This is clearly\n \n$$\\label{typeIscaling}\nt\\_p = - {1\\over \\lambda\\_0} \\ln|p-p\\_\\*| + {\\rm const.}$$ \nType I critical phenomena occur when a mass scale in the field equations becomes dynamically relevant. (This scale does not necessarily set the mass of the critical solution absolutely: there could be a family of critical solutions selected by the initial conditions.) Conversely, as the type II power law is scale-invariant, type II phenomena occur in situations where either the field equations do not contain a scale, or this scale is dynamically irrelevant. Many systems, such as the massive scalar field, show both type I and type II critical phenomena, in different regions of the space of initial data.\nCoordinate choices for the dynamical systems picture\n----------------------------------------------------\nThe time evolution of Cauchy data in GR can only be considered as a dynamical system if the ADM evolution equations are complemented by a prescription for the lapse and shift. To realise the phase space picture of Sec. [section:universality], the critical solution must be a fixed point or limit cycle. We have seen how coordinates adapted to the self-similarity can be constructed, but is there a prescription of the lapse and shift for *arbitrary* initial data, such that, given initial data for the critical solution, the resulting time evolution actively drives the metric to a form ([DSScoordinates]) that explicitly displays the self-similarity?\nGarfinkle and Gundlach have suggested several combinations of lapse and shift conditions that leave CSS spacetimes invariant and turn the Choptuik DSS spacetime into a limit cycle. (See for partial successes.) Among these, the combination of maximal slicing with minimal strain shift has been suggested in a different context but for related reasons. Maximal slicing requires the initial data slice to be maximal (*K**a**a* = 0), but other prescriptions, such as freezing the trace of *K* together with minimal distortion, allow for an arbitrary initial slice with arbitrary spatial coordinates.\nAll these coordinate conditions are elliptic equations that require boundary conditions, and will turn CSS spacetimes into fixed points (or DSS into limit cycles) only given correct boundary conditions. Roughly speaking, these boundary conditions require a guess of how far the slice is from the accumulation point *t* = *t*\\*, and answers to this problem only exist in spherical symmetry. Appropriate boundary conditions are also needed if the dynamical system is extended to include the lapse and shift as evolved variables, turning the elliptic equations for the lapse and shift into hyperbolic or parabolic equations.\nTurning a CSS or stationary spacetime into a fixed point of the dynamical system also requires an appropriate choice of the phase space variables *Z*(*x**i*). To capture CSS (or DSS) solutions, one needs scale-invariant variables. Essentially, these can be constructed by dimensional analysis. The coordinates *x**i* and *τ* are dimensionless, *l**e*− *τ* has dimension length, and *g**μ**ν* has dimension *l*2. The scaling for the ADM and any matter variables follows.\nEven with a prescription for the lapse and shift in place, a given spacetime does not correspond to a unique trajectory in phase space. Rather, for each initial slice through the same spacetime one obtains a different slicing of the entire spacetime. A possibility for avoiding this ambiguity would be to restrict the phase space further, for example by restricting possible data sets to maximal or constant extrinsic curvature slices.\nAnother open problem is that in order to talk about attractors and repellers on the phase space we need to define a norm on a suitable function space which includes both asymptotically flat data and data for the exact critical solution. The norm itself must favour the central region and ignore what is further out and asymptotically flat if all black holes of the same mass are to be considered as the same endstate.\nApproximate self-similarity and universality classes\n----------------------------------------------------\nThe field equations for the massless scalar field coupled to the Einstein equations are scale-free. Realistic matter models introduce length scales, and the field equations then do not allow for exactly self-similar solutions. They may however admit solutions which are CSS or DSS asymptotically on small spacetime scales as the dimensionful parameters become irrelevant, including type II critical solutions. This can be explored by a formal expansion in powers of the small parameter *L**e*− *τ*, where *L* is a parameter with dimensions length in the evolution equations. The zeroth order of the expansion is the self-similar critical solution of the system with *L* = 0. A similar ansatz can be made for the linear perturbations of the resulting background. The zeroth order of the background expansion determines Δ exactly and independently of *L*, and the zeroth order term of the linear perturbation expansion determines the critical exponent 1/*λ*0 exactly, so that there is no need in practice to calculate any higher orders in *L* to make predictions for type II critical phenomena where they occur. (With *L* ≠ 0, the basin of attraction of the type II critical solution will depend on *L*, and type I critical phenomena may also occur, see Sec. [section:typeI]. A priori, there could also be more than one type II critical solution for *L* = 0, although this has not been observed.)\nThis procedure has been carried out for the Einstein-Yang-Mills system and for massless scalar electrodynamics. Both systems have a single length scale 1/*e* (in geometric units *c* = *G* = 1), where *e* is the gauge coupling constant. All values of *e* can be said to form one universality class of field equations represented by *e* = 0. This notion of universality classes is fundamentally the same as in statistical mechanics. Other examples include modifications to the perfect fluid equation of state that do not affect the limit of high density. A simple example is that any scalar field potential *V*(*ϕ*) becomes dynamically irrelevant compared to the kinetic energy ∣∇*ϕ*∣2 in a self-similar solution, so that all scalar fields with potentials are in the universality class of the free massless scalar field. Surprisingly, even two different models like the *S**U*(2) Yang-Mills and *S**U*(2) Skyrme models in spherical symmetry are members of the same universality class .\nIf there are several scales *L*0, *L*1, *L*2 etc. present in the problem, a possible approach is to set the arbitrary scale in ([xtau]) equal to one of them, say *L*0, and define the dimensionless constants *l**i* = *L**i*/*L*0 from the others. The scope of the universality classes depends on where the *l**i* appear in the field equations. If a particular *L**i* appears in the field equations only in positive integer powers, the corresponding *l**i* appears only multiplied by *e*− *τ*, and will be irrelevant in the scaling limit. All values of this *l**i* therefore belong to the same universality class. From the example above, adding a quartic self-interaction *λ**ϕ*4 to the massive scalar field gives rise to the dimensionless number *λ*/*m*2 but its value is an irrelevant (in the language of renormalisation group theory) parameter.\nContrary to the statement in , we conjecture that massive scalar electrodynamics, for any values of *e* and *m*, is in the universality class of the massless uncharged scalar field in a region of phase space where type II critical phenomena occur. Examples of dimensionless parameters which do change the universality class are the *k* of the perfect fluid, the *κ* of the 2-dimensional sigma model or, probably, a conformal coupling of the scalar field (the numerical evidence is weak but a dependence should be expected).\nThe analogy with critical phase transitions\n-------------------------------------------\nSome basic aspects of critical phenomena in gravitational collapse, such as fine-tuning, universality, scale-invariant physics, and critical exponents for dimensionful quantities, can also be identified in critical phase transitions in statistical mechanics (see for an introductory textbook).\nFrom an abstract point of view, the objective of statistical mechanics is to derive relations between macroscopic observables *A* of the system and macroscopic external forces *f* acting on it, by considering ensembles of microscopic states of the system. The expectation values ⟨*A*⟩ can be generated as partial derivatives of the partition function\n \n$$Z(\\mu, f)=\\sum\\_{\\rm microstates} \\ e^{- H({\\rm\nmicrostate}, \\mu, f)}$$ \nHere the *μ* are parameters of the Hamiltonian such as the strength of intermolecular forces, and *f* are macroscopic quantities which are being controlled, such as the temperature or magnetic field.\nPhase transitions in thermodynamics are thresholds in the space of external forces *f* at which the macroscopic observables *A*, or one of their derivatives, change discontinuously. We consider two examples: the liquid-gas transition in a fluid, and the ferromagnetic phase transition.\nThe liquid-gas phase transition in a fluid occurs at the boiling curve *p* = *p**b*(*T*). In crossing this curve, the fluid density changes discontinuously. However, with increasing temperature, the difference between the liquid and gas density on the boiling curve decreases, and at the the critical point (*p*\\* = *p**b*(*T*\\*), *T*\\*) it vanishes as a non-integer power:\n \n$$\\rho\\_{\\rm liquid}-\\rho\\_{\\rm gas}\\sim (T\\_\\*-T)^\\gamma.$$ \nAt the critical point an otherwise clear fluid becomes opaque, due to density fluctuations appearing on all scales up to scales much larger than the underlying atomic scale, and including the wavelength of light. This indicates that the fluid near its critical point is approximately scale-invariant (for some range of scales between the size of molecules and the size of the container).\nIn a ferromagnetic material at high temperatures, the magnetisation ${\\bf m}$ of the material (alignment of atomic spins) is determined by the external magnetic field ${\\bf B}$. At low temperatures, the material shows a spontaneous magnetisation even at zero external field. In the absence of an external field this breaks rotational symmetry: the system makes a random choice of direction. With increasing temperature, the spontaneous magnetisation ${\\bf m}$ decreases and vanishes at the Curie temperature *T*\\* as\n \n$$|{\\bf m}|\\sim (T\\_\\*-T)^\\gamma.$$ \nAgain, the correlation length, or length scale of a typical fluctuation, diverges at the critical point, indicating scale-invariant physics.\nQuantities such as $|\\bf m|$ or $\\rho\\_{\\rm liquid}-\\rho\\_{\\rm gas}$ are called order parameters. In statistical mechanics, one distinguishes between first-order phase transitions, where the order parameter changes discontinuously, and second-order, or critical, ones, where it goes to zero continuously. One should think of a critical phase transition as the critical point where a line of first-order phase transitions ends as the order parameter vanishes. This is already clear in the fluid example. In the ferromagnet example, at first one seems to have only the one parameter *T* to adjust. But in the presence of a very weak external field, the spontaneous magnetisation aligns itself with the external field ${\\bf B}$, while its strength is to leading order independent of ${\\bf B}$. The function ${\\bf m}({\\bf\nB},T)$ therefore changes discontinuously at ${\\bf B}=0$. The line ${\\bf B}=0$ for *T* < *T*\\* is therefore a line of first order phase transitions between directions (if we consider one spatial dimension only, between ${\\bf m}$ up and ${\\bf m}$ down). This line ends at the critical point $({\\bf B}=0,T=T\\_\\*)$ where the order parameter $|{\\bf\nm}|$ vanishes. The critical value ${\\bf B}=0$ of ${\\bf B}$ is determined by symmetry; by contrast *p*\\* depends on microscopic properties of the material"},"data_id":{"kind":"string","value":"arxiv_0181681"}}},{"rowIdx":45398,"cells":{"content":{"kind":"string","value":"3D Galactic dust extinction mapping with multi-band photometry\n==============================================================\n[firstpage]\nWe present a method to simultaneously infer the interstellar extinction parameters *A*0 and *R*0, stellar effective temperature $T\\_{\\rm eff}$, and distance modulus *μ* in a Bayesian framework. Using multi-band photometry from SDSS and UKIDSS, we train a forward model to emulate the colour-change due to physical properties of stars and the interstellar medium for temperatures from 4 000 to $\\unit{9\\,000}{K}$ and extinctions from 0 to $\\unit{5}{mag}$. We introduce a Hertzsprung-Russel diagram prior to account for physical constraints on the distribution of stars in the temperature-absolute magnitude plane. This allows us to infer distances probabilistically. Influences of colour information, priors and model parameters are explored. Residual mean absolute errors (MAEs) on a set of objects for extinction and temperature are $\\unit{0.2}{mag}$ and $\\unit{300}{K}$, respectively, for *R*0 fixed to 3.1. For variable *R*0, we obtain MAEs of $\\unit{0.37}{mag}$, $\\unit{412.9}{K}$ and 0.74 for *A*0, $T\\_{\\rm eff}$ and *R*0, respectively. Distance moduli are accurate to approximately $\\unit{2}{mag}$. Quantifying the precisions of individual parameter estimates with 68% confidence interval of the posterior distribution, we obtain $\\unit{0.05}{mag}$, $\\unit{66}{K}$, $\\unit{2}{mag}$ and 0.07 for *A*0, $T\\_{\\rm eff}$, *μ* and *R*0, respectively, although we find that these underestimate the accuracy of the model. We produce two-dimensional maps in extinction and *R*0 that are compared to previous work. Furthermore we incorporate the inferred distance information to compute fully probabilistic distance profiles for individual lines of sight. The individual stellar AP estimates, combined with inferred 3D information will make possible many Galactic science and modelling applications. Adapting our method to work with other surveys, such as Pan-STARRS and Gaia, will allow us to probe other regions of the Galaxy.\nsurveys: SDSS, UKIDSS – stars: fundamental parameters, distances, statistics – ISM: dust, extinction – Galaxy: stellar content – methods: data analysis, statistical\nIntroduction\n============\nFrom our vantage point within the Galaxy, most astronomical observations are affected by interstellar dust, which attenuates light from distant objects as a variable function of wavelength. This is a central problem for many applications, as we typically want to figure out the intrinsic properties of the source, rather than how its emitted photons are influenced by the intervening matter distribution.\nThe leftovers from nuclear burning of stars, be they emitted through winds or, more energetically through supernovae explosions, produce dust grains that get mixed and reprocessed in interstellar space to eventually form new stars and planets. Dust typically affects electromagnetic radiation from ultraviolet to far-infrared wavelengths, absorbing these photons and re-emitting them as thermal radiation.\nEarly dust maps used the correlation between the dust column density and the distribution of neutral hydrogen. This data was supplanted by the dust maps produced by, who used full sky microwave data made available by the *IRAS* satellite and the *DIRBE* instrument on the *COBE* mission. Mapping the dust column densities via the calibrated dust temperature, the extinction maps, assuming a standard reddening law, were shown to be at least twice as accurate as those of.\nWith the advent of large photometric and spectroscopic datasets and the evolution of computing power and data analysis techniques, extinction can now be computed for millions of stars in a reasonable amount of time. Several different methods have been used to do this to date. One can for example determine the reddening vector for a star by computing the shift in colour-magnitude diagrams with respect to unreddened data according to an adopted extinction law. use the blue tip of the stellar locus to measure the colour shift of the main-sequence turnoff, which is a good proxy for extinction. In principle interstellar extinction can also be estimated using the spectral energy distribution (SED) of stellar templates directly, often performed via a *χ*2 fit to the data. The strong degeneracy between extinction and effective temperature of a star in optical and near-infrared bands affects such approaches, as they typically do not take into account the covariance of the data, conditioned on the APs.\nbase their maps on measuring the mean $(J-K\\_{\\rm s})$ colour of red clump giants and compare that to the colour of stars measured in Baade’s Window for fixed *E*(*B* − *V*). This way they achieve a high resolution map of the central bulge that is insensitive to differential extinction. use a Galaxy model for intrinsic colours and distances and then base their calculations on the near-infrared colour-excess method (*NICE*, and *NICER*, ). Despite unavoidable model dependencies, their results are quite robust to moderate changes in their assumptions.\nIn our method we use Bayes’ theorem to probabilistically characterise the degeneracy between extinction and effective temperature, taking into account covariances in colour space. We estimate the combined probability distribution (PDF) of extinction, effective temperature, distance modulus and *R*0 for each star individually from which we determine the three-dimensional spatial distribution of dust in the volume probed by our data. This approach expands on a previous study by, who used known parallaxes from *Hipparcos* and effective temperatures from spectroscopy to infer temperature and line-of-sight extinction for a smaller set of nearby stars.\nMethodologically, our approach is most similar to, who uses a hierarchical Bayesian model to determine the mean distance-extinction relationship for individual lines of sight as well as estimating further stellar parameters. Our method assumes a less complex model to map extinction and distances, thereby avoiding several potential systematic errors resulting from poor choices of priors.\nThis paper is organised as follows. Section [sec:theory] covers the basic theory of our approach and describes the Bayesian method used, introducing the individual factors that make up the posterior probability. We illustrate the practical model fitting procedure in Section [sec:model], detailing selection and features of the data. We quantify the performance of the method in Section [sec:validation], contrasting overall model accuracy with individual precisions. We apply our method to a large sample of data with unknown parameters in Section [sec:application] in order to investigate the distribution of dust in the Galaxy. We discuss our results and conclude in Section [sec:summary].\nTheory\n======\nOur aim is to determine the full probability density function (PDF) over the astrophysical parameters of a star given a set of photometric colours, $\\bmath{p}$. Interstellar extinction, $A\\_{\\rm 0}$ (see definition below), is of particular interest. Predicting this parameter requires simultaneous estimation of the effective temperature due to their degeneracy in respect to colour. By introducing a Hertzsprung-Russel diagram (HRD) prior we can also infer an estimate of the distance modulus, which in a band *X* is *μ* = *m**X* − *M**X* − *A**X* ( = 5log*d* − 5), where *d* is the distance in $\\rm parsec$, *m**X*, *M**X* and *A**X* are apparent magnitude, absolute magnitude and extinction, respectively. Using this, we build a forward model to infer the APs $\\{A\\_0, T\\_{\\rm eff}, M\\_{r}\\}$, i.e. estimating *r*-band absolute magnitude, from which we can then compute the distance. We can also estimate the parameter *R*0, which is the ratio of total to selective extinction. The derivation of an expression for the posterior PDF $P(\\mathbf\\Theta | \\bmath{p}$), where $\\mathbf\\Theta$ is the list of APs, is presented in this section. This work builds on and adapts the method introduced in.\nInterstellar extinction\n-----------------------\nFollowing, the extinction in a narrow band at wavelength *λ* can be parametrised by the extinction parameter $A\\_{\\rm 0}$ and $R\\_{\\rm 0}$ as\n \n$$A\\_{\\lambda} = A\\_{\\rm 0} \\cdot ( a\\_{\\lambda} + b\\_{\\lambda} / R\\_{\\rm 0} )\\,$$ \nwhere *a**λ* and *b**λ* are fixed polynomials. Note that we use $A\\_{\\rm 0}$ to characterise the extinction solely as a property of the interstellar medium rather than as a function of the spectral energy distribution of a star, which is strongly dependent on the intrinsic parameters of a particular star.\nWe introduce artificial reddening using the extinction curves from. Generally, the parameter $R\\_{\\rm 0}$ can be fixed, often at $R\\_{\\rm 0} = 3.1$, which in reality is only a mean value for the diffuse interstellar medium, rather than any individual line of sight in the Galaxy. As we will see in section [subsec:r] it can be advantageous to keep this parameter variable, and sample over it instead by including it in $\\mathbf\\Theta$.\nExtinction in a band *X* is then computed by integrating over the spectral energy distribution of a star, $F\\_{\\lambda}\\equiv F\\_{\\lambda}(T\\_{\\rm eff})$, with fixed effective temperature\n \n$$A\\_X = -2.5\\log\\left(\\frac{\\int\\! F\\_{\\lambda} h\\_{\\lambda,X} 10^{-0.4A\\_{\\lambda}}\\,{\\rm d}\\lambda}{\\int\\! F\\_{\\lambda} h\\_{\\lambda,X}\\,{\\rm d}\\lambda}\\right) \\,$$ \nwhere *h**λ*, *X* is the pass band function of filter *X*. We use this to model (predict) the changes in magnitude due to extinction and effective temperature in any band for which we have the pass band.\nPosterior\n---------\nWe now derive the posterior PDF. The full posterior $P(A\\_0, T\\_{\\rm eff}, M\\_{r}|\\, \\bmath{p}, m\\_{\\rm r}, H)$ quantifies the probability of the parameters, given the measured colours $\\bmath{p}$, the apparent *r*-band magnitude $m\\_{\\rm r}$, and HRD *H*. Using Bayes’ theorem we can write it as ($T=T\\_{\\rm eff}$, $M=M\\_{\\rm r}$, $m=m\\_{\\rm r}$)\n \n$$P(A\\_0, T, M | \\bmath{p}, m, H) = \\frac{P(\\bmath{p}, m | A\\_0, T, M, H)\\,P(A\\_0, T, M | H)}{P(\\bmath{p}, m | H)} \\.$$ \nThe first term is the likelihood, the second term summarises the priors on extinction, temperature and absolute magnitude. We do not need to quantify $P(\\bmath{p}, m)$ (the evidence). As it does not discriminate between APs, it is essentially a normalisation constant in the present context. The other terms are described in the following sections.\nNot all of the parameters and measurements are dependent on each other, this allows us to rewrite the posterior as\n \n$$\\begin{aligned}\n &&P(A\\_0, T, M |\\, \\bmath{p}, m, H) \\nonumber \\\\\n &&= \\frac{P(\\bmath{p} | A\\_0, T, M, H)\\,P(m | A\\_0, T, M, H)\\,P(A\\_0, T, M)}{P(\\bmath{p}, m | H)} \\nonumber \\\\\n &&= P(\\bmath{p} | A\\_0, T)\\, \\frac{P(A\\_0, T, M)}{P(\\bmath{p}, m)}\\,P(m | A\\_0, T, M, H) \\.\\end{aligned}$$ \nAll terms bar the final one are straight forward to understand, as we have only removed independent variables. The final term we further decompose in order to introduce explicit dependence on absolute magnitude *M* and build in the HRD. It formalises the influence of the HRD prior as the marginalisation over *M*:\n \n$$\\begin{aligned}\n&&P(m | A\\_0, T, M, H) \\nonumber \\\\\n&&= \\int\\limits\\_{M}\\! P(m | M, T, H)\\, P(M | T, H)\\,{\\rm d}M \\nonumber \\\\\n&&= \\int\\limits\\_{M}\\! P(m | M, T, H)\\, \\frac{P(M, T | H)}{P(T)}\\,{\\rm d}M \\,\n\\label{eq:hrdpriorint}\\end{aligned}$$ \nand does not explicitly depend on extinction. *A*0 drops out of the first term because *m* is independent of it once conditioned on *M*. The full posterior function is now complete and can be written as\n \n$$\\begin{aligned}\n&&P(A\\_0, T, M | \\bmath{p}, m, H) = \\nonumber \\\\\n&&P(\\bmath{p} | A\\_0, T)\\frac{P(A\\_0)}{P(\\bmath{p}, m)}\\! \\int\\limits\\_{M}\\! P(m | M) P(M, T | H) \\,{\\rm d}M.\n\\label{eqn:post1}\\end{aligned}$$ \nIndividually, we only impose a uniform prior on absolute magnitude within the parameter boundaries of the HRD. This is implicitly included in the integral already, hence we take it out of the equation here. The other APs have regular uniform priors.\nWe can also build another forward model which also takes $R\\_{\\rm 0}$ into account. The corresponding posterior function takes the analogous form\n \n$$\\begin{aligned}\nP(A\\_0, T, R\\_0, M | \\bmath{p}, m, H) = \\nonumber \\\\\n P(\\bmath{p} | A\\_0, T, R\\_0)\\frac{P(A\\_0, R\\_0)}{P(\\bmath{p}, m)} \\ \\int\\limits\\_{M}\\! P(m | M) P(M, T | H) \\,{\\rm d}M \\,\n\\label{eqn:post2}\\end{aligned}$$ \nwhere the parameter formally enters in the likelihood and possible priors. In our case, we impose a uniform prior on $R\\_{\\rm 0}$.\nForward model\n-------------\nThe forward model predicts the star’s colours given a set of astrophysical parameters. It is calculated by fitting a thin plate spline (a multidimensional analogue of a one-dimensional spline) as a function of $A\\_{\\rm 0}$ and $T\\_{\\rm eff}$ to each colour in $\\bmath{p}$, $\\bmath{f}(A\\_{\\rm 0}, T)=\\bmath{p}'$. Although we infer absolute magnitudes with our method, we do not include it in the forward model.\nWe can furthermore train a forward model which includes *R*0, expanding the dimensionality of the spline and characterising the variation over all three parameters, altering our function to $\\bmath{f}'(A\\_{\\rm 0}, T, R\\_{\\rm 0})$. The posterior in the previous section remains largely unchanged.\nLikelihood\n----------\nThe likelihood of the colours $\\bmath{p}$, given the forward model predictions of the colours, $\\bmath{f}$ (or $\\bmath{f}'$), and assuming Gaussian errors on the measurements, is \n$$P ( \\bmath{p} | A\\_0, T) = k \\cdot \\exp\\left(-0.5\\,(\\bmath{p}-\\bmath{f})^{T} \\bmath{C}^{-1}(\\bmath{p}-\\bmath{f})\\right) \\,$$ \n where $k = (2\\pi)^{-n/2} |\\bmath{C}|^{-1/2}$ is the normalisation factor, dependent on the dimensionality *n* of the colour vector $\\bmath{p}$. The covariance matrix $\\bmath{C}$ is determined from the errors on the measurements (magnitudes), taking into account the fact that the errors on consecutive colours are not independent. Taking just the five SDSS bands for illustration, the covariance matrix for the resulting colours $\\bmath{p} = (u-g, g-r, r-i, i-z)$ has the following form\n \n$$\\bmath{C} = \\left( \\begin{array}{cccc}\n\\sigma\\_u^2 + \\sigma\\_g^2 & -\\sigma\\_g^2 & 0 & 0 \\\\\n-\\sigma\\_g^2 & \\sigma\\_g^2 + \\sigma\\_r^2 & -\\sigma\\_r^2 & 0 \\\\\n0 & -\\sigma\\_r^2 & \\sigma\\_r^2 + \\sigma\\_i^2 & -\\sigma\\_i^2 \\\\\n0 & 0 & -\\sigma\\_i^2 & \\sigma\\_i^2 + \\sigma\\_z^2 \\end{array} \\right) \\,,$$ \nwhere *σ**X*2 is the variance in band *X*. For additional colours, the matrix expands analogously. A standard *χ*2 approach would ignore the off-diagonals, treating the colours as independent, which is obviously not correct.\nHRD Prior\n---------\nThe prior *P*(*M*, *T*∣*H*) is constructed to quantify the relative probability of finding a star in a particular region of the HRD. Due to the nature of stellar evolution, the HRD is inhomogeneously populated. Fixing one parameter constrains the other. This allows us to extract valuable information, even for a crude representation of the HRD (see section [subsec:buildfm] for details on how it is constructed).\nTo achieve this, we need to construct an equation that links the apparent and absolute magnitudes and distance. By independently estimating extinction and effective temperature from the colours und using the measured $m\\_{\\rm r}$ and the HRD, we can in principle determine the absolute magnitude and from that estimate the distance modulus. To effectively constrain the absolute magnitude from noisy measurements and the finite width of the HRD, we introduce a noise model to determine the probability distribution. We define the random variable\n \n$$\\kappa = m\\_{\\rm r} - M\\_{\\rm r} - \\Delta \\,$$ \nwhere we define Δ as the difference between the true (but unknown) apparent and estimated absolute magnitudes. For error-free measurements, *κ* is zero. By connecting apparent and absolute magnitudes in this way, Δ can be considered a proxy for the distance modulus, which we are trying to infer.\nFor the noise model $P(\\kappa | M\\_{\\rm r})$ we select a one-dimensional Gaussian in *κ*, $\\mathcal{N}(0, \\sigma\\_{m\\_{\\rm r}})$ which has an expectation value of zero and a standard deviation $\\sigma\\_{\\Delta} = \\sigma\\_{m\\_{\\rm r}}$, the uncertainty in the *r*-band magnitude measurement for the observed star. The absolute magnitude is not measured and as such has no error, Δ is sampled. For a particular measurement of $m\\_{\\rm r}$ the probability is\n \n$$P(\\kappa|M\\_{\\rm r}) = \\mathcal{N}(m\\_{\\rm r}-M\\_{\\rm r}-\\Delta,\\,\\sigma\\_{m\\_{\\rm r}}) \\.$$ \nAs $m\\_{\\rm r}$ is the only noisy quantity in *κ*, this term can also be written as $P(m\\_{\\rm r} | M\\_{\\rm r})$. This represents the first term of the integral in equations [eqn:post1] and [eqn:post2]. So, for any measure of Δ, obeying *κ* = 0, the Gaussian describes the scatter around this value. The full HRD prior probability is then calculated by integrating the Gaussian for a Δ over the probability distribution of the HRD at a given $T\\_{\\rm eff}$, as formalised in Equation [eq:hrdpriorint].\nUsing the HRD gives us a self-consistent means to estimate distances from photometric information. A stronger constraint on APs can be constructed if distance information is available. The original work by uses Hipparcos parallaxes to constrain the APs in this way.\nModel fitting\n=============\nIn this section we outline the practical implementation of the method and describe the data we use to build and validate the forward model and priors.\nData\n----\nTo perform the analysis described in Section [sec:theory], a set of photometric data is required, for which astrophysical parameters have been independently determined. We use stars selected from the *UKIRT Infrared Deep Sky Survey*, UKIDSS DR9 Plus and *Sloan Digital Sky Survey*, SDSS DR8. We use data from the UKIDSS *Large Area Survey* (LAS) which is designed to overlap with the footprint of SDSS. By using real data, we are able to use the proper photometric errors and intrinsic scatter to build the forward model, instead of depending on synthetic estimates. We perform a crossmatch to generate a catalogue of stars with photometry in nine bands. The bands *ugriz* are from SDSS and *YJHK* are covered by UKIDSS LAS. In principle we can also use (or combine) other surveys, as our method can easily be trained on further photometric data.\nTraining data for fitting forward model\n---------------------------------------\nWe need to simulate the effects of extinction to build an extended catalogue of stars with known APs. We therefore need to construct a catalogue of (ideally) zero extinction stars. To build our training data we select only stars at high Galactic latitudes, *b* > 70∘, which avoids the Galactic plane and most of the interstellar dust. Additionally these stars are required to be dwarfs (surface gravity log*g* > 4). This ensures that the catalogue stars only have minimal variations in log*g*, which improves the temperature estimates due to the tight correlation with colour for fixed extinction (as in Figure [fig:teffcolfixa0]). Furthermore, effective temperatures determined with the *SEGUE Stellar Parameter Pipeline*, SSPP must be available, with errors on these of less than $\\unit{200}{K}$.\nWe use a weighted average of several spectroscopic and photometric temperature estimates (called `ADOP` in the SSPP papers) for this work. When using instead the purely spectroscopic estimate, e.g. the `ANNRR` routine, we see no significant differences in our inferred extinction and distance estimates. The temperature estimates do differ, with a standard deviation of about $\\unit{80}{K}$. But as we are not primarily interested in inferring temperatures, we can in principle use either of the estimates, though using the spectroscopic estimator avoids circular arguments.\nWe chose the *r*-band magnitudes to be *r* < 19, which typically results in photometric errors approaching the systematic limit of approximately $\\unit{0.02}{mag}$. This also generally forces the photometry in the other bands to be complete and of good quality. To further ensure selection of sources with high quality data, we impose the following selection criteria on the SDSS flags: we select only unresolved sources (`sdss.type` = 6) with clean photometry (`sdss.clean` = 1) and general quality flags set (`sdss.’4295229440’` = 0, `sdss.mode` = 1). Similarly, we select objects with clean photometry from UKIDSS (`ukidss.*ppErrBits` < 65 536, one for each band) that are classified as stellar objects (`ukidss.mergedClass` = − 1).\nThe data (SDSS and UKIDSS) are queried and cross-matched using the *WFCAM Science Archive*, requiring complete photometry in all bands. In total, we obtain roughly 3 200 stars that fulfil all the above criteria. Compared to the hundreds of thousands of stars analysed in SEGUE, we retrieve comparatively few. Aside from completeness and crossmatching reasons, this is mainly influenced by the magnitude, surface gravity and temperature limits we set. In spite of this, this does not introduce systematic biases. Our selection is representative of the full SEGUE dataset (see below). The photometry is then corrected for the small estimated extinction from SFD, where the assumption holds that at high Galactic latitudes the stars are behind all the layers of dust.\nOf the selected data, a randomly sampled set of one thousand stars is chosen, covering a temperature range of approximately $\\unit{4000 - 9000}{K}$. A histogram of the temperature distribution for the full training set is shown in Figure [fig:teffhist], showing a peak around $\\unit{5600}{K}$. Only $\\unit{9}{\\%}$ of the stars have effective temperatures above $\\unit{6500}{K}$. This is representative for the whole SEGUE dataset.\n![Distribution of effective temperatures of approximately 3\\,200 crossmatched stars. Temperatures are estimated by SSPP with mean uncertainties of \\unit{58.5}{K}. Of these stars, one thousand are randomly chosen to build the extended dataset.](images/teffhist \"fig:\") [fig:teffhist]\nWhen later applying the method to data with unknown stellar parameters, we still create a selection with the quality flags used above, though we naturally impose no requirement on the existence of temperature estimates or the position in the Galaxy (low extinction).\nAdding artificial extinction\n----------------------------\nUsing the filter response functions for all filters and an extinction law, the change in magnitude for a given extinction *A*0 in each band is computed for a range synthetic SEDs. Owing to the fact that we are using broad band filters, the extinction in a band varies smoothly with temperature and we do not need to do this for every unique temperature present in our data sample. Instead we use eleven *PHOENIX* model SEDs evenly spaced in the temperature range from $\\unit{4000}{K}$ to $\\unit{9000}{K}$. This covers the temperature range in our catalogue. For each band we fit a set of one-dimensional quadratic functions in $T\\_{\\rm eff}$ for fixed $A\\_{\\rm 0}$, covering the whole temperature range noted above and extinction in steps of $\\unit{0.25}{mag}$ from 0 to 5 magnitudes.\nKnowing how the magnitudes vary with *A*0 and $T\\_{\\rm eff}$, we expand our initial dataset by adding artificial extinction in 21 discrete steps over the range noted above for each of the 1 000 stars. The photometric errors are adapted to take into account the change in magnitudes by following the average magnitude-error relation in the data. Our expanded dataset (*training set*) now comprises 21 000 stars with known artificial extinction and effective temperatures.\nBuilding the forward model\n--------------------------\nIn Figure [fig:teffcolfixa0] we show the effect of temperature and extinction on four different colours. We do not convert the magnitudes in both surveys to a common system (UKIDSS uses Vega magnitudes, for example), because the colours are merely affected by a constant offset with respect to that of the other survey, and we are only interested in changes in colours. As long as the training data and the final datasets are handled consistently, we need not take into account this offset.\n![Effects of effective temperature and extinction on four selected colours. Purple points show a subset of the training data with no extinction as a function of effective temperature. Green dots show the effects for A_0 = \\unit{2}{mag}. Each sample consists of 1\\,000 stars.](images/teffcolfixa0 \"fig:\") [fig:teffcolfixa0]\nFrom the training set we select a random subset of 4 000 stars to create the forward model itself, with the aim of being able to predict colours given temperature and extinction (see section [subsec:likelihood]). For each of the eight reduced colours we fit a two-dimensional thin-plate spline to both APs. A good fit for all colours is achieved when giving the splines 20 degrees of freedom.\nIn Figure [fig:fmtpsresiduals] we show the direct performance of the splines predicting colours given some input APs. For four example colours we plot the residuals of colour (*p**X*, *t**r**u**e* − *p**X*, *s**p**l**i**n**e*) for the ranges of temperature and extinction present in the model as contours. This is not the final result of the full parameter inference, instead it illustrates the intrinsic inaccuracies of the model which derive from the scatter in the real data. We do not see any systematic variation, and measure a typical scatter in the order of $\\unit{0.03-0.06}{mag}$ for all colours, which is roughly $\\sqrt{2}$ larger than for the magnitudes in a band. The forward model captures the intrinsic variance, as this is the same order of magnitude or within a small factor of the reported uncertainties of the survey data magnitudes. In Section [subsec:accapest] we look in detail at how assuming larger photometric errors affects the AP estimation.\n![Residuals in the forward model. This shows the difference between the true colours and the colours predicted from the forward model thin plate spline as functions of true effective temperature and extinction, for four colours. The contour levels shown incorporate 68\\%, 90\\% and 99\\% of the data points. In the top right of each panel the corresponding colour is named, with the x-axis depicting the respective parameter ranges. All panels are scaled to show the same range on the y-axis. Typical scatter is in the order of \\unit{0.03-0.06}{mag} for all colours.](images/fm_tps_residuals \"fig:\") [fig:fmtpsresiduals]\nWe characterise extinction using *A*0 to indicate that this is a parameter that solely depends on the physics of the interstellar medium, rather than on the type of star. When presenting results and maps later on we will use extinction in SDSS *r*-band to allow more convenient comparisons to other work. Note that the conversion from *A*0 to $A\\_{\\rm r}$ has a small dependence on temperature. For *R*0 = 3.1 they roughly convert as $\\left<A\\_{\\rm r}/A\\_0\\right> = 0.837$. The standard deviation from this value over the parameter range $T\\_{\\rm eff} = \\unit{4000-9000}{K}$ and $A\\_0 = \\unit{0-5}{mag}$ is only 0.005. The model values for extinction in other bands, e.g. the relative extinctions $A\\_X/A\\_{\\rm r}$ are listed in Table [tab:modelextval] for three different values of *R*0. Values reported are averages over the above range. In particular, changes occur for the short wavelength optical bands. When varying *R*0, we see that the relative extinction in the other, redder, bands changes only slightly. In Figure [fig:extr0teffva] these changes are illustrated for all the bands used and for three different effective temperatures.\n[tab:modelextval]\nccccccccc *R*0 & *u* & *g* & *i* & *z* \n2.1 & 2.393 & 1.698 & 0.707 & 0.542 \n3.1 & 1.848 & 1.445 & 0.744 & 0.555 \n4.1 & 1.610 & 1.329 & 0.761 & 0.561 \n*R*0 & *Y* & *J* & *H* & *K* \n2.1 & 0.439 & 0.337 & 0.230 & 0.147 \n3.1 & 0.427 & 0.308 & 0.204 & 0.133 \n4.1 & 0.422 & 0.296 & 0.193 & 0.126 \n![Model values for extinction relative to r-band for three temperatures over a range of R_0 and the photometric bands used in this work. Small temperature dependences can be seen for the optical bands u and g at low R_0.](images/r0arteff \"fig:\") [fig:extr0teffva]\nFor the APs used to build the forward model we implicitly implement priors that are uniform over the APs’ training ranges, and have zero probability outside them. We will demonstrate the use of a prior on extinction based on SFD estimates later in Section [subsec:furtherpriors].\nHRD prior\n---------\nThe HRD prior is constructed using stellar isochrones from. The chosen stellar population comprises 100 000 stars drawn from a Salpeter initial mass function with masses ranging from $\\unit{0.2-107}{M\\_{\\sun}}$ (though we only use stars with $T\\_{\\rm eff} \\leq \\unit{9\\,000}{K}$ and the vast majority of stars have masses below $\\unit{1.6}{M\\_{\\sun}}$). Assuming a constant star formation rate over the age of the Universe ($\\unit{13.7}{Gyr}$), all stars evolve independently with solar metallicity, i.e. there is no chemical enrichment. By using the forward model, however, we are implicitly assuming that the metallicity of stars we encounter have the same mean metallicity of the SSPP stars, which are used to fit the model. This mismatch between the metallicities of the forward model and the HRD does not strongly affect the estimation of *A*0 and $T\\_{\\rm eff}$ directly, though it does limit the ability of getting good distance estimates. As we are not estimating metallicity, this mismatch will naturally occur for most stars, independently of the choice of HRD metallicity. An example of this limitation is given in Section [subsec:demo].\nThe resulting temperatures and absolute magnitudes then are used to place the stars in the HRD plane. To use the HRD as a prior, we need to convert the data representation to a probabilistic description of the distribution. We achieve this by applying a two-dimensional binned kernel density estimate with a bandwidth of $\\unit{12.5}{K}$ and $\\unit{0.0625}{mag}$ in effective temperature and *r*-band absolute magnitude, respectively. The resulting grid has the pixel dimensions of 6002, we limit the temperature range to $\\unit{3010 - 9000}{K}$ as to extend slightly beyond the limits of the forward model. This allows for the main sequence to be fully represented down to low temperatures without an artificial cutoff. The resulting magnitude range is $r \\in \\unit{(-4,12)}{mag}$. Before normalisation, a small, but non-zero, offset is added to each point to account for the regions which nominally have zero probability, but in reality may not be completely empty. These regions include white dwarfs in the lower left part of the figure or (post) AGB stars higher up the branch to the right. Statistically, these regions will be very thinly populated and we don’t expect to find many (if any) of these stars in our samples, so we don’t model these in a more sophisticated manner. A representation of the final distribution is presented in Figure [fig:hrd].\n![Density representation of the HRD used in this analysis. The integrated probability of the HRD is normalised to one, red is high number density and blue is low. The colour scale shows base 10 logarithm of the density. White areas denote regions of the parameter space with initially zero probability. A small offset is added to each point before normalisation to avoid this in the actual computation. In this case the offset is approximately 10^{-3} times the maximum density.](images/hrd \"fig:\") [fig:hrd]\nSampling and computation\n------------------------\nThe forward model, priors and data are combined to compute the posterior function, as detailed in Section [sec:posterior]. As we are interested in inferring multiple APs, we use a standard Metropolis-Hastings Markov-Chain Monte-Carlo (MCMC) routine to efficiently sample the parameter space and to compute the posterior probability distribution. The parameter space is explored in logarithmic units of the respective quantity, forcing the estimates to remain positive and physical. The sampling matrix is diagonal with variances of $\\unit{0.1}{dex}$ in each variable (AP). We assume no correlation between the variables. Convergence is sufficient with a chain length and burn in of 10 000 steps each. This does not need to be adjusted if we also infer *R*0.\nTo compute the likelihoods we can either use the actual spline function or build a lookup table. Considering the numerical speed-up we achieve by tabulating results in advance, we generally use this approach. For this, we build a grid in extinction and effective temperature (and later in $R\\_{\\rm 0}$ too) and calculate the predicted colours. We use the same parameter ranges as for training (avoiding extrapolation into regions not covered by the data), creating a grid with *A*0 from $\\unit{0-5}{mag}$ in steps of $\\unit{0.001}{mag}$ and $T\\_{\\rm eff}$ from $\\unit{3000 - 9000}{K}$ in steps of $\\unit{10}{K}$ (effectively the same as for the HRD). For *R*0 we use a coarser stepsize of $\\unit{0.2}{}$ in the range from 2.1 − 5.9. The stepsizes are chosen as to be significantly smaller than the mean absolute error (MAE) of the residuals when using the full spline function (see Section [subsec:effectcolours].)\nWhen assessing the performance in the APs and computing further quantities, we only use those stars whose inferred parameters are within the boundaries of the lookup table (i.e. within our training space). As indicated in section [subsec:training], at lower temperatures and smaller wavelengths the linear relation between colour and temperature for fixed extinction no longer holds, therefore extrapolation is not advisable. This avoids skewing the results due to boundary effects, like MCMC chains sticking to the edge of the grid in one or more parameters.\nWe apply this filter after parameter inference as we cannot rule out including stars that have true APs outside of our training range (we do not apply any colour-cuts to the data selection). If we were to instead force a prior over the same ranges during sampling we would artificially force those stars to have incorrect parameters.\nTypically this filter throws out roughly $\\unit{5}{\\%}$ of stars when estimating extinction, effective temperature and distance modulus. When including the *R*0 parameter, this increases to close to $\\unit{30}{\\%}$, due to the additional variation in the model.\nValidation\n==========\nIn this section we first demonstrate the viability of the model and then analyse the precision and accuracy of the method, as well as the colours. The following analysis is performed on a different set of 4 000 stars (*validation set*), of which none where used to train the model.\nSimulated photometry\n--------------------\nIn this first step we aim to show the expected performance of the method and illustrate its limitations with respect to metallicity (see Figure [fig:demo]). For this purpose we set up two extinction clouds at $\\unit{500}{pc}$ ($\\mu\\approx\\unit{8.5}{mag}$) and $\\unit{1500}{pc}$ ($\\mu\\approx\\unit{10.9}{mag}$) with extensions of $\\unit{100}{pc}$. This true extinction distribution with *μ* is shown as the solid black line. We then place 500 stars along the line of sight, generating effective temperatures and absolute magnitudes from the HRD prior (solar metallicity). According to their effective temperatures, extinctions and distances, we simulate appropriate photometry and photometric errors, which are characteristic of the SDSS/UKIDSS sample we use. Finally we use our method to infer these parameters. These are plotted as points in the figure.\nWe infer the APs for two cases. In (a) we use the same solar metallicity HRD prior (*Z* = 0.019) as for selecting the stars (red dots). In this case, we can see that our method recovers the distances and extinctions very well, with points closely tracing the extinction profile. The few points that lie furthest from the true profile typically have bad temperature estimates, resulting in erroneous distance moduli. In (b) we use an HRD with a tenth of that metallicity (*Z*ʹ = 0.0019, blue dots) to infer parameters. Although the individual estimates for extinction and temperature are compatible, the mismatched stars are systematically shifted by Δ*μ* = − 1 to closer distances. Analogously, the opposite shift occurs when reversing the mismatch. Therefore the accuracy of distance estimates is strongly limited by the unknown metallicity of each individual star.\n![Inferred distance moduli \\mu and extinction A_0 for 500 stars. Red dots indicate results for which matching HRD metallicity (Z=0.019) is used for sampling and inference. Blue dots show results for mismatched HRD (Z' = 0.0019).](images/demo \"fig:\") [fig:demo]\nIt is difficult to estimate metallicity using only broad band photometry, as used here, when neither temperature nor extinction are fixed. As shown in for simulated Gaia data, given spectroscopic information it is possible to estimate metallicity simultaneously, resulting in more accurate distance estimates.\nEffects of Colours and NIR data\n-------------------------------\nConsidering that we are using photometry from two surveys covering optical and near-infrared bands, it is instructive to compare how inclusion (or exclusion) of certain bands affects the parameter estimation. To examine this we train the forward model using (a) all nine bands, resulting in the eight colours *u-g*, *g-r*, *r-i*, *i-z*, *z-Y*, *Y-J*, *J-H* and *H-K*, (b) using only the five SDSS bands and thus only four colours and (c) using all but the SDSS *u* band magnitudes. Generally, one would expect the use of more colours (more information) to yield better results, particularly when taking into account that NIR wavelengths are less affected by dust attenuation than shorter ones.\nAs expected, the performance of the method is significantly better when using the NIR data (using only NIR data is also much worse). Table [tab:residuals] summarises the differences between the two cases. We characterise the performance with three key statistics. The first is the bias, which is purely the mean value of the difference between predicted and true parameter for all stars. The second is the mean absolute error (MAE), which similarly is the mean of the absolute difference and the third is the root mean square (RMS). For temperatures below $\\unit{6500}{K}$ the performance is generally better than for higher temperatures. This results from the combination of having fewer stars in the higher temperature range and the scatter in the data being larger there (see Figures [fig:teffhist] and [fig:teffcolfixa0]).\n[tab:residuals]\nccccccc & & \n& (a) & (b) & (c) & (a) & (b) & (c) \nbias & − 0.08 & − 0.41 & − 0.29 & − 67 & − 295 & − 269 \nMAE & 0.23 & 0.59 & 0.41 & 300 & 628 & 440 \nRMS & 0.45 & 1.05 & 0.68 & 586 & 1028 & 712 \nWe note that the bias is negative in both cases and parameters, though much closer to zero in case (a). Intrinsic scatter in the data also precludes higher residual accuracy. In Figure [fig:residuals] we plot the results for the validation set of stars. Towards negative values in both parameters we see two distinct tails. Stars in the tails tend to have hotter true temperatures and larger intrinsic scatter. Only a few percent of stars are effected by such a large error in the parameter estimation. Without these tails, the performance would improve only minimally in MAE and RMS.\nWhen excluding only the *u* band, case (c), we find that the performance is still better than case (b) although significantly worse than (a). The bias, MAE and RMS for *A*0 are $\\unit{-0.29}{mag}$, $\\unit{0.41}{mag}$ and $\\unit{0.68}{mag}$, respectively. For $T\\_{\\rm eff}$ these are $\\unit{-269}{K}$, $\\unit{440}{K}$ and $\\unit{712}{K}$.\nIn temperature and extinction we see no significant systematic effect due to the metallicity of the HRD. As indicated in the previous section, this mainly affects the distance estimator.\n![Residuals in effective temperature and extinction for case (a) using all eight colours. Plotted are the 68\\%, 90\\% and 99\\% density contours of the distribution of points in the plot. The differences are computed as \\Delta\\Theta = \\Theta_{\\rm estimated} - \\Theta_{\\rm true} for A_0 and T_{\\rm eff}.](images/residuals \"fig:\") [fig:residuals]\nFigure [fig:residuals2] shows the residuals for both parameters as a function of the true parameters. We see no systematic behaviour, though we see a general bias towards negative values, which is evident in the values reported in Table [tab:residuals].\n![Residuals in effective temperature and extinction as functions of the true values in both parameters, as applied to the validation set of data. All eight colours are used. The true variation is shown for the same data as in Figure [fig:residuals]. A few dozen points are cut off by the selection of the axis range.](images/residuals2 \"fig:\") [fig:residuals2]\nAs we do not have any true distances to compare the inferred distances to, we can not reference any results for distance modulus here. We must rely on the other parameters to be accurate in order to believe the distance results we obtain. We can however roughly estimate distances using the true extinctions and temperatures and finding the corresponding $\\Delta = m\\_{\\rm r} - M\\_{\\rm r}$ via the HRD. Doing this we compute a bias of $\\unit{-0.23}{mag}$, MAE of $\\unit{2.29}{mag}$ and RMS of $\\unit{3.31}{mag}$ when using all bands. Approximately $\\unit{15}{\\%}$ of the estimates are better than $\\unit{0.2}{mag}$. In this case, the distance is poorly constrained by the combination of photometric data and our broad HRD prior, when $T\\_{\\rm eff}$ and *A*0 also have to be estimated from the data. The bias we encounter is primarily due to the mismatch of metallicities of the HRD and training data.\nPrecision of AP estimation\n--------------------------\nBeyond AP estimation accuracy (of the method), we are also interested in the precision, i.e. what are our parameter uncertainties for any given star. We quantify this using the width of the 68% confidence interval (CI68) of the posterior PDF (which is equivalent to the 1*σ* variation), using a Gaussian kernel density estimation with the mean as the central point. The distribution is not generally symmetric, so we obtain marginally different values for the left and right confidence bounds. In Figure [fig:accuracy] we show the histograms of the average CI68 bounds for both extinction and effective temperature, as well as Δ. Mean values of the confidence intervals are $\\left<\\text{CI}\\_{68}(A\\_0)\\right>=\\unit{0.04}{mag}$, $\\left<\\text{CI}\\_{68}(T\\_{\\rm eff})\\right>=\\unit{58}{K}$ and $\\left<\\text{CI}\\_{68}(\\Delta)\\right>=\\unit{2.0}{mag}$. We can therefore be confident that our parameter estimates are generally precise in a statistical sense.\n![Histograms of 68\\% confidence intervals for extinction and temperature using photometric errors in the covariance matrix. The purple vertical lines indicate the mean values in each case. For A_0 this corresponds to \\unit{0.04}{mag}, for T_{\\rm eff} it is \\unit{58}{K} and for \\Delta it is \\unit{2.0}{mag}.](images/prec \"fig:\") [fig:accuracy]\nThese values of precision are significantly smaller than the accuracy of the model (Section [subsec:effectcolours]), which could be taken to mean that the parameter inference is overconfident compared to the information content of the data. The intrinsic scatter of the spline fit is of the same order of magnitude as the photometric errors of the data (see Section [subsec:buildfm] and Figure [fig:fmtpsresiduals]), therefore properly representing the variation and uncertainties of the model.\nAverage photometric errors in all nine bands vary from 0.01 to $\\unit{0.03}{mag}$, the mean temperature uncertainty in the training data is $\\unit{60}{K}$, uncertainties in extinction can only arise from an erroneous correction of the SFD estimates, unless the reddening law is wrong. These, though, are typically only $A\\_{\\rm SFD, r}=\\unit{0.03}{mag}$. Our precision therefore is in the order of the parameter errors. If we fix $\\sigma\\_{p\\_X} = \\unit{0.1}{mag}$, which is roughly five times larger than the photometric errors, we obtain precisions of $\\left<\\text{CI}\\_{68}(A\\_0)\\right>=\\unit{0.36}{mag}$ and $\\left<\\text{CI}\\_{68}(T\\_{\\rm eff})\\right>=\\unit{402}{K}$. These values are in the order of $\\unit{25}{\\%}$ larger than the MAE reported in Table [tab:residuals]. The corresponding biases, mean absolute errors and root mean squares only change by a few percent in respect to the tabulated results. Therefore, a more accurate account for model limitations could imply using significantly larger errors on photometry, although on average the statistical performance remains compatible in either case.\nTo illustrate the output from the Monte Carlo sampling, Figure [fig:contour1] shows the contours and one-dimensional PDFs of the three parameters extinction, effective temperature and Δ for a single example star. In the extinction-temperature plane (bottom left panel) we clearly see the degeneracy between the two parameters. In the one-dimensional PDF for Δ (middle panel) we see the typical bimodal distribution, which is the result of sampling over the parts in the HRD that represent the main sequence and the giant branch. One peak is usually higher, which is why we use the mode to quantify this parameter rather than the mean. In most cases the higher peak relates to the main sequence (though not in this example), which is a reasonable justification to use it, as we have trained our models using photometry from dwarf stars.\nIt must be noted that the confidence intervals for Δ are large, though the inner 68% are more tightly constrained. We can reduce this spread by either running a longer Monte-Carlo chain, which adds more weight to the non-background regions in the HRD, or by directly decreasing the background level by a few orders of magnitude when computing it in the first place. Both options result in nearly unchanged parameter estimates but improved precision in Δ. The second option is clearly more appealing, as it does not increase the computation time.\n![Contours of the posterior PDF for a star with true APs of A_{\\rm 0, true}=\\unit{4}{mag} and T_{\\rm eff, true} = \\unit{5392}{K}. In each panel the red cross or line shows the position of the estimated parameter value (A_0 = \\unit{4.06}{mag}, T_{\\rm eff} = \\unit{5446}{K}, \\Delta = \\unit{15.9}{mag}), whereas the purple plus or line is the true (constructed) value. Contours show 68\\%, 90\\% and 99\\% confidence intervals. The smaller side panels show the marginalised one-dimensional PDFs for the single parameters, as aligned with the respective axes. The peak densities are scaled to 1 in each case. The indicated true value of distance modulus is estimated using the true temperature and the HRD from section [subsec:hrdprior]. The best estimate values for temperature and extinction are computed using the mean of the distribution, whereas the mode is used for distance modulus.](images/contour1 \"fig:\") [fig:contour1]\nEffects of $\\bmath{R\\_0}$\n-------------------------\nThe results shown so far were for a fixed *R*0 = 3.1. In reality, this value is only the mean value for the diffuse interstellar medium, whereas denser regions will typically be characterised by a higher value of *R*0, arising from a change in size distribution and composition of the dust grains responsible for extinction in optical bands. This does not, however, equate high line-of-sight extinction with large *R*0; it simply describes properties of the local dust.\nWe train a new forward model in which we simulate the change of colours due to extinction and *R*0 for stars with known temperatures. We expand our parameter grid and again use 4 000 stars to train and validate the model. The general setup remains the same as before. We use *R*0 in the range from 2.1 to 5.9. The model performance we achieve like this is presented in Table [tab:residualsr].\n[tab:residualsr]\n| | *A*0/mag | $T\\_{\\rm eff}$/K | *R*0 |\n| --- | --- | --- | --- |\n| bias | − 0.22 | − 206 | 0.32 |\n| MAE | 0.39 | 438 | 0.73 |\n| RMS | 0.69 | 771 | 0.95 |\nComparing case (a) in Table [tab:residuals] with Table [tab:residualsr], we see that modelling *R*0 slightly deteriorates the performance in the other two parameters, the standard deviation and MAE in the residuals are increased by roughly 50% for temperature and extinction. This comes as no surprise, as we are now extracting more parameters, but with the same initial data, noting that the observed change in colours are quite small for larger differences in this parameter. In general we are able to roughly infer *R*0, though the accuracy is not sufficient to confidently differentiate between, say, 3.1 and 3.8. We therefore build the lookup table using somewhat larger steps of 0.2 in *R*0 for the range noted above, and the other parameters remaining the same.\nIn Figure [fig:accuracyr] we present the precision of the predicted parameters as histograms, where the purple lines again depict the mean values. We obtain $\\left<\\text{CI}\\_{68}(A\\_0)\\right>=\\unit{0.05}{mag}$, $\\left<\\text{CI}\\_{68}(T\\_{\\rm eff})\\right>=\\unit{66}{K}$, $\\left<\\text{CI}\\_{68}(R\\_0)\\right>=0.07$ and $\\left<\\text{CI}\\_{68}(\\Delta)\\right>=\\unit{1.6}{mag}$. The first two values are minimally larger than in the previous case, though it is to be expected considering that we have added an additional degree of variation to the model. Again, the confidence intervals are significantly smaller than the model accuracy.\n![Histogram of 68\\% confidence intervals of extinction, temperature and extinction factor, which are estimates of the uncertainty on each parameter we infer. The purple vertical lines indicate the mean values in each case. For A_0 this corresponds to \\unit{0.05}{mag}, for T_{\\rm eff} it is \\unit{66}{K}, for R_{0} it is 0.07 and for \\Delta it is \\unit{1.6}{mag}.](images/precr \"fig:\") [fig:accuracyr]\nApplication to multiple fields\n==============================\nWe have so far summarised the accuracy and precision of the model using data with known stellar parameters. We now apply the model to estimate APs for large areas of the sky where individual estimates are not available. Querying the *WFCAM Science Archive* with the flags detailed in section [subsec:training], but imposing no constraints on SSPP information, we obtain a total of 4 191 659 unique stellar objects, of which 3 055 954 and 1 135 705 are north and south of the Galactic equator, respectively. We also pick out a section of the southern sky with *l* > 180∘, which we use to illustrate the variations of the model, namely using only photometry from SDSS, using all photometry, and also including *R*0 in the parameter estimation.\nWe first show the full projected maps of the northern and southern part of the surveys in Figures [fig:fullsky]. Here we use a Mollweide projection in Galactic longitude and latitude to obtain equal area pixels, using a resolution of $\\unit{11.48}{arcmin/pixel}$ in both coordinates. This corresponds to an average of 29 stars per pixel, depending on the region observed, with extremes of 3 and 250 stars per pixel.\nThe interstellar medium towards the Galactic poles typically hosts diffuse gas and lines of sight with low extinction. Nevertheless, the area jointly probed by the two surveys does cover some regions with higher average extinction. Generally we can pick out larger structures that are also visible in SFD maps of the same resolution. Towards the right of the southern part of the sky one can start to make out parts of a larger structure, with extinction estimates in the range of 1 to $\\unit{1.5}{mag}$ in the *r*-band.\n![A Mollweide equal area projection of the mean extinction in the full crossmatched portion of the sky, centred on the Galactic anti-centre. The stars are binned using a resolution of \\unit{11.48}{arcmin/pixel} in l and b. The non-cartesian grid in latitude and longitude is overplotted. The colour scale shows the mean extinction A_{\\rm r} in any given pixel. White areas are not jointly covered in the surveys.](images/northsouth \"fig:\") [fig:fullsky]\nTo more quantitatively assess the credibility of the presented map, we compare the computed extinction values with those from SFD. For this we project the data in an identical manner and calculate the difference between the SFD extinction values and our own estimates. This is plotted in Figures [fig:southsfddiff] for the southern part of the sky. We refrain from showing the equivalent image for the northern part, as it qualitively shows the same behaviour. In Figure [fig:sfddiffhist] we show the histograms for the differences between SFD extinction estimates and our own for both the southern and northern part of the sky, based on the binned data. In the south the mean difference and standard deviation are $\\unit{-0.02}{mag}$ and $\\unit{0.13}{mag}$, respectively. In the north we obtain $\\unit{-0.06}{mag}$ and $\\unit{0.11}{mag}$.\n![Difference between SFD extinction estimates and our own for stars in the southern part of the sky, using a Mollweide projection with \\unit{11.48}{arcmin/pixel} resolution. The colour coding in the range of \\Delta A_{\\rm r} = -1 to \\unit{1}{mag} shows regions where SFD predicts a higher extinction in yellow/red (positive), and where our model has a higher estimate in blue (negative).](images/south_sfd_diff \"fig:\") [fig:southsfddiff]\n![Histograms of differences between SFD extinction estimates and our own for stars (\\Delta A_{\\rm r} = A_{\\rm r, SFD} - A_{\\rm r}) in the southern part of the sky (left) and in the northern (right).](images/northsouthdiffhist \"fig:\") [fig:sfddiffhist]\nThe general trend favours very small differences, with the mean $\\Delta A\\_{\\rm r}$ being close to zero. Some regions indicate that SFD predicts higher extinction than we do. This can have several reasons, one being the fact that we compute extinction along a line of sight to individual stars, rather all the way to the edge of the Galaxy. Then again, we are looking at high (absolute) latitudes, where we assume extinction in general to be small, allowing us to cover most of the stars. Selection effects and limitations set by our data retrieval could affect this assumption. The few pixels that have a blue hue, indicating that our model predicts higher extinction, are mostly artificial effects that arise from binning the data, leaving only very few stars in those bins. If these stars happen to have (erroneously) high extinction estimates, they may not be averaged out.\nThis is visualised in Figure [fig:southsd], where we show the standard deviation of $A\\_{\\rm r}$ for each pixel over the southern part of the sky. We see a large range of estimates in each pixel, independent of the actual mean extinction in them and the position on the sky. There is a noticeable correlation between the standard deviation and the (absolute) difference between our extinction estimate and that of SFD, $\\Delta\\_{\\rm r} = |A\\_{\\rm r} - A\\_{\\rm r, SFD}|$, see Figure [fig:southsdvsresid]. Red dots are computed using the mean extinction estimates for $A\\_{\\rm r}$ in the equation above. Black dots use the maximum extinction value in each pixel, i.e. $A\\_{\\rm r, max}$ (see Section [subsec:furtherpriors] and Figure [fig:southl180upsmax] for further details). In this case the range of residuals is naturally larger. In both cases, though, the standard deviation in a pixel is a function of the difference between the two extinction estimates.\n![Standard deviation \\sigma_{\\rm r} of r-band extinction over the southern sky using nine photometric bands and fixed R_0. The colour axis is constrained from 0 to 1, only \\unit{0.5}{\\%} of the cells have higher values.](images/south_sd \"fig:\") [fig:southsd]\n![Standard deviation \\sigma_{\\rm r} of r-band extinction as a function of \\Delta_{\\rm r} = |A_{\\rm r} - A_{\\rm r, SFD}|, the difference between our estimates and SFD values for each pixel in the southern sky using nine photometric bands and fixed R_0. Red dots are used for mean extinction estimates, whereas black dots use the maximum extinction value in each pixel (see Figure [fig:southl180upsmax]).](images/south_resid_vs_sd \"fig:\") [fig:southsdvsresid]\nOn the whole, the results are consistent with how we expect the model to function, with results obtained from SFD or other data that deliver the integrated extinction to the edge of the Galaxy. On average, we slightly underestimate the mean extinction in a pixel due to the fact that we are averaging over multiple stars with varying degrees of precision in their individual estimates.\nExtinction prior\n----------------\nA simple prior to implement is using SFD estimates to constrain maximum extinction along any line of sight. Though the SFD maps do have several systematic problems, overestimating extinction particularly towards the Galactic plane and in regions of high extinction, at the latitudes probed here the corrections are only on the percent level. By constructing a broader prior we can safely ignore this.\nIn practice we implement the prior as a step function with the probability dropping from 1 to 0 at $1.3\\cdot A\\_{\\rm r, SFD}$ with $A\\_{\\rm r, SFD}$ being the converted SFD *r*-band extinction. This factor of 1.3 allows for enough range in the extinction estimates to account for errors in the reference value and smoothing of the data, whilst also not being too restrictive. To compare the performance of a model using this prior we use the known, simulated extinction. We see only negligible changes to the values in Table [tab:residuals] when adding the prior. Naturally, more stars remain within the training grid, as maximal values in extinction are constrained. Considering average parameter estimates do not change much, we can be confident that the model can properly characterise the degeneracy intrinsic to the data, independent of stronger priors.\nWhen applying this updated model to data (i.e. with unknown APs), we do notice some changes in respect to average extinction estimates. In Figure [fig:southl180ups] we show the differences for a region of the southern sky. The data are projected using a Mollweide projection and colour-coded according to the mean extinction in each pixel. The left panel depicts the results without priors (fixed *R*0 and eight colours). In the right panel, the equivalent map of SFD estimates is shown, and the centre panel that of our model including the SFD-based prior on extinction. Clearly, inclusion of the prior reconstructs more closely the SFD reference map, whereas the alternative produces a less smooth map, although it still closely follows the depicted distribution on a whole, picking out the high extinction regions towards the top-right, as well as the general trends. Individual pixels are not always accurate though. This is due to the fact that we estimate the APs for every star individually, whereas the smoothed SFD map is a proxy for the average extinction along a line of sight.\n![Extinction maps for a subset of the data in the southern sky in a Mollweide projection. The non-cartesian grid in latitude and longitude is overplotted. The data are binned to a resolution of \\unit{11.48}{arcmin/pixel}. The left panel shows the mean r-band extinction in each pixel for the case of using no extinction prior, the middle panel that for a prior based on SFD estimates. In both cases eight colours are used with fixed R_0. The right panel shows for comparison the same region using the SFD estimates for the direction of each pixel directly. White areas are either not covered by the data or are missed due to binning (due to post-processing, the panels do not necessarily use an identical set of stars, therefore some pixels appear white in the left panel but not in the right two.)](images/south_l180_uzYK_prior_sfd \"fig:\") [fig:southl180ups]\nThis issue becomes more pronounced when we use the maximal value in any bin to create the map. This is shown in Figure [fig:southl180upsmax], where we project the maximum values for the case of using the standard model (left) and including the prior (right). The equivalent map for SFD values is identical to the right panel in Figure [fig:southl180ups] as the reported values are smoothed already. With this prior we limit the maximum value for each line of sight. The resulting map is almost identical to the SFD reference map, with slight differences visible particularly at the edges of the footprints due to strongly varying stellar number densities. Ignoring the prior we compute varying maximum values of extinctions, as we have no low boundary limit. As can be seen in Figure [fig:southl180ups], when using the mean these extreme values tend to average out, even though individual estimates may still miss the true (or expected) value.\n![Region and resolution as in Figure [fig:southl180ups]. The left panel shows the maximum r-band extinction in each pixel for the case of using no extinction prior, the right panel that for a prior based on SFD estimates. In both cases eight colours are used with fixed R_0. Note that the colour scale has changed to cover the parameter range.](images/south_l180_uzYK_prior_sfd_max \"fig:\") [fig:southl180upsmax]\nExtinction profiles\n-------------------\nIn addition to extinction and temperature we have inferred distances to all stars. We also have full probabilistic information available for each parameter which we can use to compute profiles of extinction as a function of distance for individual lines of sight. The weighted mean $\\left< A\\_{\\rm r} \\right>$ and standard deviation $\\sigma\\_{A\\_{\\rm r}}$ for each distance modulus point *μ* are computed as\n \n$$\\left< A\\_{\\rm r}(\\mu) \\right> = \\frac{\\sum A\\_{{\\rm r}, i} w\\_i}{\\sum w\\_i} \\quad, \\quad\n\\sigma\\_{A\\_{\\rm r}} = \\sqrt{\\frac{\\sum w\\_i (A\\_{{\\rm r}, i}-\\left< A\\_{\\rm r} \\right>)^2}{\\frac{N-1}{N}\\sum w\\_i}}.\n\\label{eq:weightedmean}$$ \nAll stars whose 68% confidence intervals extends to the distance modulus selected are included. The standard deviation of each individual measurement is an order of magnitude smaller and barely affects the size of the total standard deviation compared to that of the generic weighted mean stated above. The weight *w**i* for each star is computed using the difference between *μ**i* and *μ*. We compute the probability for the normalised, asymmetric Gaussian described by the mode and left and right 68% confidence intervals at that difference. With *U* = *μ**i* − *μ* we obtain\n \n$$w\\_i = \\frac{2}{\\sqrt{2\\pi}(\\sigma\\_1+1)}\\cdot\\exp\\left( -\\frac{U^2}{2\\sigma\\_1^2}\\right) \\,\n\\label{eq:weights}$$ \nwhere *σ*1 is the smaller of the two confidence intervals. Stars that have a more precise estimate of distance modulus (small *σ*1) are weighted more strongly than those with imprecise estimates (large *σ*1). We repeat this for every (arbitrary) value of distance modulus we would like to use. This way we can compute extinction profiles using the full PDF of each star instead of merely averaging the data directly. Similarly this can be done for *A*0 or proper distance too.\n![Mean r-band extinction as function of distance modulus for 520 stars in the region l \\in \\unit{(189.56, 190.88)}{^\\circ} and b \\in \\unit{(-37.35, -36.35)}{^\\circ}. Mean values and error bars are computed as per Equations [eq:weightedmean] and [eq:weights]. The average SFD extinction for this field is A_{\\rm r, SFD} = \\unit{1.16}{mag}. Both profiles use all nine bands, purple points show the case of no prior and green points including a prior on extinction (Section [subsec:furtherpriors]). Top axis shows the corresponding distances d = 10^{\\mu/5+1} in parsec.](images/distance_profile \"fig:\") [fig:distprofile]\nAn example is shown in Figure [fig:distprofile] for the region $l \\in \\unit{(189.56, 190.88)}{^\\circ}$ and $b \\in \\unit{(-37.35, -36.35)}{^\\circ}$ (top right section of Figure [fig:southl180ups]). We have chosen this region because it has higher extinction and as such can demonstrate the application of Equations [eq:weightedmean] and [eq:weights]. The figure shows two profiles, both using the nine band forward model, one without prior (purple) and one including a prior on extinction as in Section [subsec:furtherpriors] (green). The profiles are similar in the sense that they both are essentially flat, possibly with a slight increase towards further distances, though the error bars on extinction are compatible with both interpretations. The errors on distance modulus are implicitly included in the way the mean extinction and its error are computed (see Equations [eq:weightedmean] and [eq:weights]). The average precision of distance modulus estimates varies from approximately $\\unit{0.5}{mag}$ to $\\unit{1.5}{mag}$ when going from the smallest to the largest value of *μ* in Figure [fig:distprofile]. The SFD extinction for this field is $A\\_{\\rm r, SFD} = \\unit{1.16}{mag}$, which is covered by the asymptotic error bars. The green curve tends to be slightly smaller, in line with conclusions from the previous section, where the maximal extinction estimates are constrained by the prior. However, the values are compatible within their uncertainties. Despite imposing no constraints on distance modulus when inferring the APs (apart from fitting into the parameter space of the HRD) and when computing the profiles, the variation of extinction with distance is physically consistent with extinction not decreasing with increasing distance. Furthermore, the shapes of the profiles are in agreement with the assumption that the stars at high latitudes reside behind the dust layers. Given the bright magnitude limit of the surveys used, we do not expect to find stars with distance moduli less than $\\mu = \\unit{6}{mag}$. Indeed, our method estimates very few stars to have smaller values of *μ*.\nLines of sight in regions with lower extinction behave in a similar manner, though naturally with lower asymptotic values of the extinction. In particular, cells at small distances show larger scatter, which can be attributed to low sampling statistics at these values, with only a few stars being computed to be very close by. In contrast, at larger distances the extinction in each cell is computed using hundreds of stars (owing to large uncertainties in the distance estimation and roughly *d*3 growth in the volume of a cell with distance).\nInferring *R*0\n--------------\nAs detailed in Section [subsec:r] we are able to infer the *R*0 parameter with moderate accuracy, in addition to extinction, temperature and Δ. Using the corresponding forward model, we reran the method on the same small field of stars as before. We refrain from showing the projected extinction map as it does not differ qualitatively from that with constant *R*0 (Figure [fig:southl180ups] left and centre panels). We instead show in Figure [fig:l180r0] the mean distribution of *R*0 in the same projection.\n![Distribution of the parameter R_0 over a subsection of the southern sky with an average over the whole field of \\left< R_0 \\right> = 3.75\\pm0.19. The same resolution as in previous maps is used.](images/south_l180_r0 \"fig:\") [fig:l180r0]\nGenerally we see quite some scatter around the mean $\\left< R\\_0 \\right> = 3.81\\pm0.20$ in this field. Using the raw data (i.e. not binned) we obtain $\\left< R\\_0 \\right>\\_{\\rm raw} = 3.82\\pm0.47$, which has a slightly larger standard deviation about the mean. Noting that we average over a large region with varying degrees of extinction and that the model accuracy is only MAE(*R*0) = 0.74, this should not be taken as a strong statement concerning the average *R*0 of the diffuse ISM.\nNonetheless, this supports results by, who finds *R*0 to be quite large and to have significant variations at high latitudes and low extinctions. In comparison, find *R*0 = 3.0 with 0.1 random and systematic uncertainties, although they argue that for $A\\_{\\rm r} < \\unit{2}{mag}$, estimates on *R*0 are very unreliable.\nLooking at the relation between $A\\_{\\rm r}$ and *R*0 (for raw and binned data) we see in Figure [fig:l180r0vsa0] that there is a trend ($R\\_0 = (-0.45\\pm0.2)A\\_{\\rm r}\\text{[mag]} + 4.0$), again consistent with the previous statement in the limit of low extinctions.\n![For each pixel of the map in Figure [fig:l180r0] we plot R_0 vs. A_{\\rm r}. Black points are the raw, unbinned data, red are for the binned cells. We see the scatter about the mean \\left< R_0 \\right> = 3.75\\pm0.19 (red) and \\left< R_0 \\right> = 3.75\\pm0.47 (black).](images/south_l180_r0vsa0 \"fig:\") [fig:l180r0vsa0]\nUsing the same technique as in Section [subsec:distanceprofiles] we compute the profiles of *R*0 as a function of distance for a small region at the top right of Figure [fig:l180r0vsa0] (as in Figure [fig:distprofile]). This is shown in Figure [fig:l180distprofiler0] in red (left axis), where the mean extinction has been replaced with mean *R*0 in the equations. In green the extinction profile for the same data is plotted (right axis). As in the fixed *R*0 case, we see a slight increase in extinction towards larger distances, though it is more pronounced now. At the same time *R*0 increases quite quickly, although the amplitude of variations are roughly the size of the error bars and well within the model accuracy. To be able to further analyse this interesting aspect, we need to probe regions with higher values of extinction (not covered by the crossmatched SDSS/UKIDSS data) or obtain more informative data from other sources. As it stands, we can only make the general statements mentioned above.\n![Mean R_0 (red, left axis) as function of distance modulus for a subset of stars from Figure [fig:l180r0]. The same technique as in Section [subsec:distanceprofiles] is used to compute the mean and standard deviation. In green the extinction profile is plotted (right axis). Distances in parsec are given in the top axis.](images/distance_profile_r0 \"fig:\") [fig:l180distprofiler0]\nConclusions\n===========\nCombining large optical and near infrared surveys of stars, such as SDSS and UKIDSS, is an effective means of constraining interstellar extinction as well as addressing the degeneracy between effective temperature $T\\_{\\rm eff}$ and extinction *A*0 for individual stars. To achieve this, we construct a forward model trained on colour changes of real data due to extinction, effective temperature and *R*0 in combination with self-consistent use of an HRD prior to infer distance information. Incorporating these physical constraints, we infer astrophysical parameters (APs) in a Bayesian framework using an MCMC algorithm to efficiently sample over the posterior distribution. This way we are able to naturally extract full probability distribution (PDF) information in each AP. Testing with synthetic data has shown that this method produces accurate results, with the caveat of not fully incorporating the systematic effect metallicity has on distance estimation.\nWe have investigated the impact of the data and assumptions on the model, in particular the effect of removing photometric bands, inferring *R*0 in addition, and the effect of an incorrectly assumed metallicity on the results. When using nine bands (five from SDSS, four from UKIDSS) we obtain a mean absolute error of the residuals of the model for *A*0 of $\\unit{0.23}{mag}$ and $\\unit{300}{K}$ in $T\\_{\\rm eff}$. These values are up to 50% better than just using SDSS photometry. The MAE for $\\Delta = m\\_{\\rm r} - M\\_{\\rm r}$ is $\\unit{2.2}{mag}$. With the same set of data we can also achieve an accuracy of 0.73 in *R*0, with acceptable changes to the performance in other parameters. Using the 68% confidence intervals (CI) to quantify the individual precision of the AP estimates, we obtain the following values: $\\left<\\text{CI}\\_{68}(\\Delta)\\right> = \\unit{1.6}{mag}$, $\\left<\\text{CI}\\_{68}(A\\_0)\\right> = \\unit{0.05}{mag}$, $\\left<\\text{CI}\\_{68}(T\\_{\\rm eff})\\right> = \\unit{66}{K}$, $\\left<\\text{CI}\\_{68}(R\\_0)\\right> = 0.07$. When estimating APs for fixed *R*0 the precision improves slightly. Accuracies of distance modulus (or Δ) estimates are strongly dependent on the correct assignment of metallicity to the stellar population being analysed. This can either be achieved by matching the metallicity of the HRD prior, or by directly inferring metallicity along with the other parameters. This, however, requires higher quality and more informative data or spectroscopic observations of the stars.\nAlthough we estimate APs for stars individually, our method is able to trace the intermediate-size dust structures visible in e.g. SFD dust maps of the same regions. In addition to line of sight extinction estimates (either averaged or per star) we can use the full PDFs to compute probabilistic profiles of *A*0 and *R*0 as a function of distance, indicating how dust is distributed along the line of sight. We generally find the expected result that, at these Galactic latitudes, the observed stars are behind the layers of dust.\nWe have limited this work to stars at high Galactic latitudes common to SDSS and UKIDSS Large Area Survey, allowing us to obtain photometry in nine bands. In principle this method can use fewer bands, with the decrease in accuracy detailed above, or use data from other surveys, such as Pan-STARRS or 2MASS. This would allow us to probe high extinction regions towards the Galactic plane and further address the performance in respect to *R*0.\nAs analysed in, the method can be expanded to combine distance estimates, such as parallaxes, with the HRD prior in order to provide a more accurate estimate of APs. Furthermore, metallicity and surface gravity may be estimated as well, although, as noted above, this requires higher quality data. But if we could estimate them using spectroscopy or more photometric bands, then this would alleviate the strong assumption made by implementing an HRD with fixed metallicity, and thus potentially improve the accuracy of distance estimates.\nCurrent work for the Gaia data processing pipeline, as summarised in, illustrate AP estimation performance using Gaia spectra and photometry. By combining current multiband surveys with Gaia parallaxes we expect to be able to increase the accuracy of the method and precision of the parameter estimates significantly.\nAcknowledgments\n===============\nWe thank E. F. Schlafly and R. Shetty for helpful comments and discussions. We also thank an anonymous referee for detailed comments on the paper. We would also like to thank an anonymous referee for helpful comments.\nThis project is funded by the Sonderforschungsbereich SFB881 ’The Milky Way System’ (subproject B5) of the German Research Foundation (DFG). RJH is member of the International Max-Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD) and the Heidelberg Graduate School of Fundamental Physics (HGSFP).\nThis work makes use of the ninth data release of UKIDSS. The UKIDSS project is defined in. UKIDSS uses the UKIRT Wide Field Camera.\nFunding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.\nSDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.\n99 Aihara H. et al., 2011, ApJS, 193, 29 Bailer-Jones C. A. L., 2011, MNRAS, 411, 435 Bailer-Jones C. A. L. et al., 2013, A&A, in press Berry M., 2012, ApJ, 757, 166 Burstein D., Heiles C., 1978, ApJ, 225, 40 Cardelli J. A, Clayton G. C., Mathis J. S., 1988, ApJL, 329, 33 Casali M. et al. 2007, A&A, 467, 777 Fitzpatrick E., 1999, PASP, 111, 63 Gontcharov G. A., 2012, AstL, 38, 12 Gontcharov G. A., 2012, AstL, 38, 87 Gonzalez, O. A. et al. 2011, A&A, 534, A3 Gonzalez, O. A. et al. 2012, A&A, 543, A13 Hauschildt P. H., Allard F., Baron E., 1999, ApJ, 512, 377 Kaiser N., 2002, SPIE, 4836, 154 Lada C. J. et al., 1994, ApJ, 429, 694 Lawrence A. et al, 2007, MNRAS, 379, 1599 Lee Y. et al., 2008, AJ, 136, 2022 Lombard M., Alves J., 2001, A&A, 377, 1023 Marshall D. J. et al., 2006, A&A, 453, 635 Perryman M. A. C. et al., 2001, A&A, 369, 339 Re Fiorentin P. et al., 2007, A&A, 467, 1373 Sale, S. E. 2012, MNRAS, 427, 2119 Savage B. D., Mathis J. S., 1979, ARA&A, 17, 73 Seaton M. J., 1979, MNRAS, 187, 73 Schlafly E. F. et al., 2010, ApJ, 725, 1175 Schlafly E. F., Finkbeiner, D. P., 2011, ApJ, 737, 103 Schlegel D., Finkbeiner D. P., Davis M., 1998, ApJ, 500, 525 Skrutskie, M. F. et al. 2006, AJ, 131, 1163 Sordo R. et al, 2010, Ap&SS, 328, 331 Yanny B. et al, 2009, AJ, 137, 4377\n[lastpage]\n---\n1. E-mail: hanson@mpia.de[↩](#fnref1)\n\n3D Galactic dust extinction mapping with multi-band photometry\n==============================================================\n[firstpage]\nWe present a method to simultaneously infer the interstellar extinction parameters *A*0 and *R*0, stellar effective temperature $T\\_{\\rm eff}$, and distance modulus *μ* in a Bayesian framework. Using multi-band photometry from SDSS and UKIDSS, we train a forward model to emulate the colour-change due to physical properties of stars and the interstellar medium for temperatures from 4 000 to $\\unit{9\\,000}{K}$ and extinctions from 0 to $\\unit{5}{mag}$. We introduce a Hertzsprung-Russel diagram prior to account for physical constraints on the distribution of stars in the temperature-absolute magnitude plane. This allows us to infer distances probabilistically. Influences of colour information, priors and model parameters are explored. Residual mean absolute errors (MAEs) on a set of objects for extinction and temperature are $\\unit{0.2}{mag}$ and $\\unit{300}{K}$, respectively, for *R*0 fixed to 3.1. For variable *R*0, we obtain MAEs of $\\unit{0.37}{mag}$, $\\unit{412.9}{K}$ and 0.74 for *A*0, $T\\_{\\rm eff}$ and *R*0, respectively. Distance moduli are accurate to approximately $\\unit{2}{mag}$. Quantifying the precisions of individual parameter estimates with 68% confidence interval of the posterior distribution, we obtain $\\unit{0.05}{mag}$, $\\unit{66}{K}$, $\\unit{2}{mag}$ and 0.07 for *A*0, $T\\_{\\rm eff}$, *μ* and *R*0, respectively, although we find that these underestimate the accuracy of the model. We produce two-dimensional maps in extinction and *R*0 that are compared to previous work. Furthermore we incorporate the inferred distance information to compute fully probabilistic distance profiles for individual lines of sight. The individual stellar AP estimates, combined with inferred 3D information will make possible many Galactic science and modelling applications. Adapting our method to work with other surveys, such as Pan-STARRS and Gaia, will allow us to probe other regions of the Galaxy.\nsurveys: SDSS, UKIDSS – stars: fundamental parameters, distances, statistics – ISM: dust, extinction – Galaxy: stellar content – methods: data analysis, statistical\nIntroduction\n============\nFrom our vantage point within the Galaxy, most astronomical observations are affected by interstellar dust, which attenuates light from distant objects as a variable function of wavelength. This is a central problem for many applications, as we typically want to figure out the intrinsic properties of the source, rather than how its emitted photons are influenced by the intervening matter distribution.\nThe leftovers from nuclear burning of stars, be they emitted through winds or, more energetically through supernovae explosions, produce dust grains that get mixed and reprocessed in interstellar space to eventually form new stars and planets. Dust typically affects electromagnetic radiation from ultraviolet to far-infrared wavelengths, absorbing these photons and re-emitting them as thermal radiation.\nEarly dust maps used the correlation between the dust column density and the distribution of neutral hydrogen. This data was supplanted by the dust maps produced by, who used full sky microwave data made available by the *IRAS* satellite and the *DIRBE* instrument on the *COBE* mission. Mapping the dust column densities via the calibrated dust temperature, the extinction maps, assuming a standard reddening law, were shown to be at least twice as accurate as those of.\nWith the advent of large photometric and spectroscopic datasets and the evolution of computing power and data analysis techniques, extinction can now be computed for millions of stars in a reasonable amount of time. Several different methods have been used to do this to date. One can for example determine the reddening vector for a star by computing the shift in colour-magnitude diagrams with respect to unreddened data according to an adopted extinction law. use the blue tip of the stellar locus to measure the colour shift of the main-sequence turnoff, which is a good proxy for extinction. In principle interstellar extinction can also be estimated using the spectral energy distribution (SED) of stellar templates directly, often performed via a *χ*2 fit to the data. The strong degeneracy between extinction and effective temperature of a star in optical and near-infrared bands affects such approaches, as they typically do not take into account the covariance of the data, conditioned on the APs.\nbase their maps on measuring the mean $(J-K\\_{\\rm s})$ colour of red clump giants and compare that to the colour of stars measured in Baade’s Window for fixed *E*(*B* − *V*). This way they achieve a high resolution map of the central bulge that is insensitive to differential extinction. use a Galaxy model for intrinsic colours and distances and then base their calculations on the near-infrared colour-excess method (*NICE*, and *NICER*, ). Despite unavoidable model dependencies, their results are quite robust to moderate changes in their assumptions.\nIn our method we use Bayes’ theorem to probabilistically characterise the degeneracy between extinction and effective temperature, taking into account covariances in colour space. We estimate the combined probability distribution (PDF) of extinction, effective temperature, distance modulus and *R*0 for each star individually from which we determine the three-dimensional spatial distribution of dust in the volume probed by our data. This approach expands on a previous study by, who used known parallaxes from *Hipparcos* and effective temperatures from spectroscopy to infer temperature and line-of-sight extinction for a smaller set of nearby stars.\nMethodologically, our approach is most similar to, who uses a hierarchical Bayesian model to determine the mean distance-extinction relationship for individual lines of sight as well as estimating further stellar parameters. Our method assumes a less complex model to map extinction and distances, thereby avoiding several potential systematic errors resulting from poor choices of priors.\nThis paper is organised as follows. Section [sec:theory] covers the basic theory of our approach and describes the Bayesian method used, introducing the individual factors that make up the posterior probability. We illustrate the practical model fitting procedure in Section [sec:model], detailing selection and features of the data. We quantify the performance of the method in Section [sec:validation], contrasting overall model accuracy with individual precisions. We apply our method to a large sample of data with unknown parameters in Section [sec:application] in order to investigate the distribution of dust in the Galaxy. We discuss our results and conclude in Section [sec:summary].\nTheory\n======\nOur aim is to determine the full probability density function (PDF) over the astrophysical parameters of a star given a set of photometric colours, $\\bmath{p}$. Interstellar extinction, $A\\_{\\rm 0}$ (see definition below), is of particular interest. Predicting this parameter requires simultaneous estimation of the effective temperature due to their degeneracy in respect to colour. By introducing a Hertzsprung-Russel diagram (HRD) prior we can also infer an estimate of the distance modulus, which in a band *X* is *μ* = *m**X* − *M**X* − *A**X* ( = 5log*d* − 5), where *d* is the distance in $\\rm parsec$, *m**X*, *M**X* and *A**X* are apparent magnitude, absolute magnitude and extinction, respectively. Using this, we build a forward model to infer the APs $\\{A\\_0, T\\_{\\rm eff}, M\\_{r}\\}$, i.e. estimating *r*-band absolute magnitude, from which we can then compute the distance. We can also estimate the parameter *R*0, which is the ratio of total to selective extinction. The derivation of an expression for the posterior PDF $P(\\mathbf\\Theta | \\bmath{p}$), where $\\mathbf\\Theta$ is the list of APs, is presented in this section. This work builds on and adapts the method introduced in.\nInterstellar extinction\n-----------------------\nFollowing, the extinction in a narrow band at wavelength *λ* can be parametrised by the extinction parameter $A\\_{\\rm 0}$ and $R\\_{\\rm 0}$ as\n \n$$A\\_{\\lambda} = A\\_{\\rm 0} \\cdot ( a\\_{\\lambda} + b\\_{\\lambda} / R\\_{\\rm 0} )\\,$$ \nwhere *a**λ* and *b**λ* are fixed polynomials. Note that we use $A\\_{\\rm 0}$ to characterise the extinction solely as a property of the interstellar medium rather than as a function of the spectral energy distribution of a star, which is strongly dependent on the intrinsic parameters of a particular star.\nWe introduce artificial reddening using the extinction curves from. Generally, the parameter $R\\_{\\rm 0}$ can be fixed, often at $R\\_{\\rm 0} = 3.1$, which in reality is only a mean value for the diffuse interstellar medium, rather than any individual line of sight in the Galaxy. As we will see in section [subsec:r] it can be advantageous to keep this parameter variable, and sample over it instead by including it in $\\mathbf\\Theta$.\nExtinction in a band *X* is then computed by integrating over the spectral energy distribution of a star, $F\\_{\\lambda}\\equiv F\\_{\\lambda}(T\\_{\\rm eff})$, with fixed effective temperature\n \n$$A\\_X = -2.5\\log\\left(\\frac{\\int\\! F\\_{\\lambda} h\\_{\\lambda,X} 10^{-0.4A\\_{\\lambda}}\\,{\\rm d}\\lambda}{\\int\\! F\\_{\\lambda} h\\_{\\lambda,X}\\,{\\rm d}\\lambda}\\right) \\,$$ \nwhere *h**λ*, *X* is the pass band function of filter *X*. We use this to model (predict) the changes in magnitude due to extinction and effective temperature in any band for which we have the pass band.\nPosterior\n---------\nWe now derive the posterior PDF. The full posterior $P(A\\_0, T\\_{\\rm eff}, M\\_{r}|\\, \\bmath{p}, m\\_{\\rm r}, H)$ quantifies the probability of the parameters, given the measured colours $\\bmath{p}$, the apparent *r*-band magnitude $m\\_{\\rm r}$, and HRD *H*. Using Bayes’ theorem we can write it as ($T=T\\_{\\rm eff}$, $M=M\\_{\\rm r}$, $m=m\\_{\\rm r}$)\n \n$$P(A\\_0, T, M | \\bmath{p}, m, H) = \\frac{P(\\bmath{p}, m | A\\_0, T, M, H)\\,P(A\\_0, T, M | H)}{P(\\bmath{p}, m | H)} \\.$$ \nThe first term is the likelihood, the second term summarises the priors on extinction, temperature and absolute magnitude. We do not need to quantify $P(\\bmath{p}, m)$ (the evidence). As it does not discriminate between APs, it is essentially a normalisation constant in the present context. The other terms are described in the following sections.\nNot all of the parameters and measurements are dependent on each other, this allows us to rewrite the posterior as\n \n$$\\begin{aligned}\n &&P(A\\_0, T, M |\\, \\bmath{p}, m, H) \\nonumber \\\\\n &&= \\frac{P(\\bmath{p} | A\\_0, T, M, H)\\,P(m | A\\_0, T, M, H)\\,P(A\\_0, T, M)}{P(\\bmath{p}, m | H)} \\nonumber \\\\\n &&= P(\\bmath{p} | A\\_0, T)\\, \\frac{P(A\\_0, T, M)}{P(\\bmath{p}, m)}\\,P(m | A\\_0, T, M, H) \\.\\end{aligned}$$ \nAll terms bar the final one are straight forward to understand, as we have only removed independent variables. The final term we further decompose in order to introduce explicit dependence on absolute magnitude *M* and build in the HRD. It formalises the influence of the HRD prior as the marginalisation over *M*:\n \n$$\\begin{aligned}\n&&P(m | A\\_0, T, M, H) \\nonumber \\\\\n&&= \\int\\limits\\_{M}\\! P(m | M, T, H)\\, P(M | T, H)\\,{\\rm d}M \\nonumber \\\\\n&&= \\int\\limits\\_{M}\\! P(m | M, T, H)\\, \\frac{P(M, T | H)}{P(T)}\\,{\\rm d}M \\,\n\\label{eq:hrdpriorint}\\end{aligned}$$ \nand does not explicitly depend on extinction. *A*0 drops out of the first term because *m* is independent of it once conditioned on *M*. The full posterior function is now complete and can be written as\n \n$$\\begin{aligned}\n&&P(A\\_0, T, M | \\bmath{p}, m, H) = \\nonumber \\\\\n&&P(\\bmath{p} | A\\_0, T)\\frac{P(A\\_0)}{P(\\bmath{p}, m)}\\! \\int\\limits\\_{M}\\! P(m | M) P(M, T | H) \\,{\\rm d}M.\n\\label{eqn:post1}\\end{aligned}$$ \nIndividually, we only impose a uniform prior on absolute magnitude within the parameter boundaries of the HRD. This is implicitly included in the integral already, hence we take it out of the equation here. The other APs have regular uniform priors.\nWe can also build another forward model which also takes $R\\_{\\rm 0}$ into account. The corresponding posterior function takes the analogous form\n \n$$\\begin{aligned}\nP(A\\_0, T, R\\_0, M | \\bmath{p}, m, H) = \\nonumber \\\\\n P(\\bmath{p} | A\\_0, T, R\\_0)\\frac{P(A\\_0, R\\_0)}{P(\\bmath{p}, m)} \\ \\int\\limits\\_{M}\\! P(m | M) P(M, T | H) \\,{\\rm d}M \\,\n\\label{eqn:post2}\\end{aligned}$$ \nwhere the parameter formally enters in the likelihood and possible priors. In our case, we impose a uniform prior on $R\\_{\\rm 0}$.\nForward model\n-------------\nThe forward model predicts the star’s colours given a set of astrophysical parameters. It is calculated by fitting a thin plate spline (a multidimensional analogue of a one-dimensional spline) as a function of $A\\_{\\rm 0}$ and $T\\_{\\rm eff}$ to each colour in $\\bmath{p}$, $\\bmath{f}(A\\_{\\rm 0}, T)=\\bmath{p}'$. Although we infer absolute magnitudes with our method, we do not include it in the forward model.\nWe can furthermore train a forward model which includes *R*0, expanding the dimensionality of the spline and characterising the variation over all three parameters, altering our function to $\\bmath{f}'(A\\_{\\rm 0}, T, R\\_{\\rm 0})$. The posterior in the previous section remains largely unchanged.\nLikelihood\n----------\nThe likelihood of the colours $\\bmath{p}$, given the forward model predictions of the colours, $\\bmath{f}$ (or $\\bmath{f}'$), and assuming Gaussian errors on the measurements, is \n$$P ( \\bmath{p} | A\\_0, T) = k \\cdot \\exp\\left(-0.5\\,(\\bmath{p}-\\bmath{f})^{T} \\bmath{C}^{-1}(\\bmath{p}-\\bmath{f})\\right) \\,$$ \n where $k = (2\\pi)^{-n/2} |\\bmath{C}|^{-1/2}$ is the normalisation factor, dependent on the dimensionality *n* of the colour vector $\\bmath{p}$. The covariance matrix $\\bmath{C}$ is determined from the errors on the measurements (magnitudes), taking into account the fact that the errors on consecutive colours are not independent. Taking just the five SDSS bands for illustration, the covariance matrix for the resulting colours $\\bmath{p} = (u-g, g-r, r-i, i-z)$ has the following form\n \n$$\\bmath{C} = \\left( \\begin{array}{cccc}\n\\sigma\\_u^2 + \\sigma\\_g^2 & -\\sigma\\_g^2 & 0 & 0 \\\\\n-\\sigma\\_g^2 & \\sigma\\_g^2 + \\sigma\\_r^2 & -\\sigma\\_r^2 & 0 \\\\\n0 & -\\sigma\\_r^2 & \\sigma\\_r^2 + \\sigma\\_i^2 & -\\sigma\\_i^2 \\\\\n0 & 0 & -\\sigma\\_i^2 & \\sigma\\_i^2 + \\sigma\\_z^2 \\end{array} \\right) \\,,$$ \nwhere *σ**X*2 is the variance in band *X*. For additional colours, the matrix expands analogously. A standard *χ*2 approach would ignore the off-diagonals, treating the colours as independent, which is obviously not correct.\nHRD Prior\n---------\nThe prior *P*(*M*, *T*∣*H*) is constructed to quantify the relative probability of finding a star in a particular region of the HRD. Due to the nature of stellar evolution, the HRD is inhomogeneously populated. Fixing one parameter constrains the other. This allows us to extract valuable information, even for a crude representation of the HRD (see section [subsec:buildfm] for details on how it is constructed).\nTo achieve this, we need to construct an equation that links the apparent and absolute magnitudes and distance. By independently estimating extinction and effective temperature from the colours und using the measured $m\\_{\\rm r}$ and the HRD, we can in principle determine the absolute magnitude and from that estimate the distance modulus. To effectively constrain the absolute magnitude from noisy measurements and the finite width of the HRD, we introduce a noise model to determine the probability distribution. We define the random variable\n \n$$\\kappa = m\\_{\\rm r} - M\\_{\\rm r} - \\Delta \\,$$ \nwhere we define Δ as the difference between the true (but unknown) apparent and estimated absolute magnitudes. For error-free measurements, *κ* is zero. By connecting apparent and absolute magnitudes in this way, Δ can be considered a proxy for the distance modulus, which we are trying to infer.\nFor the noise model $P(\\kappa | M\\_{\\rm r})$ we select a one-dimensional Gaussian in *κ*, $\\mathcal{N}(0, \\sigma\\_{m\\_{\\rm r}})$ which has an expectation value of zero and a standard deviation $\\sigma\\_{\\Delta} = \\sigma\\_{m\\_{\\rm r}}$, the uncertainty in the *r*-band magnitude measurement for the observed star. The absolute magnitude is not measured and as such has no error, Δ is sampled. For a particular measurement of $m\\_{\\rm r}$ the probability is\n \n$$P(\\kappa|M\\_{\\rm r}) = \\mathcal{N}(m\\_{\\rm r}-M\\_{\\rm r}-\\Delta,\\,\\sigma\\_{m\\_{\\rm r}}) \\.$$ \nAs $m\\_{\\rm r}$ is the only noisy quantity in *κ*, this term can also be written as $P(m\\_{\\rm r} | M\\_{\\rm r})$. This represents the first term of the integral in equations [eqn:post1] and [eqn:post2]. So, for any measure of Δ, obeying *κ* = 0, the Gaussian describes the scatter around this value. The full HRD prior probability is then calculated by integrating the Gaussian for a Δ over the probability distribution of the HRD at a given $T\\_{\\rm eff}$, as formalised in Equation [eq:hrdpriorint].\nUsing the HRD gives us a self-consistent means to estimate distances from photometric information. A stronger constraint on APs can be constructed if distance information is available. The original work by uses Hipparcos parallaxes to constrain the APs in this way.\nModel fitting\n=============\nIn this section we outline the practical implementation of the method and describe the data we use to build and validate the forward model and priors.\nData\n----\nTo perform the analysis described in Section [sec:theory], a set of photometric data is required, for which astrophysical parameters have been independently determined. We use stars selected from the *UKIRT Infrared Deep Sky Survey*, UKIDSS DR9 Plus and *Sloan Digital Sky Survey*, SDSS DR8. We use data from the UKIDSS *Large Area Survey* (LAS) which is designed to overlap with the footprint of SDSS. By using real data, we are able to use the proper photometric errors and intrinsic scatter to build the forward model, instead of depending on synthetic estimates. We perform a crossmatch to generate a catalogue of stars with photometry in nine bands. The bands *ugriz* are from SDSS and *YJHK* are covered by UKIDSS LAS. In principle we can also use (or combine) other surveys, as our method can easily be trained on further photometric data.\nTraining data for fitting forward model\n---------------------------------------\nWe need to simulate the effects of extinction to build an extended catalogue of stars with known APs. We therefore need to construct a catalogue of (ideally) zero extinction stars. To build our training data we select only stars at high Galactic latitudes, *b* > 70∘, which avoids the Galactic plane and most of the interstellar dust. Additionally these stars are required to be dwarfs (surface gravity log*g* > 4). This ensures that the catalogue stars only have minimal variations in log*g*, which improves the temperature estimates due to the tight correlation with colour for fixed extinction (as in Figure [fig:teffcolfixa0]). Furthermore, effective temperatures determined with the *SEGUE Stellar Parameter Pipeline*, SSPP must be available, with errors on these of less than $\\unit{200}{K}$.\nWe use a weighted average of several spectroscopic and photometric temperature estimates (called `ADOP` in the SSPP papers) for this work. When using instead the purely spectroscopic estimate, e.g. the `ANNRR` routine, we see no significant differences in our inferred extinction and distance estimates. The temperature estimates do differ, with a standard deviation of about $\\unit{80}{K}$. But as we are not primarily interested in inferring temperatures, we can in principle use either of the estimates, though using the spectroscopic estimator avoids circular arguments.\nWe chose the *r*-band magnitudes to be *r* < 19, which typically results in photometric errors approaching the systematic limit of approximately $\\unit{0.02}{mag}$. This also generally forces the photometry in the other bands to be complete and of good quality. To further ensure selection of sources with high quality data, we impose the following selection criteria on the SDSS flags: we select only unresolved sources (`sdss.type` = 6) with clean photometry (`sdss.clean` = 1) and general quality flags set (`sdss.’4295229440’` = 0, `sdss.mode` = 1). Similarly, we select objects with clean photometry from UKIDSS (`ukidss.*ppErrBits` < 65 536, one for each band) that are classified as stellar objects (`ukidss.mergedClass` = − 1).\nThe data (SDSS and UKIDSS) are queried and cross-matched using the *WFCAM Science Archive*, requiring complete photometry in all bands. In total, we obtain roughly 3 200 stars that fulfil all the above criteria. Compared to the hundreds of thousands of stars analysed in SEGUE, we retrieve comparatively few. Aside from completeness and crossmatching reasons, this is mainly influenced by the magnitude, surface gravity and temperature limits we set. In spite of this, this does not introduce systematic biases. Our selection is representative of the full SEGUE dataset (see below). The photometry is then corrected for the small estimated extinction from SFD, where the assumption holds that at high Galactic latitudes the stars are behind all the layers of dust.\nOf the selected data, a randomly sampled set of one thousand stars is chosen, covering a temperature range of approximately $\\unit{4000 - 9000}{K}$. A histogram of the temperature distribution for the full training set is shown in Figure [fig:teffhist], showing a peak around $\\unit{5600}{K}$. Only $\\unit{9}{\\%}$ of the stars have effective temperatures above $\\unit{6500}{K}$. This is representative for the whole SEGUE dataset.\n![Distribution of effective temperatures of approximately 3\\,200 crossmatched stars. Temperatures are estimated by SSPP with mean uncertainties of \\unit{58.5}{K}. Of these stars, one thousand are randomly chosen to build the extended dataset.](images/teffhist \"fig:\") [fig:teffhist]\nWhen later applying the method to data with unknown stellar parameters, we still create a selection with the quality flags used above, though we naturally impose no requirement on the existence of temperature estimates or the position in the Galaxy (low extinction).\nAdding artificial extinction\n----------------------------\nUsing the filter response functions for all filters and an extinction law, the change in magnitude for a given extinction *A*0 in each band is computed for a range synthetic SEDs. Owing to the fact that we are using broad band filters, the extinction in a band varies smoothly with temperature and we do not need to do this for every unique temperature present in our data sample. Instead we use eleven *PHOENIX* model SEDs evenly spaced in the temperature range from $\\unit{4000}{K}$ to $\\unit{9000}{K}$. This covers the temperature range in our catalogue. For each band we fit a set of one-dimensional quadratic functions in $T\\_{\\rm eff}$ for fixed $A\\_{\\rm 0}$, covering the whole temperature range noted above and extinction in steps of $\\unit{0.25}{mag}$ from 0 to 5 magnitudes.\nKnowing how the magnitudes vary with *A*0 and $T\\_{\\rm eff}$, we expand our initial dataset by adding artificial extinction in 21 discrete steps over the range noted above for each of the 1 000 stars. The photometric errors are adapted to take into account the change in magnitudes by following the average magnitude-error relation in the data. Our expanded dataset (*training set*) now comprises 21 000 stars with known artificial extinction and effective temperatures.\nBuilding the forward model\n--------------------------\nIn Figure [fig:teffcolfixa0] we show the effect of temperature and extinction on four different colours. We do not convert the magnitudes in both surveys to a common system (UKIDSS uses Vega magnitudes, for example), because the colours are merely affected by a constant offset with respect to that of the other survey, and we are only interested in changes in colours. As long as the training data and the final datasets are handled consistently, we need not take into account this offset.\n![Effects of effective temperature and extinction on four selected colours. Purple points show a subset of the training data with no extinction as a function of effective temperature. Green dots show the effects for A_0 = \\unit{2}{mag}. Each sample consists of 1\\,000 stars.](images/teffcolfixa0 \"fig:\") [fig:teffcolfixa0]\nFrom the training set we select a random subset of 4 000 stars to create the forward model itself, with the aim of being able to predict colours given temperature and extinction (see section [subsec:likelihood]). For each of the eight reduced colours we fit a two-dimensional thin-plate spline to both APs. A good fit for all colours is achieved when giving the splines 20 degrees of freedom.\nIn Figure [fig:fmtpsresiduals] we show the direct performance of the splines predicting colours given some input APs. For four example colours we plot the residuals of colour (*p**X*, *t**r**u**e* − *p**X*, *s**p**l**i**n**e*) for the ranges of temperature and extinction present in the model as contours. This is not the final result of the full parameter inference, instead it illustrates the intrinsic inaccuracies of the model which derive from the scatter in the real data. We do not see any systematic variation, and measure a typical scatter in the order of $\\unit{0.03-0.06}{mag}$ for all colours, which is roughly $\\sqrt{2}$ larger than for the magnitudes in a band. The forward model captures the intrinsic variance, as this is the same order of magnitude or within a small factor of the reported uncertainties of the survey data magnitudes. In Section [subsec:accapest] we look in detail at how assuming larger photometric errors affects the AP estimation.\n![Residuals in the forward model. This shows the difference between the true colours and the colours predicted from the forward model thin plate spline as functions of true effective temperature and extinction, for four colours. The contour levels shown incorporate 68\\%, 90\\% and 99\\% of the data points. In the top right of each panel the corresponding colour is named, with the x-axis depicting the respective parameter ranges. All panels are scaled to show the same range on the y-axis. Typical scatter is in the order of \\unit{0.03-0.06}{mag} for all colours.](images/fm_tps_residuals \"fig:\") [fig:fmtpsresiduals]\nWe characterise extinction using *A*0 to indicate that this is a parameter that solely depends on the physics of the interstellar medium, rather than on the type of star. When presenting results and maps later on we will use extinction in SDSS *r*-band to allow more convenient comparisons to other work. Note that the conversion from *A*0 to $A\\_{\\rm r}$ has a small dependence on temperature. For *R*0 = 3.1 they roughly convert as $\\left<A\\_{\\rm r}/A\\_0\\right> = 0.837$. The standard deviation from this value over the parameter range $T\\_{\\rm eff} = \\unit{4000-9000}{K}$ and $A\\_0 = \\unit{0-5}{mag}$ is only 0.005. The model values for extinction in other bands, e.g. the relative extinctions $A\\_X/A\\_{\\rm r}$ are listed in Table [tab:modelextval] for three different values of *R*0. Values reported are averages over the above range. In particular, changes occur for the short wavelength optical bands. When varying *R*0, we see that the relative extinction in the other, redder, bands changes only slightly. In Figure [fig:extr0teffva] these changes are illustrated for all the bands used and for three different effective temperatures.\n[tab:modelextval]\nccccccccc *R*0 & *u* & *g* & *i* & *z* \n2.1 & 2.393 & 1.698 & 0.707 & 0.542 \n3.1 & 1.848 & 1.445 & 0.744 & 0.555 \n4.1 & 1.610 & 1.329 & 0.761 & 0.561 \n*R*0 & *Y* & *J* & *H* & *K* \n2.1 & 0.439 & 0.337 & 0.230 & 0.147 \n3.1 & 0.427 & 0.308 & 0.204 & 0.133 \n4.1 & 0.422 & 0.296 & 0.193 & 0.126 \n![Model values for extinction relative to r-band for three temperatures over a range of R_0 and the photometric bands used in this work. Small temperature dependences can be seen for the optical bands u and g at low R_0.](images/r0arteff \"fig:\") [fig:extr0teffva]\nFor the APs used to build the forward model we implicitly implement priors that are uniform over the APs’ training ranges, and have zero probability outside them. We will demonstrate the use of a prior on extinction based on SFD estimates later in Section [subsec:furtherpriors].\nHRD prior\n---------\nThe HRD prior is constructed using stellar isochrones from. The chosen stellar population comprises 100 000 stars drawn from a Salpeter initial mass function with masses ranging from $\\unit{0.2-107}{M\\_{\\sun}}$ (though we only use stars with $T\\_{\\rm eff} \\leq \\unit{9\\,000}{K}$ and the vast majority of stars have masses below $\\unit{1.6}{M\\_{\\sun}}$). Assuming a constant star formation rate over the age of the Universe ($\\unit{13.7}{Gyr}$), all stars evolve independently with solar metallicity, i.e. there is no chemical enrichment. By using the forward model, however, we are implicitly assuming that the metallicity of stars we encounter have the same mean metallicity of the SSPP stars, which are used to fit the model. This mismatch between the metallicities of the forward model and the HRD does not strongly affect the estimation of *A*0 and $T\\_{\\rm eff}$ directly, though it does limit the ability of getting good distance estimates. As we are not estimating metallicity, this mismatch will naturally occur for most stars, independently of the choice of HRD metallicity. An example of this limitation is given in Section [subsec:demo].\nThe resulting temperatures and absolute magnitudes then are used to place the stars in the HRD plane. To use the HRD as a prior, we need to convert the data representation to a probabilistic description of the distribution. We achieve this by applying a two-dimensional binned kernel density estimate with a bandwidth of $\\unit{12.5}{K}$ and $\\unit{0.0625}{mag}$ in effective temperature and *r*-band absolute magnitude, respectively. The resulting grid has the pixel dimensions of 6002, we limit the temperature range to $\\unit{3010 - 9000}{K}$ as to extend slightly beyond the limits of the forward model. This allows for the main sequence to be fully represented down to low temperatures without an artificial cutoff. The resulting magnitude range is $r \\in \\unit{(-4,12)}{mag}$. Before normalisation, a small, but non-zero, offset is added to each point to account for the regions which nominally have zero probability, but in reality may not be completely empty. These regions include white dwarfs in the lower left part of the figure or (post) AGB stars higher up the branch to the right. Statistically, these regions will be very thinly populated and we don’t expect to find many (if any) of these stars in our samples, so we don’t model these in a more sophisticated manner. A representation of the final distribution is presented in Figure [fig:hrd].\n![Density representation of the HRD used in this analysis. The integrated probability of the HRD is normalised to one, red is high number density and blue is low. The colour scale shows base 10 logarithm of the density. White areas denote regions of the parameter space with initially zero probability. A small offset is added to each point before normalisation to avoid this in the actual computation. In this case the offset is approximately 10^{-3} times the maximum density.](images/hrd \"fig:\") [fig:hrd]\nSampling and computation\n------------------------\nThe forward model, priors and data are combined to compute the posterior function, as detailed in Section [sec:posterior]. As we are interested in inferring multiple APs, we use a standard Metropolis-Hastings Markov-Chain Monte-Carlo (MCMC) routine to efficiently sample the parameter space and to compute the posterior probability distribution. The parameter space is explored in logarithmic units of the respective quantity, forcing the estimates to remain positive and physical. The sampling matrix is diagonal with variances of $\\unit{0.1}{dex}$ in each variable (AP). We assume no correlation between the variables. Convergence is sufficient with a chain length and burn in of 10 000 steps each. This does not need to be adjusted if we also infer *R*0.\nTo compute the likelihoods we can either use the actual spline function or build a lookup table. Considering the numerical speed-up we achieve by tabulating results in advance, we generally use this approach. For this, we build a grid in extinction and effective temperature (and later in $R\\_{\\rm 0}$ too) and calculate the predicted colours. We use the same parameter ranges as for training (avoiding extrapolation into regions not covered by the data), creating a grid with *A*0 from $\\unit{0-5}{mag}$ in steps of $\\unit{0.001}{mag}$ and $T\\_{\\rm eff}$ from $\\unit{3000 - 9000}{K}$ in steps of $\\unit{10}{K}$ (effectively the same as for the HRD). For *R*0 we use a coarser stepsize of $\\unit{0.2}{}$ in the range from 2.1 − 5.9. The stepsizes are chosen as to be significantly smaller than the mean absolute error (MAE) of the residuals when using the full spline function (see Section [subsec:effectcolours].)\nWhen assessing the performance in the APs and computing further quantities, we only use those stars whose inferred parameters are within the boundaries of the lookup table (i.e. within our training space). As indicated in section [subsec:training], at lower temperatures and smaller wavelengths the linear relation between colour and temperature for fixed extinction no longer holds, therefore extrapolation is not advisable. This avoids skewing the results due to boundary effects, like MCMC chains sticking to the edge of the grid in one or more parameters.\nWe apply this filter after parameter inference as we cannot rule out including stars that have true APs outside of our training range (we do not apply any colour-cuts to the data selection). If we were to instead force a prior over the same ranges during sampling we would artificially force those stars to have incorrect parameters.\nTypically this filter throws out roughly $\\unit{5}{\\%}$ of stars when estimating extinction, effective temperature and distance modulus. When including the *R*0 parameter, this increases to close to $\\unit{30}{\\%}$, due to the additional variation in the model.\nValidation\n==========\nIn this section we first demonstrate the viability of the model and then analyse the precision and accuracy of the method, as well as the colours. The following analysis is performed on a different set of 4 000 stars (*validation set*), of which none where used to train the model.\nSimulated photometry\n--------------------\nIn this first step we aim to show the expected performance of the method and illustrate its limitations with respect to metallicity (see Figure [fig:demo]). For this purpose we set up two extinction clouds at $\\unit{500}{pc}$ ($\\mu\\approx\\unit{8.5}{mag}$) and $\\unit{1500}{pc}$ ($\\mu\\approx\\unit{10.9}{mag}$) with extensions of $\\unit{100}{pc}$. This true extinction distribution with *μ* is shown as the solid black line. We then place 500 stars along the line of sight, generating effective temperatures and absolute magnitudes from the HRD prior (solar metallicity). According to their effective temperatures, extinctions and distances, we simulate appropriate photometry and photometric errors, which are characteristic of the SDSS/UKIDSS sample we use. Finally we use our method to infer these parameters. These are plotted as points in the figure.\nWe infer the APs for two cases. In (a) we use the same solar metallicity HRD prior (*Z* = 0.019) as for selecting the stars (red dots). In this case, we can see that our method recovers the distances and extinctions very well, with points closely tracing the extinction profile. The few points that lie furthest from the true profile typically have bad temperature estimates, resulting in erroneous distance moduli. In (b) we use an HRD with a tenth of that metallicity (*Z*ʹ = 0.0019, blue dots) to infer parameters. Although the individual estimates for extinction and temperature are compatible, the mismatched stars are systematically shifted by Δ*μ* = − 1 to closer distances. Analogously, the opposite shift occurs when reversing the mismatch. Therefore the accuracy of distance estimates is strongly limited by the unknown metallicity of each individual star.\n![Inferred distance moduli \\mu and extinction A_0 for 500 stars. Red dots indicate results for which matching HRD metallicity (Z=0.019) is used for sampling and inference. Blue dots show results for mismatched HRD (Z' = 0.0019).](images/demo \"fig:\") [fig:demo]\nIt is difficult to estimate metallicity using only broad band photometry, as used here, when neither temperature nor extinction are fixed. As shown in for simulated Gaia data, given spectroscopic information it is possible to estimate metallicity simultaneously, resulting in more accurate distance estimates.\nEffects of Colours and NIR data\n-------------------------------\nConsidering that we are using photometry from two surveys covering optical and near-infrared bands, it is instructive to compare how inclusion (or exclusion) of certain bands affects the parameter estimation. To examine this we train the forward model using (a) all nine bands, resulting in the eight colours *u-g*, *g-r*, *r-i*, *i-z*, *z-Y*, *Y-J*, *J-H* and *H-K*, (b) using only the five SDSS bands and thus only four colours and (c) using all but the SDSS *u* band magnitudes. Generally, one would expect the use of more colours (more information) to yield better results, particularly when taking into account that NIR wavelengths are less affected by dust attenuation than shorter ones.\nAs expected, the performance of the method is significantly better when using the NIR data (using only NIR data is also much worse). Table [tab:residuals] summarises the differences between the two cases. We characterise the performance with three key statistics. The first is the bias, which is purely the mean value of the difference between predicted and true parameter for all stars. The second is the mean absolute error (MAE), which similarly is the mean of the absolute difference and the third is the root mean square (RMS). For temperatures below $\\unit{6500}{K}$ the performance is generally better than for higher temperatures. This results from the combination of having fewer stars in the higher temperature range and the scatter in the data being larger there (see Figures [fig:teffhist] and [fig:teffcolfixa0]).\n[tab:residuals]\nccccccc & & \n& (a) & (b) & (c) & (a) & (b) & (c) \nbias & − 0.08 & − 0.41 & − 0.29 & − 67 & − 295 & − 269 \nMAE & 0.23 & 0.59 & 0.41 & 300 & 628 & 440 \nRMS & 0.45 & 1.05 & 0.68 & 586 & 1028 & 712 \nWe note that the bias is negative in both cases and parameters, though much closer to zero in case (a). Intrinsic scatter in the data also precludes higher residual accuracy. In Figure [fig:residuals] we plot the results for the validation set of stars. Towards negative values in both parameters we see two distinct tails. Stars in the tails tend to have hotter true temperatures and larger intrinsic scatter. Only a few percent of stars are effected by such a large error in the parameter estimation. Without these tails, the performance would improve only minimally in MAE and RMS.\nWhen excluding only the *u* band, case (c), we find that the performance is still better than case (b) although significantly worse than (a). The bias, MAE and RMS for *A*0 are $\\unit{-0.29}{mag}$, $\\unit{0.41}{mag}$ and $\\unit{0.68}{mag}$, respectively. For $T\\_{\\rm eff}$ these are $\\unit{-269}{K}$, $\\unit{440}{K}$ and $\\unit{712}{K}$.\nIn temperature and extinction we see no significant systematic effect due to the metallicity of the HRD. As indicated in the previous section, this mainly affects the distance estimator.\n![Residuals in effective temperature and extinction for case (a) using all eight colours. Plotted are the 68\\%, 90\\% and 99\\% density contours of the distribution of points in the plot. The differences are computed as \\Delta\\Theta = \\Theta_{\\rm estimated} - \\Theta_{\\rm true} for A_0 and T_{\\rm eff}.](images/residuals \"fig:\") [fig:residuals]\nFigure [fig:residuals2] shows the residuals for both parameters as a function of the true parameters. We see no systematic behaviour, though we see a general bias towards negative values, which is evident in the values reported in Table [tab:residuals].\n![Residuals in effective temperature and extinction as functions of the true values in both parameters, as applied to the validation set of data. All eight colours are used. The true variation is shown for the same data as in Figure [fig:residuals]. A few dozen points are cut off by the selection of the axis range.](images/residuals2 \"fig:\") [fig:residuals2]\nAs we do not have any true distances to compare the inferred distances to, we can not reference any results for distance modulus here. We must rely on the other parameters to be accurate in order to believe the distance results we obtain. We can however roughly estimate distances using the true extinctions and temperatures and finding the corresponding $\\Delta = m\\_{\\rm r} - M\\_{\\rm r}$ via the HRD. Doing this we compute a bias of $\\unit{-0.23}{mag}$, MAE of $\\unit{2.29}{mag}$ and RMS of $\\unit{3.31}{mag}$ when using all bands. Approximately $\\unit{15}{\\%}$ of the estimates are better than $\\unit{0.2}{mag}$. In this case, the distance is poorly constrained by the combination of photometric data and our broad HRD prior, when $T\\_{\\rm eff}$ and *A*0 also have to be estimated from the data. The bias we encounter is primarily due to the mismatch of metallicities of the HRD and training data.\nPrecision of AP estimation\n--------------------------\nBeyond AP estimation accuracy (of the method), we are also interested in the precision, i.e. what are our parameter uncertainties for any given star. We quantify this using the width of the 68% confidence interval (CI68) of the posterior PDF (which is equivalent to the 1*σ* variation), using a Gaussian kernel density estimation with the mean as the central point. The distribution is not generally symmetric, so we obtain marginally different values for the left and right confidence bounds. In Figure [fig:accuracy] we show the histograms of the average CI68 bounds for both extinction and effective temperature, as well as Δ. Mean values of the confidence intervals are $\\left<\\text{CI}\\_{68}(A\\_0)\\right>=\\unit{0.04}{mag}$, $\\left<\\text{CI}\\_{68}(T\\_{\\rm eff})\\right>=\\unit{58}{K}$ and $\\left<\\text{CI}\\_{68}(\\Delta)\\right>=\\unit{2.0}{mag}$. We can therefore be confident that our parameter estimates are generally precise in a statistical sense.\n![Histograms of 68\\% confidence intervals for extinction and temperature using photometric errors in the covariance matrix. The purple vertical lines indicate the mean values in each case. For A_0 this corresponds to \\unit{0.04}{mag}, for T_{\\rm eff} it is \\unit{58}{K} and for \\Delta it is \\unit{2.0}{mag}.](images/prec \"fig:\") [fig:accuracy]\nThese values of precision are significantly smaller than the accuracy of the model (Section [subsec:effectcolours]), which could be taken to mean that the parameter inference is overconfident compared to the information content of the data. The intrinsic scatter of the spline fit is of the same order of magnitude as the photometric errors of the data (see Section [subsec:buildfm] and Figure [fig:fmtpsresiduals]), therefore properly representing the variation and uncertainties of the model.\nAverage photometric errors in all nine bands vary from 0.01 to $\\unit{0.03}{mag}$, the mean temperature uncertainty in the training data is $\\unit{60}{K}$, uncertainties in extinction can only arise from an erroneous correction of the SFD estimates, unless the reddening law is wrong. These, though, are typically only $A\\_{\\rm SFD, r}=\\unit{0.03}{mag}$. Our precision therefore is in the order of the parameter errors. If we fix $\\sigma\\_{p\\_X} = \\unit{0.1}{mag}$, which is roughly five times larger than the photometric errors, we obtain precisions of $\\left<\\text{CI}\\_{68}(A\\_0)\\right>=\\unit{0.36}{mag}$ and $\\left<\\text{CI}\\_{68}(T\\_{\\rm eff})\\right>=\\unit{402}{K}$. These values are in the order of $\\unit{25}{\\%}$ larger than the MAE reported in Table [tab:residuals]. The corresponding biases, mean absolute errors and root mean squares only change by a few percent in respect to the tabulated results. Therefore, a more accurate account for model limitations could imply using significantly larger errors on photometry, although on average the statistical performance remains compatible in either case.\nTo illustrate the output from the Monte Carlo sampling, Figure [fig:contour1] shows the contours and one-dimensional PDFs of the three parameters extinction, effective temperature and Δ for a single example star. In the extinction-temperature plane (bottom left panel) we clearly see the degeneracy between the two parameters. In the one-dimensional PDF for Δ (middle panel) we see the typical bimodal distribution, which is the result of sampling over the parts in the HRD that represent the main sequence and the giant branch. One peak is usually higher, which is why we use the mode to quantify this parameter rather than the mean. In most cases the higher peak relates to the main sequence (though not in this example), which is a reasonable justification to use it, as we have trained our models using photometry from dwarf stars.\nIt must be noted that the confidence intervals for Δ are large, though the inner 68% are more tightly constrained. We can reduce this spread by either running a longer Monte-Carlo chain, which adds more weight to the non-background regions in the HRD, or by directly decreasing the background level by a few orders of magnitude when computing it in the first place. Both options result in nearly unchanged parameter estimates but improved precision in Δ. The second option is clearly more appealing, as it does not increase the computation time.\n![Contours of the posterior PDF for a star with true APs of A_{\\rm 0, true}=\\unit{4}{mag} and T_{\\rm eff, true} = \\unit{5392}{K}. In each panel the red cross or line shows the position of the estimated parameter value (A_0 = \\unit{4.06}{mag}, T_{\\rm eff} = \\unit{5446}{K}, \\Delta = \\unit{15.9}{mag}), whereas the purple plus or line is the true (constructed) value. Contours show 68\\%, 90\\% and 99\\% confidence intervals. The smaller side panels show the marginalised one-dimensional PDFs for the single parameters, as aligned with the respective axes. The peak densities are scaled to 1 in each case. The indicated true value of distance modulus is estimated using the true temperature and the HRD from section [subsec:hrdprior]. The best estimate values for temperature and extinction are computed using the mean of the distribution, whereas the mode is used for distance modulus.](images/contour1 \"fig:\") [fig:contour1]\nEffects of $\\bmath{R\\_0}$\n-------------------------\nThe results shown so far were for a fixed *R*0 = 3.1. In reality, this value is only the mean value for the diffuse interstellar medium, whereas denser regions will typically be characterised by a higher value of *R*0, arising from a change in size distribution and composition of the dust grains responsible for extinction in optical bands. This does not, however, equate high line-of-sight extinction with large *R*0; it simply describes properties of the local dust.\nWe train a new forward model in which we simulate the change of colours due to extinction and *R*0 for stars with known temperatures. We expand our parameter grid and again use 4 000 stars to train and validate the model. The general setup remains the same as before. We use *R*0 in the range from 2.1 to 5.9. The model performance we achieve like this is presented in Table [tab:residualsr].\n[tab:residualsr]\n| | *A*0/mag | $T\\_{\\rm eff}$/K | *R*0 |\n| --- | --- | --- | --- |\n| bias | − 0.22 | − 206 | 0.32 |\n| MAE | 0.39 | 438 | 0.73 |\n| RMS | 0.69 | 771 | 0.95 |\nComparing case (a) in Table [tab:residuals] with Table [tab:residualsr], we see that modelling *R*0 slightly deteriorates the performance in the other two parameters, the standard deviation and MAE in the residuals are increased by roughly 50% for temperature and extinction. This comes as no surprise, as we are now extracting more parameters, but with the same initial data, noting that the observed change in colours are quite small for larger differences in this parameter. In general we are able to roughly infer *R*0, though the accuracy is not sufficient to confidently differentiate between, say, 3.1 and 3.8. We therefore build the lookup table using somewhat larger steps of 0.2 in *R*0 for the range noted above, and the other parameters remaining the same.\nIn Figure [fig:accuracyr] we present the precision of the predicted parameters as histograms, where the purple lines again depict the mean values. We obtain $\\left<\\text{CI}\\_{68}(A\\_0)\\right>=\\unit{0.05}{mag}$, $\\left<\\text{CI}\\_{68}(T\\_{\\rm eff})\\right>=\\unit{66}{K}$, $\\left<\\text{CI}\\_{68}(R\\_0)\\right>=0.07$ and $\\left<\\text{CI}\\_{68}(\\Delta)\\right>=\\unit{1.6}{mag}$. The first two values are minimally larger than in the previous case, though it is to be expected considering that we have added an additional degree of variation to the model. Again, the confidence intervals are significantly smaller than the model accuracy.\n![Histogram of 68\\% confidence intervals of extinction, temperature and extinction factor, which are estimates of the uncertainty on each parameter we infer. The purple vertical lines indicate the mean values in each case. For A_0 this corresponds to \\unit{0.05}{mag}, for T_{\\rm eff} it is \\unit{66}{K}, for R_{0} it is 0.07 and for \\Delta it is \\unit{1.6}{mag}.](images/precr \"fig:\") [fig:accuracyr]\nApplication to multiple fields\n==============================\nWe have so far summarised the accuracy and precision of the model using data with known stellar parameters. We now apply the model to estimate APs for large areas of the sky where individual estimates are not available. Querying the *WFCAM Science Archive* with the flags detailed in section [subsec:training], but imposing no constraints on SSPP information, we obtain a total of 4 191 659 unique stellar objects, of which 3 055 954 and 1 135 705 are north and south of the Galactic equator, respectively. We also pick out a section of the southern sky with *l* > 180∘, which we use to illustrate the variations of the model, namely using only photometry from SDSS, using all photometry, and also including *R*0 in the parameter estimation.\nWe first show the full projected maps of the northern and southern part of the surveys in Figures [fig:fullsky]. Here we use a Mollweide projection in Galactic longitude and latitude to obtain equal area pixels, using a resolution of $\\unit{11.48}{arcmin/pixel}$ in both coordinates. This corresponds to an average of 29 stars per pixel, depending on the region observed, with extremes of 3 and 250 stars per pixel.\nThe interstellar medium towards the Galactic poles typically hosts diffuse gas and lines of sight with low extinction. Nevertheless, the area jointly probed by the two surveys does cover some regions with higher average extinction. Generally we can pick out larger structures that are also visible in SFD maps of the same resolution. Towards the right of the southern part of the sky one can start to make out parts of a larger structure, with extinction estimates in the range of 1 to $\\unit{1.5}{mag}$ in the *r*-band.\n![A Mollweide equal area projection of the mean extinction in the full crossmatched portion of the sky, centred on the Galactic anti-centre. The stars are binned using a resolution of \\unit{11.48}{arcmin/pixel} in l and b. The non-cartesian grid in latitude and longitude is overplotted. The colour scale shows the mean extinction A_{\\rm r} in any given pixel. White areas are not jointly covered in the surveys.](images/northsouth \"fig:\") [fig:fullsky]\nTo more quantitatively assess the credibility of the presented map, we compare the computed extinction values with those from SFD. For this we project the data in an identical manner and calculate the difference between the SFD extinction values and our own estimates. This is plotted in Figures [fig:southsfddiff] for the southern part of the sky. We refrain from showing the equivalent image for the northern part, as it qualitively shows the same behaviour. In Figure [fig:sfddiffhist] we show the histograms for the differences between SFD extinction estimates and our own for both the southern and northern part of the sky, based on the binned data. In the south the mean difference and standard deviation are $\\unit{-0.02}{mag}$ and $\\unit{0.13}{mag}$, respectively. In the north we obtain $\\unit{-0.06}{mag}$ and $\\unit{0.11}{mag}$.\n![Difference between SFD extinction estimates and our own for stars in the southern part of the sky, using a Mollweide projection with \\unit{11.48}{arcmin/pixel} resolution. The colour coding in the range of \\Delta A_{\\rm r} = -1 to \\unit{1}{mag} shows regions where SFD predicts a higher extinction in yellow/red (positive), and where our model has a higher estimate in blue (negative).](images/south_sfd_diff \"fig:\") [fig:southsfddiff]\n![Histograms of differences between SFD extinction estimates and our own for stars (\\Delta A_{\\rm r} = A_{\\rm r, SFD} - A_{\\rm r}) in the southern part of the sky (left) and in the northern (right).](images/northsouthdiffhist \"fig:\") [fig:sfddiffhist]\nThe general trend favours very small differences, with the mean $\\Delta A\\_{\\rm r}$ being close to zero. Some regions indicate that SFD predicts higher extinction than we do. This can have several reasons, one being the fact that we compute extinction along a line of sight to individual stars, rather all the way to the edge of the Galaxy. Then again, we are looking at high (absolute) latitudes, where we assume extinction in general to be small, allowing us to cover most of the stars. Selection effects and limitations set by our data retrieval could affect this assumption. The few pixels that have a blue hue, indicating that our model predicts higher extinction, are mostly artificial effects that arise from binning the data, leaving only very few stars in those bins. If these stars happen to have (erroneously) high extinction estimates, they may not be averaged out.\nThis is visualised in Figure [fig:southsd], where we show the standard deviation of $A\\_{\\rm r}$ for each pixel over the southern part of the sky. We see a large range of estimates in each pixel, independent of the actual mean extinction in them and the position on the sky. There is a noticeable correlation between the standard deviation and the (absolute) difference between our extinction estimate and that of SFD, $\\Delta\\_{\\rm r} = |A"},"data_id":{"kind":"string","value":"arxiv_0181683"}}},{"rowIdx":45399,"cells":{"content":{"kind":"string","value":" mass for a sample of 466 MSF clumps for which we have a radius and that have been detected in one of the three ammonia surveys discussed in Sect.[sect:ammoniaobs]. In a recent paper, have shown that the critical virial parameter ($\\alpha\\_{\\rm{cr}}$) for an isothermal sphere that is in hydrostatic equilibrium (i.e., a Bonnor-Ebert sphere; ) that is not supported by a magnetic field is $\\alpha\\_{\\rm{cr}}$ = 2. Clumps with *α* > 2 are subcritical and unless pressure confined by their local environment will expand. Clumps with *α* < 2 are supercritical and unstable to collapse unless supported by a strong magnetic field. The solid black horizontal line shown in Fig.[fig:virialmass] indicates the *α* = 2 locus and the grey shaded area shows the region of parameter space where clumps are gravitationally unstable. These virial ratios are a useful measure of a clump’s stability and although not definitive for any particular clump they do provide some measure of the stability of the population as a whole.\nEven allowing for an equal amount of magnetic and kinetic support (i.e., $\\alpha\\_{\\rm{cr}}$ = 1; ) we find that the vast majority of the clumps are likely to be unstable and in a state of gravitational collapse. Since all of these clumps are in the process of forming massive stars we should expect smaller regions to be collapsing locally. However, the fact that parsec scale clumps are unstable implies that they are also undergoing global collapse.\nThe distribution of the whole sample in Fig.[fig:virialmass] reveals a strong trend for decreasing values of *α* with increasing clump masses, which implies that the most massive clumps are also the least gravitationally stable. A log-log linear regression fit to the distribution gives a slope of − 0.53 ± 0.16, which is similar to values reported by from their analysis of massive star formation studies in the literature (e.g., ). As noted by the trend for decreasing virial ratios with increasing clump mass provides a relatively simple explanation to the relative sparsity of very massive pre-stellar clumps, of which very few have been found outside the Galactic centre (). The larger values of *α* found for lower mass clumps may indicate that feedback from the embedded stars is able to play a bigger role in stabilising their natal environment on smaller scales through the injection of turbulence and heating of the circumstellar envelope.\nComparing these observed values of *α* (i.e., *α* ≪ 2) found them to be inconsistent with the monolithic accretion model, which requires *α* ≥ 2. They concluded that magnetic fields would need to be included but that this would represent a major modification to the original model. Conversely, the observed values of *α* do satisfy the conditions required for competitive accretion to be viable for a large proportion of the MSF clumps (where *α* < 1; ). Parsec-sized, globally collapsing clumps provide the flow of material that is funneled into the centre of the proto-cluster and onto the competing protostars (e.g., ). However, as point out, this does not rule out the monolithic accretion model, and there are a number of lower-mass clumps (i.e., < 1000M⊙; ) that are unlikely to form a stellar cluster and therefore competitive accretion is not viable for all clumps.\nThe distributions of the four subsamples presented in Fig.[fig:virialmass] reveal a clear difference between the MYSOs and H ii regions with the former generally having lower clump masses and higher values of *α* compared to the latter. However, these are largely due to the significant difference in clump mass of the two subsamples that has already been discussed and so we will not dwell further on this difference here.\nLuminosities\n------------\n### RMS Bolometric luminosities\nWe calculate the luminosities of RMS sources using the model spectral energy distribution (SED) fitter developed by and applying a similar method to that discussed in. For this process, we start with near-infrared photometry from the 2MASS, UKIDSS or Vista-VVV surveys and mid-infrared photometry taken from MSX and WISE — see for a discussion on the preferred choice of values when two or more are available for a particular wavelength. We complement these near- and mid-infrared measurements with *Herschel* far-infrared photometry provided by the Hi-GAL survey, and submillimetre data from the ATLASGAL CSC and distances as discussed in Sect.[sect:distances].\nThe fluxes are input into the SED fitter along with the allowed range of distances which we set to the RMS distance assuming an uncertainty of ± 1kpc. The fitter searches through 20000 models which probe a range of properties of both the central (proto)star and surrounding material, each of which can be viewed from 10 different viewing angles, resulting in possible 200000 SEDs. The fitter also takes into account foreground extinction and the different beam sizes of the various observations, returning the best fit and all models within a given tolerance through chi-squared minimisation. We calculate the luminosity and uncertainty using chi-squared weighting as discussed in. The mean chi-squared value for the best fit is ∼ 12 with a mean delta-chi-squared value for the best 10 fits of ∼ 3. Fig.[fig:sedexample] shows the measured flux values and the model fit obtained from the SED fitter for G028.6874+00.1772, as a typical example. We are primarily concerned with obtaining a good fit to these data and an accurate estimate of the bolometric luminosity and so the precise details of the models are somewhat irrelevant and may be inappropriate anyway.\n[fig:sedexample] Spectral energy distribution for the RMS source G028.6874+00.1772. The near-infrared fluxes (2MASS, VVV, or UKIDSS) are shown in cyan, WISE and MSX mid-infrared fluxes are shown in blue and green, respectively, *Herschel* Hi-GAL PACS/SPIRE fluxes are shown in red and the integrated ATLASGAL flux is shown in black. The best-fit model obtained from the SED fitter is plotted as a solid black curve while the 9 next best fits are shown as solid grey curves.\nIn a number of cases where there are multiple MYSOs or H iiregions associated with a particular MSX source the individual embedded sources are not resolved in the longer wavelength surveys (i.e., *MSX*, *Herschel* 250$\\umu$m, 350$\\umu$m, 500$\\umu$m and ATLASGAL). For these sources we only use the longer wavelength photometric measurements to estimate the total bolometric luminosity of all the embedded sources. The luminosity is then split between the sources present using the ratio of their relative mid/far-infrared fluxes measured from the images.[6](#fn6) The *Herschel* fluxes have been used to fit SEDs to 845 RMS sources, which corresponds to ∼ 88percent of the RMS sample considered here. The full results of this process, along with details of flux determination and analysis of the infrared colours will be presented in Mottram et al. (2014). Bolometric luminosities for the other 119 RMS sources are taken from.\n \nFig.[fig:bollum] shows the luminosity distribution of all matched RMS sources as well the MYSO and H ii-region subsamples. The distribution peaks at 1-2 × 104L⊙. However, the distributions of the two subsamples are significantly different, with the MYSO and H ii-region distributions peaking at ∼ 1 × 104L⊙ and ∼ 4 × 104L⊙, respectively. This difference in the luminosity function of these two samples and the lack of very luminous MYSOs has been previously noted and discussed by. The latter show that MYSOs with masses larger than ∼ 30M⊙ have a Kelvin-Helmholtz contraction time that is shorter than their accretion timescale, resulting in the rapid formation of an H ii region. It is interesting to note that there are very few MYSOs with luminosities above 105L⊙and none that have luminosities larger than 2 × 105L⊙, which corresponds to a zero age main sequence (ZAMS) star with a mass of ∼ 30M⊙ . This is consistent with the models of and described in the introduction of this paper and, indeed, it is clear from the plots shown in Fig.[fig:bollum] that the MYSOs are approximately an order of magnitude less luminous than the more evolved compact H ii regions. This also means that we are less sensitive to MYSOs with increasing distance. A KS test of the distance-limited sample reveals that the luminosity distributions of these two subsamples are significantly different (*r* ≪ 0.01), but this is also nicely explained by the Hosokawa et al. models and to be expected as the MYSOs transition into H ii regions.\n### Methanol maser luminosities\nBolometric luminosities are not currently available for the majority of the methanol-maser associated embedded sources. However, following the work of we are able to obtain a rough estimate by scaling up their infrared luminosity. Since it is likely that the methanol masers are more embedded than the MYSOs and H ii regions, we use far-infrared wavelengths where the extinction is less likely to affect our analysis. We have therefore compared the bolometric fluxes of a large sample of MYSOs and H ii regions identified by the RMS survey with their *Herschel* 70-$\\umu$m fluxes. In Fig.[fig:pacs70ratio] we show the results of this comparison in the form of a histogram showing the distribution of ratio values obtained. The distribution is skewed and so we take the peak of the distribution as the conversion factor and fit an exponential function to each side of the peak to estimate the asymmetric uncertainties; this gives a $F\\_{\\rm{bol}}/F\\_{\\rm{PACS\\,70\\,\\umu m}}$ conversion factor 9.1− 3+ 16. Note that in applying this conversion we are implicitly assuming that the $F\\_{\\rm{bol}}/F\\_{\\rm{PACS\\,70\\,\\umu m}}$ is approximately the same for both methanol maser sources and MYSOs/H iiregions.\n[fig:pacs70ratio] Comparison of the RMS bolometric fluxes obtained from SED fits to the infrared and submillimetre photometry and their PACS 70$\\umu$m flux. The red curve show the results of exponential fits to either side of the peak of the distribution from which we estimate the uncertainties. The bin size is 0.1dex. \nWe have used the publicly available level 2.5 70$\\umu$m tiles of the Hi-GAL open-time Key Project () obtained from the *Herschel* Science Archive.[7](#fn7) We have used the high pass filtered maps which have been calibrated and reduced with SPG v10.3.0 and combine both scan directions (parallel and orthogonal). From these tiles we have extracted postage-stamp images for all the methanol masers and performed aperture photometry to estimate their PACS 70$\\umu$m emission.The main objective here is not to provide a complete set of reliable bolometric luminosities for the methanol masers (this is being done by the Hi-Gal team and will be published separately) but to provide a reasonable indication of their range and distribution.\nWe have used the 70-$\\umu$m fluxes to estimate the bolometric luminosities for ∼ 200 methanol-maser associated clumps and the distribution of values is shown in Fig.[fig:bollum]. Although the luminosity of any particular methanol maser is less reliable than those of RMS sources, the overall luminosity distribution should be robust (i.e., the sample size is large enough that the mean should be well determined). The luminosities cover the same range as those of the MYSO subsample but the overall distribution is significantly flatter. A KS test on the distance-limited subsamples is unable to reject the null hypothesis that the methanol masers and MYSOs are drawn from the same population at the required 3*σ* confidence level (*r* = 0.003). We do find a significant difference between the luminosity distribution of the H ii regions and that of the methanol masers (*r* ≪ 0.001). We note that many of the methanol masers are located above the MSX 21-$\\umu$m sensitivity limit (or would be if they had an MYSO like SED) but do not have an MSX counterpart, which is approximately 70percent of the sample. This is consistent with methanol masers being associated with very luminous objects that are more embedded than the MYSOs and H ii region subsamples, and therefore likely to precede these two more evolved and less embedded stages.\nEmpirical Relationships\n=======================\nAs previously noted here and in PapersI and II, all of the associated clumps appear to be rather spherical in structure, are centrally condensed and the vast majority are associated with a single massive star-formation tracer. Even in cases where multiple tracers are found, these tend to be clustered towards the centre of their host clump (e.g., see lower panels of Fig.2). These clumps typically have masses of a few 103M⊙ and radii of ∼ 1pc, which are similar to the physical sizes and masses expected to form stellar clusters. It is therefore likely that the majority of clumps are in the process of forming a stellar cluster. It follows that the methanol masers, mid-infrared point sources and radio continuum emission are associated with the most massive members of a young proto-cluster. The derived clump properties (e.g., mass, density, radius etc) and the bolometric luminosities of the embedded sources are therefore much more likely to be related to the embedded cluster rather than to a single star.\nBefore we begin investigating these relationships we caution that the mere existence of any correlations between the observed properties of clumps does not imply causation (i.e., post hoc ergo propter hoc). Most are simply the result of the trivial fact that most massive clumps tend to have more extremes within them. So more massive clumps tend to be larger and contain more luminous stars and this alone may not be meaningful and care must therefore be taken not to over-interpret these correlations. There is, however, potentially useful information about physics in the gradients of any correlations and in the positions of sources in the relevant parameter space.\nMass-Luminosity relation\n------------------------\n[fig:massrelations] Clump mass-bolometric luminosity relationship for all MSF clumps and their associated subsamples. The orange shaded region indicates the area in which 90percent of the Monte-carlo simulations of ZAMS stellar clusters are located, while the region highlighted in grey indicates the area where the most massive star of each simulated cluster is located. The lower, middle and upper diagonal dash-dot lines indicate the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$=1, 10 and 100L⊙/M⊙, respectively. The solid black curves, running respectively left to right, show the model evolutionary tracks calculated by for stars with final masses of 6.5, 13.5 and 35M⊙, respectively. The long-dashed diagonal red and dark green lines show the results of a log-log, outlier-resistant, linear fit to the MYSO and H ii-region associated clumps and methanol-maser associated clumps, respectively.\nIn this subsection we compare the clump masses and the bolometric luminosities of the embedded proto-clusters and examine the relationship between the physical properties of the clumps and their associated star formation. Mass-luminosity (*M*$\\_{\\rm{clump}}$-*L*$\\_{\\rm{bol}}$) diagrams have previously been used in studies of low-mass () and high-mass star-forming regimes () and are taken to be a useful diagnostic tool with which to separate different evolutionary stages. Fig.[fig:massrelations] shows the mass-luminosity distribution of all associated clumps for which a distance and luminosity are available. We use different colours and symbols to show the distributions of the methanol-maser, MYSO and H ii-region associated clouds and those associated with multiple evolutionary stages (green, red, blue and magenta, respectively). In cases where a clump is associated with multiple RMS sources we have integrated the luminosity of each embedded source to obtain a total bolometric luminosity for each clump. In Fig.[fig:lumcolden] we present a similar plot showing the correlation between luminosity and column density. It is clear from this plot and the *M*$\\_{\\rm{clump}}$-*L*$\\_{\\rm{bol}}$ diagram in Fig.[fig:massrelations] that the most massive stars are forming in the most massive and dense clumps.\n developed a simple model for the formation of massive stars based on the observational evidence that their formation is a scaled-up version of the inside-out collapse model that has been successfully applied to low-mass star formation. Using the turbulent core-collapse model with accelerating accretion rates proposed by as their starting point, they calculated how the clump mass and source bolometric luminosity should change as the protostar evolves, producing evolutionary tracks for stars with final masses between ∼ 6 and 35M⊙. These tracks consist of a vertical and horizontal component that refer to as the *main accretion* and the *envelope clean-up* phases, which we will also adopt here. Note that the efficiency with which the clump mass is converted into stellar mass determines the position of the apex in mass-luminosity space, but is not predicted by the model and are included for illustrative purposes only and no attempt is made to constrain the models.\n[fig:lumcolden] Comparison of peak column density and bolometric luminosities for the four subsamples of massive star-forming clumps. The dashed vertical and dash-dotted horizontal cyan lines indicate the nominal column density sensitivity of the ATLASGAL survey ( ∼ 1022cm2) and the luminosity section cut applied to the RMS sources ( > 1000L⊙). The long-dashed red line shows the result of a linear fit to the data.\nAssuming that the accretion rate increases with the mass of the protostar (as favoured by the population synthesis model; ), and that this accelerates towards the end of its vertical track it will reach the end of its main accretion phase with increasing rapidity. However, once the main accretion phase has ended it takes a long time before the embedded star begins to disrupt its natal clump sufficiently to make much progress along its horizontal track. A consequence of this is that a source will spend a relatively large proportion of its embedded lifetime at the apex where these two tracks intersect and we therefore expect to find that a significant fraction of embedded sources will collect along the locus of apices.\nThe MYSO, H ii region and the multi-phase subsamples all combine to form a continuous distribution that is concentrated towards the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ = 10L⊙/M⊙ locus that extends over approximately 3 orders of magnitude in both axes. There is a broad spread in the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ values but the upper and lower envelopes of this range are well fitted by value of 100 and 1L⊙/M⊙, respectively. These three lines of constant *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ (i.e., 1, 10 and 100L⊙/M⊙) are overlaid on Fig.[fig:massrelations]. We evaluate the correlation between the luminosity and clump masses of these three subsamples using a partial Spearman correlation test and obtain a correlation coefficient of *r* = 0.64 with a significance of *t* ≪ 0.001. The strong correlation observed is therefore statistically significant and not the result of distance-related biases. A log-log linear regression fit to these data has a slope of 1.03 ± 0.05. To check that the slope is not affected by a distance bias we repeated the fit for the distance-limited sample and obtained a slope of 1.09 ± 0.13. These values are slightly lower but consistent with the slope reported by ( ∼ 1.27) from a similar fit to a sample of 27 infrared-bright sources and consistent with the results of a similar analysis presented by. Our analysis builds on the results of these previous studies confirming their main findings and showing this correlation holds for masses and luminosities an order of magnitude larger than previously considered.\nThe methanol-maser associated sample covers a similar range of masses as the MYSO, H ii-region and multi-phase subsamples, however, there is a clear offset to lower bolometric luminosities for the same clump mass than for the other three subsamples. On average, the methanol-maser associated clumps are approximately a factor of two less luminous than the other associated clumps. The distribution is also significantly more scattered. However, the mass and luminosity of these clumps are still correlated with each other although not as strongly (*r* = 0.58 with a significance of *t* ≪ 0.001). From the log-log linear fit to these parameters we determine the slope of 1.06 ± 0.07, which is consistent with the slope found for the MYSOs and H ii regions.\n### Evidence for an evolutionary sequence?\n[tbl:suumaryofderivedclumppara]\nlrcrcrcrcr & & && & \n& & & & & & & && \nMethanol masers & 314 & 6.8 ± 0.2 & 298 & 4.3 ± 0.3 & 197 & 3.7 ± 0.2 & 269 & 6.8 ± 0.7 & 195 \nMYSO & 210 & 5.6 ± 0.2 & 210 & 4.2 ± 0.3 & 210 & 3.2 ± 0.2 & 210 & 15.7 ± 1.3 & 210 \nH ii regions & 375 & 8.2 ± 0.2 & 373 & 4.8 ± 0.3 & 343 & 3.7 ± 0.2 & 373 & 16.4 ± 1.1 & 343 \nMulti-phase & 231 & 7.2 ± 0.3 & 230 & 5.0 ± 0.4 & 219 & 3.9 ± 0.3 & 230 & 15.4 ± 1.1 & 219 \nAll MSF clumps &1130 & 7.1 ± 0.1 & 1111 & 4.8 ± 0.2 & 772 & 3.7 ± 0.1 & 1082 & 16.0 ± 0.7 & 772 \n \n \nIn this section we explore the hypothesis that the methanol masers, MYSOs and H ii regions represent distinct evolutionary stages of massive star formation (as outlined in Sect.1.1). We will put to one side the multi-phase clumps as these contain multiple evolutionary signatures and therefore cannot provide any insight into changes in the clump properties as the formation procedes. As previously mentioned, the luminosity-to-mass ratio has been proposed as a diagnostic tool for distinguishing between different evolutionary states and therefore, if the methanol masers, MYSOs and H ii regions are distinct stages we should expect to find a trend of increasing *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ ratios for these subsamples.\nFig.[fig:lummasshisto] shows the distribution of luminosity-mass ratios for the whole sample of MSF clumps and for the clumps associated with the methanol maser, MYSO and H ii region subsamples and present a summary of their properties in Table[tbl:summaryofderivedclumppara]. There is a clear difference between the methanol-maser associated subsample and the MYSO and H ii-region associated clumps. The latter have almost the same distribution in *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ with approximately the same mean value ( ∼ 16L⊙/M⊙). The *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ distribution for the methanol masers is shifted to lower values with a significantly lower mean of ∼ 7L⊙/M⊙. Methanol masers are widely thought to trace an earlier stage in the massive star-formation process than the mid-infrared-bright MYSOs and H ii regions and their significantly lower *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ ratio is consistent with this hypothesis.\nDifferences in the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ ratio for H ii regions and IR-selected HMPOs have been previously reported in the literature (e.g., ). However, no clear difference between the MYSO and H ii-region associated subsamples is seen in Fig.[fig:lummasshisto] and the KS test is unable to reject the null hypothesis that they are drawn from the same parent population (*r* = 0.026). The similarities in the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ distributions suggest that these two subsamples cover a similar range of evolutionary stages. As previously noted, the MYSOs and H ii regions are concentrated around the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$= 10L⊙/M⊙ locus, which, according to the scenario outlined by, is likely to correspond to the apex between the main accretion and envelope clean-up phases. This would suggest that the majority of the MYSOs and H ii regions are nearing the end of their main accretion phase, or that the accretion has already been halted. arrived at a similar conclusion independently by modeling the Galactic star-formation rate from the RMS source counts. They found that an intermediate radio-quiet phase was required between the end of the main accretion phase and the formation of the H ii region to obtain a satisfactory qualitative fit to the lower-luminosity MYSO distribution. This may also provide an explanation for the relatively low association rate of MYSOs and H ii regions with methanol masers that are thought to be radiatively pumped in the accretion disc ( ∼ 30 and 10percent for MYSOs and H ii regions, respectively; see Sect.3.3).\nAlthough the MYSOs and H ii regions have similar *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ ratios and form a continuous distribution in the M-L diagram (Fig.[fig:massrelations]), the MYSOs dominate the lower clump-mass and luminosity end of the parameter space while the H ii regions occupy the higher end of both parameter ranges. However, there is no obvious break in the combined distribution of the two subsamples that might be expected if the luminosities of the two groups originated in different physical processes.\n[fig:lummasshisto] Ratio of total bolometric luminosity (integrated luminosity of all objects embedded in the clumps) to clump mass for all ATLASGAL clumps associated a methanol maser, MYSO or H ii region (filled grey histogram). Green, red and blue hatched histograms show the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ ratio distributions of the methanol-maser, MYSO and H ii-region associated clumps, respectively. The bin size is 0.2dex.\nIn previous sections we have found that the MYSO population covers a lower luminosity range than the H ii regions and, since the most massive clumps tend to be located at larger distances, the lower number of MYSOs found to be associated with the more massive clumps may simply reflect this. In this case we might expect there to be a population of massive clumps associated with MYSOs that have yet to be found, however, there are a few points that limit this possibility. First, comparing distance-limited subsamples we still find that the MYSOs are associated with clumps that have significantly lower mass and column density clumps than the H ii regions, which suggests that the difference is not a selection effect. Second, in Sect.[sect:colden] we found that there are practically no high column-density clumps in the Galactic disc that have not already been associated with massive star formation. We also found a strong positive correlation between clump mass and peak column density with more massive clumps being associated with high column densities (see Fig.[fig:masscolden]). We are therefore also very likely to already have identified the vast majority if not all of the most massive (i.e., $M\\_{\\rm{clump}} \\gtrsim 1000$M⊙ out to 20kpc), compact clumps located within the inner Galactic disc, which means there is limited scope for the existence of large population of MYSOs embedded in more massive clumps in this part of the Galaxy. Finally, we find a similar correlation and slope to that in Fig. [fig:massrelations] if we consider the distance-limited samples of MYSOs and H ii regions (the slope is 1.08 ± 0.13 and correlation coefficient is 0.63) and so this is unlikely to be due to limited flux sensitivity.\nThere are no MYSOs associated with very massive clumps, however, it is likely that a significant number of the methanol masers found to be associated with massive clumps are actually MYSOs given that ∼ 20percent of methanol masers are coincident with an MYSO ( < 2; as discussed in Sect.[sect:multiplicity]). Therefore a number of the more distant methanol maser sources are likely to be tracing luminous MYSOs that fall below the MSX 21$\\umu$m detection threshold. However, even taking this into account it does not change the differences in the overall distribution of the MYSO and H ii region subsamples seen in Fig.[fig:massrelations], which are still present in the distance-limited analysis. Furthermore, we have found significant differences in the clump mass, column density and kinetic temperature of clumps associated with MYSOs and H ii regions with the former having lower values for all parameters. There are clearly some differences between the clump properties of these two subsamples.\nThe simplest explanation is that the massive stars forming in the most massive clumps evolve much faster than those embedded in lower mass clumps, which is consistent with the decreasing timescale with increasing MYSO luminosity found by. In this case, the lack of very luminous MYSOs associated with the most massive clumps would be because such sources evolve into an H ii region so quickly that we are unlikely to observe the MYSO stage. This would also explain why the methanol maser associated clumps appear to have similar intermediate properties to both the H ii region and MYSO associated clumps, if they are tracing an earlier evolutionary stage of both the MYSOs and H ii regions.\nIt is clear from the strong correlations between bolometric luminosity and both clump mass and column density that the most massive stars are forming in the most massive and dense clumps. derived a formula for the accretion rate as a function of time for a free-falling envelope:\n \n$$\\dot{m}\\_\\star \\; = \\; 4.6 \\times 10^{-4} \\left(\\frac{m\\_{\\star\\rm{f}}}{30\\,{\\rm{M\\_\\odot}}}\\right)^{3/4} \\Sigma\\_{\\rm{clump}}^{3/4} \\left(\\frac{m\\_\\star}{m\\_{\\star\\rm{f}}}\\right)^{0.5} \\; \\left(\\frac{{\\rm{M\\_\\odot}}}{{\\rm{yr}}}\\right)$$ \nwhere $\\dot{m}\\_\\star$ is the time-dependent accretion rate, *m*⋆ and $m\\_{\\star\\rm{f}}$ are the current and final mass of the star in M⊙, respectively, and $\\Sigma\\_{\\rm{clump}}$ is the clump surface density in gcm− 2. For a particular $m\\_{\\star\\rm{f}}$ the accretion rate is proportional to $\\Sigma\\_{\\rm{clump}}^{3/4}$ and therefore stars forming in the most massive and dense clumps will accrete mass more rapidly than stars forming in a lower mass clump and will arrive on the main sequence much sooner. For example, if we consider two stars with the same final mass but forming in clumps where the surface density is different by an order of magnitude. In this case the star in the higher surface density clump will arrive on the main sequence ( ∼ 30M⊙) a factor of 3–5 faster than the one embedded in the lower surface density clump.\nDifferences of bf two orders of magnitude in the peak surface densities are seen in our data (i.e., Fig.[fig:colden]) and therefore such large variations in timescale seem plausible. Furthermore, the lower mass clumps appear to be forming less massive stars and since the accretion rate is also dependent on the final mass of the star, the difference in the time taken to arrive on the main sequence is likely to be significantly longer than the factor of 3-5 estimated for stars evolving in the lower mass clumps.\nThe faster evolution times for stars forming in the most massive and dense clumps means they arrive on the main sequence and form an H ii region in a fraction of the time taken for less massive stars forming in lower-mass clumps. The lifetime of the MYSO stage is therefore anti-correlated with clump mass. This nicely explains all of the differences seen in the properties of the MYSOs and H ii-region associated clumps and the similarities of both of these subsamples with those of the methanol-maser associated clumps.\nAn important implication is that the MYSO stage for the most massive stars may be so short due to fast accretion rates and very high extinction that we are unlikely to observe it. All of the MYSOs identified will go on to form H ii regions, however, and if the Molinari model is correct then they are already near the end of their main accretion phase and their final luminosity is unlikely to increase significantly from its current value. This sample of MYSOs are therefore unlikely to be the progenitors of the most massive early O-type stars. However, there are still a significant number of MYSOs associated with the multi-phase clumps that have not been considered in the comparison presented here. Some of these multi-phase clumps are also among some of the most luminous and massive in the Galactic disc and may host MYSOs that are possible precursors to early O-type stars. There are also a number of very massive clumps that are associated with methanol masers that might also contain the earliest stages of the most massive stars in the Galaxy. Higher-resolution interferometric observations of these MYSOs and methanol masers are required to investigate these potential early O-type star precursors in more detail.\n### Comparison with expected cluster luminosity\nThe orange and grey shaded regions shown in Fig.[fig:massrelations] indicate the possible range of bolometric luminosities of an embedded stellar cluster and of its most massive star, respectively, forming in a clump of a given mass. These have been generated using a Monte-Carlo simulation to randomly sample a standard IMF () with a stellar-mass range of 0.1 to 120M⊙ and assuming ZAMS luminosities and a star-formation efficiency (SFE) of 10percent (see PaperII for a more detailed description of the model).\nComparing the measured source luminosities to the model values, we find good agreement for MYSOs and H ii regions associated with clumps with masses between ∼ 100 and a few thousand M⊙. However, the match is significantly poorer for clumps above and below this range, with the lower-mass clumps observed to have excess luminosity and the higher-mass clumps being under-luminous compared to the model predictions. A similar disparity was seen by from their analysis of HMPOs. This would suggest that the instantaneous SFE decreases as a function of increasing clump mass and is much lower for the more massive clumps than the 10percent SFE assumed in our model. In previous studies, low SFE has been associated with the least evolved clusters and is thought to increase as it evolves with an upper limit of roughly 30percent ( and references therein).\nIf we assume that the low- and high-star formation in a given cluster is coeval, then we would expect the most massive stars to evolve into H ii regions on short timescales but that the rest of the associated cluster members, particularly the lower-mass stars, will take much longer before they begin to make a significant contribution to the total cluster luminosity. The higher accretion rates for the more massive stars results in a similar formation time for the whole cluster regardless of the large range of masses of the individual stellar members; approximately 3 ± 1 × 105yr (). However, as previously mentioned the most massive stars in the cluster reach the main sequence and form an H ii region while still deeply embedded and continue to accrete material for a significant period before attaining their final mass. The MYSO lifetime for the most massive stars ( > 30M⊙) is estimated to be of order ∼ 1 × 105yr after which they join the main sequence and begin to form a H ii region ().\nSo although the overall formation time is similar for all cluster members the most massive star is likely to reach the main sequence significantly ahead of the lower mass stars and dominate the cluster luminosity. For a typical cluster consisting of 1500M⊙ of stars we would expect to find 1 × 60M⊙, 1-2 × 30M⊙, 5-10 × 15M⊙, 10-20M⊙ with the rest of the mass residing in low- and intermediate mass stars (cf. Table2 of ). When fully formed approximately 50percent of the cluster luminosity will be generated by the most massive 60M⊙ star, with majority of the other high-mass stars contributing the majority of the rest of the cluster’s luminosity. If we now consider their contribution to the cluster’s luminosity at the time the most massive star joins the main sequence (i.e., when it has a stellar mass of ∼ 30M⊙). At this stage, due to the flattening off of the *L*/*M* ratio with increasing stellar mass, the most massive star is producing approximately a third of its final mass luminosity, however, since luminosity is a strong function of stellar mass the rest of the clusters massive star population are only producing approximately 10percent of their final mass luminosity. As a consequence, at the time the most massive star joins the main sequence it is contributing up to 75percent of the cluster’s total luminosity and result in a lower SFE than would otherwise be expected.\nWe also find that the more massive stars are forming in the most massive clumps and are therefore likely to join the main sequence faster than less massive stars forming in lower mass clumps. The difference between the time taken for the most massive stars to reach the main sequence and cluster formation will be larger for the most massive stars (i.e., clusters are less evolved). Since the most massive stars are forming in the most massive clumps we would expect the SFE to decrease with increasing clump mass, because the contribution from the lower mass cluster members is likely to be much smaller (as discussed in the previous paragraph). Furthermore, the cluster formation times are estimated assuming that the surface density is constant for a give cluster, however, this is unlikely to be true as there is likely to be strong density gradients. We have already found that the most massive stars are forming in the highest column density regions towards the centre of the clumps where the gravitational potential well is deepest. The majority of the lower mass stars are therefore likely to be forming in lower density regions, which would lead to longer formation times and further lower the measured SFE.\nThe difference in cluster formation timescale and the time taken for the most massive stars to join the main sequence would explain the low SFE seen in our data. Support for this hypothesis comes from the fact that, although the luminosities measured for the most massive clumps fall well short of the expected ZAMS cluster luminosities, however, there is a better match between the expected luminosities of the most massive stars in a given cluster and the measured bolometric luminosity. Furthermore, in PaperII we found that there is a good correlation between the bolometric luminosities and the luminosities derived independently from the measured radio-continuum emission from the compact H ii regions. Since the ionizing flux traced by the radio emission in a cluster is totally dominated by the most massive star, it would strongly suggest that the bolometric luminosity measured for the most massive clumps is also dominated by the most massive stars and that the clusters are less evolved and the rest of the lower mass members of the cluster are yet to make a significant contribution. Conversely, the lower mass clumps have the highest SFE and are much less strongly bound, which may indicate that these are much more evolved.\nThese results therefore suggest a scenario where the most massive stars form towards the centre of very massive gravitationally unstable clumps with the lower mass stars forming either coevally or at later times as the proto-cluster is fed from the massive globally collapsing clump. However, more detailed modeling of the early stages of cluster evolution are need to investigate this further; this work is ongoing and will be reported in a subsequent publication.\nMass-Radius relation\n--------------------\n[fig:massradiusdistribution] The mass-size relationship of ATLASGAL clumps associated with methanol masers, MYSOs and H ii regions. The colours and symbols are explained in the legend, with the exception of the cyan stars which indicate the distribution of the MPC candidates found towards the Galactic centre (). The beige shaded region shows the part of the parameter space found to be devoid of massive star formation that satisfies the relationship *m*(*r*) ≤ 580M⊙ $(R\\_{\\rm{eff}}/{\\rm{pc}})^{1.33}$ (cf. ). The orange shaded area towards the top of the diagram indicates the region of parameter space where young massive cluster progenitors are expected to be found (i.e., ). The long-dashed red line shows the result of a linear power-law fit to the whole sample of associated clumps. The dashed cyan line shows the sensitivity of the ATLASGAL survey ($N\\_{\\rm{H\\_2}} \\sim 10^{22}$cm− 2) and the upper and lower dot-dashed lines mark surface densities of 1gcm− 2 and 0.05gcm− 2, respectively. The diagonal light blue band fills the gas surface density ($\\Sigma(\\rm{gas})$) parameter space between 116-129M⊙pc− 2 suggested by and, respectively, to be the threshold for “efficient” star formation.\nIn Fig.[fig:massradiusdistribution] we present a mass-radius (*M*$\\_{\\rm{clump}}$-*R*$\\_{\\rm{eff}}$) diagram of the whole sample of ATLASGAL sources for which a distance has been determined. We presented similar diagrams in PapersI and II and found that the methanol-maser and H ii-region associated samples were strongly correlated with each other. Here we build on that work with increased sample sizes for the latter and inclusion of the MYSO-associated clumps (the combined sample consists of ∼ 1000 massive star-forming clumps). As seen in the two previous papers, there is a strong correlation between these parameters. Again using a partial Spearman correlation test to remove any dependence of the correlation on distance yields a coefficient value of 0.85 with a *t*-value ≪ 0.001. We fit these parameters using a power-law which yields log($M\\_{\\rm{clump}}) = 3.42\\pm0.01 + (1.67\\pm0.025)\\times {\\rm{Log}}(R\\_{\\rm{eff}}$); the long-dashed red line shows the fit to these data. The fit agrees within the uncertainty with those determined in PapersI and II and we find no significant differences between the different subsamples, which all combine to form a continuous distribution over two orders of magnitude in radius and almost 4 orders of magnitude in clump mass. Interestingly, the slope is similar to that found for cluster mass as a function of radius as determined by (i.e., 1.71 ± 0.07) suggesting a fairly constant SFE regardless of Galactic location and environment, or clump mass ( ∼ 15percent).\nIn our earlier work we found nearly all of our associated sources were located within a relatively narrow range of surface densities and this is also the case for the larger and more complete data set presented here. The diagonal dashed-dotted lines overlaid on Fig.[fig:massradiusdistribution] trace the lines of constant surface density, $\\Sigma({\\rm{gas}})$, of 1gcm− 2 and 0.05gcm− 2, respectively; these provide a reasonable empirical fit to the upper and lower range of our sample of MSF clumps. We also show the minimum threshold of ∼ 116-129M⊙pc− 2 proposed by and (hereafter LH threshold) for “efficient” star formation in nearby molecular clouds (d ≤ 500pc); this range is indicated by the thick light blue band on the *M*$\\_{\\rm{clump}}$-*R*$\\_{\\rm{eff}}$ diagram, however, the 0.05gcm− 2 that effectively traces the lower envelope of the ATLASGAL clumps is approximately twice the LH threshold. This suggests that a surface density of 0.05gcm− 2 might better represent a minimum threshold of “efficient” massive star formation.\nThe shaded regions shown in the lower-right part of Fig.[fig:massradiusdistribution] correspond to the regions of parameter space where massive stars are not typically found (empirically determined by ). Comparing the distribution of our sources with the threshold (i.e., *m*(*r*) ≥ 580M⊙ $(R\\_{\\rm{eff}}/{\\rm{pc}})^{1.33}$)[8](#fn8) we find good agreement with all but a handful of sources located below this level, however, we note that the empirical fit to the lower envelope of our distribution (i.e., 0.05gcm− 2) provides a tighter constraint for clumps larger than ∼ 0.5pc and more massive than 500M⊙.\n### Massive proto-cluster candidates\n[tbl:MPCderivedpara]\nlcc......c & && & & & & & & \n& && & & & &&& \n \nAGAL010.472+00.027 & GAL 010.46+00.02 & MM/HII & 8.6 & 2.3 & 4.55 & 23.94 & 5.54 & 9.73 & 1,2,4 \nAGAL012.208 − 00.102 & & MM/HII & 13.6 & 2.8 & 4.58 & 23.46 & & & 1,4*a* \nAGAL019.609 − 00.234 & & MM/HII & 12.7 & 2.8 & 4.60 & 23.68 & 5.66 & 11.54 & 1,3,4*a* \nAGAL032.797+00.191 & G32.80 & HII & 13.3 & 2.5 & 4.49 & 23.49 & 5.55 & 11.65 & 1,4*a* \nAGAL043.148+00.014 & W49A & MM/HII & 11.1 & 2.3 & 4.52 & 23.16 & 5.62 & 12.63 & 1,3 \nAGAL043.164 − 00.029 & W49A & HII & 11.1 & 3.1 & 4.77 & 23.53 & 5.68 & 8.22 & 1,3,4 \nAGAL043.166+00.011 & W49A & MM/HII & 11.1 & 4.2 & 5.34 & 24.19 & 6.04 & 5.04 & 1,3,4 \nAGAL043.178 − 00.011 & W49A & MM & 11.1 & 2.2 & 4.51 & 23.16 & & & 1 \nAGAL049.472 − 00.367 & W51 (A&B) & MM & 5.4 & 2.2 & 4.61 & 23.67 & & & 1,3,4 \nAGAL049.482 − 00.402 & W51 (A&B) & MM & 5.4 & 2.2 & 4.49 & 23.59 & & & 1 \nAGAL049.489 − 00.389 & W51 (A&B) & MM/HII & 5.4 & 1.4 & 4.70 & 24.30 & 5.57 & 7.33 & 1,3,4 \nAGAL328.236 − 00.547 & & MM & 11.4 & 4.9 & 4.99 & 23.40 & 5.16 & 1.47 & 1,2 \nAGAL329.029 − 00.206⋆ & & MM & 12.0 & 3.3 & 4.69 & 23.38 & 4.78 & 1.21 & 1,2 \nAGAL330.954 − 00.182 & GAL 331.03-00.15 & MM/HII & 9.3 & 2.2 & 4.72 & 24.07 & 5.75 & 10.84 & 1 \nAGAL337.704 − 00.054 & & MM/HII & 12.3 & 2.4 & 4.53 & 23.53 & 5.16 & 4.21 & 1 \nAGAL350.111+00.089 & & MM & 11.4 & 2.1 & 4.55 & 23.27 & & & 1,2 \n \nG000.253+00.016 & “The Brick” & Starless? & 8.4 & 2.8 & 5.1 &23.39&&& 5,6 \nAGAL000.411+00.051 (d)*b* & & Starless? & 8.4 & 3.5 & 4.9 &23.23&&& 6 \nAGAL000.476 − 00.007 (e)*b* & & Starless? & 8.4 & 4.5 &5.2 &23.53&&& 6 \nAGAL000.494+00.019 (f)*b* & & MM & 8.4 & 2.7& 4.8 &23.26&&& 6 \n \n \nReferences: (1) this work, (2) PaperI, (3) PaperII, (4), (5), (6) \nNotes: *a* Identified by as a massive clump but the derived mass was slightly less than the 3 × 104M⊙ required for inclusion in their sample of MPC candidates. *b* The source names have been taken from the ATLASGAL CSC () while the letters in parentheses are the nomenclature used by. ⋆ The distance to this source is uncertain and should be used with caution.\nThe shaded area towards the top of Fig.[fig:massradiusdistribution] indicates the region of the parameter space where massive proto-cluster (MPC) candidates are thought to be found (see for details). Comparing the upper end of the observed mass–radius distribution, we find 16 clumps satisfying the MPC criteria that are potential precursors to the next generation of young massive clusters (YMCs; i.e., ); these are broadly defined as having stellar masses $\\gtrsim$10000M⊙ and younger than 100Myr ( and references therein). The ATLASGAL region covers approximately 70percent of the Galactic disc (assuming the star formation is distributed between 1-15kpc; ) and therefore this sample should include a large fraction of the total Galactic population of MPC candidates. We find similar numbers of MPC candidates in the northern and southern Galactic plane, which is to be expected since the division is arbitrary. Table[tbl:MPCderivedpara] lists the derived properties for these ATLASGAL clumps and also includes four additional MPC candidates associated with the Galactic centre region identified by and. These objects are also plotted in Fig.[fig:massradiusdistribution] as cyan stars.\nIn PaperI, two additional clumps were identified as potential MPC candidates (i.e., AGAL351.774 − 00.537 and AGAL352.622 − 01.077). However, the distances to these sources are no longer considered reliable and they have therefore been excluded from this updated version of the table. A maser velocity of 1.3kms− 1 was used to determine the distance to AGAL351.774 − 00.537, placing it in the outer Galaxy, but the velocity from an ammonia measurement of − 3kms− 1 and correlation with H i data suggest that a near distance is more likely; this source has been studied in detail by and is thought to be located at a distance of 1.8kpc (). For AGAL352.622 − 01.077, a positive radial velocity would place this source in the outer Galaxy at a distance of ∼ 19kpc, however, this would also locate it too far from the Galactic mid-plane, and so a near distance is more likely in this case as well; this was also noted by.\nThe MPC candidates located in the northern Galactic plane have been previously identified by and so can be considered reasonably robust. We have checked the distances assigned for the southern Galactic-plane candidates and find them to be reliable, with the exception of AGAL329.029 − 00.206, where the distance looks a little less certain. Inspection of the mid-infrared images towards this clump reveals that it is associated with some infrared extinction, which may indicates an association with an infrared-dark cloud and therefore possibly located at the near kinematic distance. place this clump at the far distance with a high level of confidence and so we include it in Table[tbl:MPCderivedpara], with a footnote to indicate that its distance is less robust than the others.\nIn the previous section we used the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ ratio as a means to evaluate the current evolutionary state of the embedded star formation, with higher values being associated with more evolved objects. However, this ratio can also be used as a proxy for the SFE of a given clump. Given that some of the clumps identified as MPC are associated with complexes previously described as ``mini-starbursts\" (e.g., W49 and W51) it is somewhat surprising to find that their SFE values are rather less than spectacular. We might have expected these mini-starbursts to stand out from the general population of MSF clumps and have much higher *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ ratios. compared *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{^{13}CO}}$ ratios for RMS sources matched with molecular clouds identified by the Galactic Ring Survey (GRS; ) and found a ratio for W51 similar to that estimated in this paper from the sub-millimetre continuum (7.6 ± 2.3L⊙/M⊙). In W49, however, they found a value of 32 ± 6L⊙/M⊙, approximately 4 times larger than the ratio derived from the dust emission. Taken at face value this implies that the W49 molecular clouds have a significantly larger fraction of mass in high-density clumps but that the SFE and YSO luminosity distribution within those clumps is not much different from that in other regions.\nThe distribution of the Galactic-centre and disc samples occupy approximately the same region of parameter space in Fig.[fig:massradiusdistribution] and, comparing the clump properties in Table[tbl:MPCderivedpara], we find them to be very similar. The column densities and sizes are very closely matched but the Galactic-centre MPC candidates are marginally more massive. However, a KS test comparing these three parameters reveals no significant differences (*r* = 0.02, 0.23 and 0.01, respectively). In contrast, there is a big difference in the level of star-formation activity associated with the two samples, with very little evidence for *any* star formation taking place within the Galactic centre MPC candidates (), while all of the candidates in the disc are extremely active star-forming regions.\nWe have identified our sample of massive star-forming clumps by matching the ATLASGAL survey results with catalogues of star-formation tracers and so it is not surprising that all of the MPC candidates that we have identified in the disc are star-forming. Two recent detailed studies by and searched for massive starless clumps in the BGPS and in a subsample of the ATLASGAL survey (i.e., $10\\degr < \\ell < 20\\degr$) and, although identified a handful of MPC candidates (e.g., W43 and W49; see Table[tbl:MPCderivedpara] for full list), they found none that are starless. Since a detailed study of all potential MPC candidates has not yet been made of the southern Galactic plane, similar to that performed by in the northern Galactic plane, it is possible that there are some starless MPC candidates located in the disc that have yet to be discovered, the number is likely to be very small and perhaps even zero.\nIt is currently unclear why the observed level of star formation is so very different for the Galactic centre and disc populations, particularly since their clump properties are so similar. This would not be expected if the formation processes were so very different. The most significant difference is that the velocity dispersion is significantly larger and gas temperatures are higher for the Galactic-centre population (a factor or a few) compared to those in the disc. It seems more plausible that the formation mechanism is the same but that the much more extreme Galactic-centre environment is somehow impeding the star formation in its local MPC candidate population. The difference in star-formation activity has led to suggestions that the mode of star formation is fundamentally different in the Galactic centre compared to the disc (see for a detailed discussion).\nSummary and conclusions\n=======================\nThis is the third in a series of papers that use the ATLASGAL survey to conduct a detailed and comprehensive investigation of massive star-formation environments. The main aim of these papers is to use the unbiased coverage and uniform sensitivity of the ATLASGAL survey to map the dust emission over the inner Galactic Plane to connect the results derived from samples selected using different high-mass star formation tracers. We have combined the ATLASGAL and RMS surveys to identify a large sample of massive, submillimetre-continuum-traced clumps associated with the formation of massive stars. These MYSO and H ii-region associated clumps are complemented with similar clumps identified through their association with either methanol masers or compact radio-continuum emission (for details see PapersI and II). We have matched the ATLASGAL sources with additional methanol masers identified outside the MMB region to obtain a more complete sample covering the whole inner Galactic plane. In total we have identified ∼ 1700 embedded massive stars with ∼ 1300 clumps, which is the largest sample of massive star-forming clumps yet compiled and includes a large fraction of all embedded massive stars in the Galaxy. This represents ∼ 15percent of the whole catalogue of clumps identified by ATLASGAL in the regions in which the surveys are complete ($280\\degr < \\ell < 350\\degr$ and $10\\degr < \\ell < 60\\degr$).\nUsing distances drawn from the literature, we derive clump masses and column densities, bolometric luminosities, physical sizes and virial masses for the majority of these associated clumps. These clumps have typical sizes and masses of ∼ 1pc and a few thousand M⊙, respectively. The embedded sources have luminosities of ∼ 104L⊙ and appear to be warming their local environments with typical gas temperatures of ∼ 20K.\nWe find that approximately a third of all clumps are associated with two or more massive-star formation tracers and are therefore likely to host multiple evolutionary stages. These clumps present a problem when trying to attribute derived physical properties to a specific evolutionary stage and so have been excluded from the detailed analysis. We have identified ∼ 100 methanol masers that are positionally coincident ( < 2) with a MYSO or H ii region; this angular size corresponds to less than 0.1pc at 10kpc and therefore these sources are located in the same core within the larger clump and very likely to be tracing the same embedded object. These “multi-phase” sources may represent interesting evolutionary transition objects but also reveal that methanol masers are associated with a wide range of evolutionary stages. Approximately 20percent of the methanol masers are positionally associated with a MYSO and a further ∼ 10percent are associated with an H ii region and therefore care needs to be taken to identify those associated with the pre-MYSO and H ii region stages. For the purposes of our analysis, if a methanol maser is found to be coincident with a MYSO or H ii region it has been classified as a MYSO or H ii region, respectively.\nWe separate the sample of the clumps into four observationally distinct subsamples; these are methanol-maser, MYSO, H ii-region and multi-phase associated clumps. We compare statistical properties of these subsamples to look for evidence to support the commonly assumed evolutionary sequence: methanol maser → MYSO → H ii region. Our main findings are as follows:\n1. Clumps associated with a massive-star formation tracer are more spherical in morphology and centrally condensed than the rest of the quiescent population of ATLASGAL clumps. We find that the young massive stars identified are tightly correlated with the position of the peak of the submillimetre emission found towards the centres of their host clumps, with typical projected separations of ∼ 0.14pc. The most massive stars are therefore predominately found towards the centres of spherical, centrally condensed clumps.\n2. The massive star-forming clumps identified are strongly associated with the highest column-density clumps in the Galaxy. Essentially *a*ll clumps with column densities in excess of 2–3 × 1023 cm− 2 are host to current massive star formation. There is also a significant correlation between column density and clump mass with the most massive clumps also having the highest column densities. Our samples of MSF clumps therefore include all of the most massive clumps located in the Galactic disc.\n3. No significant difference is found between the *Y*-factors, aspect ratios or relative location within the clump, with respect to the submm peak, for any of the MSF subsamples, which would suggest that the clump structure changes little as the embedded objects evolve from the protostellar, through the MYSO to the compact H ii-region stages. There are clearly differences between the clumps currently associated with massive star formation and the more quiescent population of clumps, which suggests that the structure of the pre-stellar clumps must evolve significantly before star formation begins but, once underway, it proceeds faster than the clump can evolve. This is consistent with a fast star-formation process ().\n4. Almost all of the subsample of massive star forming clumps for which we have ammonia linewidths have a virial parameter *α* < 2 and, unless supported by strong magnetic fields, are gravitationally unstable against collapse. These clumps are thus likely to be undergoing global infall. Furthermore, we find a strong anti-correlation between clump mass and the virial parameter, with the virial parameters decreasing with increasing clump mass (cf. ). The most massive clumps are therefore the least stable against gravity, which provides a simple explanation as to why no massive pre-stellar clumps have yet been identified in the Galactic disc.\n5. We find a very strong correlation between clump mass and the bolometric luminosity of the embedded methanol-maser sources (*r* = 0.58) and MYSOs and H ii regions (*r* = 0.64). The bolometric luminosity for all embedded sources is linearly dependent on clump mass. We therefore conclude that the most massive stars are forming in the most massive and highest column-density clumps.\n6. The MYSOs, H ii regions and multi-phase subsamples form a continuous distribution over three orders of magnitude in mass and four orders of magnitude in luminosity. All of these clumps are clustered at approximately *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ = 10L⊙/M⊙, which, according to the scenario outlined by, is the transition point the main accretion and envelope clean-up phases in the formation of massive stars. This suggests that the MYSOs and H ii regions occupy a similar range of evolutionary stages with the majority near the end of their main accretion phase.\n7. Despite the close correspondence of the *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ distributions of clumps containing MYSOs and H ii regions, these two subsamples dominate different ends of the mass-luminosity parameter space. The MYSOs dominate the lower-mass and lower-luminosity part of the distribution while the H ii regions occupy the more massive and luminous end. However, these two subsamples together form a continuous distribution with no break of slope, which might be expected if different underlying processes were producing the observed luminosity.\n8. MYSOs are associated with significantly lower mass and column-density clumps compared with the H ii region subsample. This is also true for a distance-limited subset of the data and so is unlikely to stem from any distance-related sensitivity bias. However, the differences between these two subsamples are likely to result from the higher column densities associated with the most massive clumps as these are theoretically predicted to lead to higher accretion rates and faster evolution times. We conclude that the MYSOs and H ii regions cover a similar range of evolutionary stages, and the main difference between them is due to a decreasing MYSO lifetime with increasing clump mass as also suggested by and. A consequence of the faster evolutionary times may be that the MYSO stage may not be observable for the most massive stars.\n9. The properties of the clumps hosting methanol masers and H ii regions are much more closely matched. The KS test is unable to distinguish between their clump-mass and column-density distributions. However, the methanol masers are significantly less luminous than the H ii regions. The *L*$\\_{\\rm{bol}}$/*M*$\\_{\\rm{clump}}$ ratio for the methanol masers is ∼ 7L⊙/M⊙, which is approximately half the value found for the MYSOs or H ii region associated clumps and so these are likely to be in an earlier evolutionary stage. The methanol masers associated with the most massive clumps are therefore a promising sample of early O-type star precursor candidates.\n10. The bolometric luminosities determined for the most massive clumps fall far below those expected for an embedded stellar cluster for a fixed SFE at all clump masses, but are more consistent with the luminosity expected from the most massive stars in a cluster. They are also consistent with ZAMS luminosities derived independently from the radio emission from the compact H ii regions. This suggests that the observed luminosities are primarily coming from the most massive embedded stars in the forming cluster and that the rest of the lower-mass stars have yet to make a significant contribution to the cluster luminosity.\n11. We compare the clump masses and radii to the empirical criterion derived by for high-mass star formation and those derived by and for “efficient” low-mass star formation. We find a strong correlation (*r* = 0.85) for all associated clumps that covers 2 orders of magnitude in radius and four orders of magnitude in clump mass. We find no significant difference between the different subsamples. All sources are above the threshold for “efficient” low-mass star formation and all but a handful of sources satisfy the criterion and so these clumps have the potential to form massive stars.\n12. We have identified a sample of 16 MPC candidates located in the Galactic disc. This sample is likely to consist of a large fraction of the whole Galactic population of these objects. We compare the properties with those of a similar sample found towards the Galactic centre and find most of their properties to be similar (mass, size and densities). However, while the MPC candidates found in the disc are very actively forming massive stars, with a few being classified as mini-starbursts, the Galactic-centre candidates are effectively starless.\nFinally ATLASGAL provides a uniform set of measurements that can be used to put the result of these previous studies in a more Galaxy-wide context and in the developing evolutionary sequence for the formation of massive stars. This sample of massive-star-forming clumps we have compiled should provide an ideal starting point for more detailed studies in future, particularly with ALMA.\nAcknowledgments\n===============\nWe thank Steve Longmore, Adam Ginsburg and Padelis Papadopoulos for discussions concerning the MPC candidate sources, and Mark Heyer and Rosie Chen for some interesting discussions on an earlier draft of the paper. We would also like to thank the referee for their comments and suggestions that have helped to significantly improve this manuscript. The ATLASGAL project is a collaboration between the Max-Planck-Gesellschaft, the European Southern Observatory (ESO) and the Universidad de Chile. This research has made use of the SIMBAD database operated at CDS, Strasbourg, France. This work was partially funded by the ERC Advanced Investigator Grant GLOSTAR (247078) and was partially carried out within the Collaborative Research Council 956, sub-project A6, funded by the Deutsche Forschungsgemeinschaft (DFG). This paper made use of information from the Red MSX Source survey database at http://rms.leeds.ac.uk/cgi-bin/public/RMS\\_DATABASE.cgi which was constructed with support from the Science and Technology Facilities Council of the UK. L. B. acknowledges support from CONICYT project Basal PFB-06.\n---\n1. E-mail: jurquhart@mpifr-bonn.mpg.de (MPIfR)[↩](#fnref1)\n2. The full version of Tables1, 3, 6 and 7, and Fig.3 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/MNRAS/.[↩](#fnref2)\n3. The radio emission from these extended H ii regions is either filtered out by the interferometric radio continuum observations (i.e., ) or their surface brightness, which decreases as they expand, has fallen below the sensitivity of these observations (see discussion in PaperII for more details).[↩](#fnref3)\n4. The aspect ratio and compactness parameter are discussed in the Sect.3.2 while derivation and analysis of the rest of the parameters (i.e., distance, clump mass etc) is given in Sect.4.[↩](#fnref4)\n5. The *Y*-factor is defined as the ratio of integrated to peak submillimetre fluxes and provides an estimate of how centrally condensed the emission is towards the centre of the clump.[↩](#fnref5)\n6. Note this is an important difference to the method previously used () where the mid/far-infrared fluxes were apportioned before the SEDs were fitted.[↩](#fnref6)\n7. http://herschel.esac.esa.int/Science\\_Archive.shtml.[↩](#fnref7)\n8. Note we have reduced the value of the mass coefficient given by by 1.5 to match the dust opacity value used in this paper to estimate clump masses[↩](#fnref8)\n\nATLASGAL — towards a complete sample of massive star forming clumps[2](#fn2)\n============================================================================\n### Accepted??. Received??; in original form??\n[firstpage]\nBy matching infrared-selected, massive young stellar objects (MYSOs) and compact HII regions in the RMS survey to massive clumps found in the submillimetre ATLASGAL survey, we have identified ∼ 1000 embedded young massive stars between $280\\degr$ < ℓ < $350\\degr$ and $10\\degr$ < ℓ < $60\\degr$ with $|\\,b\\,|<1.5\\degr$. Combined with an existing sample of radio-selected methanol masers and compact HII regions, the result is a catalogue of ∼ 1700 massive stars embedded within ∼ 1300 clumps located across the inner Galaxy, containing three observationally distinct subsamples, methanol-maser, MYSO and HII-region associations, covering the most important tracers of massive star formation, thought to represent key stages of evolution. We find that massive star formation is strongly correlated with the regions of highest column density in spherical, centrally condensed clumps. We find no significant differences between the three samples in clump structure or the relative location of the embedded stars, which suggests that the structure of a clump is set before the onset of star formation, and changes little as the embedded object evolves towards the main sequence. There is a strong linear correlation between clump mass and bolometric luminosity, with the most massive stars forming in the most massive clumps. We find that the MYSO and HII-region subsamples are likely to cover a similar range of evolutionary stages and that the majority are near the end of their main accretion phase. We find few infrared-bright MYSOs associated with the most massive clumps, probably due to very short pre-main sequence lifetimes in the most luminous sources.\nStars: formation – Stars: early-type – ISM: clouds – ISM: submillimetre – ISM: H ii regions.\nIntroduction\n============\nMassive stars ( > 8M⊙ and 103L⊙) play an important role in many astrophysical processes, from the formation of the first solid material in the early Universe () to their substantial influence upon the evolution of their host galaxies and future generations of star formation (). These stars have a profound impact on their local environment through powerful outflows, strong stellar winds, optical/far-UV radiation and supernovae, shaping the interstellar medium (ISM) and regulating star formation. Our understanding of how galaxies evolve over cosmological timescales depends critically on various assumptions that are made about massive star formation, such as how the feedback introduced by massive stars affects the star-formation rate, or the universality of the initial mass function (IMF). However, the underlying physical processes involved in the formation of massive stars are still poorly defined and this presents a major hurdle to furthering understanding of the massive star formation itself and how galaxies evolve (e.g., and ).\nOutside the Milky Way, we are generally restricted to studies of global star-formation properties integrated over entire star-forming complexes or even groups of complexes, which renders it impossible to study either feedback processes or the environmental dependence of star formation (). Within the Milky Way we are able to probe massive star-forming regions in far greater detail, which provides the best opportunity of understanding the processes involved in massive star formation. Furthermore, the Milky Way consists of a large range of environments including the Galactic centre with its extreme UV-radiation and cosmic-ray fluxes, intense star-forming regions found in the spiral arms (e.g., W43, W49 & W51; often described as “mini-starbursts”) and the reduced metallicity of the outer Galaxy (). Many of these environments have extragalactic analogues and an understanding of how they affect star formation in the Milky Way will provide crucial insights into star formation in the era of galaxy formation.\nDespite their importance, our understanding of the initial conditions required and processes involved in the formation and early evolution of massive stars is still rather poor in comparison to low-mass stars. Massive stars are rare and the regions where they form are significantly more distant than low-mass star forming regions. Massive stars form almost exclusively in clusters (), making it hard to distinguish between the properties of the cluster and individual members. Furthermore, the Kelvin-Helmholtz contraction time is shorter than the accretion time for massive stars. Consequently, massive protostars reach the main sequence while still deeply embedded in their natal environment and so all of the earliest stages can only be probed at far-infrared and (sub)millimetre wavelengths.\nIt is still not yet clear how massive stars obtain their final mass as much of this needs to be acquired after they have reached the main sequence when radiation pressure would be expected to halt accretion. this problem can be overcome through accretion via a disc, however, the mechanism by which this happens is still an open question. Currently, there are two competing accretion models by which an embedded protostar can acquire the mass required to evolve into a massive star; these are the “monolithic accretion” and “competitive accretion” models (e.g., and, respectively).\nOur ability to make significant progress in this field has been dramatically enhanced in recent years with the completion of a large number of Galactic-plane surveys that cover the whole wavelength range from the near-infrared to the radio (e.g., the UKIDSS GPS, ; GLIMPSE, ; MSX, ; MIPSGAL, ; *Herschel* Hi-GAL, ); APEX Telescope Large Area Survey of the Galaxy (ATLASGAL), ; Bolocam Galactic Plane Survey (BGPS), ; methanol multibeam (MMB) survey, ; and the Co-ordinated Radio and Infrared Survey for High-Mass Star Formation (CORNISH), and ). These surveys probe the large spatial volumes required to address the major problem in studying Galactic massive star formation — its intrinsic rarity. At last we can identify and study statistically significant samples over the full range of evolutionary stages, from the pre-stellar through to the post-compact HII region stage when the star emerges from its natal environment.\nEvolutionary sequence\n---------------------\nIn spite of the observational challenges associated with massive star formation, an outline of an evolutionary sequence has emerged over the past decade from a combination of observations and numerical modeling (e.g., ). This sequence can be roughly divided into four distinct stages:\n1. Pre-stellar: consisting of a massive, dense clump that is gravitationally bound but shows no evidence of being associated with any embedded point source at infrared wavelengths or other tracers.\n2. Clump-core collapse: the clumps collapse and fragment into cores that form a protostar surrounded by an accretion disc. As the protostar gains mass it begins to warm its natal envelope, exciting class II methanol-maser emission through radiative pumping in the disc ().\n3. Protostellar swelling and contraction: Numerical simulations with high accretion rates have been performed by and. These models reveal that the high accretion rates lead to a dramatic swelling up of the protostar from a few solar radii to ∼ 100R⊙ as its stellar mass increases from 5 to 10M⊙; the increased volume of the protostar results in lower internal densities and temperatures, which delays the onset of core hydrogen burning (see also ). Above 10M⊙ the Kelvin-Helmholtz timescale is lower than the accretion timescale and as a consequence the protostar begins to contract and the internal temperature gradually increases. When the temperature exceeds 107K core hydrogen burning begins and the star joins the main sequence with a mass of ∼ 30M⊙. This implies there are no stars more massive than this at ZAMS onset and accretion must continue post-ZAMS for larger stars. Observationally, protostars that have not yet begun to form a H ii region will have luminosities between 103 and 105L⊙; we will refer to these objects as massive young stellar objects (MYSOs; ). The RMS survey has identified ∼ 600 MYSOs located throughout the Galaxy (). The most luminous of these correspond to the range predicted by the Hosokawa et al. models (i.e., ≤ 105L⊙).\n4. Formation of the H ii region: once core hydrogen burning has begun and the star has joined the main sequence it will begin to emit a sufficient amount of ionizing radiation and a H ii region will form around it. This will begin as a hyper-compact H ii region but will rapidly expand, transitioning through the ultra-compact (UC) and compact H ii region phases before breaking out of its natal clump and evolving into the more classical H ii regions that are observable at optical wavelengths. The thermal bremsstrahlung emission from the ionized gas within the H ii regions can be readily detected at radio wavelengths (e.g., and ). Radio continuum observations can therefore be used to distinguish between MYSOs and H ii regions ().\nIn order to test this evolutionary sequence we need to compile a large sample of massive star-forming regions with a sufficient number of sources that the statistical properties of each stage can be determined.\nIdentifying an evolutionary sample of massive star forming clumps\n-----------------------------------------------------------------\nAll of the earliest stages of massive star formation take place while the forming stars are deeply embedded in dense clumps (), where they may remain hidden, even at mid-infrared wavelengths. However, the thermal emission from dust at submm wavelengths is optically thin and is therefore an excellent tracer of column density and total mass of a clump (). The ATLASGAL survey is a blind 870-$\\umu$m survey covering 420 square degrees of the inner Galactic plane. Its unbiased coverage and fairly uniform sensitivity provides an ideal resource for identifying a large and representative sample of massive, star-forming structures (, Urquhart et al. 2014b). Furthermore, ATLASGAL is complete to all dense clumps (Log(*N*(H2)) > 22cm− 2) with masses ($M\\_{\\rm{clump}}$) > 1000M⊙ located within the inner Galactic disc at heliocentric distances up to ∼ 20kpc (; hereafter PaperI) and provides a uniform set of measurements that can be used to test the proposed evolutionary sequence.\nIn two previous studies we have used the ATLASGAL Compact Source Catalogue (CSC; ) as a starting point and cross-matched the positions of dense clumps with those of methanol masers identified by the MMB survey (PaperI) and compact H ii regions identified by the CORNISH VLA 5GHz continuum survey (; hereafter PaperII). These two studies identified a large sample of ∼ 800 massive star-forming clumps. This was used to derive a minimum surface density threshold for massive star formation and provided some of the strongest evidence that methanol masers trace one of the earliest stages in the formation of massive stars. Furthermore, we used the Galactic distribution of the methanol masers to estimate the star-formation efficiency (SFE) across the inner Galaxy and found it to be approximately three times lower in the Galactic Centre than in the disc (cf. ).\nIn this paper, the final in this series, we investigate the properties of the ATLASGAL clumps associated with MYSOs and H ii regions identified by the Red MSX Source (RMS) survey (). Inclusion of this sample of young massive stars will produce the largest and most comprehensive sample of massive star-forming clumps to date and cover the whole accretion process from the earliest, most deeply embedded stage to the creation and expansion of an ionized nebula that disrupts the natal environment, halting accretion and setting the final mass of the star.\nThe structure of this paper is as follows: in Sect.2 we give a brief summary of the individual surveys used and describe the matching procedure and reliability checks performed. In Sect.3 we discuss the correlation of RMS-associated clumps with the samples of methanol masers and H ii regions presented in PapersI and II and look at the Galactic distribution of the whole sample. In this section we also distinguish between clumps associated with two or more of the different tracers from those associated with a single tracer (i.e., methanol-maser, mid-infrared bright and compact radio emission) in order to identify relatively clean samples with which to look for evolutionary trends that might be present in these data. In Sect.4 we derive properties of the associated clumps and compare the statistical properties of the observationally defined subsamples. In Sect.5 we investigate the empirical relationships between clump mass, bolometric luminosity and radius with respect to the different subsamples and discuss their implications. We present a summary of our main findings and give an overview of this series of papers in Sect.6.\nSurvey descriptions and matching statistics\n===========================================\n[fig:galdistributionnodata] Schematic of the Galactic disc as viewed from the Northern Galactic Pole showing the positions of the Galactic bar and the spiral arms. The orange wedges show the coverage of the ATLASGAL survey assuming a mass sensitivity of 1,000M⊙ at a distance of 20kpc. The brighter wedge shows the inner most 10 in longitude not included in the RMS survey. The spiral arms are labeled in black and Galactic quadrants are given by the roman numerals in the corners of the image. The Galactic longitude covered by the AMGPS, MMB and CORNISH surveys are indicated by red, blue and green curves, respectively. The Sun is located at the apex of the wedges and is indicated by the ⊙ symbol. The dot-dashed cyan circle shows the Solar circle and the inner most thick green circle shows the location of the 5kpc molecular ring where most of the molecular gas in the Galaxy resides. Background image courtesy of NASA/JPL-Caltech/R. Hurt (SSC/Caltech).\nATLASGAL\n--------\nATLASGAL is one of the first systematic surveys of the inner Galactic plane in the submillimetre band. The survey was carried out with the Large APEX Bolometer Camera (LABOCA; ), an array of 295 bolometers observing at 870$\\umu$m (345GHz). The APEX telescope has a full-width half-maximum (FWHM) beam size of 19.2 at this frequency and has a positional accuracy of ∼ 2. The initial survey covered a Galactic longitude region of $300\\degr < \\ell < 60\\degr$ and $|b| < 1.5\\degr$ but this was later extended to include $280\\degr < \\ell < 300\\degr$ and $-2\\degr < b < 1\\degr$ to take account of the warp present in this part of the Galactic plane (). This extension increasing the overall coverage of the survey to 420sq.degrees and includes the Sagittarius spiral arm tangent and the Carina star-forming complex.\nUsing the source extraction algorithm `SExtractor` () a total of 10000 dust clumps have been identified in the ATLASGAL emission maps. An initial catalogue of the inner 150sq. degrees of the Galactic plane and a detailed description of the method and consistency checks are presented in while a description of the rest of the catalogue is presented in a subsequent research note (Urquhart et al., 2014). The whole catalogue is 99percent complete at ∼ 6*σ*, which corresponds to a flux sensitivity of 0.3-0.5Jybeam− 1 (see Fig.1 of for survey sensitivity as a function of Galactic longitude). The effective radius of the clumps range from 20-100, with a mean of and median value of ∼ 30, and so the majority of sources are extended with respect to the APEX beam.\nWe present a schematic of the Galactic plane in Fig.[fig:galdistributionnodata] showing the ATLASGAL coverage of the inner Galactic disc. This survey covers ∼ 2/3 of the surface area of the Galactic disc (assuming the molecular disc has a Galactocentric radius, $R\\_{\\rm{GC}}$, < 15kpc) and 97percent of the surface area located inside the Solar circle and all of the 5kpc molecular ring where the vast majority of the molecular gas in the Galaxy is located (i.e., 4kpc $< R\\_{\\rm{GC}} < 6$kpc; ). The full ATLASGAL compact source catalogue (CSC) therefore provides a comprehensive census of dense dust clumps located in the inner Galaxy and is complete to all potential massive star-forming compact clumps with masses greater than 1000M⊙ out to a heliocentric distance of 20kpc (see Sect.4.5 and PaperI for details).\n \nRMS Survey\n----------\nThe RMS survey is a Galaxy-wide sample of massive young stellar candidates selected by comparing near- and mid-infrared colours to those of previously identified MYSOs and H ii regions (). Although near-infrared images were used to help aid in the colour selection a detection in this wavelength range was not required since many will be deeply embedded and do not have a near-infrared counterpart. This initially, mid-infrared-selected sample has been followed-up by a comprehensive multi-wavelength programme of observations. These observations were carefully chosen to distinguish between genuine embedded MYSOs and H ii regions from other dust-enshrouded contaminating sources (e.g., evolved stars, planetary nebula and nearby low-mass YSOs). These included radio continuum observations (), a mixture of targeted mid-infrared imaging (e.g., ) and archival images obtained by the *Spitzer* GLIMPSE legacy project (), near-infrared spectroscopy (e.g., ) and molecular-line observations to obtain kinematic distances (e.g., ).\nA detailed description of these follow-up observations and discussion of the properties of the final catalogue are presented in. The RMS survey has identified approximately ∼ 1700 MYSOs and H ii regions and represents the largest and best-characterised sample compiled to date. Distances are available for ∼ 90percent of the catalogue; these are a compilation of maser parallax (e.g., ), spectrophotometric (e.g., ) and kinematic distances taken from the literature (e.g., ) and the RMS project’s own analysis (see and for more details).\n[tbl:atlasrmsmatches]\nll.l.c & && & & \n& && & & \nAGAL010.323 − 00.161 & G010.3208 − 00.1570B & 3.5 & YSO & 4.4 & no \nAGAL010.439+00.009 & G010.4413+00.0101 & 7.6 & H ii region & 4.4 & no \nAGAL010.472+00.027 & G010.4718+00.0256 & 1.1 & H ii region & 5.5 & yes \nAGAL010.624 − 00.384 & G010.6235 − 00.3834 & 2.6 & H ii region & 5.6 & yes \nAGAL010.884+00.122 & G010.8856+00.1221 & 3.5 & YSO & 3.8 & no \nAGAL010.957+00.022 & G010.9592+00.0217 & 3.2 & H ii region & 5.1 & yes \nAGAL011.034+00.061 & G011.0340+00.0629 & 6.0 & H ii region & 5.0 & yes \nAGAL011.171 − 00.064 & G011.1723 − 00.0656 & 4.9 & H ii region & 4.4 & yes \nAGAL011.497 − 01.485 & G011.5001 − 01.4857 & 7.0 & YSO & 3.3 & no \nAGAL011.756 − 00.149 & G011.7588 − 00.1502 & 7.3 & H ii region & 4.1 & no \n \n \nNotes: Only a small portion of the data is provided here, the full table is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/MNRAS/.\nATLASGAL-RMS associations\n-------------------------\nOf the ∼ 1700 MYSOs and H ii regions identified by the RMS survey 1232 are located in the region covered by ATLASGAL; this corresponds to approximately 80percent of the RMS catalogue. Following the same matching procedure developed for the CORNISH comparison we have used the ATLASGAL emission masks to map these embedded RMS sources to their host dust emission clumps (see Sect.3.1 of PaperII for more details). The emission masks are produced by the extraction algorithm and consist of an image the same dimensions as the original emission map but where the pixel values have been changed to integer values that relate each pixel to the catalogue identification number. Matching the source position with its associated pixel therefore provides an unambiguous match to the ATLASGAL source, even in cases where the clump morphology is irregular. We have inspected the mid-infrared images, dust emission and positions of the RMS sources for all detections and identified a small number of sources (22) where the match did not appear to be reliable. These tended to have the largest angular separation (between ∼ 40-250; see Fig.[fig:angularoffset] for typical angular offsets) between the dust and the RMS source and were found to be more extended H ii regions that have broken out of their natal environment. Any properties derived for their associated clump are unlikely to provide any reliable information about the relationship between the host clump and its associated star formation and so these have been removed from the sample.\nAfter the exclusion of this relatively small number of less reliable matches we are left with a sample of 1056 MYSO and H ii regions for which distances and luminosities have been determined. We further filter this sample to exclude sources with luminosities lower than 1000L⊙ (see and Sect.[sect:bollum] for details on the luminosity determination); this is to ensure that the sample contains only embedded sources that are going to become massive stars. This reduces the sample to 964, which corresponds to ∼ 90percent of the RMS catalogue located within the ATLASGAL region and includes approximately two-thirds of the whole RMS sample of these objects for which a distance is available. These RMS sources are associated with 768 ATLASGAL clumps, with 151 clumps harbouring two or more embedded massive stars. We failed to find a match for 154 RMS sources but the majority of these were found to be more extended H ii regions that have already disrupted their natal environment to the extent that the dust emission falls below the ATLASGAL sensitivity or lower luminosity YSOs. Of the other 92 RMS source matched to an ATLASGAL clump (i.e., 1056 − 964), distances are not available for 75 matches and 17 have bolometric luminosities below the threshold (i.e., < 1000L⊙). See Table[tbl:atlasrmsmatchesbreakdown] for a summary of these ATLASGAL-RMS matches.\n[tbl:atlasrmsmatchesbreakdown]\nlr & \nTotal & 1056 \nLuminosity > 1000L⊙& 964 \nLuminosity < 1000L⊙& 17 \nNo distance available& 75 \n \n \nThe associated RMS sample consists of 361 MYSOs, 25 MYSO/H ii regions and 578 H ii regions. The MYSO/H ii regions classification is used when there is insufficient data to distinguish between these two stages. We checked the matches for a possible systematic offset between the RMS and ATLASGAL matches and although there is evidence of a subtle effect this is small compared to the APEX pointing uncertainty and so can be neglected.\nThe RMS sample of H ii regions consists of a mixture of compact and UC H ii regions that are associated with compact radio emission and more extended H ii regions that are radio-quiet but can be classified from their mid-infrared morphology.[3](#fn3) It is therefore useful to further distinguish between the compact and extended H ii regions depending on their association or absence of radio emission; in this way we classify 339 H ii regions as compact (associated with radio emission) and 239 H ii regions as extended. In Table[tbl:atlasrmsmatches] we present a complete list of ATLASGAL-RMS matches and the angular offset between the embedded object and the peak of the dust emission, the RMS source classification, bolometric luminosity and a flag to indicate the RMS sources that are associated with compact radio emission.\nIn the left panel of Fig.[fig:angularoffset], we plot the surface density of sources as a function of the angular offset of the tracer position from the peak of the host-clump submillimetre emission, for all matches with separations less than 40. The distribution is clearly peaked towards smaller offsets, revealing a strong correlation between the positions of the embedded MYSOs and H ii regions and the submillimetre emission peak. Approximately, 70percent of the matches have angular separations less than 10 and over 85percent have separations less than 20. Only 34 matches have angular separations between ∼ 40 and 80arcseconds. The majority of these are classified as radio quiet H ii regions (20), which suggests they are more evolved and may be close to breaking out of their natal clump. Of the remaining fourteen large-separation matches, five are compact H ii regions and nine are MYSOs; these appear to be geniune associations and their location on the edges of these clumps may be the result of an interaction between the clump and its environment (e.g., ).\nIn PapersI and II we found similar strong positional correlations between the peak of the dust emission and both the methanol masers and radio-continuum traced compact H ii regions. The right panel of Fig.[fig:angularoffset] shows the cumulative distribution of the angular separations of the RMS sources, together with those of the previous two source samples. It is clear that there is no significant difference in the distribution of separations between the three samples which are all strongly correlated with the position of the peak dust emission. The young embedded massive stars pin-pointed by these tracers are preferentially found towards the centre of their host clumps, where the highest column and volume densities are likely to be found. Furthermore, finding little difference between the different samples, which are expected to cover all of the important embedded stages, we suggest that the relative position of the forming massive star within its host clump does not change significantly over the course of its evolution.\nAll three distributions are, however, significantly different to that of the radio-quiet (extended) H ii regions.1 This is consistent with our hypothesis that the radio-quiet H ii regions are more evolved than the more compact radio-loud H ii regions. As the H ii region expands it is likely to have a disruptive influence on the surrounding molecular gas and dust, resulting in the peak of the dust emission being pushed away from the centre of the H ii region (see for detailed discussion of these processes). If we exclude these extended H ii regions then we find that ∼ 85percent of the embedded objects are located within 10 and that ∼ 95percent of sources are located within 20 of the sub-millimetre peak.\nMethanol-maser associations between ℓ=20-60\n-------------------------------------------\nIn PaperI we presented a detailed comparison between methanol masers identified by the MMB survey and the properties of their host clumps. This study was restricted to the MMB survey alone as it provides an large sample of sources and unbiased coverage of a large region of the Galaxy and uniform sensitivity, make it ideal for conducting a detailed statistical study. However, while the the MMB catalogue covers the all of the southern Galactic plane (∣*b*∣ < 2) it only covers the innermost 20 of the northern Galactic plane (i.e., $\\ell < 20\\degr$) and in order to examine the larger-scale Galactic distribution we need to expand our sample of methanol masers and their host clumps to include the whole of the northern Galactic plane covered by ATLASGAL.\nArecibo Methanol Maser Galactic Plane Survey (AMGPS; ) is a blind survey covering approximately 20 square degrees of the northern Galactic plane (ℓ = 35.2-53.7 and ∣*b*∣ < 0.41). The majority of the detected masers have been followed up with MERLIN to obtain accurate positions (). We have used these updated positions to determine their association with ATLASGAL sources and, where a match has been identified, to calculate the angular offset between the peak of the submillimetre emission and the maser. We have matched 59 of the 86 methanol masers reported by ; these are associated with 57 ATLASGAL clumps with three masers found to be associated with a single clump.\nFor the regions not covered by the MMB or AMGPS surveys we have used the catalogue of methanol masers compiled by. We note that this catalogue has been compiled from a mixture of interferometric and single-dish data and, as a consequence, the positional accuracy is not as consistent as the MMB and AMGPS surveys. As methanol masers are thought to be intimately associated with the star-formation process one would expect them to be coincident with an embedded thermal source and, although the extinction will be high at mid-infrared and shorter wavelengths, these objects should be detected in the far infrared. We have therefore cross-matched these with *Herschel* 70 and 160$\\umu$m HiGAL images () and removed any matches where the methanol maser is not coincident with a far-infrared point source. This insures that only methanol masers with accurate positions are included in the analysis that follows. In total 53 methanol masers from the catalogue are matched with 53 ATLASGAL clumps.\n### Combined methanol maser sample\nIn total, we have matched an additional 112 methanol masers from these two catalogues that are associated with 110 ATLASGAL clumps in the ℓ = 20-60 region not covered by the MMB survey. This brings the total number of methanol-maser-associated clumps to 740 and represents the largest sample of methanol masers and their host clumps. However, this sample includes 146 methanol masers located within 10 of the Galactic Centre (i.e., ∣ℓ∣ < 10), which is not covered by the RMS or CORNISH surveys and so these are not included in the analyse presented in this paper. This reduces the number of methanol maser associations considered here to 594.\nSurvey sensitivities and coverage\n---------------------------------\nWe have previously mentioned that the ATLASGAL is complete to all compact clumps with masses in excess of 1000M⊙ to a heliocentric distance of 20kpc across the inner Galactic disc. This is shown by the orange shaded region in Fig.1. The only significant feature beyond 20kpc in this longitude range is the farside segment of the Scutum-Centaurus arm, however, this is known to be rising out of the plane () in this direction and is unlikely to be covered in the ATLASGAL survey.\n estimated that the RMS survey is ∼ 90percent complete for all MYSOs and compact H ii regions with luminosities over a few 104L⊙ to also to a distance of 20kpc; this is roughly equivalent to a zero age main sequence (ZAMS) star of spectral type of B0 or earlier. The CORNISH survey has a 5*σ* sensitivity of 2mJybeam− 1, which is sufficient to detected an unresolved, optically thin, H ii region powered by a ZAMS star with a spectral type of B0.5 or earlier to the same distances as the RMS and ATLASGAL surveys (). Finally, the MMB survey has a flux sensitivity of 0.17Jybeam− 1, which corresponds to a luminosity of ∼ 1000Jykpc2 at 20kpc (see Fig.12 of PaperI). The AMGPS has a similar flux sensitivity as the MMB and is therefore likely to have a similar luminosity sensitivity.\nWith the exception of some of the outer Galactic, warped disc, all the constituent surveys are therefore effectively complete, above the relevant limits and within the observed longitude ranges, out to the edge of the Galaxy. However, to avoid introducing a distance bias when comparing the properties of the clumps associated with the different tracers we will use a distance-limited sample that we define as having heliocentric distances between 3 and 5kpc; this range has been chosen as it includes the highest source densities of all three tracers and covers a large range of clump properties (masses, column densities and sizes) and therefore facilitates statistical robust comparisons between different subsamples (see Sect.4.1 for more details).\nAlthough the surveys have comparable completeness the overlap in Galactic longitude and latitude between them is not homogeneous (see Fig.1 for longitude range covered by each survey) and so care needs to be taken to avoid introducing a selection bias. We have a uniform coverage of our sample of MYSOs and H ii region over the whole ATLASGAL region where ∣ℓ∣ > 10, however, we only have unbiased coverage for the methanol masers in the regions covered by the MMB and AMGPS (i.e., $280\\degr < \\ell < 20\\degr$ and $35\\degr < \\ell < 54\\degr$). It is this more limited range in longitude that will be used in the analysis to come when comparing the relative number of each type of association.\nMassive star forming clumps\n===========================\nCombining the associations discussed in the previous section with those presented in PapersI and II, we have identified ∼ 1700 embedded sources that are associated with 1256 ATLASGAL clumps. Of these clumps, 1131 are located within the $280\\degr<\\ell < 350\\degr$ and $10\\degr<\\ell < 60\\degr$ region where the overlap between the different surveys is reasonably uniform, which corresponds to approximately 15percent of the 7255 of ATLASGAL clumps within this region see Fig.[fig:galdistributionnodata] for survey coverage and source distribution. The embedded sources located in the overlapping survey region break down as follows: 594 methanol masers, 361 MYSOs, 611 H ii regions and 25 YSO/H ii regions, where the distinction between the MYSOs and H ii regions comes from the RMS and CORNISH classifications. This represents an increase in the number of methanol masers discussed in PaperI by ∼ 112 and a threefold increase in the number of H ii regions discussed by PaperII as well as expanding the study to include the MYSO stage. This sample of massive star forming (MSF) clumps is the largest of its kind and incorporates a large fraction of the whole Galactic population of these objects (see Sect.2.1 for discussion of survey coverage) and also includes many of the sources used in previous studies of massive star formation presented in the literature (e.g., radio continuum studies of UCH ii regions — and ; mid-infrared bright samples of high-mass protostellar objects (HMPO) — ; IRAS colour selected samples — and ; and methanol masers — ). ATLASGAL is therefore able to put all of these studies into a larger global framework.\nIn Fig.[fig:iracimages] we present three-colour, mid-infrared images that show examples of the massive star-forming environments and position of the embedded object with respect to the dust distribution of the clump. Approximately a third of the matched ATLASGAL clumps are associated with two or more of the tracers and therefore are likely to host objects with a range of evolutionary stages; this is discussed in more detail in Sect.[sect:multiplicity]. In the upper panels we show examples of clumps associated with a single observational tracer, while in the lower panels we show examples of clumps hosting two or more of the different tracers. We will refer to these as multi-phase clumps. Many of the most intense star-forming regions in the Galaxy fall into the multi-phase subsample (e.g., W43, W49 and G305). For the analysis that follows we will define four distinct subsamples: methanol maser, MYSO, H ii region and multi-phase associated clumps. The properties for the individual clumps are given in Table[tbl:derivedclumppara] while the statistical properties of the four subsamples and combined sample of MSF clumps are given in Tables[tbl:derivedpara] and [tbl:derivedparaall], respectively.[4](#fn4)\n \n[tbl:derivedclumppara]\nlc............ & & & & & & & &&& & & & \n& & & & & & & &&& & & & \nAGAL028.244+00.012 & HII & 2.1 & 1.04 & 12.11 & 11.6 & 106.5 & 8.5 & 4.2 & 5.06 & 22.42 & 3.7 & 4.42 & 3.07 \nAGAL028.288 − 00.362 & HII & 1.4 & 2.74 & 25.73 & 9.4 & 48.8 & 11.6 & 5.8 & 7.72 & 22.84 & 4.3 & 5.53 & 3.67 \nAGAL028.301 − 00.382 & MM/YSO & 1.8 & 1.56 & 17.05 & 10.9 & 85.5 & 9.8 & 4.6 & 5.68 & 22.59 & 4.0 & 4.03 & 3.64 \nAGAL028.336+00.117 & YSO & 1.6 & 1.92 & 9.99 & 5.2 & 81.0 & 5.0 & 4.8 & 2.14 & 22.68 & 3.1 & 3.94 & 2.91 \nAGAL028.608+00.019 & HII & 1.5 & 2.93 & 20.21 & 6.9 & 101.2 & 8.7 & 4.3 & 5.29 & 22.87 & 3.9 & 4.69 & 3.50 \nAGAL028.649+00.027 & HII & 1.1 & 3.10 & 13.53 & 4.4 & 103.6 & 8.6 & 4.2 & 3.31 & 22.89 & 3.7 & 4.59 & 3.33 \nAGAL028.687+00.177 & HII & 1.1 & 0.71 & 1.85 & 2.6 & 102.9 & 9.8 & 4.7 & 0.91 & 22.25 & 3.0 & 4.35 & 2.58 \nAGAL028.816+00.366 & HII & 1.6 & 2.73 & 8.02 & 2.9 & 86.4 & 9.6 & 4.6 & 3.25 & 22.83 & 3.6 & 4.13 & 3.26 \nAGAL028.861+00.066 & MM/YSO & 1.6 & 3.82 & 23.13 & 6.1 & 102.8 & 8.6 & 4.3 & 5.51 & 22.98 & 4.0 & 5.11 & 3.73 \nAGAL029.436 − 00.174 & YSO & 1.6 & 0.91 & 3.22 & 3.5 & 85.6 & 5.2 & 4.7 & 1.28 & 22.36 & 2.7 & 3.77 & 2.52 \n \n \nNotes: Only a small portion of the data is provided here, the full table is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/MNRAS/.\n[tbl:derivedpara]\nlc...... & & & & & & & \n \nMethanol masers & 269&7.61&0.25 & 4.16 & 7.70 & 0.02 & 19.62 \nMYSOs & 210&5.57&0.23 & 3.29 & 4.51 & 1.12 & 15.28 \nH ii regions & 373&8.21&0.21 & 4.06 & 8.47 & 1.12 & 18.47 \nMulti-phase & 230&7.20&0.25 & 3.82 & 5.91 & 1.65 & 17.08 \n \nMethanol masers & 223&1.17&0.06 & 0.97 & 0.91 & 0.00 & 5.59 \nMYSOs & 164&0.79&0.04 & 0.54 & 0.64 & 0.15 & 2.83 \nH ii regions & 320&1.25&0.04 & 0.78 & 1.07 & 0.17 & 5.01 \nMulti-phase & 210&1.40&0.06 & 0.93 & 1.19 & 0.20 & 5.23 \n \nMethanol masers & 546&0.27&0.02 & 0.41 & 0.14 & 0.00 & 3.55 \nMYSOs & 359&0.23&0.02 & 0.29 & 0.13 & 0.01 & 2.53 \nCompact H ii regions & 339&0.28&0.02 & 0.37 & 0.19 & 0.00 & 3.36 \nExtended H ii regions & 236&0.55&0.04 & 0.64 & 0.33 & 0.03 & 4.14 \n \nMethanol masers & 105&2.38&0.08 & 0.79 & 2.20 & 1.30 & 5.00 \nMYSOs & 105&1.87&0.06 & 0.64 & 1.80 & 0.60 & 4.80 \nH ii regions & 173&2.53&0.08 & 1.06 & 2.40 & 0.70 & 11.00 \nMulti-phase & 129&2.54&0.08 & 0.90 & 2.50 & 0.90 & 5.90 \n \nMethanol masers & 98&23.21&0.61 & 6.03 & 22.19 & 13.85 & 42.42 \nMYSOs & 95&20.71&0.46 & 4.53 & 20.07 & 13.85 & 33.64 \nH ii region & 151&23.70&0.40 & 4.93 & 22.51 & 15.65 & 41.15 \nMulti-phase & 123&24.42&0.46 & 5.12 & 23.46 & 14.52 & 42.08 \n \nMethanol masers & 264&3.24&0.04 & 0.65 & 3.28 & 1.03 & 4.99 \nMYSOs & 210&2.91&0.04 & 0.51 & 2.91 & 1.52 & 4.21 \nH ii regions & 373&3.40&0.03 & 0.52 & 3.45 & 1.72 & 4.77 \nMulti-phase &230&3.52&0.04 & 0.56 & 3.50 & 1.93 & 5.34 \n \nMethanol masers & 314&22.61&0.02 & 0.38 & 22.54 & 21.94 & 23.81 \nYSOs & 210&22.56&0.02 & 0.33 & 22.51 & 22.04 & 23.68 \nH ii regions & 375&22.66&0.02 & 0.39 & 22.62 & 21.92 & 24.14 \nMulti-phase & 231&22.90&0.03 & 0.46 & 22.84 & 21.96 & 24.30 \n \nMethanol masers & 88&0.72&0.11 & 1.02 & 0.53 & 0.04 & 9.29 \nYSOs & 91&0.77&0.05 & 0.47 & 0.67 & 0.18 & 2.49 \nH ii regions& 163&0.65&0.05 & 0.61 & 0.48 & 0.11 & 4.00 \nMulti-phase & 124&0.58&0.04 & 0.40 & 0.51 & 0.06 & 3.02 \n \nMethanol masers & 199&3.70&0.07 & 0.96 & 3.84 & -2.40 & 5.28 \nYSOs & 361&3.91&0.03 & 0.53 & 3.87 & 3.01 & 5.66 \nH ii regions & 578&4.39&0.03 & 0.61 & 4.40 & 3.02 & 6.38 \n \nMethanol masers & 199&6.92&0.72 & 10.10 & 4.23 & 0.12 & 109.61 \nYSOs & 210&15.74&1.35 & 19.60 & 9.37 & 0.63 & 143.01 \nH ii regions & 343&16.45&1.07 & 19.80 & 10.99 & 0.71 & 184.54 \nMulti-phase & 219&12.72&1.58 & 23.36 & 10.81 & 0.56 & 106.82 \n \n[tbl:derivedparaall]\nlc...... & & & & & & & \nAngular offset (′′) & 1695 &8.90&0.26 & 10.71 & 5.40 & 0.16 & 82.58 \nAspect Ratio & 1130&1.50&0.01 & 0.48 & 1.38 & 1.00 & 8.97 \n*Y*-factor & 1130&5.67&0.12 & 3.92 & 4.59 & 1.14 & 46.05 \nDistance (kpc) & 1082&7.33&0.12 & 4.00 & 6.02 & 0.02 & 19.62 \nEffective radius (pc) & 917&1.18&0.03 & 0.85 & 0.95 & 0.00 & 5.59 \nPhysical Offset (pc) & 1480&0.31&0.01 & 0.44 & 0.18 & 0.00 & 4.14 \nLine width (kms− 1) & 512&2.37&0.04 & 0.93 & 2.20 & 0.60 & 11.00 \nKinetic Temperature (K) & 467&23.18&0.25 & 5.30 & 22.26 & 13.85 & 42.42 \n < Log[Clump Mass (M⊙)] > & 1077&3.29&0.02 & 0.60 & 3.33 & 1.03 & 5.34 \n < Log[H2 column density (cm− 2)] > & 1130&22.67&0.01 & 0.41 & 22.61 & 21.92 & 24.30 \nViral Ratio & 466&0.67&0.03 & 0.64 & 0.53 & 0.04 & 9.29 \n < Log[Bol. lum (L⊙)] > & 1163&4.11&0.02 & 0.72 & 4.11 & -2.40 & 6.38 \n*L*$\\_{\\rm{bol}}$-*M*$\\_{\\rm{clump}}$ ratio (L⊙/M⊙) & 971&14.11&0.57 & 17.68 & 8.91 & 0.12 & 184.54 \n \n \nClump structure and morphology\n------------------------------\nThe clump aspect ratios (i.e., $\\theta\\_{\\rm{maj}}/\\theta\\_{\\rm{min}}$) and the ratio of the integrated to peak flux densities (compactness parameter or *Y*-factor) are two observationally derived clump parameters that can be used to obtain some insight into the structure of this sample of massive star forming clumps.[5](#fn5) Moreover, these are relatively distance-independent parameters, allowing us to compare the properties of the massive star-forming clumps with the quiescent ATLASGAL clump population.\n \nFig.[fig:clumpstructure] shows the cumulative distributions of the aspect ratios and compactness parameter (*Y*-factor) for the four subsamples and the unassociated clumps. The distributions of both parameters in the associated clumps are noticeably different to those of the general population of ATLASGAL clumps, with the former being significantly more centrally condensed and spherical (cf. ). The mean and median values for the associated clumps are 1.50 ± 0.01 and 1.38 for the aspect ratio and 5.67 ± 0.12 and 4.59 for the *Y*-factor, compared to 1.70 ± 0.01 and 1.56 and 7.68 ± 0.07 and 6.31 obtained for the quiescent clumps in the ATLASGAL CSC, respectively. Using a Kolmogorov-Smirnov (KS) test to compare these two samples, we find that the null hypothesis that these are drawn from the same parent population can be rejected for the aspect ratio and *Y*-factor with greater than 3*σ* confidence (*r* ≪ 0.001 for both parameters, where *r* is the probability that the two samples are similar).\nA KS test confirms that there is no significant difference between any of the methanol maser, MYSO or H ii-region associated subsamples. The only difference between the four subsamples is found for the aspect ratio of the multi-phase clumps where a trend to slightly larger values is observed. The star formation taking place within these clumps is more complicated and is likely to have had an affect on the clump structure.\nAn important caveat is that the location of the peak emission and its intensity are likely to be affected by the presence of the embedded source itself as it warms its local environment making heated clumps more strongly peaked. Comparisons of the temperatures of star forming and quiescent clumps find average temperatures of ∼ 20 and 15K, respectively (e.g., ) and that the clump temperature is correlated with the luminosity of the embedded object with warmer clumps being associated with the more luminous and evolved sources ().\nThe lack of any significant differences between the three subsamples suggests that the structure of the clump is not sensitive to the evolutionary stage of the embedded object. Furthermore, the quiescent clumps are themselves centrally condensed with aspect ratios ∼ 2, just not to the same extent as the MSF clumps. The presence of an embedded thermal source might lower the aspect ratio and *Y*-factor, but does not alter the fact that the massive stars are forming towards the centre of their host clumps.\nThe similarities between the compactness and shape of the MSF clumps lead us to conclude that the structure of the clump *envelope* does not evolve significantly during the early embedded stages in the massive star-formation process. This is consistent with the results of previous studies (e.g., HMPO; and UCH ii regions; ), but for a much larger sample which strengthens their findings. This result suggests that, in order for the quiescent clumps to form massive stars, they need either to be initially, or to become much more centrally condensed and spherical in structure. However, once the star formation begins, it proceeds more rapidly than the clump can adjust and therefore the morphological clump properties change little as it proceeds. This suggests a fast star-formation process as explored by.\nMultiplicity\n------------\nAs mentioned above, we have found ∼ 300 massive star-forming clumps that are associated with two or more of the three tracers ( ∼ 30percent of the MSF clumps). As discussed in the next paragraph in approximately a third of cases the tracers are in fact associated with the same embedded sources. However, in the remaining two-thirds this multiplicity of tracers present in a clump is likely to indicate that multiple evolutionary stages may be present. This is not altogether surprising, given that most of the observational evidence supports the hypothesis that all massive stars form in clusters and there is no reason to expect all of the star formation taking place in a clump to be coeval. It is still an open question whether the low-mass stars form before, after or at the same time as their high-mass counterparts in a cluster. However, this may have serious implications when trying to attribute observed properties to a particular evolutionary stage. This problem may be particularly acute in cases where a particular sample has been selected using a single observational tracer (e.g., radio continuum or the presence of a methanol maser) as there may be multiple evolutionary stages present in the same clump that remain hidden.\n[fig:vendiagram] Venn diagram showing the distribution of the methanol masers, MYSOs and H ii regions identified with their associated ATLASGAL clumps with distances between 3 and 5kpc (i.e., distance-limited sample). For the multi-phase clumps that are associated with a methanol maser an additional number is given in parenthesis; this gives the number of clumps where the embedded MYSO or H ii region is found to be positionally coincident with the methanol maser (angular separation $<2\\arcsec$) and likely to be tracing the same object.\nTo investigate the issue of multiplicity further we present a Venn diagram in Fig.[fig:vendiagram] showing how the methanol masers, MYSOs, H ii regions are distributed amongst the associated ATLASGAL clumps. We use a distance-limited sample of all massive star-forming clumps located in the MMB and AMGPS survey regions (see Sect.[sect:sensitivity]). This sample consists of 241 clumps and avoiding any potential bias due to the different completeness as a function of distance of the tracers used (see Sect.[sect:distances] for more details). Furthermore, we exclude the clumps associated with the 25 RMS sources classified as MYSO/H ii as some ambiguity about their nature remains. We will focus the following statistical analysis on this distance-limited sample. Before starting this analysis we note that methanol masers have been found to be positionally coincident sometimes with MYSOs and H ii regions (e.g., ). We therefore need to determine whether the detection of a methanol maser indicates the presence of a distinct embedded object or is likely to be physically associated with an embedded MYSO or H ii region.\n[tbl:rmsmmbmatches]\nllll. & & & & \n& & & & \nAGAL010.323 − 00.161 & G010.3208 − 00.1570B & G010.323 − 00.160 & YSO & 0.35 \nAGAL010.472+00.027 & G010.4718+00.0256 & G010.472+00.027 & H ii region & 0.40 \nAGAL010.957+00.022 & G010.9592+00.0217 & G010.958+00.022 & H ii region & 0.58 \nAGAL011.034+00.061 & G011.0340+00.0629 & G011.034+00.062 & H ii region & 0.98 \nAGAL011.990 − 00.271 & G011.9920 − 00.2731 & G011.992 − 00.272 & YSO & 1.48 \nAGAL012.024 − 00.031 & G012.0260 − 00.0317 & G012.025 − 00.031 & YSO & 0.36 \nAGAL012.198 − 00.034 & G012.1993 − 00.0342B & G012.199 − 00.033 & YSO & 0.32 \nAGAL012.888+00.489 & G012.8909+00.4938C & G012.889+00.489 & YSO & 0.79 \nAGAL015.029 − 00.669 & G015.0357 − 00.6795 & G015.034 − 00.677 & H ii region & 0.99 \nAGAL017.637+00.154 & G017.6380+00.1566 & G017.638+00.157 & YSO & 1.16 \n \n \nNotes: Only a small portion of the data is provided here, the full table is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/MNRAS/.\nThe methanol masers, MYSOs and H ii regions in this sample have astrometry accurate to an arcsec or better and so we are able to distinguish between genuine coincidences, in which more than one of the three tracers is found in the same core, from cases where two tracers are associated with separate cores in the same clump. Using a matching radius of 2, we find a total 113 cases in the whole sample of 246 clumps where the methanol maser is considered coincident with the MYSO or H ii region present in the same clump. There are 39 of these positional coincidences in the distance limited sample, which corresponds to approximately half of all of the methanol masers associated with the multi-phase clumps (76 in total). As a point of reference, 2 is the typical extent of the spread of groups of methanol-maser spots and corresponds to a physical size of 0.1pc at 10kpc. This is also the radius of a typical core that is likely to form either a single star or small group of no more than a few stars. These positional associations are therefore likely to be genuine and may indicate that an evolutionary transition of some sort is taking place. Table[tbl:rmsmmbmatches] lists these positional coincidences and the angular offset between them.\nIn total, 141 methanol masers are included in the distance-limited sample. Of these, we find 29 that are positionally coincident with an MYSO and 10 are positionally associated with an H ii region; these correspond to ∼ 20 and 6 percent of the associated methanol masers. found that the radio continuum emission at 5 and 8GHz towards six of these H ii regions is unresolved with a mean spectral index of ∼ 1 (values range between 0.3 and 2, with a standard error on the mean of 0.3 and standard deviation of 0.7). The H ii regions are therefore optically thick and making these sources good candidate hypercompact (HC) H ii regions, which is the stage prior to the formation of the UC region. The radio continuum emission seen towards the other four H ii regions is also found to be unresolved (), however, these have only been observed at one frequency and their spectral indices cannot be estimated. The association of these compact radio sources with methanol masers may indicate that accretion continues after the formation of the H ii regions through ionized infall () or through the partial shielding of the ionizing radiation by the accretion flow (). In the full sample we have found a total of 34 sources where the position the methanal is coincident with the H ii region and these may provide basis for a more statistical study of HC H ii regions.\nThe proportions of methanol masers associated with MYSOs and H ii regions are similar to those reported by and, however, the analysis presented here on a distance limited sample with accurate astrometry is more robust. Methanol masers are therefore present over a large range of the early evolutionary stages, from the beginning of accretion to after the formation of the H ii region. However, ∼ 74percent of this sample of methanol masers are not within 2 of either an MYSO or H ii region and therefore they are likely to be tracing a stage that precedes MYSO’s and H ii region stages. For completeness we estimate the corresponding number of MYSOs and H ii regions that are coincident with a methanol maser to be approximately 28 and 10percent, respectively.\nSince a methanol maser coincident with a MYSO or H ii region is not considered to be tracing a separate stage (i.e., angular offsets < 2), it is reasonable to combine these clumps into the MYSO and H ii region subsamples. In the distance-limited sample (i.e., between 3 and 5kpc; see Sect.4.1.2 for details) there are 65 methanol maser only clumps, 60 MYSO only clumps (41 MYSOs + 19 MYSOs coincident with a methanol maser) and 63 H ii region clumps (54 H ii regions + 9 H ii regions coincident with a methanol maser), 17 clumps that host a MYSO and H ii region, 18 clumps that host a methanol maser and H ii region, 9 clumps that are associated with a methanol maser and MYSO, and 9 clumps that host all three tracers. The methanol maser, MYSO, H ii region and multi-phase subsamples each include ∼ 25percent of the clumps, with approximately a third of all tracers being associated with one of the multi-phase clumps. This highlights the need for careful analysis of high-resolution multi-wavelength data in order to properly distinguish between clumps containing multiple evolutionary stages and those genuinely associated with a single, fairly well-defined stage.\nWe have now identified three relatively clean subsamples of methanol-maser, MYSO and H ii-region only associated clumps that will be the primary focus of this paper. In the analysis that follows, we will not refer to specific survey matches but to the subsamples they have been used to define.\nDerived clump properties\n========================\nIn this section, we describe the procedures used to derive parameters for all of the associated clumps and present the analysis of these parameters. As well as discussing the statistical properties of the whole sample we will also make comparisons of the clump properties of the methanol maser, MYSO and H ii region subsamples. We will restrict our analysis to comparisons primarily of these three subsamples as these provide the best opportunity to identify differences that may be linked to the evolutionary stage of the embedded objects. However, we include multi-phase clumps in all of the plots for completeness and to facilitate comparison with the other subsamples.\nDistances\n---------\n### RMS-MMB distance comparison\n[fig:atlasdistancehist] Heliocentric distance distribution for all MSF-tracer associated ATLASGAL clumps and subsamples. The vertical long-dashed cyan lines indicate the range (3 – 5kpc) from which our distance-limited sample is drawn. The bin size is 1kpc.\nDistances have already been obtained for the vast majority of the ATLASGAL-RMS matched sources ( ∼ 90percent; see Urquhart et al. 2014 for details). Although parallax and spectrophotometric distances have been adopted where available, a large number are kinematically derived distances estimated using a source’s radial velocity (as measured from NH3 or 13CO observations) and a model of the Galactic rotation curve. The near/far kinematic distance ambiguities (KDAs) have been resolved in an analysis of archival H i spectra (e.g., ). The MMB survey has conducted a similar analysis using the same H i data and the velocity of the peak maser component as a proxy for the velocity of the natal cloud.\nIn total there are 120 ATLASGAL clumps that are associated with both an RMS and MMB source that have an assigned kinematic distance. For 95 of these matches, the kinematic distances are in reasonable agreement. This corresponds to ∼ 80percent of this sample, which is similar to the reliability estimated from other studies (e.g., ). In Table[tbl:rmsmmbdvlsr] we provide a list of the 25 sources and their associated parameters for which the distance discrepancy is larger than 1.5kpc.\nDifferences of only a few kms− 1 in the velocity of the local standard of rest (V$\\_{\\rm{LSR}}$) can be enough to change the kinematic distance solution from near to far or vice versa, moving the source velocity so that it aligns with an absorption feature in the H i spectrum or breaking the alignment between the source velocity and an absorption feature, leading to an incorrect distance solution. Methanol-maser emission often consists of a number of strong peaks spread over several kms− 1 in velocity (7.2 ± 5.3kms− 1, where the uncertainty is the standard deviation, estimated from the combined MMB catalogue) with the strongest peak changing over time and, although normally close to the systemic velocity of the source, can differ sufficiently to produce errors in the distance solution. Molecular-line observations of thermally excited transitions provide a more reliable measurement of the clump systemic velocity and we have therefore adopted the RMS distances in cases where the MMB and RMS kinematic distances disagree.\n[fig:galdistribution] Galactic distribution of all MSF clumps with masses larger than 1000M⊙. Different symbols and colours are used to indicate the different subsamples (see legend for details). The background image is the same as presented in Fig.1. The orange wedges show the ATLASGAL longitude coverage our to a distance of 20kpc where is has a clump mass sensitivity of ∼ 1000M⊙. The smaller of the two cyan dot-dashed circles represent the locus of tangent points, while the larger circle traces the Solar circle.\n[tbl:rmsmmbdvlsr]\nll.....cc & & & & & & & & \n& & & & & & & & \nG011.0340+00.0629 & G011.034+00.062 & 15.4 & 20.5 & 5.1 & 14.2 & 2.9 & F & N \nG013.6562 − 00.5997 & G013.657 − 00.599 & 47.4 & 51.2 & 3.8 & 4.1 & 12.0 & N & F \nG014.3909 − 00.0216 & G014.390 − 00.020 & 23.1 & 26.9 & 3.8 & 2.5 & 13.5 & N & F \nG018.4608 − 00.0034 & G018.460 − 00.004 & 52.2 & 49.4 & -2.8 & 12.1 & 3.9 & F & N \nG281.7059 − 01.0986 & G281.710 − 01.104 & -1.0 & 0.9 & 1.9 & 1.7 & 4.2 & N & F \nG294.5117 − 01.6205 & G294.511 − 01.621 & -15.3 & -11.9 & 3.4 & 2.6 & 1.0 & F & N \nG311.2292 − 00.0315 & G311.230 − 00.032 & 29.2 & 24.8 & -4.4 & 5.5 & 13.6 & N & F \nG314.3197+00.1125 & G314.320+00.112 & -48.9 & -43.5 & 5.4 & 3.6 & 8.8 & N & F \nG318.9480 − 00.1969 & G318.948 − 00.196 & -33.8 & -34.6 & -0.8 & 2.4 & 10.5 & N & F \nG319.8366 − 00.1963 & G319.836 − 00.197 & -13.2 & -9.2 & 4.0 & 11.7 & 0.6 & F & N \n \n \nNotes: Only a small portion of the data is provided here, the full table is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/MNRAS/.\n### Distance and Galactic distribution\nThe kinematic distances presented in PapersI and II were calculated using the Galactic rotation curve, while the RMS distances are calculated using the rotation curve. To maintain consistency with the results presented in the previous two papers we have recalculated the RMS kinematic distances using the model. However, for the vast majority of sources, the differences in distance given by the two rotation curves is smaller than their associated uncertainty and so the statistical results are robust against the choice of model.\nWe have collected together distances for 1082 MSF clumps and show the distribution of these distances for all the associated clumps and the four associated subsamples in Fig.[fig:atlasdistancehist] while in Fig.[fig:galdistribution] we show the Galactic distribution as viewed from above the northern Galactic pole. The overall distribution seen in Fig.[fig:atlasdistancehist] consists of two prominent peaks centred at approximately 4.5 and 11.5kpc. Looking at the Galactic distribution shown in Fig.[fig:galdistribution] it is clear the 4.5kpc peak coincides with the inner part of the Scutum-Centaurus arm and the northern end of the Galactic bar at $\\ell \\sim 30\\degr$, while the 11.5kpc peak is associated with the far segment of the Perseus arm between ℓ = 10 − 60 and the location where the latter coincides with the southern end of the Galactic bar also at ℓ ∼ 330. The presence of the dip between these two peaks is due to a combination of several factors: 1) the low number of clumps located within 3-4kpc of the Galactic centre; 2) the rotation curves tend to locate sources away from the tangent point; and 3) there is a general lack of any prominent features of Galactic structure in these bins. A detailed discussion of the Galactic distribution of the methanol masers, MYSOs, H ii regions has been presented in a number of recent papers (i.e.,, PapersI and II and ) and so will not be discussed here.\nComparing the distance distributions of the four subsamples we find that the H ii-region, methanol-maser and multi-phase associated clumps have a similar distribution to each other. KS tests of combinations of these do not reveal any significant differences, however, the MYSO distribution is shown to be significantly different from the other three subsamples, with a much higher fraction being located within 6kpc. In Sect.4.6 we find that the MYSOs are systematically less luminous than H ii regions and so the lower number of more distant MYSOs found is likely to be due to the sensitivity of the 21$\\umu$m MSX band (). As mentioned in Sect.[sect:sensitivity], we define a distance-limited sample to be all sources located between 3 and 5kpc. This is indicated by the vertical cyan lines shown in Fig.[fig:atlasdistancehist]; this distance range has been selected as it includes the peaks in the distributions of all"},"data_id":{"kind":"string","value":"arxiv_0181684"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":453,"numItemsPerPage":100,"numTotalItems":46503,"offset":45300,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjIyOTA0OSwianRpIjoiNGYwY2M2YTUtZmRjNC00MGRhLWEyY2EtZGQ4OGRiMjNiYzRjIiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODYyMzI2NDksImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.K4rrAyXsfXP-0VA-lRhtpQcUqtH4NBfTwAXlkbASk5hMh_-cio3IlljHtlCLluOivvyHwiwPuASazRAdMqp4DA","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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Categoricity Properties forComputable Algebraic Fields
======================================================
[1](#fn1)
We examine categoricity issues for computable algebraic fields. We give a structural criterion for relative computable categoricity of these fields, and use it to construct a field that is computably ... | arxiv_0181352 |
Dynamical Evolution of Multiple-Population Globular Clusters
============================================================
[firstpage]
We have carried out a set of Monte Carlo simulations to study a number of fundamental aspects of the dynamical evolution of multiple stellar populations in globular clusters with differe... | arxiv_0181355 |
A supermartingale approach to Gaussian process based sequential design of experiments
=====================================================================================
,
**.** Gaussian process (GP) models have become a well-established framework for the adaptive design of costly experiments, and notably of compute... | arxiv_0181358 |
Relational Quantum Mechanics and Probability
============================================
We present a derivation of the third postulate of Relational Quantum Mechanics (RQM) from the properties of conditional probabilities. The first two RQM postulates are based on the information that can be extracted from interactio... | arxiv_0181369 |
We calculate the Sivers function in semi-inclusive deep inelastic scattering (SIDIS) and in the Drell-Yan process (DY) by employing the quasi-classical Glauber–Mueller/ McLerran–Venugopalan approximation. Modeling the hadron as a large “nucleus” with non-zero orbital angular momentum (OAM), we find that its Sivers func... | arxiv_0181370 |
*V**σ* is *monotone* if for each *σ*, *V**σ* is prefix-free and contains only proper extensions of *σ*. We define the code-tree *S* generated by a monotone c.e. map *σ* ↦ *V**σ* from strings to sets of strings, inductively as follows. The empty string *λ* is in *S* and has depth 0. If *σ* ∈ *S* and has depth ℓ, then a... | arxiv_0181371 |
We do not observe nitrate test results but observing claimed tests still gives us an indication of physicians private information relative to the machine learning algorithm. Further, the claims data allow us to identify pregnant women from mandatory pregnancy-associated examinations. This is important as pregnant wome... | arxiv_0181379 |
particular node in the Fock space graph. However, the effective potential is in general an analytically intractable object to calculate, because it requires one to invert the Hamiltonian. We would thus like to calculate corrections to the effective potential perturbatively, much like Anderson has done using the famous... | arxiv_0181382 |
Information travels in massless fields in 1+1 dimensions where energy cannot
============================================================================
It has been demonstrated that, in (1+1) spacetime dimensions, massless fields can be used for information transmission even between parties that are time-like rather ... | arxiv_0181384 |
Device-Independent Bit Commitment based on the CHSH Inequality
==============================================================
**Bit commitment and coin flipping occupy a unique place in the device-independent landscape, as the only device-independent protocols thus far suggested for these tasks are reliant on tripartit... | arxiv_0181386 |
Introduction
============
In this note, we are interested in studying singularity formation in the mean curvature flow in higher codimension. There is little hope to classify singularity models in general, even in codimension one. However, for suitable assumptions on initial data, this problem becomes more feasible. Fo... | arxiv_0181396 |
Assume but Verify: Deductive Verification of Leaked Information in Concurrent Applications (Extended Version)
=============================================================================================================
We consider the problem of specifying and proving the security of non-trivial, concurrent programs t... | arxiv_0181398 |
biological materials. From a measurement standpoint, this type of material can be problematic because the apparent viscosity or modulus can vary over a large range with the frequency. Correspondingly, we may expect relaxation times to depend appreciably on the timescale utilized to fit the relaxation data when the rh... | arxiv_0181401 |
universal behavior of critical phenomena. Many significant breakthroughs in statistical physics are attributed to the study of the model from either its computational or its theoretical perspective. It is well known that the Ising model has no phase transition in one dimension; however, it undergoes a second-order pha... | arxiv_0181405 |
Quasi-Triangularization of Matrix Polynomialsover Arbitrary Fields [5](#fn5)
============================================================================
In, Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial *P*(*λ*) over an algebraically closed field is spectrally equivalent to a triangular matrix... | arxiv_0181407 |
Data-driven Emotional Body Language Generation for Social Robotics
==================================================================
In social robotics, endowing humanoid robots with the ability to generate bodily expressions of affect can improve human-robot interaction and collaboration, since humans attribute, and ... | arxiv_0181408 |
Forking paths in financial economics
====================================
We argue that spanning large numbers of degrees of freedom in empirical analysis allows better characterizations of effects and thus improves the trustworthiness of conclusions. Our ideas are illustrated in three studies: equity premium predictio... | arxiv_0181409 |
is outward migration of the Jupiter-Saturn system with the two planets maintaining the 3:2 resonance and sharing a common gap. Here, the migration reversal occurs at *t* ∼ 2500 orbits, when *m**J* ∼ 0.6 *M**J* and *m**S* ∼ 0.3 *M**J*. These values are in reasonable agreement with Masset & Snellgrove (2001) who estimat... | arxiv_0181410 |
26 & − 49
The results for the 11 clusters are plotted in Figure [fig:clusterzio]. The *g* − *i* cluster color errors SE*g**i* were estimated as the standard error on the mean *g* − *i* color of the red sequence galaxies we selected, and cluster redshift errors were taken to be SE*g**i*/3, taken from the slope of the... | arxiv_0181416 |
**o**f**t*, *h**a**r**d*}
Error value scaling to compute dataset weights & cat & see text
[tab:selector:space]
#### Meta-Features.
To train our and to select a policy, as well to use the backup strategy, we use meta-features describing all meta-train datasets (TR5) and new datasets (TE1). To avoid the problems dis... | arxiv_0181418 |
Non-negative Matrix Factorization via Archetypal Analysis
=========================================================
Given a collection of data points, non-negative matrix factorization (NMF) suggests to express them as convex combinations of a small set of ‘archetypes’ with non-negative entries. This decomposition is u... | arxiv_0181428 |
The Century Survey Galactic Halo Project III: A Complete 4300 deg2 Survey of Blue Horizontal Branch Stars in the Metal-Weak Thick Disk and Inner Halo
=====================================================================================================================================================
We present a complet... | arxiv_0181430 |
Schwinger mechanism in linear covariant gauges
==============================================
In this work we explore the applicability of a special gluon mass generating mechanism in the context of the linear covariant gauges. In particular, the implementation of the Schwinger mechanism in pure Yang-Mills theories hin... | arxiv_0181431 |
itaev materials as a platform to explore the electric field physics discussed in this work remains an intriguing possibility. We note that several recent theoretical works have explored the (related) effect of tunnelling in such heterostructures through the effect of *localized* electric fields.
While each of these co... | arxiv_0181432 |
*)-set agreement problem in a system with up to *k* failures, and (2) the (*k* + 1, *k*)-set agreement problem is known to be impossible in presence of *k* failures, it follows that there is no *k*-resilient (*n*, *k*)-set agreement algorithm.
#### Content of the paper: on the BG-simulation side
As already indicated, t... | arxiv_0181435 |
}$$
where *ε* = *M**E**l**k**o*2/*k*2. It is obvious that the asymptotic behavior is the same with ([KKmodeinRSmode]) and the general solution is
$$\begin{aligned}
\alpha\_{n}(z)&=&C\_1H(z)M\_{\frac{1}{2},-\sqrt{5+2\epsilon}}\left(\text{i}\frac{2m\_{n}}{k}(k|z|+1)\right)\nonumber\\
&+&C\_2H(-z)M\_{\frac{1}{2},\sq... | arxiv_0181436 |
CERN-PH-TH/2009-039
Parameter dependence of magnetized CMB observables
Massimo Giovannini [1](#fn1)
Abstract
Pre-decoupling magnetic fields affect the scalar modes of the geometry and produce observable effects which can be constrained also through the use of current (as opposed to forthcoming) data stemming fro... | arxiv_0181438 |
ssrhist] shows the resultant *r*-band distribution. While the majority of AGN candidates are well detected in the SDSS imaging, their optical brightness distribution peaks at *r* ∼ 19.5, making them fainter than the typical spectroscopic limits of SDSS, *i* ∼ 19.1 for quasars and *r* ∼ 17.8 for galaxies (as discussed i... | arxiv_0181439 |
\}$ and ${\cal A}\_\text{Rx} = \left\{0,\frac{2\pi}{2^{N\_\text{Q,Rx}}},\ldots,\frac{2\pi (2^{N\_\text{Q,Rx}}-1)}{2^{N\_\text{Q,Rx}}}\right\}$. The number of quantization bits is set to *N*Q,Tx = *N*Q,Rx = 4. The number of OFDM sucarriers is set to *K* = 256, and a Zero-Prefix (ZP) of length *Z*P = 64 is assumed to rem... | arxiv_0181442 |
Radial and azimuthal gradients of the moving groups in Gaia DR3: The slow/fast bar degeneracy problem
=====================================================================================================
### Received YYY; accepted XXX
We aim to identify and characterise the dynamical moving groups across the MW disc, a... | arxiv_0181443 |
,\frac{0.0673}{Z}\ +\ 1.049647\ -\ 0.000842\, Z\, {\nonumber}\\
\beta\_F &\ =\ -\, \frac{0.466383}{Z}\ +\ 0.919016\ +\ 0.001578\,Z\, {\nonumber}\\
\gamma\_F &\ =\ -\,\frac{0.704459\ +\ 0.752684\,Z\ -\ 1.52103\,Z^2}
{4.278577\ -\ 4.017493\,Z\ -\ 0.388753\,Z^2}\, {\nonumber}\\
a\_F &\ =\ (4 c\_{e,e}\ -\ {\alpha}\_F\ ... | arxiv_0181444 |
8.7)circle(15pt); (-8.6,-5)circle(15pt); (8.6,-5)circle(15pt); at (-20,0) ⇒ ;
[jjm]
Let the graph we obtained be *G* and containing *n* vertices. Notice that *G* contains *k* snowflakes and 3 diamond holders.
Now we count the number of vertices of *G* in terms of *k*. Notice the number of vertices of a snowflake excep... | arxiv_0181446 |
*, with *N*0 = 64 and Δ*N* = 16, are presented in figure [fig:heat1]. We observe from this figure that the mean $\bar{{\EuScript{G}}}$ of the cost function barely decays with increasing number of training samples. Moreover, the standard deviation is also unacceptably high, even for a very high number of training sample... | arxiv_0181447 |
A Multitask, Multilingual, Multimodal Evaluation of ChatGPTon Reasoning, Hallucination, and Interactivity
=========================================================================================================
=1
This paper proposes a framework for quantitatively evaluating interactive LLMs such as ChatGPT using publ... | arxiv_0181449 |
Inexact Proximal Gradient Methods for Non-convexand Non-smooth Optimization
===========================================================================
In machine learning research, the proximal gradient methods are popular for solving various optimization problems with non-smooth regularization. Inexact proximal gradi... | arxiv_0181454 |
Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
======================================================================================================
We present systematic constructions of tensor-network wavefunctions for bosonic symmetry protected topological (SP... | arxiv_0181455 |
Lucey, J. R., 2017, MNRAS, 468, 1594 Alton, P. D., Smith, R. J., Lucey, J. R., 2018, MNRAS, 478, 4464 Asplund, M., Grevesse, N., Sauval, A. J., Scott, P., 2009, ARA&A, 47, 481 Baldwin, J. R., Danziger, I. J., Frogel, J. A., Persson, S. E., 1973, ApJL, 14, 1 Baldwin, C., McDermid, R. M., Kuntschner, H., Maraston, C., C... | arxiv_0181461 |
Learning Human Pose Models from Synthesized Data for Robust RGB-D Action Recognition
====================================================================================
### Received: date / Accepted: date
We propose Human Pose Models that represent RGB and depth images of human poses independent of clothing textures, ... | arxiv_0181462 |
mathcal{S}^{\rm c}$ unambiguously.[↩](#fnref3)
4. Note that although notions of causal correlations and causal Bayesian networks both share the “causal” qualifier, they are distinct concepts: a causal correlation is *not* simply one that can be obtained from any particular causal Bayesian network.[↩](#fnref4)
5. For CI... | arxiv_0181463 |
the unknown distribution functions by *F*1 and *F*2 by the empirical distribution functions calculated from the subsample before and after the estimated change-point.
In order to analyze size and power, we choose sample sizes *n* = 200, 500, 1000 with serially independent random variables, N(0, 1)-distributed in the f... | arxiv_0181470 |
Cyber Loss Model Risk Translates to Premium Mispricing and Risk Sensitivity
===========================================================================
We focus on model risk and risk sensitivity when addressing the insurability of cyber risk. The standard statistical approaches to assessment of insurability and potent... | arxiv_0181471 |
Operadic multiplications in equivariant spectra, norms, and transfers
=====================================================================
[1](#fn1)
[2](#fn2)
We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define *N*∞ operads, equivariant generalizations of *E*∞ ... | arxiv_0181482 |
in phases with different pairing patterns. Our main conclusion is that the relative correction to the critical temperature is to a good accuracy proportional to the mean-field ratio of the critical temperature and the chemical potential. As a consequence, it is significant even rather deep in the BCS regime, where phe... | arxiv_0181483 |
}{2}^-$&1655&165&$\surd$
*Y*1\* &*S*01(1405)&0&$\frac{1}{2}^-$& 1406.5 & 50 & -
*Y*2\* &*S*01(1670)&0&$\frac{1}{2}^-$& 1670 & 35 & -
*Y*3\* &*S*01(1800)&0&$\frac{1}{2}^-$& 1800 & 300 & $\surd$
*Y*4\* &*S*11(1750)&1&$\frac{1}{2}^-$& 1750 & 90 & -
Numerical Results
-----------------
The particles which might co... | arxiv_0181484 |
we believe that the ranking results of 16-spectral radius reflects the significance of vertices individually.
1.0c|ll|ll|ll & & &
& Num. & Val. & Num. & Val. & Num. & Val.
1 & 39 & 0.4082483175 & 41 & 0.4081204985 & 1& 0.1709715830
2 & 38 & 0.4082482858 & 39 & 0.4081204985 & 31 & 0.1678396311
3 & 31 & 0.40824... | arxiv_0181486 |
Tate’s thesis in the de Rham setting
====================================
We calculate the category of *D*-modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local ... | arxiv_0181487 |
Designer Flat Bands: Topology and Enhancement of Superconductivity
==================================================================
We construct quasi one-dimensional topological and non-topological three-band lattices with tunable band gap and winding number of the flat band. Using mean field (MF) and exact density ... | arxiv_0181489 |
15.cmlc@r &
& & 30 October 2017
&&
Several principal aspects of a theoretical approach to the theory of faster-than-light particles (tachyons) are considered in this note. They concern the resolution of such problems of tachyon theory as the causality violation by tachyons, the stability of the tachyon vacuum,... | arxiv_0181490 |
wavelength with a 100 × objective. Scanning electron microscopy (SEM) images were taken with a Hitachi S-3400N instrument with a field emission gun and beam voltage.
Sheet resistance was measured in the substrate plane with a collinear four-point probe directly contacting the film. The sheet resistance was derived by... | arxiv_0181491 |
Arbitrary Transmission Power in the SINR Model:Local Broadcasting, Coloring and MIS
===================================================================================
Introduction
============
One of the most fundamental problems in wireless ad hoc networks is to enable efficient communication between neighboring node... | arxiv_0181492 |
Temporal Action Segmentation:An Analysis of Modern Techniques
=============================================================
Introduction
============
action segmentation (TAS) is a video understanding task that segments, in time, a temporally untrimmed video sequence. Each segment is labeled with one of a finite set of... | arxiv_0181504 |
frequency spectrum is dominated by low-frequency noise with a strong red-noise component. The most significant frequencies are at 0.31 and 1.86. The frequency at 3.72 seems to be the first harmonic of the 1.86 frequency. There are two stable high frequencies at 4.91 and 5.64. Several short-lived significant frequencie... | arxiv_0181510 |
^e\_p}{\hbar}
\frac{\eta\_{ev}}{\hbar\omega} I \nonumber\end{aligned}$$
with *ɛ* = (ℏ*ω* − *E**g**Q**W*)*m**v*/(*m**c* + *m**v*) being the kinetic energy of the photoexcited electrons and $y' \parallel [0
0 \bar{1}]$. In contrast to Eq. ([fz’x]), this contribution depends on the polarization plane of the incident lig... | arxiv_0181512 |
BTZ extensions of globally hyperbolic singular flat spacetimes
==============================================================
[t]0.5 *Laboratoire de Mathématiques d’Avignon
Université d’Avignon et des pays du Vaucluse*
[t]0.35 **Léo Brunswic**
0.7
---
---
Minkowski space, namely $\R^3$ endowed with the quadradic form... | arxiv_0181513 |
The First Systematic Study of Type Ibc Supernova multi-band light curves
========================================================================
We present detailed optical photometry for 25 Type Ibc supernovae (SNe Ibc) within *d* ≈ 150 Mpc obtained with the robotic Palomar 60-inch telescope in 2004-2007. This study ... | arxiv_0181520 |
E. Kröner, Zeit. f. Phys. **136**, 402 (1953) P. De and R.A. Pelcovits, Phys. Rev. B **35**, 8609 (1987) K. Franzrahe, P. Nielaba and S. Sengupta, Phys. Rev. E **82**, 016112 (2010) J.D. Eshelby, Proc. R. Soc. London A **241**, 376 (1957) K. Garikipati, M. Falk, M. Bouville, B. Puchala, and H. Narayanan, J. Mech. Phys... | arxiv_0181528 |
2] ∈ *H*2(*M*, Z2) be the homology class. If [*c*2] = 0, then *c*2 is a boundary, and *U*(*c*2) is a product of ${\cal B}\_{c\_3}$ operators for individual 3-cells, which all project to 1 in the ground state subspace. On the other hand if [*c*2] is non-trivial, then *c*2 = *f* + ∂*c*3, *i.e.* *c*2 is the sum of a leaf ... | arxiv_0181530 |
ac *d*log*Y**i* ∧ *d*log*Y**j* = ⟨*α**i*, *α**j*⟩Dirac *d*log*R**i* ∧ *d*log*R**j*.
The rational map *Y**i* → *R**i* is simply the classical limit of the monodromy action, from which we may recover the full quantum action by taking the normal–order prescription for the operators (this is true for all simply–laced, *... | arxiv_0181534 |
LAMP: The Long-term Accretion Monitoring Program of T Tauri stars in Chamaeleon-I
=================================================================================
[firstpage]
We present the results of a variability study of accreting young stellar objects in the Chameleon I star-forming region which is based on ∼ 30... | arxiv_0181537 |
Radio properties of H2O maser host galaxies
===========================================
### received ; accepted
Introduction
============
Since the first detection of an extragalactic H2O maser towards M33 (Churchwell et al. 1977), 22GHz (*λ*=1.3cm) H2O maser surveys have covered ∼ 3000 galaxies, with reported detecti... | arxiv_0181545 |
30% in the most massive galaxies.
identified 7824 radio sources in the NVSS in the six-degree Field Galaxy Survey associated with galaxies brighter than K = 12.75 mag, spanning the 0.003 < *z* < 0.3 range. They separated radio sources powered by star formation and AGNs and found that radio-loud AGNs are preferentiall... | arxiv_0181546 |
Spectroscopic and thermodynamic properties in a four-band model for pnictides
=============================================================================
In this paper we provide a comprehesive analysis of different properties of pnictides both in the normal and superconducting state, with a particular focus on the o... | arxiv_0181547 |
**Cooperative resonances in light scattering from two-dimensional atomic arrays**
=================================================================================
We consider light scattering off a two-dimensional (2D) dipolar array and show how it can be tailored by properly choosing the lattice constant of the order... | arxiv_0181551 |
at a disjoint split with additional axioms. On the contrary, the results we obtain here are inline with ideas explored in concerning axiomatizations of classical logic, and fit well with our running track of research on extracting calculi for PNmatrices in an automated way, by taking advantage of monadicity requireme... | arxiv_0181555 |
− max*u* ∈ [0, 1]*h*ʹ(*u*) ≤ *c* ≤ *c**f*\* − min*u* ∈ [0, 1]*h*ʹ(*u*).
In view of the sign of *y*ʹ and the monotonicity properties of the solutions with respect to *c*, we immediately have that *c* can be admissible only if *c* + max*u* ∈ [0, 1]*h*ʹ(*u*) > 0; moreover, it has to be *c* < *c**f*\*. Now, if there exis... | arxiv_0181567 |
ExoplANNET: A deep learning algorithm to detect and identify planetary signals in radial velocity data
======================================================================================================
### Received TBC; accepted TBC
Introduction
============
The study of extrasolar planets is a relatively new field... | arxiv_0181571 |
Completeness in Polylogarithmic Time and Space[1](#fn1)
=======================================================
### Received: date / Accepted: date
Complexity theory can be viewed as the study of the relationship between computation and applications, understood the former as complexity classes and the latter as problem... | arxiv_0181573 |
the limsup or liminf of their average payoff on a prefix and the least priority seen infinitely often, which is consistent with the usual order on the payoff component (i.e. getting more payoff while keeping the same priority is always better). A *ɛ*-Nash equilibrium is one where no player can improve by changing the ... | arxiv_0181574 |
Optimal spectral shrinkage and PCA with heteroscedastic noise
=============================================================
Introduction
============
Singular value shrinkage and eigenvalue shrinkage are popular methods for denoising data matrices and covariance matrices. Singular value shrinkage is performed by comput... | arxiv_0181583 |
Implications of a non-zero Poynting flux at infinity sans radiation reaction for a uniformly accelerated charge
===============================================================================================================
### Received: date / Accepted: date
We investigate in detail the electromagnetic fields of a uni... | arxiv_0181585 |
Towards systematic grid selection in LES:iterative identification of the coarse-graining length scale by minimizing the solution sensitivity
============================================================================================================================================
The accuracy of a large eddy simulatio... | arxiv_0181586 |
Maurice Janet’s algorithms on
systems of linear partial differential equations
---
height 1.5pt
Kenji IoharaPhilippe Malbos
14cm **Abstract –** This article describes the emergence of formal methods in theory of partial differential equations (PDE) in the French school of mathematics through Janet’s work in the perio... | arxiv_0181588 |
array of such structures. For a system with *p* processors and *N* initial stars, we will first consider the scenario where we try to achieve ideal load balance by naively distributing the data of *N*/*p* stars to each processor. Fo simplicity, we will assume here that *N* is divisible by *p*, and analyze the data dep... | arxiv_0181592 |
Metabasins - a State Space Aggregation for highly disordered Energy Landscapes
==============================================================================
Glass-forming systems, which are characterized by a highly disordered energy landscape, have been studied in physics by a simulation-based state space aggregation... | arxiv_0181596 |
Towards Understanding Asynchronous Advantage Actor-critic: Convergence and Linear Speedup
=========================================================================================
Asynchronous and parallel implementation of standard reinforcement learning (RL) algorithms is a key enabler of the tremendous success of mo... | arxiv_0181598 |
ROPS: A New Search for Habitable Earths in the Southern Sky
===========================================================
### Submitted March 2012; Accepted May 2012
[firstpage]
We present the first results from our Red Optical Planet Survey (ROPS) to search for low mass planets orbiting late type dwarfs (M5.5V-M9V) in t... | arxiv_0181600 |
Deconvolution with unknown noise distribution is possible for multivariate signals
==================================================================================
This paper considers the deconvolution problem in the case where the target signal is multidimensional and no information is known about the noise distrib... | arxiv_0181603 |
. P. 1998. Process and policy: resource-bounded nondemonstrative reasoning. Computational Intelligence, **14**:1–38.
Makinson, D. 1988. General Theory of cumulative inference. M. Reinfrank, J. de Kleer, M. L. Ginsberg and E. Sandewall (editors). Proceedings of the second International Workshop on Non-monotonic Reasonin... | arxiv_0181604 |
7 (2005). J. M. Gerard and E. Kou, Phys. Lett. B **616**, 85 (2005). Q. Zhao, Phys. Lett. B **697**, 52 (2011). J. Fu, H. B. Li, X. Qin, and M. Z. Yang, Mod. Phys. Lett. A **27**, 1250223 (2012). H. E. Haber and J. Perrier, Phys. Rev. D **32**, 2961 (1985). P. Ball, J. M. Frere and M. Tytgat, Phys. Lett. B **365**, 367... | arxiv_0181608 |
Drug resistance revealed by in silico deep mutational scanning and mutation tracker
===================================================================================
#### Abstract
As COVID-19 enters its fifth year, it continues to pose a significant global health threat, with the constantly mutating SARS-CoV-2 virus ... | arxiv_0181612 |
Valuation equations for stochastic volatility models
====================================================
### December 12, 2011
[1](#fn1)
[2](#fn2)
[3](#fn3)
We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion c... | arxiv_0181615 |
field of Sgr A\*, the possibility that IRS13E hosts an intermediate-mass black hole was previously discussed by Maillard et al. ().
In order to examine this scenario in more detail, we tested whether stars within a radius of 0.7” are kinematically connected with the three central sources. We extracted proper motions f... | arxiv_0181618 |
064, [[arXiv:1109.1983](http://xxx.lanl.gov/abs/1109.1983)].
---
1. For reviews of these topics we invite the reader to consult the textbooks or reviews.[↩](#fnref1)
2. We use this term even when *c*eff is not associated with either anomalies or a central term in any algebra.[↩](#fnref2)
3. The mass of string states ... | arxiv_0181620 |
from a natural language processing perspective. are designing operational frameworks for HS, annotating corpora with several semantic frameworks, and automatic classifiers based on supervised machine learning models . Most of the classifiers proposed in literature for hate speech detection are based on supervised mach... | arxiv_0181622 |
edge stretching in the vertical and horizontal mid plane has been chosen such that the ratio of the largest and smallest cell edge length equals four.
 [fig:skewe... | arxiv_0181635 |
Advancing MINNLO to diboson processes: production at NNLO+PS
============================================================
Introduction
============
Lacking clear hints for new-physics phenomena, particle phenomenology at the Large Hadron Collider (LHC) has entered the precision era. The accurate measurement of Standard... | arxiv_0181636 |
Intergranular normal stress distributions in untextured polycrystalline aggregates
==================================================================================
From a general point of view, InterGranular Stress-Corrosion Cracking (IGSCC) results from the interplay between mechanical loading and grain boundaries o... | arxiv_0181638 |
lexinvariantLM. To do this, we train a small two-layer MLP probe on top of a frozen trained lexinvariantLM. For each training sequence, we first embed the input sequence with a randomly sampled token embedding *E* as described in section [sec:construction] and obtain the hidden activation of the final layer generated ... | arxiv_0181639 |
Stabilization of Age-Structured Chemostat Hyperbolic PDE with Actuator Dynamics
===============================================================================
Introduction
============
Motivation: Chemostat as a benchmark for nonlinear control and for control in epidemiology
-------------------------------------------... | arxiv_0181644 |
]
================================================
An important property of Algorithm [alg:uav-positioning] is that the segment of search trajectory in Step [enu:alg-two-phase-update-loop]) does not “turn back”, and so does that in Step [enu:alg-two-phase-upadte-loop-left]).
[Monotonicity][lem:monotonicity] Step [enu:a... | arxiv_0181646 |
radius 12 pc and is surrounded by 11 Bright-Rimmed clouds with embedded IRAS point sources. SFO 38, also known as IC 1396N, is located at a projected distance of ∼ 11 pc to the north of HD 206267 with the bright rim corresponding to the ionization front facing the O6.5V star. SFO 38 is located in the Cep OB2 associa... | arxiv_0181647 |
kbi)..controls +(0.1,0) and +(-0.1,0).. (km’) ;
[xshift=-1.5cm] (kh) at (0,1.8) ; (km) at (0,1.5) ; (kb) at (0,1.2) ; (khi) at (0.5,2) ; (kbi) at (1,1) ; (kh’) at (1.5,1.8) ; (km’) at (1.5,1.5) ; (kb’) at (1.5,1.2) ; (kh)..controls +(0.1,0) and +(-0.1,0).. (kb’) ; (km)..controls +(0.1,0) and +(-0.1,0).. (khi) ; (khi)..... | arxiv_0181651 |
Topological insulators and superconductors: ten-fold way and dimensional hierarchy
==================================================================================
It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Withi... | arxiv_0181660 |
A quadratically convergent semismooth Newton method for nonlinear semidefinite programming without the subdifferential regularity
=================================================================================================================================
### This version: 2024/2/21
We introduce a quadratically con... | arxiv_0181665 |
Zeta-regularized Lattice Field Theory with Lorentzian background metrics
========================================================================
Lattice field theory is a very powerful tool to study Feynman’s path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined... | arxiv_0181671 |
desc> <conceptsignificance>500</conceptsignificance> </concept> </ccs2012>
Introduction
============
Dataset discovery, or identifying secondary data useful for research, business, or individual purposes, is a vital activity in contemporary life. This is increasingly true for not only data experts, but also for more ca... | arxiv_0181675 |
On Gauduchon connections with Kähler-like curvature
===================================================
We study Hermitian metrics with a Gauduchon connection being ``Kähler-like", namely, satisfying the same symmetries for curvature as the Levi-Civita and Chern connections. In particular, we investigate 6-dimensional ... | arxiv_0181679 |
Critical phenomena in gravitational collapse
============================================
As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and s... | arxiv_0181681 |
3D Galactic dust extinction mapping with multi-band photometry
==============================================================
[firstpage]
We present a method to simultaneously infer the interstellar extinction parameters *A*0 and *R*0, stellar effective temperature $T\_{\rm eff}$, and distance modulus *μ* in a Bayesian... | arxiv_0181683 |
mass for a sample of 466 MSF clumps for which we have a radius and that have been detected in one of the three ammonia surveys discussed in Sect.[sect:ammoniaobs]. In a recent paper, have shown that the critical virial parameter ($\alpha\_{\rm{cr}}$) for an isothermal sphere that is in hydrostatic equilibrium (i.e., ... | arxiv_0181684 |
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